Virtual Photons in Magnetic
Resonance
FRANK ENGELKE
Bruker Elektronik GmbH, Akazienweg 2, Rheinstetten 76287, Germany
ABSTRACT: Magnetic resonance often relies on a semi-classical picture in which the
spin particles are submitted to quantum theory and the electromagnetic field is treated
as a classical field. Although in many applications there are very good reasons to work
within this theoretical framework, it appears worthwhile either for educational purposes,
or for studies in magnetic resonance with microscopically small samples or very weak rf
fields as well as for other applications that may seem exotic today, to ask how to gain a
unified view when comparing the concepts and methods of quantum electrodynamics
(QED) with those of classical electrodynamics commonly used in magnetic resonance.
The present article attempts to develop such a unified view for electromagnetic interactions
in magnetic resonance by focusing on the concept of virtual photon exchange based
on the Feynman propagator technique and by exploring the cross links between basic
aspects of ‘‘semi-classical magnetic resonance’’ and the same basic aspects of magnetic
resonance as seen through the frame of QED. Ó 2010 Wiley Periodicals, Inc. Concepts
Magn Reson Part A 36A: 266–339, 2010.
KEY WORDS: magnetic resonance; quantum electrodynamics; Feynman propagator; virtual
photon; asymptotically free photon
INTRODUCTION
Magnetic resonance is a phenomenon that originates
from the interaction between low-energy particles
with spin and low-energy electromagnetic fields. In
ordinary matter surrounding us, mainly liquids and
solids, we usually have in mind either the electron
spin or the nuclear spin, thus we speak of electron
spin resonance (ESR) or nuclear magnetic resonance
(NMR), respectively. A more exotic field in physics
is concerned with muon spin resonance (mSR) based
on the spin of the muon, a particle similar to the electron,
but heavier in mass and unstable—a free muon
decays in a time on the order of microseconds. Magnetic
resonance techniques have found widespread
applications in condensed matter physics, material
science, organic and inorganic chemistry, biochemistry,
molecular biology, and as an imaging technique
in medical diagnostics and other fields. In these areas
of concern the spin particles of interest are situated
in atoms and molecules which are part of a bulk macroscopic
sample. So, normally we care about manyspin
systems with their manifold interactions: interactions
among spin particles themselves, couplings
with other physical degrees of freedom present in
liquids and solids, and the response of such multispin
systems to external homogeneous static fields,
or pulsed or continuous-wave time-harmonic fields,
or pulsed gradient fields. The various interactions
Received 28 March 2010; revised 22 July 2010;
accepted 29 July 2010
Correspondence to: Frank Engelke; E-mail: frank.engelke@bruker.de;
Concepts in Magnetic Resonance Part A, Vol. 36A(5) 266–339 (2010)
Published online in Wiley Online Library (wileyonlinelibrary.com)
DOI 10.1002/cmr.a.20166
Ó 2010 Wiley Periodicals, Inc.
266
display themselves in the characteristics of magnetic
resonance spectra and are the source of information
that spectroscopists find valuable when studying the
structures and dynamics in their samples.
The spin of the electron or of an atomic nucleus is
a quantum mechanical attribute. The need to switch
to quantum theory to characterize spin particles in
magnetic resonance appears when we talk about
interactions between spins: the direct dipole–dipole
coupling among nuclear spins and among electron
spins, as well as between nuclear and electron spins,
the coupling between electron spins and electron orbital
momentum, and the coupling between nuclear
spins and electron orbital momentum. Again, it is often
possible to employ a qualitative classical picture
as a more or less accurate approximation and as a
mnemonic device to view phenomena reflected in the
magnetic resonance spectra. As an example, the phenomenon
of chemical shift or magnetic shielding in
NMR—originating from the interaction of a given
nuclear spin with the orbital momenta of surrounding
electrons in a constant external field—can be understood
by visualizing the moving electrons in an atom
or molecule as currents that generate local magnetic
fields which weakly shield the external constant field
at the site of the nuclear spin, hence slightly changing
its resonance frequency. As this local shielding is
very sensitive to the local electron distribution
around the nucleus, e.g., the chemical bonds, it provides
information about the local chemical structure
of the atom or molecule—hence the name for this
shift of the resonance frequency: chemical shift. To
explore such phenomena in greater depth and arrive
at quantitative results agreeing with experimental
data we have to turn to quantum theory of atoms and
molecules, i.e., we use computational quantum chemistry.
As it turns out for heavy elements with many
electrons, one even has to take into account relativistic
corrections in the quantum chemical calculations
(1, 2) to explain experimental results. Quantum
theory including effects that belong to the realm of
special relativity leads us to relativistic quantum
mechanics, which is one precursor of quantum elec-
trodynamics.
Another point of contact with the field of relativistic
phenomena emerges when considering the interaction
between the nuclear spin with the surrounding
distribution of an electron being in an s state, i.e., an
electron orbital state with vanishing electron orbital
momentum (3). The probability for an s electron to
be found at the site of a nucleus at the center of this
electron orbital is different from zero. But how do
we calculate the interaction energy (for electron–nucleus
Coulomb interaction and electron spin-nuclear
spin dipolar coupling) in the case of vanishing distance
between electron and nucleus? Here, we face
two problems: (a) classically for point particles and
vanishing distance between them the Coulomb interaction
energy diverges and (b) the point-dipole
approximation that allows us to treat the spin–spin
coupling as direct dipole–dipole coupling breaks
down. Case (a) with energy terms diverging is a signature
for relativistic effects: the interaction energy
of the particles can be on the order of or may exceed
the rest energy of the particles. So, at least we have
to take relativistic effects into account as higher
order corrections in perturbative computations. The
case of the coupling between a nuclear spin and an s
electron, both in the same atom, leads us to the Fermi
contact interaction, a phenomenon well known in
magnetic resonance.
So far we have spoken briefly about quantum features
of the spin-carrying particles. We have not yet
mentioned quantum field theory and we have not yet
addressed the subject of quantizing the electromagnetic
field. Quantum electrodynamics is the quantum
field theory for electromagnetic interactions and in
its full extension it encompasses quantization of the
electromagnetic field and the field quantization of the
particles with nonzero rest mass that interact via
quantized electromagnetic fields. The quantization of
the electromagnetic field with particular attention for
the case of magnetic resonance will be the subject of
the present article. In the past, until recently, treating
the electromagnetic field explicitly as a quantized
field has found only relatively modest attention in the
magnetic resonance literature. Instead, in the majority
of the published articles electromagnetic fields are
treated classically. What is the reason for treating the
electromagnetic fields as classical? In magnetic resonance,
we use macroscopic devices—resonators,
coils, and circuits—to generate time-harmonic magnetic
fields external to the spins in our sample at low
frequencies, either in the radiofrequency range (meter
to centimeter wavelength) or in the millimeterwave
range: As a consequence, the energy of single
photons is small and with state-of-the-art static-fieldgenerating
cryomagnets, rf or microwave power
transmitters, and circuits or resonators it is not difficult
to generate either static or time-harmonic field
amplitudes that formally correspond to an astronomically
large number of photons. Hence, according to
the argument often brought forward, electromagnetic
fields can be treated classically.
As we know from experimental evidence, the
time-harmonic macroscopic fields have well defined
field amplitudes and phases. From the quantum point
of view, however, field amplitude (or equivalently,
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number of photons in a given field mode) and phase
are complementary variables (4): the uncertainty of
each cannot be made arbitrarily small without affecting
the other; they undergo Heisenberg’s uncertainty
relation. From that argument it becomes obvious that
low-energy electromagnetic fields generated by macroscopic
classical devices cannot be in a quantum
state, which is an eigenstate of the photon number
operator for a given mode or an eigenstate of the
phase operator. Rigorously speaking, if the field were
in a number eigenstate, the phase of the field would
be entirely uncertain, and vice versa, if the field were
in an eigenstate of the phase operator, the amplitude
would be entirely uncertain. However, there is a class
of superpositions of number states, referred to as
coherent states (4, 5), where both amplitude and
phase uncertainty are at a minimum simultaneously,
consistent with Heisenberg’s relation, and where the
relative uncertainty of both becomes negligibly small
for large average photon numbers, i.e., in the classical
limit, the expectation values of observables (e.g.,
amplitude and phase) calculated with these coherent
states lead to the familiar classical equations such
that, for example, the electric and magnetic fields
obey Maxwell’s equations.
The electromagnetic field has certain distinct and
unique characteristics that warrant special attention.
First, electromagnetic interactions propagate with
the speed of light—hence electromagnetic fields are
subject to special relativity. Even though in the 19th
century Faraday, Ampere, Maxwell, Hertz, and
others developed the electromagnetic theory prior to
the development of the theory of special relativity by
Einstein, Schwarzschild, and Lorentz at the beginning
of the 20th century, and even though classical
electrodynamics is thus often presented in a technical
form with definitions and equations that are not
form-invariant under Lorentz transformations (see
Appendix A), electromagnetic fields are relativistic
entities. For theoretical studies, the relativistic point
of view has been taken into account by finding formulations
of classical electrodynamics that satisfy
form-invariance under Lorentz transformations (for
the sake of brevity we say: the equations can be written
in a form that is Lorentz invariant or covariant).
We will use some parts of this covariant formalism
throughout this article at places where it is either necessary
or convenient: the covariant notation is quite
compact and for certain formal chains of arguments
it is a very economic one to use. When quantizing
the electromagnetic field one finds photons as the elementary
excitations of this field and as a consequence
of the relativistic nature of the electromagnetic field
one discovers that photons are entities with rest mass
zero carrying a spin equal to 1 (in units of "h). Hence,
photons are bosons, in contrast to spin-1/2 particles
like electrons, protons, and neutrons, which are
fermions.
Second, if one introduces electromagnetic potentials
(the well-known scalar potential f and the vector
potential A), one finds that the (electric and magnetic)
fields as well as Maxwell’s equations are forminvariant
under gauge transformations (see Appendix
B) of these potentials: the fields are gauge invariant.
In classical electromagnetism one can adopt the attitude
that the electromagnetic potentials are auxiliary
quantities and that the ‘‘real physics’’ lies in the electric
and magnetic field strengths, so one might say
that gauge invariance is just a mathematical playground
without physical implication. This attitude
cannot be kept anymore when we turn to the quantum
theoretical point of view. Here, gauge invariance
in electromagnetism and invariance of the Schro¨dinger,
Pauli, or Dirac equation (see Appendix E)
under local phase transformations of the wave function
of particles interacting with an electromagnetic
field are intimately linked to each other leading, e.g.,
to the Aharonov-Bohm effect (6–8), which also has
been verified experimentally (9). In other words,
electromagnetic potentials as well as gauge transformations
gain physical significance as soon as we turn
to quantum theory (10). Gauge and phase invariance
can be seen as a special case of geometric phases
(11, 12), the latter have been investigated also by
NMR (13–19). Although we will not treat these
topics here, they implicitly affect our approach, for
example, when choosing the proper (i.e., covariant)
gauge condition for the electromagnetic field.
The Dirac equation for an electron in an external,
classical electromagnetic field suggests that the electron
is a spin-1/2 particle (a fermion) with the gyromagnetic
ratio ge ¼ ge=2me, with g as the Lande´ factor
for the electron. For exactly this case, i.e., an
electron with Dirac wave function c in an external
electromagnetic field, where the latter is treated classically,
it turns out that the Dirac theory yields
exactly g ¼ 2. However, as one knows beyond doubt,
for example, from high-energy electron scattering
experiments, in reality g is slightly larger than 2,
leading to the so-called anomalous magnetic moment
of the electron (20–22). For the electron, g ¼
2.00231930436 (22). Is there an explanation arising
from a deeper quantum theory that also treats the
electromagnetic field as a quantum field? The answer
is affirmative and the elucidation for the electron as a
fermion particle characterized by g . 2 is one of the
early and certainly most impressive triumphs of
quantum electrodynamics (QED) (20, 21). Because g
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determines the gyromagnetic ratio of the electron, it
becomes clear that it establishes the Larmor frequency
of an electron in an electromagnetic field,
where we note that this is an electron not bound in
electron orbitals of an atom or molecule, in this sense
this is the g factor for a ‘‘free electron.’’
Let us turn the focus to nuclear spins. The simplest
example is the proton, a spin-1/2 particle like
the electron whose gyromagnetic ratio can also be
written, analogous to the electron case, as
gp ¼ gpe=2mp. Here, mp stands for the proton rest
mass and gp denotes the proton g factor. However, it
was found out experimentally that for the proton gp
is quite different from 2. The reason is that although
the proton possesses spin 1/2, it is, strictly spoken,
not a Dirac particle, i.e., it does not obey Dirac’s
equation. The fact that gp = 2 cannot be deduced
from QED. The raison d’eˆtre for the proton’s defiance
lies in the fact that it is a composite particle, not
an elementary particle like the electron. The proton
as composite is facing the external world as a stable
particle and it is interacting via its electric charge
and magnetic dipolar moment with electromagnetic
fields. Internally, it has a quite complex structure that
is governed by strong nuclear forces, which are the
subject of quantum chromodynamics (QCD), not
QED. Thus, to explain the origin of gp, one needs to
step outside QED and treat more advanced quantum
field theories like QCD (23, 24). It is also interesting
to recognize that the proton, treated simply as a spinbearing
particle in 1
H NMR, is still under active
experimental study in the field of high-energy
physics nowadays, where the goal is to learn more
about the detailed origin of the proton spin from the
constituents and interactions in nuclear matter (25).
For the case of low energies (where ‘‘low’’ means
energies small compared to the binding energy of the
quarks bound together in the proton by strong, nonelectromagnetic
interactions), and for energies small
compared to the rest energy mpc2
of the proton, and
for the purpose of exploring QED phenomena in
magnetic resonance, we can treat the proton although
not as a Dirac particle, but as a Dirac-like particle for
which we simply have to measure the value of gp. It
turns out that gp ¼ 5.585692. Therefore, in the lowenergy
limit, the proton becomes very similar to the
electron, in principle. Both, electron and proton are
spin-1/2 particles (fermions), and their respective
electric charges –e and þe, masses me and mp, and
gyromagnetic factors ge ¼ ge=2me and gp ¼
gpe=2mp determine the sizes of their respective magnetic
dipole moments.
How can we treat nuclear spin particles different
from the proton? First of all, the composite nature of
the nuclei is evident. Moreover, there are nuclei like
13
C, 15
N, 29
Si that carry spin-1/2, whereas others
may have spin 0 or 1 or spin 3/2 or 5/2, etc. All these
spin quantum numbers mentioned here refer to the
respective ground state of the nucleus—we do not
consider excited nuclear states. The half-integer spin
particles are fermions, while the particles with integer
spin are bosons. In a quantum statistical ensemble
of indistinguishable particles, the former are governed
by Fermi-Dirac statistics with wave functions
of the ensemble being antisymmetric under particle
permutation, while the latter obey Bose-Einstein statistics,
which yield symmetric wave functions. The
impact of the quantum statistics on the physical
behavior of spin-1/2 particle ensembles becomes
noticeable in atomic or molecular electron systems
governed by the Pauli exclusion principle, but also in
nuclear spin systems like the dihydrogen state known
as parahydrogen. In parahydrogen molecules, the two
proton spins form a spin-singlet state (26–29) as
opposed to orthohydrogen, where the two proton
spins appear in triplet states depending on which
rotational state is occupied by the two-atom
molecule.
The spin of a nucleus as well as its associated gn
factor originate from the complex internal nuclear
structure (bound protons, neutrons, which in turn are
composites of quarks, undergoing strong interactions
mediated by gluons). Nevertheless, in the very-low
energy regime we are allowed to focus upon the electromagnetic
nature of nuclei only, represented by the
nucleus’ charge, its spin, and its electric quadrupole
moment (the latter is zero for the proton). In the following
we will not discuss nuclear-spin particles different
from the proton, i.e., in the present article we
focus only on low-energy QED including spin-1/2
particles like electrons and, as ‘‘nuclear spin prototypes,’’
protons. We are allowed to treat protons as
Dirac-like particles with an empirical gp factor at low
energies, and in such a way we can develop a formalism
that treats electron spins and proton spins alike.
For example, as we will see, the current density associated
with the Dirac equation is easily derivable for
electrons. We can take over an analogous expression
of the current density for protons. The current density
in general can be submitted to the Gordon decomposition
(Appendix F) to extract the spin part of the current
density, where the latter is needed to formulate
the spin interactions in quantum electrodynamics.
In the present article, we will study electromagnetic
interactions with particular attention to magnetic
resonance. Interaction may mean interaction
between two current densities, such as a spin current
density interacting with a conduction current density
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(the latter, for example, represented by the time-harmonic
current in a macroscopic piece of wire). We
may also say that interaction occurs between a spin
particle and an external electromagnetic field, generated
by a source that is not necessarily specified yet
in detail and that is different from the spin particle.
But taking this source of the external field as another
current density, then we see that we return to the case
of two interacting current densities. So, one of the
central questions will be to express the electromagnetic
current-to-current interaction in quantum electrodynamics.
The answer to this question will
directly lead us to the concept of virtual photons as a
general notion in quantum electrodynamics and thus
also as a vehicle for describing electromagnetic
couplings in magnetic resonance.
Why should we take the effort to quantize the
electromagnetic field in magnetic resonance while it
has been often shown that the classical description is
generally sufficient? First, compared to classical and
semi-classical theory, QED provides a different point
of view, and comparison of the ‘‘QED language’’
with the established technical language of EPR and
NMR spectroscopists could suggest alternative solutions
to research questions. Furthermore, there are
areas where QED might be regarded as interesting or
even important, e.g., force-detected NMR microscopy
and spectroscopy (30–36). For example, Butler
(36) uses the Jaynes-Cummings approximation
[which is often used to study the coupling between a
two-level system and one mode of an electromagnetic
field in quantum optics (4)] to describe a nanoscale
spin resonator for force-detected NMR. Longitudinal
spin relaxation induced by the resonator is
studied by quantizing the mechanical oscillator and
analyzing the appropriate master equation. Another
field of interest is NMR spectroscopy and imaging
with sub-mm or micrometer size rf coils in conjunction
with samples in the volume range of picoliter to
femtoliter (37–43). In these domains and dimensions
one cannot take it for granted anymore that the average
number of photons in an electromagnetic field
mode and in a given small sample volume is astronomically
large. So, the question: does the uncertainty
in amplitude and phase of rf fields play a role
here? Furthermore, does QED contribute when analyzing
the quantum measurement process (44–47) in
magnetic resonance and when exploring quantum
computing and information processing using magnetic
resonance (48–53), taking into account not only
spin quantum states but also quantum states of the
electromagnetic field? Finally, there can be a general
educational benefit. For example, the QED viewpoint
can provide a unified picture of electromagnetic phenomena
in magnetic resonance, including the static
Zeeman coupling, the coupling between spin and rf
or microwave field, the coupling between nuclear
spin and nearby electrons, and the dipolar coupling
between two nuclear spins, to name just a few. Similarly,
it is instructive to explore the limit process
from quantum to classical fields, which may help us
also to understand better the classical and quantum
aspects of interacting spins and electromagnetic
fields.
Previous work illustrates the possibilities of incorporating
field quantization, either of electromagnetic
or other fields, in magnetic resonance. Jeener and
Henin (54) investigated a general model for the coupling
of an atom with an electromagnetic field in the
framework of quantum optics (4), where for the simplest
case of a two-level atom, expressible with pseudospin
operators, parallels with NMR have been discussed.
In a later article (55), the same authors provide
a fully quantized theory for nuclear magnetic
resonance in the framework of quasi-classical
(coherent) states of the electromagnetic field. We will
address some more details of Jeener’s and Henin’s
work in Section ‘‘A QED NMR Probe Model: Pulsed
NMR as a Scattering Process.’’ In a series of articles,
Hoult et al. (56–60) have considered NMR signal
reception as a near field phenomenon that can be
interpreted classically via Faraday induction and
quantum theoretically through virtual photon
exchange, and argued against the conception of
NMR as a radiative field phenomenon or a phenomenon
linked to coherent spontaneous emission sometimes
advocated. Boender, Vega, and de Groot (61)
propose a quantum field treatment to incorporate the
MAS rotor as a quantum rotor describing and characterizing
rotor-frequency driven dipolar recoupling
(RFDR) NMR experiments, whereas Blok (62) et al.
consider relaxation processes of 67
Zn in ZnO taking
into account the phonon field in the ZnO lattice
including zero-point fluctuations of the phonon vacuum.
Analogies between quantum optics or optical
spectroscopy and magnetic resonance have been
drawn (63, 64), whereas the photon picture has been
extensively used in the work exploring two-photon
and multi-photon transitions in EPR and NMR
(65–76). Although in the majority of the aforementioned
articles field quantization appears more or less
as a specific tool to answer certain questions, in the
present article we propose to place quantization of
the electromagnetic field interacting with spin particles
in the center and concentrate on the specific
role of photons in such interactions. So, the focus is
directed on a theoretical device that is of interest to
us, which could be used as a tool. Thus, it is the
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manner as we look at familiar effects like Larmor
precession, the nuclear spin Zeeman effect, the free
induction decay, and related phenomena. It is the
detailed physical background that is of interest while
we derive the familiar picture of phenomena from
the general view of QED.
There is a variety of equivalent formalisms to
choose from which can be used to represent QED,
e.g., the propagator formalism, path integrals, diagrammatic
techniques as an auxiliary tool, and
others. In the sequel we will choose virtual photons
as the central notion to describe electromagnetic couplings.
The formalism or mathematical vehicle to
characterize virtual photons is represented by the
Feynman propagator or Feynman-Green function for
the electromagnetic field, for brevity also referred to
as the photon propagator. To obtain a precise idea
about virtual photons as a physical concept, either in
general, or in magnetic resonance, we must first lay a
groundwork that includes some of the mathematics
involved in generalized functions (Schwartz distributions),
complex functions and functional analysis
(Appendix C) as well as field operators and their
commutators. To keep the main text readable, we
avoid the more complex mathematical definitions
and formal implications as well as the longer and tedious
derivations of formal expressions. For the
interested reader, these are assembled in appendices.
In the section subsequent to this introduction we
will provide an informal entrance and a comparison
of the semi-classical view (spin as quantum object
and the electromagnetic field as classical) with the
view point offered by QED. In Section ‘‘The Feynman
Propagator’’ we will turn to a first formal feature
by deriving an expression for the photon propagator
DF in general form using a development based on
physical arguments introduced originally by Feynman
(77). In Section ‘‘Quantization of the Electromagnetic
Interaction Field: Virtual Photons’’ we will
show that the photon propagator characterizes the
appearance of virtual photons, whereas in Section ‘‘A
QED NMR Probe Model: Pulsed NMR as a Scattering
Process’’ we will link the concepts of virtual photons
and asymptotically free photons with a model
for pulsed NMR spectroscopy. In the subsequent sections
we will use the QED view to explore phenomena
intrinsic to magnetic resonance, such as Zeeman
effect and Larmor precession, the interaction of a
spin-1/2 particles with time-harmonic fields, a singlespin
FID, and NMR radiation damping. The scope
covered by the present article is by no means comprehensive,
already the wealth of implications and
crosslinks between full quantum descriptions with
semi-classical and classical treatments is overwhelmingly
large. Even if the present article covers some of
the basic aspects in quantum electrodynamics, it cannot
replace the study of textbooks on QED—the
interested reader is referred to the literature, e.g.,
references (21, 22, 79–82, 85, 88–90, 92, 93, 95). In
the concluding section we will summarize and tentatively
show possible pathways to further explore
quantum electromagnetic fields in magnetic
resonance.
QUANTUM ELECTRODYNAMICS
IN MAGNETIC RESONANCE
Before turning to technical details necessary to
describe magnetic resonance phenomena by quantum
electrodynamics, let us gain some informal access
and overview first. We want to study low-energy particles
with spin angular momentum that interact with
low-energy electromagnetic fields. The latter classically
obey Maxwell’s equations for the electric and
magnetic vector field, or they obey d’Alembert’s
wave equation for the vector and scalar potential.
The interaction energy of a spin particle with magnetic
dipolar momentum l situated in an electromagnetic
field given by an electric field vector E and a
magnetic induction field vector B is proportional to
the scalar product lB, hence it depends on the relative
orientation of the vectors l and B. Otherwise,
the energy values lB are arbitrary and because |l|
and |B| are values from a continuous set, the classical
interaction energy lB is also from this set. Quantization
of angular momentum leads to discrete values
for the interaction energy—that means, when we
choose the direction of the classical static field vector
B to be the axis of quantization for the angular momentum,
for spin-1/2 particles, only two discrete
interaction energy values are allowed, 6g"h|B|/2.
Therefore, a definite change of energy in a transition
between these two energy levels can only be 6g"h|B|.
What does it mean actually to quantize the electromagnetic
field? To develop this idea, consider first
the free electromagnetic field, i.e., a field in absence
of any electric charge or current distribution. This is
also a field without sources, which, again, would be
charge or current distributions. The fact that such a
source-free field exists at all is ensured by Maxwell’s
equations or by d’Alembert’s wave equations, where
in these equations all charge and current distributions
are set to zero, and it can be straightforwardly shown
that non-zero solutions exist to these homogeneous
differential equations. We may think of free electromagnetic
fields as theoretical entities insofar, that
when we want to measure field quantities or we look
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at the influence of fields on charged particles, spins,
etc., we always work with interacting fields. So to
say, free fields are entities to be considered when the
interactions are going towards zero. The energy of the
classical free field is proportional to the volume integral
over |E|2
þ |B|2
. If E and B here are classical vector
quantities, i.e., if they are continuous functions of
space and time coordinates, the energy density values
are also continuous—the energy of a classical free
electromagnetic field can take arbitrary values. Quantizing
the electromagnetic field is synonymous with
giving up the paradigm of continuity in the range of
possible energy values. Instead, the possible energy
values form a discrete set (Planck’s hypothesis, 1900,
black-body radiation). In so far, the situation is similar
or analogous (although not identical) when quantizing
the angular momentum of a particle—classically
there exists a continuous set of values mB, quantum-mechanically
(for spin-1/2) only two discrete
energy values 6g"h|B|/2 remain.
Which difference do we find for the quantized
electromagnetic field? When we look at a change of
field energy between two discrete values of field
energy, we cannot speak of a free electromagnetic
field anymore. Any change of energy in the field must
be compensated by some energy uptake or energy
yield of particles in the field, for example the spin
particle with its two allowed discrete energy values,
hence these particles interact with the field. In other
words, we assume energy (and momentum) conservation
for the composite system field plus particle(s).
Like a classical electromagnetic field, the quantized
field can be submitted to Fourier decomposition
and each term in the resulting Fourier series is
referred to as a field mode characterized by some
(angular) frequency o and wave vector k. The smallest
energy change for one particular field mode is
equal to "ho, with "h denoting Planck’s constant
divided by 2p. When we consider the quantized electromagnetic
field interacting with some spin particle
placed in that field, we basically understand interaction
as some discrete change of energy (and/or momentum)
of the field balanced by the accompanying
discrete change of energy (and/or momentum) of the
spin particle: a certain number of energy quanta "ho
is being exchanged. In this way, not only the admissible
energy values for the field and for the particle
are discretized, also the interaction itself is considered
as a discrete sequence of elementary events or
processes. We may consider this ‘‘sequencing’’ or
even ‘‘discrete network formation’’ as one unique
feature of QED, because it allows us to decompose
more complex interaction scenarios occurring, for
example, in multispin systems, even though it is a
further question how to treat the multiparticle interactions
or the interactions in bound states extended
in time by analytical and perturbative schemes. We
imagine the discrete sequence of interaction events
as exchange processes of quanta between the interacting
entities. Already early in the history of quantum
theory, this quantum for the electromagnetic
field has been termed photon. A photon is the smallest
entity that can be exchanged in electromagnetic
interaction processes. The total energy of an electromagnetic
field at a given time is equal to the sum of
energies arising from all photons present in the field
at that time.
It is to be expected that the exchange of photons is
not a deterministic process. As a consequence of
quantization, the quantum probabilistic character of
the electromagnetic interaction has to be taken into
account when we turn to QED. As we know, probabilistic
behavior already appears when we describe
particles like nuclei and electrons as quantum objects
characterized by wave functions. Here, the classically
treated electromagnetic field plays the role of an
external background field and transitions between
particle quantum states do not change this external
field. In QED, the electromagnetic field becomes an
active partner that undergoes transitions or changes of
its quantum state as well when it is coupled to particles.
Both, particular transitions between quantum
states of particles and transitions between states of the
field are related, or matched to each other. A change
of the electromagnetic field state corresponds to the
emission or absorption of photons, it is accompanied
by a corresponding transition between particle
states—in magnetic resonance these are transitions
between spin states. The probabilistic nature of photon
exchange as mechanism of electromagnetic interaction
is characterized by uncertainty relations, either
for energy, momentum, space position, or time intervals
involved in the interaction (Section ‘‘Quantization
of the Electromagnetic Interaction Field: Virtual
Photons’’). For example, a photon once emitted is not
reabsorbed necessarily with certainty. If, within a
given time interval, photon absorption takes place
after the photon has been emitted, we speak of a virtual
photon: it constitutes an intermediate state of the electromagnetic
field. If, after emission, photon reabsorption
does not occur, the photon is free in the sense that
for more and more extended time intervals the probability
for reabsorption of that photon goes towards
zero—the photon appears to be asymptotically free.
Spin particles interacting with electromagnetic
fields do not require any special treatment in QED.
The tools developed in quantum field theory are
applicable, in principle. Of course, it makes sense to
272 ENGELKE
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take into account the specific boundary conditions
under which we study magnetic resonance phenomena,
like focusing on low-energy particles and fields,
thus being allowed to apply nonrelativistic approximations
(Section ‘‘Spin Current Density, Zeeman
Hamiltonian, and Larmor precession’’). On the other
side, certain tools of QED to analyze, to compare,
and to illustrate electromagnetic interactions are in
our hands. One of these tools that we will use in the
present article is the Feynman diagrammatic technique.
In Fig. 1 we have listed several Feynman diagrams
to provide a correspondence between situations
familiar from the semi-classical point of view of magnetic
resonance and the associated QED view. Feynman
diagrams are symbolic representations of rigorous
mathematical expressions. Apart from that, at
the same time they provide some kind of intuitive picture
of the elementary interaction processes for which
they stand. Without going into any details now—we
will treat some of them in later sections—let us briefly
discuss the basic phenomena illustrated in Fig. 1.
The first one, [Fig. 1(A)], represents the interaction
of a spin-1/2 particle with an external electromagnetic
field. The source of the field might be
unspecified or it might be explicitly given (e.g., a
current in a coil). In the corresponding Feynman diagram,
for example the diagram (a) in Fig. 1(A), we
draw a line with an arrow to represent the spin particle.
This directed line enters a vertex, where the discrete
‘‘interaction event’’ occurs, here it is the emission
or absorption of a virtual photon, symbolized by
a wavy line. After the interaction event the spin particle
is left in a state (drawn by an outgoing line with
arrow) different from the initial state. Virtual photon
exchange happens between the spin particle and the
(nonspecified) external source of the electromagnetic
field. We may add more details to this process by explicitly
specifying the external source current [diagram
(b)]. In that case, the virtual photon exchange
affects both interaction partners, the current density
that corresponds to the spin particle (Section ‘‘Spin
Current Density, Zeeman Hamiltonian, and Larmor
Precession’’) and the current density that represents
the external source. The two diagrams (a and b) provide
the basis to appreciate the concept of the
‘‘single-spin free induction decay’’ and the basic
mechanism of NMR radiation damping, both treated
in Section ‘‘Single-Spin FID and NMR Radiation
Damping’’ Figure 1(B) offers a glance to more details
when a spin particle interacts with an rf field, for
example, during the application of an rf pulse. Classically,
the rf current in a coil or circuit generates an
electromagnetic field that interacts with the spin
inside or nearby the coil. We call this field the near
field, characterized by a distance to the source (the
current in the coil wire) small compared to one wavelength.
A free standing coil (that means a coil not
completely surrounded by a shield) also generates an
rf field at larger distances (one wavelength and more)
which is the far field. In Fig. 1(B) in the center, the
graphical plot on top shows a snapshot of the rf far
field magnitude |B| generated by an rf current in a solenoidal
coil (6 turns, pitch angle x, limit radius r0,
cf. Ref. 87) located in the center of the field distribution.
Taking the center part of that field distribution
(field of view magnified by a factor of 5), we see the
near field close to the coil. Under the QED point of
view, we may interpret the interaction between spin
and near field as virtual photon exchange while the
far field arises from those photons that, for example,
are emitted by the rf current and that are not
absorbed within a given time interval, thus they are
asymptotically free photons. This principal situation
is shown by the Feynman diagram in Fig. 1(B), more
details will be treated in Sections ‘‘A QED NMR
Probe Model: Pulsed NMR as a Scattering Process’’
and ‘‘Interaction of a Spin-1/2 Particle with External
Time-Harmonic Fields.’’ Likewise, direct dipole–
dipole interaction, a specific example for a spin–spin
interaction illustrated in Fig. 1(C) can be understood
as virtual photon exchange. Finally, even spin-lattice
relaxation (characterized by the time constant T1) can
be cast into the QED language, Fig. 1(D). Every transition
of a spin between Zeeman levels corresponds
to a virtual photon exchange between the spin particle
and the lattice-degrees of freedom, where the
lattice represents a partner for the interaction with a
stochastic or statistically fluctuating current density,
resulting from the stochastic motion, e.g., of
molecules in a liquid. In a similar way, spin–spin
relaxation could be treated.
In the present article we focus on the basic interactions
exemplified in Fig. 1(A,B), i.e., we deal with
the interaction of a single spin (or more than one, but
then isolated spins) with the static field and with
an rf field or microwave field and we provide a QED
explanation of the electromotive force induced in a
coil or circuit by a single spin. We will call that
‘‘single-spin FID,’’ although an FID of a macroscopic
sample contains more features, for example, the
dephasing of magnetization originating from many
spins with slight resonance offsets, spin–spin relaxation,
explicit spin–spin couplings, etc. These latter
phenomena are not treated in the present article. On
the other hand, NMR radiation damping—to be
understood as the back action of the rf current generated
by the electromagnetic field originating from the
spin particle—also appears when only one spin is
VIRTUAL PHOTONS IN MAGNETIC RESONANCE 273
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Figure 1 Basic interaction processes occurring in magnetic resonance. The middle column
shows in a schematic way (A) a single spin interacting with an external classical field, (B) the
near field and the far field generated by an rf coil, (C) spin dipole-dipole interaction, and (D) a
level diagram associated with spin-lattice relaxation. The diagrams in the right column represent
Feynman diagrams for the corresponding QED elementary processes.
274 ENGELKE
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present. Admittedly, this effect is extremely small and
‘‘hard to measure’’ for a single spin. So, the scope of
the present article might be summarized as treating
the basic and elementary interaction processes relevant
for magnetic resonance, like virtual photon exchange
and emission of asymptotically free photons. These elementary
processes could be used as building blocks
to deal with more complex situations in bulk samples
including multi-spin systems characterized by interactions
between like spins and unlike spins, spin–lattice
relaxation, and spin–spin relaxation.
To finally achieve the proximity to magnetic resonance
phenomena in the form as we know it from the
NMR and ESR spectroscopy literature, we meet
approximations and assumptions: first, we start with
the simple classical expression of Coulomb interaction
energy and generalize this step by step to arrive
at a covariant expression for the Feynman propagator.
After analyzing it and deriving the associated concepts
of virtual photons and asymptotically free photons
(Sections ‘‘The Feynman Propagator’’ and
‘‘Quantization of the Electromagnetic Interaction
Field: Virtual Photons’’) we introduce a photon scattering
model for pulsed NMR (Section ‘‘A QED
NMR Probe Model: Pulsed NMR as a Scattering Process’’).
Here, we treat the current densities as nonoperator
quantities, which is allowed when we would
treat the spin particles either as classical or when we
treat them as quantum objects, in the latter case in
first quantization (with the wave function as a function).
For the spin particles we perform the transition
to the nonrelativistic realm (slow velocities, low energies,
and low momenta). Thus three cornerstone
assumptions are involved: (a) the electromagnetic
field is assumed to be a relativistic quantity
(expressed by the associated Feynman propagator),
(b) initially the spin current density is introduced as a
covariant quantity, however, for the spin particles we
apply the nonrelativistic approximation (i.e., small
velocities, small energies, and small momenta), see
Sections ‘‘A QED NMR Probe Model: Pulsed NMR
as a Scattering Process’’ and ‘‘Spin Current Density,
Zeeman Hamiltonian, and Larmor Precession,’’ and
finally (c) we consider the spin particles as quantum
objects (with a spatial and spin wave function) with
the result that the spin current density is also a function
(no second quantization of the fermion field). We
start with (a) in deriving the Feynman propagator.
THE FEYNMAN PROPAGATOR
As indicated in the introduction, we must create
some formal basis to fully appreciate the concept of
virtual photons. For edifying reasons let us begin
with an almost tautological definition of the term
interaction. Two partners (particles, fields, etc.) are
interacting with each other when partner 1 acts upon
partner 2 and partner 2 also acts upon partner 1. So,
we may explain interaction by reducing it to the concept
of action. In physics, the quantity of action is
defined as the time integral over the Lagrange function
L. In classical mechanics, the Lagrange function
is given by the difference of kinetic and potential
energy of particles. The potential energy term constitutes
the interaction of the particle with either an
external field or with an interaction partner. If for the
moment we disregard kinetic energy and turn to field
theory, the Lagrange function for the interaction
becomes (up to a sign) equal to the interaction Hamiltonian
or interaction energy. Therefore, we will
start our discourse by discussing a stepwise generalization
of formal expressions for the electromagnetic
interaction energy and then turn to the quantity of
action associated with it. We begin by considering
two point charges Q1 and Q2 at distance r. The electrostatic
interaction energy reads
EQ ¼
1
4pe0
Q1Q2
r
[1]
where e0 denotes the dielectric permittivity of the
vacuum and r is equal to the distance between the
two charges Q1 and Q2. The Coulomb interaction
energy EQ can take positive or negative values,
depending on the signs of Q1 and Q2, where for opposite
charges, i.e., attractive forces, we get EQ , 0.
We make a first step towards generalization by not
assuming point charges, but electric charge densities
r1(x) and r2(y) distributed in space given by the
position vectors x and y with r ¼ |x À y|. Thus,
EQ ¼
1
4pe0
Z
d3
x
Z
d3
y
r1ðxÞr2ðyÞ
r
[2]
where now two integrals appear over the threedimensional
volumes with volume elements d3
x and
d3
y occupied by elements of the charge densities
r1(x) and r2(y), respectively. A similar expression
can be written down for the interaction energy of two
electromagnetic currents or current densities j1(x)
and j2(x):
EC ¼ À
m0
4p
Z
d3
x
Z
d3
y
j1ðxÞj2ðyÞ
r
[3]
with m0 being equal to the magnetic permeability of
the vacuum. We may also admit charge and current
VIRTUAL PHOTONS IN MAGNETIC RESONANCE 275
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densities that explicitly depend on time and, for generality,
we may assume that charge distributions as
well as current distributions are present simultaneously.
The total interaction energy reads
EI ¼
1
4p
Z
d3
x
Z
d3
y
1
e0
r1ðx;tÞr2ðy;tÞÀm0j1ðx;tÞj2ðy;tÞ
r
¼
m0
4p
Z
d3
x
Z
d3
y
c2
r1ðx;tÞr2ðy;tÞÀj1ðx;tÞj2ðy;tÞ
r
[4]
where c2
¼ 1/e0m0. Let us introduce the four-position
vector xm
¼ (ct,x) including time and space coordinates,
where the superscript m counts the components,
with m ¼ 0 for the time coordinate (multiplied by the
speed of light c), x0
¼ ct and m ¼ 1, 2, 3 referring to
the three spatial components of the vector x. Likewise,
let us merge the scalar charge density r(x) and the
vector current density j(x) into a single four-vector jm
¼ (rc,j) and refer to it as the four-current density. For
more details on the definition of four-vectors in Minkowski
space-time, we refer to Appendix A. In Eq.
[4], let us drop the subscripts 1 and 2 labeling the
charge and density distributions because we recognize
that the space-time coordinates x and y distinguish
them already. So, Eq. [4] appears in the concise form
EI ¼
m0
4p
Z
d3
x
Z
d3
y
jm
ðxÞjmðyÞ
r
[5]
with the scalar product, jm
ðxÞjmðyÞ¼j0
ðxÞj0ðyÞ
ÀjðxÞjðyÞ, in Minkowski space-time. Note, in expressions
like am
bm we apply the sum convention: am
bm
: a0
b0 þ (a1
b1 þ a2
b2 þ a3
b3) (see Appendix A).
We recognize that until now we have made no
explicit assumptions about how fast the interaction
may propagate between jm
(x) and jm(y)—in fact, in
Eq. [5] we implicitly assumed that the interaction
propagates infinitely fast. This becomes more
obvious if we rewrite Eq. [5] as
EI ¼
m0
4p
Z
d3
x
Z
d3
y
Z
dy0 jm
ðxÞdðx0
À y0
ÞjmðyÞ
r
[6]
where we have inserted Dirac’s d function with the
time difference x0
À y0
as an argument and an additional
integral over the time coordinate y0
. Integrating
the whole expression in [6] over dy0
along the
entire time axis, we come back to Eq. [5]. Alternatively,
we could have performed the time integral
also over dx0
instead of dy0
, with the same result. In
other words, the time coordinates appearing in jm
(x)
and in jm(y) are equal, x0
¼ y0
, as dictated by the d
function—the interaction occurs instantaneously.
As stated earlier, the Lagrange function L for the
interaction equals the interaction Hamiltonian or interaction
energy EI up to a sign, i.e., we have L ¼ ÀEI.
Furthermore, the time integral over L is equal to the
action functional W1 for the interaction considered:
W1 ¼
Z
dx0
c
L ¼ À
1
c
Z
dx0
EI ¼ À
m0
4pc
Z
dx0
Z
d3
x
Â
Z
d3
y
Z
dy0 jm
ðxÞdðx0
À y0
ÞjmðyÞ
r
½7Š
A few words to explain what is meant by action
functional. A functional is a mapping that assigns a
number to a function or to a vector of functions (likewise,
a function assigns numbers to numbers or to a
vector of numbers). In the case of the action functional
[7], we have the function jm
(x)d(x0
Ày0
) jm(y)/r.
Integrating over the full range of all the variables
(i.e., entire time dimension and entire space) assigns
a number, W1 to the vector of two functions j(x) and
j(y). Now from Eq. [7] on we are not concerned with
interaction energy anymore, but more generally with
the action related to the interaction.
The integrals over the time intervals dx0
and the
three-dimensional volume elements d3
x can be formally
assembled together as one integral over the
space-time element d4
x. Likewise, the same can be
achieved for the time intervals dy0
and volume elements
d3
y to obtain d4
y. Hence, the action functional
[7] can be written with two space-time integrals as:
W1 ¼ À
m0
4pc
Z
d4
x
Z
d4
y
jm
ðxÞdðx0
À y0
ÞjmðyÞ
r
[8]
Now we introduce, for physical reasons, the
requirement that any action from jm
(x) to jm(y), or
vice versa (thus, interaction), can only be a retarded
action, i.e., it takes a certain time to propagate from
one region in ordinary space to another one at distance
r. When this propagation occurs with the speed
of light, c, the time interval for propagation is equal
to r/c. We can easily modify Eq. [8] to take into
account retardation by modifying the argument of the
d function accordingly:
W1 ¼ À
m0
4pc
Z
d4
x
Z
d4
y
jm
ðxÞdððx0
À y0
Þ À rÞjmðyÞ
r
[9]
With Eq. [9] we have obtained a first remarkable
result. To see this, let us define the function
276 ENGELKE
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Dretðx À yÞ ¼ À
m0
4pc
dððx0
À y0
Þ À rÞ
r
; r ¼ jx À yj
[10]
such that the action W1 can be written as
W1 ¼
Z
d4
x
Z
d4
yjm
ðxÞDretðx À yÞjmðyÞ
¼
Z
d4
xjm
ðxÞAR
mðxÞ
and where we have implemented the retarded potential
AR
m(x) generated by the current jm(y),
AR
mðxÞ ¼
Z
d4
yDretðx À yÞjmðyÞ
¼ À
Z
d3
y
Z
dy0
m0
4pc
dððx0
À y0
Þ À rÞ
r
jmðyÞ
¼ À
m0
4pc
Z
d3
y
jmðx0 À r; yÞ
r
½11Š
as the convolution integral of Dret( xÀ y) with the current
density jm(y). The kernel of the first integral in Eq. [11],
Dret(x À y), is referred to as the retarded Green function
associated with the inhomogeneous wave equation. It is
also called the retarded propagator (see Appendix B).
After having introduced retarded action, returning
to Eq. [9], let us perform the next step: we consider
the time ordering of events. In Eq. [9] we have done
so already in an implicit way by expressing retardation
through the d function with the argument
(x0
Ày0
)Àr. Because r is strictly positive, the integrals
in [9] are nonzero if and only if x0
Ày0
. 0, which
means x0
. y0
: the event at time instant y0
(for example,
a change of the current density element located in
space position y) is earlier than the event at time x0
(when the field change caused by the previous current
density change arrives at the current density element
in space position x)—the cause precedes its action. In
general, when we talk about interaction, we have to
admit that, in principle, also the inverse time order
may happen in conjunction with an exchange of the
two positions in space, i.e., an event at position x happens
first at time x0
, the field change travels to position
y where it arrives at a later time y0
such that, as
causality dictates again, we have x0
, y0
. Since both
event orders are allowed, the total action is given as
the arithmetically weighted sum of both:
W ¼
W1 þ W2
2
¼ À
m0
4pc
Z
d4
x
Z
d4
y
Â
jm
ðxÞ½dððx0
À y0
Þ À rÞ þ dðÀðx0
À y0
Þ À rފjmðyÞ
2r
[12]
The first d function on the right-hand side of Eq.
[12] stands for the time order x0
Ày0
. 0, while the
second one takes the reverse order into account,
x0
Ày0
, 0. The fact that both time orders must
appear is almost trivial for an interaction. j(y) acts on
j(x), so x0
Ày0
. 0 as well as j(x) acts on j(y), hence
we have x0
Ày0
, 0. We may depict this diagrammatically
as shown in Fig. 2. The two interacting current
densities are drawn as lines with arrows, the
interaction between them as a wavy line. In the diagrams
the time axis points into vertical upward direction
and Figs. 2(A,B) correspond to the two time
orderings as explained above. In Fig. 2, we have
drawn second-order Feynman diagrams symbolizing
the elementary electromagnetic interaction process
occurring between two electromagnetic current densities.
A systematic introduction into general Feynman
diagrammatic techniques can be found, for
example, in references (81, 89, 90, 124).
We observe that the d function is even, i.e., it
holds
dðÀðx0
À y0
Þ À rÞ ¼ dððx0
À y0
Þ þ rÞ [13]
such that Eq. [12] formally appears to be the sum of
a time-retarded and a time-advanced part. As we
have discussed above, the latter just reflects the alternative
time ordering (which always takes place
together with an exchange of spatial coordinates) that
we have to admit in interaction processes. The timeadvanced
part does not indicate a violation of causality
nor does it describe noncausal evolution backwards
in time, although formally, sometimes in the
literature, it is termed anticausal. So, the action functional
for the electromagnetic interaction of two current
densities reads
Figure 2 Diagrammatic representation of the interaction
between two currents, (A) where j(y) acts on j(x) and (B)
vice versa, j(x) acts on j(y), representing two opposite
time orderings. The current densities are drawn as lines
with arrows, the electromagnetic interaction between them
as wavy lines. The diagrams shown represent two examples
of second-order Feynman diagrams.
VIRTUAL PHOTONS IN MAGNETIC RESONANCE 277
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W ¼
W1 þ W2
2
¼ À
m0
4pc
Z
d4
x
Z
d4
y
Â
jm
ðxÞ½dððx0
À y0
Þ À rÞ þ dððx0
À y0
Þ þ rފjmðyÞ
2r
[14]
The difference between Eqs. [9] and [14] is that in
Eq. [9] only one fixed time order is taken into
account, while Eq. [14] is more general, because it
also allows the reverse order when simultaneously
the space coordinates are reversed. Both time orders
correspond to retarded action.
From now on, for the sake of simplicity in notation,
we adopt Heaviside-Lorentz units (which are
characterized by setting e0 ¼ 1 and m0 ¼ 1) and, in
addition, we also set c ¼ 1, "h ¼ 1. In the resulting
system of physical units it then appears that energy,
mass, linear momentum, wave number, and frequency
have the same unit: 1/meter. The corresponding units
in SI are obtained by multiplying accordingly with
c and/or "h, the SI units for the field quantities by
re-introducing e0 and m0 accordingly (78).
We take a look at the formal Fourier decomposition
of the d function,
dðx0
Þ ¼
1
2p
Zþ1
À1
expðÀiox0
Þdo; [15]
which tells us that, upon integration, it contains positive
as well as negative frequencies o. As it will turn
out, for the case of the quantized electromagnetic
field these frequencies o correspond to energies "ho
for photons.
Again for physical reasons, anticipating the photon
picture that we will adopt when we turn to QED, we
require that o is strictly positive or zero, not negative.
This is equivalent to admitting only positive (or zero)
energy values carried by a single photon. To take this
requirement into account, we have to restrict the integration
range in Eq. [15] to run over o values only
from 0 to 1 instead of the full range of À1 to 1
such that we arrive at the modified d function defined
by the ‘‘half-sided’’ Fourier integral (note here, by
definition, the different pre-factor 1/4p2
).
dþðx0
Þ ¼
1
4p2
Zþ1
0
expðÀiox0
Þdo [16]
We will investigate the detailed properties of dþ
later in Section ‘‘Quantization of the Electromagnetic
Interaction Field: Virtual Photons’’ (see also Appendix
C). For the moment it may suffice to say that the
features of dþ are central to the mathematical description
of photons appearing in interaction processes.
One remark in advance: while Dirac’s d function can
be considered as a real-valued generalized function,
i.e., its arguments are on the real line and the values
of d are, loosely speaking, either zero or infinite, but
real-valued, this ‘‘real-valuedness’’ is not the case anymore
for dþ. So, the requirement of positivity for the
photon energy with the consequence of the restricted
integration range in Eq. [16] leads to a phenomenon
called time dispersion: dþ becomes complex-valued
(with a real and an imaginary part). To understand the
consequences, let us rapidly anticipate some of the
steps that lead into QED, more thoroughly discussed
in later sections. The positivity requirement for the
energy of photons, exchanged between the current
densities, has far-reaching implications. First, the
quantity of action, W, becomes a complex-valued
quantity. Second, with the probability amplitude Z in
QED (22) defined as:
Z ¼ expðiWÞ [17]
for the photon propagation in QED, the resulting probability
for a photon propagating from one current to
the other current is equal to |Z|2
. Thus if W is real, we
get |Z|2
¼ 1 and if W appears to have an imaginary part,
we see that |Z|2
, 1. The latter statement means that
not every photon emission leads necessarily to photon
re-absorption—there is a finite probability that photons
‘‘break free.’’ We will discuss that in more detail in
Sections ‘‘Quantization of the Electromagnetic Interaction
Field: Virtual Photons’’ and ‘‘A QED NMR Probe
Model: Pulsed NMR as a Scattering Process.’’
So, if we modify Eq. [14] by replacing d by dþ,
we arrive at
W ¼ À2p
1
4p
Z
d4
x
Z
d4
y
Â
jm
ðxÞ½dþððx0
À y0
Þ À rÞ þ dþððx0
À y0
Þ þ rފjmðyÞ
2r
where the additional factor 2p in front of the two spacetime
integrals takes into account the different prefactor
in Eq. [16] as compared to the pre-factor in Eq. [15].
The following identities hold (see Appendix C)
dððx0
À y0
Þ À rÞ þ dððx0
À y0
Þ þ rÞ
2r
¼ dððx0
À y0
Þ2
À r2
Þ ¼ dððx À yÞ2
Þ
dþððx0
À y0
Þ À rÞ þ dþððx0
À y0
Þ þ rÞ
2r
¼ dþððx0
À y0
Þ2
À r2
Þ ¼ dþððx À yÞ2
Þ ½18Š
278 ENGELKE
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where (xÀy)2
is equal to the squared four-distance
between the two space-time points of the elementary interaction
events, i.e., ðx À yÞ2
¼ ðx0
À y0
Þ À jx À yj2
¼
ðx0
À y0
Þ À r2
. When introducing [18], the expression
for the electromagnetic action functional reads
W ¼ À
1
2
Z
d4
x
Z
d4
yjm
ðxÞdþððx À yÞ2
ÞjmðyÞ [19]
Let us introduce
Dmn
F ðxÞ ¼ Àgmn
dþðx2
Þ; [20]
where gmn
designates the metric tensor (cf. Appendix
A) and refer to DF as Feynman propagator or Feynman-Green
function for the electromagnetic interaction.
Similar to the retarded propagator Dret defined
by Eq. [10], the Feynman propagator DF is a further
Green function associated with the wave equation
(Appendix B). With the definition [20], the action
functional [19] can be written as
W ¼
1
2
Z
d4
x
Z
d4
yjnðxÞDmn
F ðx À yÞjmðyÞ: [21]
The expression [21] states that the electromagnetic
action (interaction) between two current densities
jn(x) and jm(y) is mediated by the Feynman
propagator DF(xÀy) for the electromagnetic field.
We remark that [21] is quite general, however, it can
be specialized for the case of spin–spin interactions
as well as for interactions between spins and resonator,
by specifying the current densities accordingly.
There is no principal restriction for the current density:
it can represent a spin current density or a current
density due to electron conductivity, microscopic
or macroscopic. We will turn to that subject in
more detail in Section ‘‘Spin Current Density, Zeeman
Hamiltonian, and Larmor Precession.’’ In analogy
to the retarded potential, Eq. [11], the integral
over d4
y is equal to the four-potential An
(x) at spacetime
point or region x generated by the current density
jm(y) at space-time point or region y,
An
ðxÞ ¼
Z
d4
yDmn
F ðx À yÞjmðyÞ: [22]
Hence,
W ¼
1
2
Z
d4
xjnðxÞAn
ðxÞ [23]
In summary, we have started our discussion with
the classical expression for the interaction energy for
two interacting current densities and introduced (a)
time retardation, (b) time ordering of events, and (c)
positivity of the energy exchanged as a quantum
between the interaction partners. As a result we
obtain the Feynman propagator DF for the electromagnetic
field. So far, we are still (almost) within the
scope of classical electrodynamics including special
relativity (retarded field propagation with finite
speed). We have not yet properly quantized the electromagnetic
field. In doing so, in the next section, we
will find an interpretation for DF in physical terms.
The general result [21] remains valid in a quantized
field theory. Focusing on magnetic resonance, it
becomes necessary to specify the general current
densities jm for spins and/or electric currents. We will
accomplish that in Section ‘‘Spin Current Density,
Zeeman Hamiltonian, and Larmor Precession.’’
QUANTIZATION OF THE
ELECTROMAGNETIC INTERACTION
FIELD: VIRTUAL PHOTONS
Quantization of the electromagnetic field involves
two formal ingredients. First (a) the field functions
Am
(x), i.e., the four-vector potential Am
ðxÞ ¼
ðA0
ðxÞ; AðxÞÞ including the scalar potential A0
(x) ¼
f/c and the vector potential A(x), become field operators.
Furthermore (b), generally, these field operators
are subject to certain commutation or anticommutation
relations that determine the basic nature of
the quantum field. This procedure related to these
two cornerstones (a) and (b) is called canonical quantization
and will be the basis for our attempt to discover
virtual photons in magnetic resonance. From a
practical point of view it is advantageous to begin
with the classical Fourier expansion of the field functions
(four-potentials) in three-dimensional momentum
space (k space), reading
AmðxÞ ¼
Z
d3
k
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2pð Þ3
2ok
q
X3
l¼0
eðlÞ
m ðkÞ
 aðlÞ
ðkÞeÀikx
þ aðlÞþ
ðkÞeikx
 
½24Š
Here k ¼ km
¼ ðk0
; kÞ denotes the contravariant
four-momentum vector (see Appendix A) with k0
proportional to the energy variable "hck0 of the field
mode k and k equal to the usual wave number vector
such that "hk is equal to the three-momentum. In
space-time there are generally four different polarization
vectors e
ðlÞ
m ðkÞ ¼ ðe
ðlÞ
0 ðkÞ; ÀeðlÞ
ðkÞÞ, enumerated
by the superscript l ¼ 0, 1, 2, 3. The Fourier coefficient
a(l)
(k) and its conjugate complex a(l)þ
(k) are
functions of the three-momentum k.
VIRTUAL PHOTONS IN MAGNETIC RESONANCE 279
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The exponentials exp(6ikx) contain the four-scalar
product kx ¼ km
xm ¼ ðk0
; kÞðx0; ÀxÞT
¼ k0
x0 À k Á x:
Summation over all polarization directions and
integration over k-space yields the four-vector Am
(x)
as a function of space coordinate vector x and time
coordinate x0
. Converting the field functions Am
(x)
into operators leads us to ask which of the constituents
in the integral expression [24] takes over the operator
role. This role is filled by the Fourier coefficients
a(l)
(k) and a(l)þ
(k), that are required to satisfy
the following canonical commutation relationships
(CCR):
½aðlÞ
ðkÞ;aðrÞþ
ðk0
ފ ¼ Àglr
d3
ðk À k0
Þ; l;r ¼ 0;1;2;3
[25]
with all other combinations of elements in the
commutator brackets different from those in [25]
yielding zero. Thus, with [24] being read as a Fourier
expansion of field operators and with the Fourier
coefficients required to satisfy the CCR [25],
we have performed the formal task of quantizing
the electromagnetic field. This allows us to calculate
commutators of field operators, [Am
(x), An
(y)],
or products of field operators Am
(x)An
(y), etc. We
may define the Lagrangian density for the free or
the interacting electromagnetic field and obtain the
associated Hamiltonian (79–82), which for the free
field turns out to be identical in form with the
Hamiltonian for a system of harmonic oscillators.
As explained in texts on quantum field theory and
quantum electrodynamics, the operators a(l)
(k) and
a(l)þ
(k) reveal themselves as annihilation and creation
operators for photons, the quanta of the electromagnetic
field.
Suppose there is a state |0i of the electromagnetic
field with no photons present in the field, also called
the ground state or the vacuum state of the field and
define
aðlÞ
ðkÞj0i ¼ 0; aðlÞþ
ðkÞj0i ¼ j1i [26]
where |1i denotes a state of the field with exactly one
photon present with momentum k and polarization
e(l)
(k). We may generalize this for n photons,
aðlÞ
ðkÞjni ¼
ffiffiffi
n
p
jn À 1i;
aðlÞþ
ðkÞjni ¼
ffiffiffiffiffiffiffiffiffiffiffi
n þ 1
p
jn þ 1i ½27Š
Applying a(l)
(k) and subsequently a(l)þ
(k) to the
field state |ni we obtain from [27]:
aðlÞþ
ðkÞaðlÞ
ðkÞjni ¼ njni [28]
which qualifies the product operator a(l)þ
(k)a(l)
(k)
as the photon number operator. The field states |ni
with a precisely defined number n of photons in a
given mode (l, k) are eigenstates of the photon number
operator and are called Fock states. These form
an orthonormal basis set of states in a state space
called Fock space, a specific example of an infinitedimensional
Hilbert space. We mentioned the Fock
states in the introduction when discussing the uncertainty
relation of field amplitude and phase and we
maintained already that Fock states cannot correspond
to states of the electromagnetic field in the
classical limit.
In quantum field theory it makes a difference
whether we say that the field contains zero photons,
i.e., it is in its vacuum state |0i, or we say that there
is no field. In the latter case, there is nothing to talk
about, in the former case, there is quite a bit to discuss.
We may calculate expectation values of operators
with field states to be compared with measurements,
so for example we might be interested in vacuum
expectation values. Interestingly, we find for
instance,
h0jAm
ðxÞj0i ¼ 0; h0jAm
ðxÞAn
ðyÞj0i 6¼ 0; [29]
which can be easily verified when taking into account
Eqs. [24–26]. While the vacuum expectation value of
the field operators vanishes [likewise this is also true
for the vacuum expectation values of the electric
field and the magnetic induction field calculated from
Am
(x)], it is not the case for the vacuum expectation
value of higher order products. Thus, the variance, or
two-point correlation, or fluctuation amplitude of
field quantities can be nonzero in the vacuum state—
a typical situation occurring in quantum field theory.
We return to the discussion we began in the previous
section concerning the two current densities
interacting with each other, i.e., the current density
jn
(x) generating the field An
(y) at the space-time position
y of the current density jm
(y) and vice versa, the
current density jm
(y) producing the field Am
(x) at the
space-time position x of the current density jn
(x). The
interaction of the two current densities has been
expressed by Eq. [21]. But consider the following:
instead of correlating the two current densities in that
way, we can also correlate the two associated fields
by forming the product Am
(x)An
(y) and calculating
expectation values for given field states—the simplest
and most straightforward would be the vacuum
expectation value. Moreover, because we saw in the
previous section that time ordering is essential in
280 ENGELKE
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describing interactions to ensure causal behavior, we
may include that in our field correlation and calculate
not just h0|Am(x)An(y) |0i, but calculate the time-ordered
vacuum expectation value h0|T(Am
(x)An
(y)) |0i,
where Dyson’s time ordering operator T has been
introduced, defined as
TðAm
ðxÞAn
ðyÞÞ ¼
Am
ðxÞAn
ðyÞ; x0
> y0
An
ðyÞAm
ðxÞ; y0
> x0
&
¼ yðx0 À y0ÞAm
ðxÞAn
ðyÞ þ yðy0 À x0ÞAn
ðyÞAm
ðxÞ
[30]
y denotes the Heaviside step function defined as
y(x0
Ày0
) ¼ 1 for x0
Ày0
. 0 and zero otherwise. We
have now all the means available to calculate
h0|T(Am(x)An(y)) |0i. As shown in detail in Appendix
C, this calculation yields
h0jTðAm
ðxÞAn
ðyÞÞj0i ¼ ÀiDmn
F ðx À yÞ
¼ igmn
dþððx À yÞ2
Þ ½31Š
Thus, we find that for a quantized electromagnetic
interaction field the Feynman propagator equals (up
to a constant prefactor) the vacuum expectation value
of the time-ordered field operator product Am
(x)An
(y),
that is, the Feynman propagator corresponds to the
two-point vacuum field correlation function. Note,
the correlation here includes time and space correlation—we
are in space-time.
Reading the expression [31] as a space-time twopoint
correlation function opens up the possibility of
an even more extended interpretation of the Feynman
propagator DF. First, we observe that when we insert
the expansion [24] for the field operators into
the expression for the vacuum expectation value
of the time-ordered field operator product,
h0jTðAm
ðxÞAn
ðyÞÞÞj0i, terms like
h0jTðaðlÞ
ðkÞeÀikx
þ aðlÞþ
ðkÞeikx
ÞðaðlÞ
ðkÞeÀiky
þ aðlÞþ
ðkÞeÀiky
Þj0i ¼
¼ h0jTaðlÞ
ðkÞeÀikðxÀyÞ
aðlÞþ
ðkÞj0i ½32Š
appear. The last line in [32] may be read from right
to left as follows: initially there is the photon vacuum
|0i. The creation operator a(l)þ
(k) creates the singlephoton
state j1 >¼ aðlÞþ
ðkÞj0i. As indicated by the
time ordering and the exponential function eÀik(xÀy)
,
the photon propagates (assuming y0 , x0) from y to
x. At space-time point x the annihilation operator
a(l)
(k) annihilates the single-photon state again,
which leaves the field in the vacuum state:
j0i ¼ aðlÞ
ðkÞj1i.
The process of creating a photon in y, letting it
propagate to x, then annihilating that photon in x,
appears in a time-ordered fashion—i.e., the photon is
first created at time y0
, and then later at time x0
it is
annihilated (left side of Fig. 3)—formally this is
taken care of by the time-ordering operator T. For the
case x0
, y0
(right diagram in Fig. 3) the photon is
created in x, propagates to y where it is annihilated
again. The photon only exists while propagating from
y to x (or from x to y when the time order is
reversed), the initial and final field states are photon
vacuum states. It is for that reason that the intermediate
single-photon state, j1i ¼ aðlÞþ
ðkÞj0i, appearing
here is referred to as occupied with a virtual photon.
Virtual photons emerge as intermediate states
between the initial and final photon vacuum state.
Therefore, the Feynman propagator, also referred to
as the photon propagator, represents the mathematical
vehicle describing virtual photons. With Eq. [31]
we could set up a new expression for the action functional
W, Eq. [21], which now has also a new interpretation
for the probability amplitude Z ¼ exp(iW),
Eq. [17], that governs the probability |Z|2
for the photon
propagation between the site of emission and the
site of absorption. Photons are being exchanged with
probability |Z|2
1. Photon propagation is thus not a
deterministic process, it exhibits some uncertainty
measure expressed by |Z|2
admitting the possibility
that an actual photon exchange happens, as illustrated
in Fig. 3, emission and subsequent reabsorption (the
appearance of a virtual photon) as well as admitting
the possibility that photons emitted are not
Figure 3 Pictorial representation of the photon propagator
with explicit time ordering according to Eq. [30]. For
x0
, y0
the photon is created in space-time point x and
propagates to space-time point y where it is annihilated
(right diagram). For y0
, x0
the photon is created in y
and propagates to x where it is annihilated (left diagram).
VIRTUAL PHOTONS IN MAGNETIC RESONANCE 281
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reabsorbed or photons absorbed have not been emitted
in the past.
In the scheme drawn here it seems that the virtual
photon apparently emerges out of nothing and disappears
into nothingness. This is, of course, not the
case, we just have isolated the interaction part—
interaction occurs between two current densities
jm
(x) and jn
(y) and the diagrams in Fig. 3 can be seen
as subdiagrams contained in the diagrams of Fig. 2.
So, it is more appropriate to say that virtual photons
are emitted by one current density and reabsorbed by
another (or the same) current density. Emission of
virtual photons changes the state of the particles that
constitute the current density, likewise the reabsorption
of virtual photons. In Fig. 2 we have indicated
that by a kink in the lines symbolizing the current
densities, appearing when emission or absorption
occurs. The particles that compose the current densities
in Eq. [20] might be conduction electrons in a
piece of metal wire being a part of a coil, or it might
be a spin particle in a specific spin state.
The photon propagator governs the electromagnetic
interaction between current densities. To see
this, we must turn back to Eqs. [20] and [31] and further
analyze the distribution dþððx À yÞ2
Þ, an essential
constituent of the Feynman propagator DF(xÀy)
and arising from Eq. [31]. As shown in lengthy detail
in Appendix C (see, Eqs. [C8] to [C17]), DF(xÀy)
turns out to be,
Dmn
F ðx À yÞ ¼ Àgmn
dþððx À yÞ2
Þ
¼ À
gmn
4p2
pdððx À yÞ2
Þ À i}
1
ðx À yÞ2
!" #
½33Š
As already mentioned in Section ‘‘The Feynman
Propagator’’ when introducing the probability amplitude
Z (Eq. [17]), we observe in Eq. [33] that
DF(x À y) is complex-valued; it consists of a real
part, given by Dirac’s d distribution, and of an imaginary
part, given by Cauchy’s principal value distribution
}(1/(x À y)2
). The appearance of both is elaborated
in Appendix C (see Eqs. [C1–C7]). As a preliminary
and very crude exposition, we can say that
the d distribution represents a singular function that
is zero everywhere except where its argument is
equal to zero—there it is divergent. The Cauchy principal
value distribution }(1/(x À y)2
) behaves like
the function 1/(x À y)2
, for (x À y)2
approaching 0 it
is also divergent. We have to suspect, that such singular
‘‘functions’’ are not simply functions in the ordinary
sense, their singular behavior and other features
warrant a whole mathematical theory—their
treatment is part of the theory of distributions as
introduced by Schwartz (83) (see also Refs. 84 and
85). It would be far outside the scope of the present article
to elaborate on this theory here; we will only pick
those bits and pieces necessary to formulate the photon
propagator and other Green functions. Functions
or distributions with singularities are only one kind of
infinity or divergence characteristic of quantum electrodynamics
or, more generally, of quantum field theories.
In QED, expressions that reveal divergent
behavior can be submitted to procedures called
renormalization or regularization, which allow us to
calculate expectation values for physical quantities
like energy, linear momentum, angular momentum,
and others, which then turn out to be finite. One
relatively simple example for such a regularization
procedure, the ‘‘taming’’ of the singularities of d(x)
and }(1/x), is carried out in Appendix C.
In four-dimensional k space, the Fourier domain
of space-time, we find for the photon propagator (see
Appendix C, Eq. [C14]),
Dmn
F ðkÞ ¼ lim
e!0
gmn
k2 þie
¼ gmn
}
1
k2
 
Àipdðk2
Þ
 
[34]
In Eq. [34] we recognize, comparing it with Eq.
[33], that the d and } distribution exchange their
roles as far as the real and imaginary parts are concerned.
We stress the point: Eqs. [33] and [34] represent
the central result of this section and will provide
us with the key characteristics for virtual photons.
For our following discussion of Eqs. [33] and [34]
it is worthwhile to introduce two technical concepts:
(a) the concept of a lightcone in space-time and (b)
the concept of a mass shell in momentum space. The
technical definition of the notions of a lightcone and
a mass shell is given in Appendix A (in particular,
see Figs. A1 and A2), where it is shown that a fourvector
u given in space-time can be one of three
kinds, depending on its norm-square u2
. If u2
is positive,
then vector u is referred to as time-like. For
coordinate vectors x this means x2
¼ c2
t2
À r2
> 0,
from which follows c2
t2
> r2
. In other words, timelike
coordinate vectors in space-time refer to propagation
over distances r with a propagation speed
below the speed of light, c, i.e., subluminal propagation,
where r , c|t|. These vectors refer to points
inside a region in space-time that forms a double
cone with its apex at the coordinate origin (Fig. A1).
There are time-like vectors in the forward cone for
which t . 0 and in the backward cone for t , 0. Furthermore,
there are vectors u in space-time, called
space-like vectors, for which u2
, 0 holds. For coordinate
vectors this becomes x2
¼ c2
t2
À r2
< 0,
consequently c2
t2
< r2
, referring to superluminal
282 ENGELKE
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propagation. Finally, there are vectors u with u2
¼ 0,
called light-like vectors. Coordinate vectors x with x2
¼ 0 are precisely those vectors that lie on the surface
of the lightcone and it holds c2
t2
¼ r2
, i.e., propagation
with luminal speed c.
The k space is characterized by the same metric
properties as space-time. Thus, the Fourier domain of
the time dimension x0
¼ ct becomes the k0
dimension,
which is frequency or energy. Likewise, the
three spatial dimensions needed to specify a threedimensional
position vector x have their Fourier domain
counterpart in three k space dimensions with
the momentum vector k. The relationship k2
¼ 0, or
equivalently o ¼ 2pc/l with frequency o and wavelength
l of a freely propagating electro-magnetic
wave or free photons (see Appendix A, Eqs. [A11,
A12]) defines a spherical shell with radius k0
(Appendix
A, Fig. A2) in three-momentum space. Those
photons that satisfy the energy-momentum relationship
k2
¼ 0 are called on-shell, otherwise off-shell.
Likewise, photons that travel exactly on the surface
of the lightcone are called to be on the lightcone,
otherwise off the lightcone.
Now let us examine in detail the expressions in
Eqs. [33] and [34]. In Eq. [33] we have the sum of
two terms. The first term contains d((xÀy)2
), which
means that it contributes only for (xÀy)2
¼ 0, (where
Dirac’s d function becomes singular) to integrals like
[21] or [22], i.e., the space-time difference vector
xÀy is light-like, hence it describes propagation of
photons on the lightcone, propagation with the speed
of light, c. The second term in Eq. [33] constitutes
Cauchy’s principal value }(1/((xÀy)2
)). This distribution
is singular for (xÀy)2
¼ 0 (i.e., on the lightcone),
but it is also different from zero for (xÀy)2
,
0 as well as for (xÀy)2
. 0, i.e., for time-like as well
as for space-like distances xÀy in space-time. Hence
this term formally describes propagation off the
lightcone! Turning our attention to Eq. [34], we find
again two terms: d(k2
) contributing only for k2
¼ 0,
i.e., corresponding to photons on-shell, and }(1/k2
)
being singular for k2
¼ 0, but contributing for k2
= 0
as well, hence allowing photons to be off-shell. We
also recognize that in Eq. [33] the real part of the
photon propagator, Re(DF(xÀy)), contains d((xÀy)2
)
while the imaginary part, Im(DF(xÀy)), contains
}(1/((xÀy)2
)). In momentum space it is just the other
way around, Re(DF(k)) gives }(1/(k2
)) while
Im(DF(k)) contains d(k2
).
These findings are illustrated in Fig. 4. In this figure
propagation on the lightcone (with d((xÀy)2
)) is
symbolized by a sharply drawn diagram. In contrast,
propagation off the lightcone [characterized by
}(1/((xÀy)2
))] is drawn as a slightly fuzzy lightcone
symbolizing that propagation may deviate from the
lightcone surface. Likewise in momentum space:
being off-shell (with }(1/(k2
))) is symbolized by a
sphere with unsharp boundary, being on-shell (d(k2
))
by a sharply drawn sphere. We further recognize that
a sharp lightcone corresponds to an unsharp momentum
sphere and vice versa, a sharp momentum sphere
has an unsharp lightcone as its Fourier counterpart.
We summarize these results once more in Table 1
where we also provide a first interpretation. As discussed
before with Eq. [32], the photon propagator
DF yields virtual photons, i.e., photons that are emitted
and reabsorbed, and between these two events
they propagate through space arbitrating the interaction
between emitting and absorbing current densities.
But now apparently we have found two kinds
of photons to which we referred to in Table 1 as
Figure 4 Pictorial illustration of the Fourier transform
relationships as expressed in Eqs. [33 and 34] and Table
1. The real part of DF(x) characterizes propagation strictly
on the lightcone indicated by a sharply drawn lightcone
surface, corresponding to off-shell positioning in k space,
as given by the real part of DF(k), symbolically indicated
by a sphere with fuzzy surface. Likewise, off-lightcone
propagation described by the imaginary part of DF(x) is
illustrated by an unsharp lightcone, and on-shell positioning
as given by the imaginary part of DF(k) pictured by a
k sphere in 3D momentum space with sharp radius k0.
Re(DF(x)) and Re(DF(k)) are Fourier transforms of each
other. The same holds for Im(DF(x)) and Im(DF(k)) also
forming a Fourier transform pair.
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virtual photons and asymptotically free photons,
either being on the lightcone or off the lightcone in
space-time, or equivalently, being off-shell or onshell
in k-space, respectively. In a general setting, the
distinction of virtual photons and asymptotically free
photons on the basis of analyzing Eqs. [33, 34] has
been suggested in a article by Castellani, et al. (94).
The peculiarity of two types of photons needs an
explanation that involves (i) Heisenberg’s uncertainty
relations for the virtual photon exchange and (ii) an
iteration of the argument related to the imaginary
part of DF(xÀy) to explain the probabilistic appearance
of asymptotically free photons.
Ad (i) In a general setting, as regards the uncertainty
relations for quantum mechanical observables
expressed by (Hermitian) operators F and G acting
on state functions f that are elements of a Hilbert
space, we have the general relationship
DFDG ! ð1=2Þjh½F; GŠij
for the standard deviations (uncertainties) DF ¼
||(FÀhFi) f || and DG ¼ ||(GÀhGi) f ||, which follows
from Schwarz’ inequality valid for state functions
in Hilbert space (e.g., Ref. 98, pp. 191). The
notation h. . .i indicates the quantum expectation
value and [F, G], as usual, denotes the commutator
of the two operators F and G. The symbol ||f|| stands
for the norm (which is a real, positive number for f
= 0) of the state function f. Hence the expression
||(FÀhFi) f || denotes the norm of the state function
that we get as a result when applying the operator
(FÀ hFi) to f. As an example, we could take the
position coordinate X (as an operator) and the
operator of three-momentum component Px and with
the commutator ½X; PxŠ ¼ i"h we arrive at the
well-known position-momentum uncertainty relation
DXDPx ! "h=2.
If we want to apply the same ‘‘recipe’’ to the physical
quantities energy and time to infer how energy and
time uncertainties are related to each other, we face a
serious obstacle: while energy E appears as an operator,
it is the Hamiltonian of the system considered,
time t is just a parameter, a coordinate in space-time.
There is no time operator, neither in orthodox nonrelativistic
quantum mechanics nor in quantum electrodynamics!
Nevertheless, the question of the validity of an
energy-time uncertainty relation DEDt ! "h=2 is
entirely reasonable as long as we can provide a physical
meaning for the time interval Dt. To further clarify
this meaning of Dt, let us return to the accepted
uncertainty relation DE DG ! (1/2) |h[H,G]i | involving
the energy uncertainty DE, with G equal to an arbitrary
Hermitian operator, and H equal to the Hamiltonian.
Denoting by _G the operator for ‘‘change of G
over time,’’ then the equation of motion reads
_G ¼ ði="hÞ½H; GŠ þ qG=qt. If we suppose that G does
not depend explicitly on time, then _G ¼ ði="hÞ½H; GŠ
and we are allowed to write DEDG ! ð"h=2Þh _Gi.
Now, due to Ehrenfest’s theorem (Ref. 98, p. 210)
it holds h _Gi ¼ dhGi=dt, such that DEDG !
ð"h=2Þjdi=dtj. Let us define the time duration
Dt ¼
DG
dhGi
dt






[35]
which can be understood as that time interval it takes
for the expectation value hGi to change in time by a
value as large as the uncertainty DG, or to change
over time and take all values within the range DG.
With this definition for the time interval Dt, it directly
follows the energy-time uncertainty relation
DEDt ! "h=2. Now with Eq. [35] providing a meaning
to the ‘‘time uncertainty’’ Dt, we may claim, for
example, that DEDtx ! "h=2 where in this case G
represents the position operator X (for one dimenTable
1 Characteristics of the Real and Imaginary Parts of the Feynman–Green Function in Space-Time and k
Space Representation and Their Interpretation in Terms of Virtual and Asymptotically Free Photons
Feynman-Green Function DF Real Part of DF Imaginary Part of DF
Minkowski space-time,
coordinates x, y
À 1
4p dððx À yÞ2
Þ does propagate
on the lightcone
þ 1
4p2 } 1
ðxÀyÞ2
 
may propagate
off the lightcone
Four-dimensional momentum
space, Coordinates k
þ}ð1=k2
Þ positioned off-shell Àpd(k2
) positioned on-shell
Interpretation Virtual photons travel on the lightcone,
i.e., (xÀy)2
¼ 0, but they may be off
zero-mass shell, k 2
= 0
Asymptotically free photons may
travel off the lightcone, i.e., (xÀy)2
= 0,
but they are on zero-mass shell, k 2
¼ 0.
Associated solution of the
wave equation
Inhomogeneous solution,
interacting fields
Homogeneous solution, free fields
284 ENGELKE
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sion), further DG ¼ r is given by a distance, such that
for a virtual photon propagating with velocity
dhGi=dt ¼ dhXi=dt ¼ c on the lightcone we arrive at
Dtx ¼ r/c and may interpret Dtx as the average lifetime
(the propagation time) between emission and
absorption of the virtual photon with the associated
energy uncertainty DE. Nonetheless we emphasize,
that this is just one possibility depending on the
choice of the operator G ¼ X. Alternatively we
may take G ¼ P with P equal to the three-momentum
(again in one dimension), then we obtain Dtp
¼ DP/Fp with the force Fp equal to the total time
derivative of the expectation value hPi for the momentum
P. As we see here, the duration Dtp refers
to the time derivative of the expectation value of
operator P and thus has a different definition and
meaning as compared to Dtx, because now it is
related to the operator P instead of X. Summarizing,
the time duration Dt appearing in the energy-time
uncertainty relation DEDt ! "h=2 depends on the
choice of the operator G, generally noncommuting
with the Hamiltonian H. For short, G specifies Dt,
see Eq. [35].
It should be clear by now that the question of the
energy uncertainty DE for a virtual photon depends
on the second observable G that we may want to consider
for a measurement, as an experimental boundary
condition. Suppose we know the precise distance
r (obtained by some measurement) between emission
and absorption, the energy uncertainty for the virtual
photon is equal to DE ! "hc=ð2rÞ. But we keep in
mind that this is only one possibility. For example, as
we will investigate later in Section ‘‘Spin Current
Density, Zeeman Hamiltonian, and Larmor Precession’’
when looking at Larmor precession through the
QED view frame, we may face the situation that, for
some additional reason, DE goes to zero, i.e., we
may know that the virtual photon has a sharp or certain
energy. For this case it follows that we are not
allowed to interpret the duration Dt as arising from a
certain propagation distance r, instead we have to
take Dtp ¼ DP/Fp with the uncertainty DP of the
photon’s three-momentum going towards infinity, or
Fp going to zero, in such a way that still
DEDtp ! "h=2 holds. The fact that here in our example
we focus on the three-momentum P as possibly
being entirely uncertain (DP becoming infinite) with
an associated time interval Dtp going towards infinity,
does not contradict the situation that the virtual
photon travels the propagation distance r during the
time interval r/c ¼ Dtr on the lightcone. It is just that
Dtp = Dtr and in the case of sharp energy E the
energy-time uncertainty relation has to be written
down with the time interval Dtp, not Dtr. As we also
can see, there is no contradiction with the statement
that virtual photons are off-shell: the photon energy
may have a sharp value, i.e., the zero-mass shell
has a sharp radius, while the three-momentum is
entirely uncertain. For a discussion on the notion of
the uncertainty with respect to the Fourier transformation
in relation to magnetic resonance we refer
to Ref. 86.
Ad (ii) As discussed earlier with Eqs. [31, 32] and
with the introduction of the probability amplitude Z,
the Feynman propagator as expressed in Eq. [33] also
includes the transition for a photon being emitted and
not being reabsorbed yet (and maybe it never will be
reabsorbed) within a given time interval, or a photon
being absorbed but with the uncertainty at which
time it has been emitted (and, perhaps, it never was).
The positivity requirement for the photon energy led
to a complex-valued action functional W. The associated
probability amplitude for photon propagation is
equal to Z ¼ exp(iW) where the probability character
is governed by the imaginary part of W. Table 1 lists
virtual photons as those which, within a given time
interval x0
Ày0
, are emitted and reabsorbed. All other
photons are asymptotically free in the sense that either
emission or absorption has not taken place yet
within this specified time interval. In the literature
these asymptotically free photons are sometimes
referred to as real photons. We prefer not to use the
term ‘‘real photon’’ here, because in real interactions
virtual photons appear to be just as real as asymptotically
free photons. Nevertheless, as pointed out in
expression [31], the Feynman propagator DF(xÀy) as
a whole characterizes the travelling path or trajectory
of photons, but these trajectories appear to be ‘‘fuzzy
trajectories,’’ the fuzziness given by the term }(1/
((xÀy)2
)), which is the imaginary part in the action
functional W. This uncertainty does not only cover
the trajectory between their beginning (emission) and
end points (absorption), it also includes these initial
and end ‘‘points’’ themselves. When discussing Eq.
[16], where we have restricted the frequency or
energy range for photons to be positive (including
zero), thus making a restriction in k0 space by reducing
the energy value range from À1 . . . þ1 to a
smaller subrange of 0 . . . þ1, the price to pay when
doing this is that we got time dispersion, the associated
uncertainty is Dt as specified by Eq. [35]. An
increasing uncertainty Dtr for the propagation time
also includes the possibility that within a given time
interval no emission or no reabsorption may take
place. If this happens to be the case for increasingly
long, or asymptotically long time intervals, then the
corresponding photons are asymptotically free, either
when looking towards the past (backward lightcone)
VIRTUAL PHOTONS IN MAGNETIC RESONANCE 285
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or looking towards the future (forward lightcone).
Thus the photon propagator DF includes or admits
the possibility, that exchanging photons escape from
their fate of otherwise being caught as truly virtual
and going forth and back between the interaction
partners. The probability for a photon to succeed in
escaping to ‘‘asymptotic freedom’’ is different from
zero and can be computed from DF and by taking the
interacting current densities into account. We will
perform the calculation of such an escape probability
(Eq. [57]) when discussing a specific NMR probe
model in the next section.
Last but not least, with the dichotomy in Table 1
we are reminded of the distinction well-known in
classical electromagnetism: near field and far field.
The near field is considered as the electromagnetic
field in a region with distances from the source below
(or small compared to) one wavelength for a given
field mode. The far field is found at distances at or
larger than one wavelength. There is no sharp boundary
between near and far field, the transition from
near to far field is gradual. Asymptotically free photons
may escape to the far field, or more precisely,
they may constitute the far field. Likewise, virtual
photons should belong to the near field. The interpretation
of virtual and asymptotically free photons outlined
above will be applied to all electromagnetic
phenomena discussed in the following.
From Table 1 we may derive a useful means of recognizing
the signature of the presence of virtual photons.
Apart from using the general definition that in
QED virtual photons mediate the electromagnetic interaction,
the following indicator points to the appearance
of virtual photons: the free-photon energy-momentum
relationship k2
¼ 0 does not hold (see Appendix A,
Eq. [A10] and Fig. A2), i.e., virtual photons are offshell.
Let us review briefly some examples:
(i) In structures with wave compression (which
means k2
, 0) like helical waveguides or solenoidal
coils (87), there it is the interaction of
one part of the structure with another part of
the same structure, for instance two neighboring
coil turns or a space-periodic repetition of
structural elements like coil turns, that signifies
an exchange of virtual photons. In this
sense, wave compression is understood as
l=c < 2p=o, i.e., we have o2
=c2
< 4p2
=l2
and consequently we arrive at
o2
=c2
À 4p2
=l2
¼ ðk0
Þ2
À jkj2
¼ k2
< 0:
Classically we may say that the geometric
boundary conditions of wave propagation are
different from those in free space. Using the
language of QED we may say that virtual photons,
appearing because of geometric boundary
conditions not present in free space, govern
the wave propagation, which is different
from the free space propagation characterized
by asymptotically free photons.
(ii) Another example where k2
= 0 occurs in
electromagnetic fields interacting with
dielectric materials. Here, we encounter
interactions between the electromagnetic
field and the bound electric charges or electric
dipoles in these materials. Macroscopically
we find again wave compression: for
example, a standing wave in an resonating
transmission line filled with a dielectric material
with er . 1 experiences a compression
of its effective wavelength.
(iii) Static interactions typically lead to k0 ? 0
(see our discussion of the Larmor precession
in Section ‘‘Spin Current Density, Zeeman
Hamiltonian, and Larmor Precession’’), i.e.,
the zero-mass shell degenerates to a point in
k space and every photon with |k|2
. 0
must be virtual.
A QED NMR PROBE MODEL: PULSED
NMR AS A SCATTERING PROCESS
In the previous two sections the basic understanding
of how to interpret the Feynman-Green function, or
synonymously, the photon propagator has been introduced.
In the following, we want to apply and illustrate
the basic concepts in a model that has relevance
for pulsed NMR spectroscopy. Among all the parts
of an NMR spectrometer it is the NMR probe whose
design is based on the interaction between the spins
in a sample and the radiofrequency (rf) electromagnetic
field produced or received by the probe. In their
2002 article (55) on pulsed magnetic resonance with
full quantization of the rf field, Jeener and Henin propose
a probe model that is well suited to the task of
describing NMR in QED terms. Their model decomposes
the rf electromagnetic field into two parts: the
free field and the bound field. The ‘‘free’’ field corresponds
to the (undamped) eigenmodes of the probe
circuitry including the transmission line connecting
probe with the transmitter or the receiver. The bound
field, in Jeener and Henin’s terms, is associated with
the NMR sample. To avoid modeling the losses that
occur in any real probe, which would lead to dissipation,
the probe and the transmission line are
286 ENGELKE
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assumed to be loss-free but with a length of the transmission
line (whose characteristic impedance acts as
a load to the circuit) being sufficiently large to ensure
a large propagation time for the pulse. As a consequence
such an incident rf pulse—propagating from
the transmitter to the NMR coil or resonator (containing
the sample), reflected there and travelling
back as an outgoing pulse—can be considered as an
event like a bypassing perturbation and, within a certain
time interval in the model, the pulse is supposed
to not re-appear in the probe. In this sense, we are
actually allowed to view the rf pulse as being scattered
by the spins in the sample with the incoming
pulse as arising from some asymptotic initial condition
in the past and the outgoing pulse disappearing
with some asymptotic final condition in the future.
This strongly idealized model is sketched in Fig. 5.
On the basis of the decomposition into a free field and
a bound field, Jeener and Henin provide a discussion
of the quantization of the electromagnetic field within
this model by setting up an ad hoc Hamiltonian that
provides a link between the fully quantized scheme
with the classical scheme based on Bloch’s equations.
Although we will make use of the above principal
probe model here as well, we will choose a different
approach to introduce the QED view into NMR. We
want to consider the following situation: first, let us
suppose that the electromagnetic four-potential associated
with the incident pulse, denoted by Am
in, is
given as a quantum field. Please remember that the
superscript m in Am
in is a contravariant index that
counts components, m ¼ 0, 1, 2, 3, while here the
subscript ‘‘in’’ indicates the incoming field.
Second, when the incoming pulse arrives at the
NMR sample region inside the NMR coil, it starts
interacting with the spins inside the sample. Third,
after the pulse travelled through the NMR coil and
sample region, it leaves as outgoing pulse, where
now, of course, the field of the outgoing pulse, symbolized
by Am
(t,x), has been changed as compared to
Am
in. It is as if the incident pulse has collided with or
has been scattered by the spins in the sample, the latter
represented by a current density jn
(y), and the
associated electromagnetic interaction has changed
the overall field. Since we have not allowed dissipation
in our model, we may claim that the scattered
field Am
(t,x) is related to the field Am
in of the incident
pulse by a unitary transformation,
Am
ðt; xÞ ¼ UðtÞAm
inUþ
ðtÞ [36]
with the field operator Ain in [36] referring to the
incoming electromagnetic pulse an (asymptotically)
long time before it arrives in the spin sample region.
The time evolution operator U(t) is given as the
unitary operator
Figure 5 Schematic model of an NMR probe for pulsed NMR similar to those proposed by
Jeener and Henin in Ref. 55. An incident rf pulse generated by the rf transmitter travels over the
transmission line, through the circuit to the NMR coil or resonator where it is reflected and travels
back through the circuit and transmission line. The NMR sample containing the spins (not shown)
is situated inside the coil or resonator and represents a spin current density which interacts with the
incident pulse. The result of this interaction is the outgoing pulse followed by the FID signal, the
latter is routed to the receiver (this usually happens in the preamplifier, which is here thought as
part of the receiver). The coil/resonator and the circuitry are enclosed by a shielding tube. The rf
electromagnetic field can only enter and exit through the long transmission line.
VIRTUAL PHOTONS IN MAGNETIC RESONANCE 287
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
UðtÞ ¼ T exp Ài
Zt
À1
dt0
Hintðt0
Þ
0
@
1
A [37]
with the interaction Hamiltonian
Hintðt0
Þ ¼
Z
d3
yAm
inðy; t0
Þjmðy; t0
Þ [38]
where now Am
inðy; t0
Þ refers to the field during the
scattering process in the region characterized by
position vectors y for the spins in the sample. Dyson’s
time-ordering operator T, defined by Eqs. [30],
appears in Eq. [37] because, in general, the interaction
Hamiltonian does not commute with itself at different
time instants. We have introduced the interaction
between spins and electromagnetic field in a
fairly general manner. The product Am
inðy; t0
Þjmðy; t0
Þ
in the integrand in Eq. [38] representing a Hamiltonian
density describes in a universal manner the electromagnetic
coupling between any current density
and the electromagnetic field, as we have convinced
ourselves in Section ‘‘The Feynman Propagator,’’ Eq.
[23]. To specialize to the case of magnetic resonance,
one has to characterize the current density jn
ðy; t0
Þ
that represents the spins. We will derive that spin
current density later on in Section ‘‘Spin Current
Density, Zeeman Hamiltonian, and Larmor Precession,’’
where we also show that jn
ðy; t0
Þ may represent
a function although we treat the spin particle
quantum mechanically. So far, we have taken
account of the action of the rf field pulse Ain(x) to the
spin system and the action of the spin system on the
rf field, during the pulse. After the pulse has propagated
through the sample region, then some pastpulse
response of the spin system—the free induction
decay, FID—appears. For the case of a single-spin
FID, we look more closely in Section ‘‘Single-Spin
FID: NMR Radiation Damping.’’
Now Eq. [36] is supposed to tell us in detail how
the electromagnetic four-potential of the incident
pulse is changed by the interaction with the spins in
the sample or, so-to-say, by the scattering process.
The time evolution in U(t) covers a time interval
from an (infinitely) far past or, in practice, a sufficiently
far past when there is not yet any interaction
between incoming pulse and spin system, yet, lasting
to a time instant labeled by t. This instant t can be situated
in the interval during which the interaction
happens, but also before or afterwards, taking into
account that the Hamiltonian density Am
inðy; t0
Þjmðy; t0
Þ
itself is dependent on space position and time. In the
following, starting from Eq. [37], we want to derive
an expression for the time evolution operator that
contains the photon propagator DF, such that we may
be set in a position to analyze the appearance of virtual
and asymptotically free photons within the probe
model for pulsed magnetic resonance. To achieve
this goal, we will proceed by focusing on the incoming
and outgoing pulse, which propagate through the
rf coil or resonator and interact with the spins.
The unitary time evolution operator U(t) transforms
the incident quantum electromagnetic field
pulse Am
in into a quantum field that includes the effect
of propagation through the probe and also includes
interactions with the current density jn
ðy; t0
Þ of the
spins. Hence, taken from Eqs. [37, 38], we begin with
UðtÞ ¼ T exp Ài
Zt
À1
dt0
Hintðt0
Þ
0
@
1
A
¼ T exp Ài
Zt
À1
dt0
Z
d3
yAn
inðy; t0
Þjnðy; t0
Þ
0
@
1
A ½39Š
To introduce the virtual-photon picture by re-establishing
the role of DF in the time evolution operator
[39], we have to consider a factorization of U(t) that
allows us to view time ordering (as signified by
Dyson’s time ordering operator T) of operators of the
electromagnetic field as retardation in the propagation
process of electromagnetic fields. After performing
this first step and familiarizing ourselves with the concept
of normal ordering of field operators, we will be
in a position to derive an expression for U(t), which
contains the photon propagator DF. These steps, beginning
with the factorization of U(t), are presented in
detail in Appendix D; the result of the first step (see
Eqs. [D1–D10]), which relates U(t) to retardation, is
UðtÞ ¼ exp Ài
Zt
À1
dx0
Z
d3
xAn
inðxÞjnðxÞ
0
@
1
A
 exp i
2
gmn
Zt
À1
dx0
Z
d3
x
0
@
Â
Zt
À1
dy0
Z
d3
yjmðxÞDretðx À yÞjnðyÞ
1
A ½40Š
One crucial assumption made in the derivation of
Eq. [40] is that the current densities jm(x) and jn(y)
are functions, not operators. This may happen, if we
would treat them as classical current densities. But
also in the case of particles obeying ordinary quantum
mechanics (first quantization), current densities
are functions as we will see later on. The assumption
288 ENGELKE
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
of current densities being functions, not operators,
restricts the generality of [40] and all expressions that
depend on it. The consequences will be discussed at
the end of this section and more thoroughly in Section
‘‘Interaction of a Spin-1/2 Particle with External
Time-Harmonic Fields.’’
Equation [40] represents a noteworthy result.
Comparing it with Eq. [39], it tells us that time-ordering,
symbolized by the time-ordering operator T in
Eq. [39], translates into a multiplicative exponential
term in Eq. [40] with an integral in the exponent containing
the retarded propagator. Thus time-ordering
transforms into time-retardation. The first exponential
term in Eq. [40] contains the unitary operator part,
with the Hermitian field operator Ain(x) in the exponent.
Furthermore, in the interpretation of Eq. [39] we
were clear about the physical meaning of the field operator
Am
in(y) and the current density jm(y). The operator
Ain(y) stood for the four-potential of the quantized
incident field pulse within the spatial region y of the
spin particles while j(y) represented the current density
of the spin particles themselves. In the second
term of Eq. [40] there are now two current densities,
j(x) and j(y), the interaction between both mediated
through the retarded propagator Dret(xÀy). Although
jm(y) still represents the spins in the sample, we now
have to admit that j(x) must signify the rf current in
the circuit and the coil of the probe. The retarded
four-potential of the incident pulse,
An;retðxÞ ¼
1
2
Zt
À1
dy0
Z
d3
yDretðx À yÞjnðyÞ; [41]
acting in the spatial region x where the rf current density
is to be found, is generated by the spin current
density j(y). Hence Eq. [40] could be written as
UðtÞ¼exp Ài
Zt
À1
dx0
Z
d3
xðAn
inðxÞÀAn
retðxÞ1ÞjnðxÞ
0
@
1
A
[42]
with the field (operator) Ain(x) generated by the rf current
and Aret(x) produced by the spin current and
being retarded. 1 denotes the unity operator. An important
observation should be made, namely, the
four-potential Aret(x) expressed by Eq. [41] is not an
operator of the electromagnetic field anymore—in
terms of field variables it represents a function. The
detailed properties of jn (except that they have to be
functions) are still left open and will be treated below.
The field operator part for the time evolution operator
characterizing the quantum evolution of the electromagnetic
field is taken over by the first exponential
term in Eq. [40].
So far we have managed to express U(t) in [40]
via the retarded propagator Dret, indicating the inherent
time-ordering requirement in Eq. [39]. Proceeding
further from Eq. [40], U(t) can be expressed via
the photon propagator DF. For that purpose we define
the shorthand notation : . . . : by
: exp Ài
Zt
À1
dx0
Z
d3
xAn
inðxÞjnðxÞ
0
@
1
A :
¼ exp Ài
Zt
À1
dx0
Z
d3
xAnÀ
in ðxÞjnðxÞ
0
@
1
A
 exp Ài
Zt
À1
dx0
Z
d3
xAnþ
in ðxÞjnðxÞ
0
@
1
A ½43Š
called normal-ordered product, where we have used (see
Eq. [24]) the positive-frequency part including the photon
annihilation operator of the four-potential, given as
A
mðþÞ
in ðxÞ ¼
Z
d3
k
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2pð Þ3
2ok
q
X3
l¼0
emðlÞ
ðkÞaðlÞ
ðkÞeÀikx
[44]
and the negative-frequency part with the photon creation
operator,
A
mðÀÞ
in ðxÞ ¼
Z
d3
k
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2pð Þ3
2ok
q
X3
l¼0
emðlÞ
ðkÞaðlÞþ
ðkÞeikx
[45]
From [44, 45] we recognize that when we expand
the two exponentials on the right-hand side of [43] in
a power series, normal ordering results in an arrangement
of creation and annihilation operators in product
terms such that all creation operators aþ
are always to
the left of all annihilation operators a appearing in
that product. Thus, products like aþ
a or aþ
aþ
a are
normal-ordered, while aaþ
or aþ
aaþ
are not.
With these definitions, as shown in detail in Appendix
D, Eqs. [D11–D17], Eq. [40] leads to
UðtÞ ¼ : exp Ài
Zt
À1
dx0
Z
d3
xAn
inðxÞjnðxÞ
0
@
1
A :
exp À
i
2
Zt
À1
dx0
Z
d3
x
Zt
À1
dy0
0
@
Â
Z
d3
yjmðxÞDmn
F ðx À yÞjnðyÞ

½46Š
VIRTUAL PHOTONS IN MAGNETIC RESONANCE 289
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Equation [46] formally achieves our goal of
expressing the time evolution operator U(t) via the
photon propagator DF(xÀy). However, as written in
Eq. [46], we have some preliminary price to pay:
U(t) on the left-hand side is supposed to be a unitary
operator. The exponential containing DF in the exponent
is not an operator or operator function related to
the electromagnetic field variables and the normal-ordered
exponential operator : . . .: taken separately by
itself is not unitary. For the sake of demonstration,
how can we convince ourselves that the entire righthand
side of Eq. [46] fulfills the condition of unitarity?
For that purpose we insert Eqs. [33, 43] into
[46], such that we see all separate factors explicitly
written out,
UðtÞ¼exp Ài
Zt
À1
dx0
Z
d3
xAnÀ
in ðxÞjnðxÞ
0
@
1
A
Âexp Ài
Zt
À1
dx0
Z
d3
xAnþ
in ðxÞjnðxÞ
0
@
1
A
Âexp þ
i
2
Zt
À1
dx0
Z
d3
x
Zt
À1
dy0
0
@
Â
Z
d3
yjmðxÞ
gmn
4p2
pdððxÀyÞ2
ÞÀi}
1
ðxÀyÞ2
!" #
jnðyÞ
!
In the third exponential we detach the real and
imaginary parts of DF into separate factors—this is
possible if j(x) and j(y) commute—such that we obtain
UðtÞ ¼ exp Ài
Zt
À1
dx0
Z
d3
xAnÀ
in ðxÞjnðxÞ
0
@
1
A
 exp Ài
Zt
À1
dx0
Z
d3
xAnþ
in ðxÞjnðxÞ
0
@
1
A
 exp þ
1
2
Zt
À1
dx0
Z
d3
x
Zt
À1
dy0
0
@
Â
Z
d3
yjmðxÞ
gmn
4p2
}
1
ðx À yÞ2
!
jnðyÞ
!
 exp þ
i
2
Zt
À1
dx0
Z
d3
x
Zt
À1
dy0
0
@
Â
Z
d3
yjmðxÞ
gmn
4p
dððx À yÞ2
ÞjnðyÞ

½47Š
We see that that the imaginary part of DF (containing
i times the principal value distribution })
gives rise to an exponential with real-valued exponent.
Introducing the amplitudes
aðk; x0
Þ ¼ ÀiemðlÞ
ðkÞJmðk; x0
Þeikx0
;
aðk; tÞ ¼
Zt
À1
dx0
aðk; x0
Þ ½48Š
containing the three-dimensional k space Fourier
transform of the four-current density,
Jmðk; x0
Þ ¼
Z
d3
x expðik Á xÞjmðx; x0
Þ [49]
and confining ourselves to the case where the currents
generate only a single mode k of the electromagnetic
field with polarization l, we may derive
from [47] (see Appendix D, Eqs. [D18–D25]
UðtÞ ¼ exp aðk; tÞaþ
ðkÞ À aÃ
ðk; tÞaðkÞð Þ
 exp
i
8p
Zt
À1
dx0
Z
d3
x
Zt
À1
dy0
0
@
Â
Z
d3
yjn
ðxÞdððx À yÞ2
ÞjnðyÞ
1
A ½50Š
where we have omitted the integral over momentum
space and where now k is referring to one specific
field mode. The restriction to one field mode is not
really a strong restriction of generality here, because
in all terms and expressions of U that contain the
amplitudes a(k,t) we may reintroduce multiple
modes by writing again the integral over d3
k, see
Appendix D, Eqs. [D24, D25]. Nevertheless writing
down the expressions only for one mode makes the
notation less bulky and easier to read. The first exponential
in Eq. [50] represents Glauber’s displacement
operator (see Refs. 4, 5, and 97),
DaðtÞ ¼ exp aðk; tÞaþ
ðkÞ À aÃ
ðk; tÞaðkÞð Þ
¼ expðaaþ
Þ expðÀaÃ
aÞ expðÀaaÃ
=2Þ ½51Š
where on the right-hand side in the second equation
of [51] we have omitted the arguments k and t. Thus
for Eq. [50] we may write
UðtÞ ¼ DaðtÞ exp
i
8p
Zt
À1
dx0
Z
d3
x
Zt
À1
dy0
0
@
Â
Z
d3
yjn
ðxÞdððx À yÞ2
ÞjnðyÞ
1
A ½52Š
290 ENGELKE
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
So starting from the general, time-ordered expression
[39], passing through Eq. [40], which provides a
link to the retarded propagator, arriving first at Eq.
[46], which expresses U(t) via the photon propagator,
we finally arrive at Eq. [52], which represents U(t) as
a product of Glauber’s displacement operator Da(t)
and an exponential that contains the real part of the
photon propagator, i.e., it contains the d function
d((xÀy)2
) indicating photon propagation on the lightcone.
The imaginary part of DF has been subsumed
into the operator Da(t). Hence we are led to the conclusion
that Glauber’s displacement operator represents
asymptotically free photons (being on-shell)
appearing in the incoming or outgoing rf pulse, while
the exponential containing the real part of DF stands
for the virtual photons being exchanged between the
current density in the circuit, resonator, or coil and
the spin current density. By power series expansion
of the displacement operator Da(t) we can easily convince
ourselves that the adjoint operator associated
with Da(t) is equal to Dþ
a ðtÞ ¼ DÀaðtÞ and from Eq.
[51] it follows then, that DaDþ
a ¼ Dþ
a Da ¼ 1, i.e.,
the displacement operator Da is, indeed, unitary. For
multiple modes this argument stays valid, because
creation and annihilation operators, belonging to different
modes, commute (Eq. [25]), which allows us
to factorize Da and each factor Dak for each mode k
is unitary by itself. Also by using the power series
expansion for the unitary operator Da, we obtain the
commutator relationships ½aþ
; DaŠ ¼ aÃ
Da; ½a; DaŠ ¼
aDa. These yield, when left-multiplying by Da
þ
:
DaðtÞaþ
Dþ
a ðtÞ ¼ aþ
À aÃ
ðtÞ;
DaðtÞaDþ
a ðtÞ ¼ a À aðtÞ ½53Š
From Eqs. [53] the name displacement operator
for Da becomes clear: performing a unitary transformation
of the photon creation operator aþ
or the photon
annihilation operator a by means of Da results in
a displacement by a* or by a, respectively. The Fourier
expansion of field operators, Eq. [24], and the
property of Da expressed by [53] lead us to conclude
DaðtÞAm
ðt;xÞDþ
a ðtÞ ¼ Am
ðtþt;xÞÀAm
a ðt;xÞ;
Am
a ðt;xÞ ¼ aðk;tÞeÀikx
þaÃ
ðk;tÞeikx
; x ¼ ðct;xÞ ½54Š
Thus time evolution via Da changes the quantum
field Am(x) by displacing the time coordinate and displacing
it by the field Am
a (x) that depends on the current densities
involved in the interaction. When we apply Da to
the photon vacuum state |0i, we create a new state (4)
jaðtÞi ¼ DaðtÞj0i [55]
In order to characterize this time-dependent state
|a(t)i, we note first that it can be expanded in a basis
of Fock states |ni as follows (4)
jaðtÞi ¼
X1
n¼0
hnjaijni ¼ eÀ1
2aaÃ
X1
n¼0
an
ðtÞ
ðn!Þ1=2
jni [56]
which can be verified by using the power series
expansion for Da(t) given in Eq. [51]. States |a(t)i
defined by Eq. [56] are called coherent states. Since
they represent a superposition of Fock states, they
are not eigenstates of the photon number operator,
i.e., coherent states are characterized by an uncertainty
of the photon number in the field. Another
interesting feature of these states is that |a(t)i represents
an eigenvector of the incident-field photon
annihilation operator a(k), aðkÞjai ¼ ajai, associated
with the time-dependent eigenvalue a(t). Note,
the photon annihilation operator a(k) is
not Hermitian and a(t) is a complex number. From
Eq. [56] we can compute the probability to find
exactly n photons in the field that is in coherent
state |ai, i.e., the square of the transition amplitude
hn|ai,
paðnÞ ¼ jhnjaij2
¼
eÀjaj2
jaj2n
n!
; jaj2
¼ aaÃ
¼ "n
[57]
We recognize in [57] a Poissonian distribution
with "n being the average number of photons in the
field being in coherent state |ai, while the width
(standard deviation) of this distribution is equal toffiffiffi
"n
p
. In mathematical statistics the Poisson distribution
appears as a limiting case of the Bernoulli or
binomial distribution
pn;z ¼
z
n
 
pn
ð1 À pÞzÀn
[58]
stating the probability pn,z for an event X to occur n
times in a number z of independent trials, where the
probability for X to occur in an individual trial is
equal to p. For the specific case where this individual
probability is very small, p ( 1, one obtains
pn;z ¼
ðzpÞn
n!
expðÀzpÞ [59]
which is identical to Eq. [57] when we set "n ¼ zp.
Hence we may interpret Eq. [57] as providing the
probability for the emission (the event X) of n photons,
e.g., in a number z of consecutive time interVIRTUAL
PHOTONS IN MAGNETIC RESONANCE 291
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
vals, or alternatively by a number z of independent
nuclear or electron spins, where the individual probability
p that one photon is emitted (in a given time
interval or by a given spin) is small compared to
unity. The average number of photons emitted is
equal to "n ¼ zp, hence this average number, according
to Eqs. [48, 49], depends on the current density in
three-dimensional momentum space. Moreover from
[57], the probability that no photons are emitted at all,
p0 ¼ expðÀ"nÞ; [60]
becomes larger, if the expectation or average value "n
of the photon number becomes smaller.
If we consider the time evolution (Eq. [36]) leading
from the incident field into the interacting and
outgoing field pulse, we realize that the exponential
in [52] containing the real part of DF commutes with
Am
inðt; xÞ, because it does not contain any operator
term that acts on electromagnetic field states. Thus
Eq. [36] becomes
Am
ðt; xÞ ¼ DaðtÞAm
inDþ
a ðtÞ [61]
Therefore Glauber’s displacement operator Da
alone governs the time evolution of the field operators
for the external incoming and outgoing field. Da
depends on the time-dependent quantity a, which in
turn depends on the k space current density Jmðk; x0
Þ,
Eqs. [48, 49], representing the current density
involved in the interaction of the incident rf pulse.
Da includes the imaginary part of the photon propagator
DF (see Eqs. [47–50]) that we connected with
on-shell and off-lightcone photons. So the field
Am
ðt; xÞ of the interacting and outgoing pulse,
according to Eq. [61], contains asymptotically free
photons emitted by either the spin current distribution
jm
(y) or by the rf current jm(x). The detailed
action of Da on the quantum field operator has been
shown already in Eq. [54]—the associated unitary
transformation leads to a displacement of the field
operator Am
in(x) by Am
a (x). Note, Am
a (x) is not an electromagnetic
field operator anymore.
Let us draw some conclusions from the findings in
the current section for the NMR probe model:
(i) The rf field of the incoming, interacting and
outgoing pulse is generated by the rf current
density in the circuit or resonator and interacting
with the spin current density located in
the sample. The time evolution operator for
the field operators is given by Eq. [39].
(ii) For non-operator current densities, j(x) and
j(y), we have found the single-mode electromagnetic
rf field pulse in a coherent state
|ai, characterized by an average photon
number "n that depends on the integral over
the current density in k space (Eqs. [48, 49])
and governed by a Poissonian probability
distribution [57].
(iii) Coherent states |ai are linear superpositions
of Fock states |ni (Eq. [56]), hence they are
not energy eigenstates; more precisely, they
are not eigenstates of the Hamiltonian for
the free electromagnetic field.
(iv) The time evolution [61] of field operators
with the electromagnetic field in a coherent
state |ai is governed by Glauber’s displacement
operator Da, Eq. [51], which contains
the normal-ordered product [43] and the
imaginary part =mðDFðx À yÞÞ of the photon
propagator (worked out in Eqs. [47–50] and
Appendix D). Therefore the action of Da as
time evolution operator for the electromagnetic
field originates from asymptotically
free photons (Section ‘‘Quantization of the
Electromagnetic Interaction Field: Virtual
Photons’’).
(v) As it turns out, coherent states |ai are
quasi-classical states. This means that the
probability distribution [57], characterized
by the average value "n and by the standard
deviation
ffiffiffi
"n
p
(Poisson distribution), leads to
a relative standard deviation offfiffiffi
"n
p
="n ¼ 1=
ffiffiffi
"n
p
which goes towards zero for
very large average photon numbers. For the
expectation value ha|(Am(x))2
|ai of the
square of the four-potential with the field in
coherent state |ai we find, according to [24]
and remembering |ai as an eigenstate of the
annihilation operator, as follows from Eq.
[56], that ha|(Am(x))2
|ai ! "n. Hence, the
relative standard deviation for Am(x) in the
coherent state is proportional to 1=
ffiffiffi
"n
p
. The
same applies to the electric field strength
and the magnetic induction field strength
computed from Am(x) (see Ref. 4). Therefore,
the field strengths take on sharp average
values, i.e., the relative standard deviations
vanish, for large average photon numbers,
i.e., in that case we are in the
classical limit.
Generalizing the above considerations to multiple
modes, we recognize that the electromagnetic field
with each mode in a coherent state |aki gives rise to
a product state jfi ¼
Q
k jaki which is still a
quasi-classical state in the sense that the relative
292 ENGELKE
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
uncertainties of observables vanish in the classical
limit, but it is not a coherent state as defined by Eq.
[56] anymore (that means that for |fi the Poisson distribution
does not apply). However, because |fi can
be written as a product of coherent states, it holds
that (a) a unitary time evolution operator can be
derived as Df ¼
Q
k Dak with Dak equal to Glauber’s
displacement operator for mode k and that (b) this
unitary operator Df acts still as the creation operator
for the quasi-classical state |fi from the vacuum state:
jfi ¼ Df j0i (Refs. 54, 55, and 97).
One major supposition made in the derivations of
the present section is that the current densities j(x)
and j(y) are functions, not operators. In our probe
model, j(x) represents the rf current density, and j(y)
represents the spin current density. More specifically:
j(x) corresponds to a current density of electrons in
an conduction band of a metal that makes up the
building material of coils or resonators and other circuit
elements, while the spin current density j(y) specifically
stands for the spin particles in the sample.
Before we address these issues in more detail in Section
‘‘Interaction of a Spin-1/2 Particle with External
Time-Harmonic Fields,’’ we will turn our focus in
the next section to the question of how to specify the
spin current density.
SPIN CURRENT DENSITY, ZEEMAN
HAMILTONIAN, AND LARMOR
PRECESSION
Although we have introduced electromagnetic fields
generated by sources, we have not treated in any
detail the sources themselves. We will begin to do so
now in the present section with particular attention to
the spin particle. As we have outlined in the introduction,
we call particles Dirac particles in the strict
sense, if their associated wave functions satisfy the
Dirac equation when acted on by an external electromagnetic
field, and if these particles do not undergo
strong interactions. In the strict sense, the electron
and the positron, the latter being the anti-particle
associated with the electron, are Dirac particles. For
these particles we have the following: (a) The Dirac
equation allows us to predict that these particles have
a magnetic moment originating from the spin angular
momentum with spin quantum number of 1/2, hence
they are fermions, (b) to within a high degree of accuracy,
the Lande´ factor g can be calculated assuming
a quantized electromagnetic field, and (c) the calculated
g factor agrees very well with the experimentally
determined value. Protons or compound nuclei
with spin 1/2 are not Dirac particles in this strict
sense. These particles have an internal structure
whose components (quarks) interact with each other
via strong nuclear forces that are not subject to quantum
electrodynamics. The field theory for particles
with strong interactions is QCD. Nevertheless, at low
energies (as compared to the nuclear rest mass and as
compared to nucleon-nucleon binding energies) and
abstracting from nuclear structure by just considering
them as nearly point-like particles with rest mass m
and a given nuclear g factor—which however then
cannot be inferred from quantum electrodynamics—
they may be considered as Dirac-like particles in processes
that describe only electromagnetic couplings
between nuclei and electrons, or electromagnetic couplings
among different nuclei. Therefore low-energy
nucleus-electron interactions can be treated by quantum
electrodynamics—such interactions appear, e.g.,
in the NMR chemical shift interaction or scalar couplings,
in EPR fine structure or hyperfine structure
couplings, or also as electromagnetic nucleus-electron
couplings with measurements mediated by the rf electromagnetic
field either generated by a macroscopic
electron current through an rf coil or resonator acting
on a nuclear spin-dipolar moment, or vice versa, the
nuclear spin-dipolar magnetic field as inducing a Faraday
voltage with a resulting electron current through
an rf coil connected to an rf circuit. Even direct dipolar
spin-spin couplings between two nuclei can be
investigated in this way. In the present article we will
not treat the Dirac equation with all its implications.
For a brief introduction including definitions we refer
to Appendices E and F, and the textbook literature
(e.g., Refs. 21, 22, 79–82, 89, 90, and 93).
Nevertheless, to continue with our discussion, we
need one essential result concerning the density of a
current of Dirac particles or Dirac-like particles—in
the following let us refer to these particles jointly as
spin-1/2 particles. As discussed above, electrons
strictly obey Dirac’s equation. Protons do not do that,
although protons can be characterized by the same
principal mathematical form of the current density as
for electrons. This general form can be derived for
electrons from Dirac’s equation by demanding conditions
that we expect to be fulfilled by any four-current
density jk of spin-1/2 particles, notably the fulfillment
of a continuity equation and the existence of
a positive-definite time-like component j0 such that it
can be interpreted as a probability density for the
spin-1/2 particle. It reads
jkðxÞ ¼ e"cðxÞgkcðxÞ [62]
For more details on how to obtain [62] we refer to
Appendices E and F, in particular to the result
VIRTUAL PHOTONS IN MAGNETIC RESONANCE 293
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
obtained in Eq. [F3]. The algebra behind Dirac’s
equation requires that the Dirac matrices gk appearing
in Eq. [62] are of dimension 4Â4 and the wave
function c(x) represents an object called bispinor,
having four components, each a function of spacetime
coordinates x. A bispinor is a mathematical
object which takes into account that the covariance
of Dirac’s equation requires to tackle (a) the Lorentz
transformation of space-time variables, and in juxtaposition,
(b) the associated unitary transformation in
Hilbert space of spin variables (21, 79, 81). An early,
comprehensive investigation on the unitary representation
of the inhomogeneous Lorentz group dates
back to Wigner (99). In the past, NMR has been used
to explicitly demonstrate the spinor character of spin-
1/2 nuclei and the rotational symmetries of spins-1/2
and spin-1 nuclei (100–103). The current density
jk(x) represents a four-vector in space-time—it is not
a bispinor. The quantity e is equal to the electric
charge of the particle. In connection with the conditions
mentioned above for the four-current density jk,
note the following two features of the current density
[62]: (a) jk(x) has vanishing four-divergence, i.e.,
qk
jk ¼ 0. Using ordinary vector notation it becomes
apparent that this represents a continuity equation
(see Appendix A, Eqs. [A16, A31, A32]). Furthermore
(b) the time-like component j0ðxÞ ¼ "cg0c ¼
cþ
c of the current density is equal to the squared
magnitude of c, hence positive definite, and can be
interpreted as a probability density for the spin-1/2
particle. For the proper definitions of the bispinors
c;"c;cþ
we refer to Appendices E and F (Eqs. [E11,
E12, F1]).
Before going into technical details, one important
remark has to be made here. For the electromagnetic
field (given by four-potential functions) we have
directly applied quantum field theory based on
CCR’s and obtained quantum field operators An. We
took the classical electromagnetic field and performed
quantization. For spin-1/2 particles the initial
situation is different ! The Dirac equation with the
Dirac wave function c as the solution of the former
is one possible relativistic generalization of the
Schro¨dinger equation for a quantum particle. Thus
here the function c does not indicate a proper classical
field function as in the case of the electromagnetic
field, rather c represents already a (multi-component)
field function for the quantum spin-1/2 particle
which allows a probabilistic interpretation. In
order to arrive at a full quantum field theory including
both, spin-1/2 particles and electromagnetic
fields, one would have to quantize the field c as well
(as if it were a classical field) by assigning operator
character to c and require the fulfillment of anticommutation
rules for these new field operators. In the
present article we do not perform this field quantization
(sometimes called second quantization) for the
spin-1/2 particle wave function c. That means we
treat spin-1/2 particles with wave functions c in analogy
to nonrelativistic Schro¨dinger quantum mechanics,
because we want to stay as close as possible to
the ‘‘orthodox’’ spin particle picture as used in magnetic
resonance. It is for that reason, that the current
density jn in Eq. [62], with c representing functions
and not operators, is a (vector) function as well. The
requirement for the current density jn representing a
function (and not an operator) is essential for the
appearance of coherent states and quasi-classical
states as discussed in Section ‘‘A QED NMR Probe
Model: Pulsed NMR as a Scattering Process.’’
A short time after Dirac introduced his equation
of motion, it could be shown that the current density
[62] can be decomposed into two physically relevant
parts. The first part represents the particle current
resulting from the spatial motion of the particle and
the second part originates from the spin of the particle.
In the literature this decomposition is commonly
referred to as Gordon decomposition (see Refs. 21,
91, 92, and Appendix F),
jk
ðxÞ ¼ e"cðxÞgk
cðxÞ ¼ À
e
2m
"cðxÞðPk
cðxÞÞ
À
ÀðPk "cðxÞÞcðxÞ À iPnð"cðxÞskn
cðxÞÞ
Á
½63Š
In Eq. [63] we have introduced the momentum
Pk
¼ pk
À eAk
of a particle situated in an electromagnetic
field with the four-potential Ak
(x) (m denoting
the rest mass of the spin-1/2 particle and pk
¼ iqk
designating the momentum operator for the free particle).
In other words, we look already at an electromagnetic
coupling between the particle and the external
field. The prescription Pk
¼ pk
À eAk
to introduce
the electromagnetic field into Dirac’s equation is
called minimum coupling condition and can be
understood as a direct consequence of gauge invariance
of the electromagnetic field equation (Appendix
B) with the concomitant phase invariance of Dirac’s
equation (see for example, Ref. 10). As we recognize
on the right-hand side of Eq. [63], the first contribution,
consisting of the first two terms involving the
linear momentum Pk
only, may be considered as a
‘‘convection’’ or ‘‘conduction’’ current, arising from
the motion of the Dirac particle in space. The second
contribution is given by the third term in Eq. [63]
and represents the spin current as signified by the
spin tensor skn
(Appendix F, Eq. [F12]).
With an explicit expression like Eq. [63] for the
current density, we are now in a position to describe
294 ENGELKE
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the interaction of a spin-1/2 particle with an external
electromagnetic field. The particle is characterized
by a Dirac wave function c (x,s), where x denotes
space-time coordinates and s stands for the spin variable.
An external electromagnetic field is represented
by the four-potential Am. The interaction Hamiltonian
density reads
Hintðx; sÞ ¼ jm
ðx; sÞAmðxÞ ¼ e"cðx; sÞgm
cðx; sÞAmðxÞ
[64]
The associated Feynman diagram, specified for
the case of spin interaction, is depicted in Fig. 6, displaying
a diagram similar to the one in Fig. 1(A). It
shows an incident particle in the state c with spin
angular momentum s ¼ L1, an outgoing particle in
the state "c with spin angular momentum s ¼ L2 and
an exchange of a photon between the source (shown
as a small circle) of the external field and the particle.
In Fig. 6 the time axis points in vertical direction
upwards, but the arrows labeled with c and "c do not
symbolize spatial motion. The diagram depicts a particle
initially in spin state cðL1Þ, then exchanging a
photon characterized by three-momentum K and
angular momentum M with the source of the external
field Am
which leaves the particle in spin state "cðL2Þ.
The energy density associated with this interaction is
given by the Hamiltonian density [64].
Focus on the spin current, more specifically, the
spin part of the current density for a single spin, as
follows from [63]
jm
spin ¼
ie
2m
Pnðcsmn
cÞ [65]
In [65] we want to separate jm
spin into its timelike
and spacelike components. Accomplishing this goal
is somewhat tedious and lengthy and can be achieved
best by calculating all components of the spin tensor
smn
(defined in Appendix F, Eq. [F12]), then assembling
all component expressions together, and finally
reading off the following result from the component
expressions:
j0
spin ¼
Àe
2m
ðp À eAÞðjÃ
sw À wÃ
sjÞ;
jspin ¼
e
2m
ðp0 À eA0Þ Â ðjÃ
sw À wÃ
sjÞ½
þiðp À eAÞ Â ðjÃ
sj À wÃ
swފ ½66Š
where ‘‘Â’’ denotes the ordinary cross product of
vector analysis and where we have formally decomposed
the bispinor c into two spinors j and w
according to
c ¼
c1
c2
c3
c4
0
B
B
@
1
C
C
A ¼
j
w
 
; j ¼
j1
j2
 
; w ¼
w1
w2
 
[67]
and s ¼ ðs1; s2; s3ÞT
denotes the ‘‘vector of Pauli
2Â2 spin matrices’’. Each of the quantities j and w is
a spinor, hence the term bispinor for c.
In order to find the bridge between the general
physical formalism of spinors and associated current
densities and the more familiar spin formalism utilized
in magnetic resonance, we need to proceed in
two steps. First, we have to make the passage to the
nonrelativistic realm, i.e., the domain of low velocities
as compared to c, low energies, low momenta,
stable particles, excluding antiparticles. Second, we
need to find a link between the notation with spinor
wave functions and the more familiar nonrelativistic
spin operator formalism that we encounter in magnetic
resonance.
Nonrelativistic Limit
In the nonrelativistic limit—a precise definition is
given in Appendix F, Eqs. [F23]—the magnitude of
the spinor w becomes small compared to that of the
spinor j (see Appendix F, Eqs. [F17] to [F25]). If in
Eqs. [66] for the components of the spin current density
we are allowed to neglect w altogether due to its
smallness, then the timelike component j0
spin of the
spin current density vanishes and the spacelike part
reads
jspin ¼
ie
2m
ðp À eAÞ Â ðjÃ
sjÞ [68]
Figure 6 Interaction of a particle with an external field
characterized by the exchange of a photon with three-momentum
vector K and angular momentum vector M
between the source of the external field (small circle) and
the spin particle with initial state c(L1) and final state
"cðL2Þ and their associated angular momenta L1 and
L2, respectively.
VIRTUAL PHOTONS IN MAGNETIC RESONANCE 295
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With the three-momentum operator p ¼ Àir we
have
jspin ¼
e
2m
r  ðjÃ
sjÞ À
ie2
2m
A Â ðjÃ
sjÞ [69]
Pauli referred to the vector
d ¼ jÃ
sj ¼
jÃ
s1j
jÃ
s2j
jÃ
s3j
0
@
1
A [70]
as spin density vector (Ref. 104, pp. 159–164). In Eqs.
[69, 70] we have connected quite diverse objects with
different algebraic character. First, d is an ordinary
three-vector. In contrast j ¼ (j1, j2)T
(the superscript
T
denotes the transpose), stands for a two-component
column vector with wave functions j1 and j2, representing
a spinor defined in a two-dimensional Hilbert
space and j* ¼ (j1*, j2*) denotes the two-component
row vector with conjugate complex elements j1* and
j2*, while s1, s2, s3 are operators (given by the Pauli
2Â2 matrices) acting on j. The spin density vector d
has the particular feature that any rotation of it in ordinary
three-dimensional coordinate space, mediated by
the 3Â3 rotation matrix R, is equivalent to a unitary
transformation, given by a 2Â2 unitary matrix U in
the two-dimensional Hilbert space of spinors j:
Rd ¼ RðjÃ
sjÞ ¼
jÃ
Us1Uþ
j
jÃ
Us2Uþ
j
jÃ
Us3Uþ
j
0
@
1
A [71]
To separate space (x) and spin (s) variables, we
take the ansatz
jðx; sÞ ¼ xðxÞtðsÞ
with the scalar wave function x(x) depending only on
space-time coordinates xm ¼ (ct, r) and the spinor t(s)
being a function of spin variables s only. This ansatz
is only consistently possible in a nonrelativistic treatment
(cf. [26], pp. 134). Thus the nonrelativistic twocomponent
spinor wave function is going to be
jðx; sÞ ¼
j1
j2
 
¼
xðxÞt1ðsÞ
xðxÞt2ðsÞ
 
[72]
The functions t1 and t2 are to be regarded as two
orthogonal wave functions for the spin-1/2 particle
and the scalar function x(x) represents the spatial part
of the wave function. For expressions like the spin
density vector d ¼ jÃ
sj as in Eqs. [69, 70], we may
write
d ¼ jÃ
sj ¼
jÃ
s1j
jÃ
s2j
jÃ
s3j
0
B
@
1
C
A ¼
jÃ
1j2 þ jÃ
2j1
ÀijÃ
1j2 þ ijÃ
2j1
jÃ
1j1 À jÃ
2j2
0
B
@
1
C
A
¼
xÃ
xðtÃ
1t2 þ tÃ
2t1Þ
ÀixÃ
xðtÃ
1t2 À tÃ
2t1Þ
xÃ
xðtÃ
1t1 À tÃ
2t2Þ
0
B
@
1
C
A ¼ xÃ
x
tT
tS
tP
0
B
@
1
C
A ½73Š
Note that in the last equation of [73] we have
introduced the spin functions tT, tS, and tP by the
definitions
tT ¼ tÃ
1t2 þ tÃ
2t1; tS ¼ ÀiðtÃ
1t2 À tÃ
2t1Þ;
tP ¼ tÃ
1t1 À tÃ
2t2 ½74Š
With d being a three-vector with each component
being a spin function, it becomes clear that the first
term r  ðjÃ
sjÞ in the spin current density jspin
(Eq. [69]) is also a vector of functions. The second
term A Â ðjÃ
sjÞ appearing in jspin contains the
three-vector potential A that, when we would take it
as field operator and not as an external classical field
function, would also cause jspin to become an operator
of the electromagnetic field. However as we will
see below (Eqs. [76, 77]), we are allowed to disregard
the term A Â ðjÃ
sjÞ because it does not contribute
to the interaction energy of the spin in the
electromagnetic field. Thus, finally taking the spin
current density as depending only on r  ðjÃ
sjÞ, it
represents a vector with functions as components.
Therefore, the spin current density fulfills the
requirement that we have emphasized in Section ‘‘A
QED NMR Probe Model: Pulsed NMR as a Scattering
Process,’’ which allowed us to arrive at the formalism
of coherent states and quasi-classical states
of the electromagnetic field.
Spin Operator Formalism
Up to and including Eq. [74] we have treated tT, tS,
and tP as spinor functions (of spin variables s). In
order to find the link to the familiar spin operator formalism
commonly used in magnetic resonance, the
components tT, tS, and tP appearing in the spin density
vector d need to be interpreted as spin operators.
This can be rapidly achieved by not relying formally
on the spin density vector d but rather directly take the
three-vector s ¼ ðs1; s2; s3ÞT
of Pauli matrices. It
becomes clear that then the resulting spin current density
becomes dependent on a spin vector operator s
and thus looses its ‘‘innocence’’ of appearing merely
as a vector of functions. With this transition from spin
wave functions to spin operators, we arrive at
296 ENGELKE
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^jspin ¼
e
2m
rðxÃ
xÞ Â s À
ie2
2m
xÃ
xðA Â sÞ [75]
where now we have written a caret sign over the symbol
of the spin current density in order to indicate the
formal difference of the function jspin in [69] and the
operator ^jspin in [75]. In both cases the spin physics is
the same: we are still in the single-particle regime and
do not consider quantum field theory of spin-1/2 particles,
we just switch from spin functions to the associated
spin operators. The only reason to write down
the spin current density in the form [75] is to establish
the connection to the common spin operator formalism
used in magnetic resonance. For the general
Hamiltonian density [64], which characterizes the
interaction between a Dirac current and external field,
we specialize for the spin interaction
Hint;spin ¼
e
2m
A Á rðxÃ
xÞ Â sÞ À iexÃ
xðA Â sÞð
¼
e
2m
A Á rðxÃ
xÞ Â sÞð ½76Š
where on the right-hand side of the last equation we
took into account the identity
A Á ðA Â sÞ ¼ 0 [77]
so the field dependent part of the spin current density
does not contribute to the energy of the spin particle
in an external field. An alternative derivation of Eq.
[76] based on the formalism in k space can be found
in Steven Weinberg’s monograph (21).
Applying to Eq. [76] a series of transformations
using well-known vector relationships (see Appendix
F, Eqs. [F26–F35]), we arrive at
Hint;spin ¼
e
2m
A Á rðxÃ
xÞ Â sð Þ
¼
e
2m
xÃ
xB Á t þ r Á ðs  ðxÃ
xAÞÞð Þ ½78Š
with the magnetic induction field B as defined by Eq.
[F35]. In order to get the Hamiltonian from the Hamiltonian
density, we need to integrate over threedimensional
space,
Hint;spin ¼
Z
V
d3
xHint;spinðxÞ [79]
Applying that to Eq. [78] we obtain
Hint;spin ¼ þ
e
2m
Z
V
d3
x xÃ
xB Á sð Þ
þ
e
2m
Z
V
d3
x r Á ðs  ðxÃ
xAÞÞð Þ; ½80Š
In [80] we recognize that the divergence term
(second term on the right-hand side) can be transformed
into a closed surface integral according to
Gauss-Ostrogradski’s theorem, i.e.,
Z
V
d3
xð ~r Á aÞ ¼
I
S
dS Á a [81]
by setting a ¼ s  ðxÃ
xAÞ,
Hint;spin ¼
e
2m
Z
V
d3
x xÃ
xBÁsð Þþ
e
2m
I
S
dSÁðsÂðxÃ
xAÞÞ
[82]
and the surface S encloses the integration volume V.
The differential dS is equal to a surface element with
associated normal vector dS.
Now consider the case of a spin particle strongly
localized in three-dimensional space. This means that
the probability density x*x in space is a very narrow
peak, i.e., in the limit of a point-like particle it is
equal to a d function at the coordinate origin if we
place the particle there:
xÃ
x ¼
1
2
gd3
ðrÞ;
Z
V
d3
xxÃ
x ¼ 1 [83]
g denotes again the Lande´ factor. With the assumption
[83] in mind we observe that the surface integral
in [82] becomes identically equal to zero,
I
S
dSðs  ðd3
ðrÞAÞÞ ¼ 0 [84]
and the volume integral term on the right-hand side
of [82] reads
Z
V
d3
x d3
ðrÞB Á s
À Á
¼ Bðr ¼ 0Þ Á s ¼ B Á s [85]
Hence
Hint;spin ¼ gB Á
1
2
s [86]
The quantity
g ¼
ge
2m
[87]
is referred to as gyromagnetic ratio. The spin operators
s1/2, s2/2, and s3/2 are equal to the components
Ix, Iy, and Iz, of the spin vector II:
VIRTUAL PHOTONS IN MAGNETIC RESONANCE 297
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Ix ¼
1
2
s1; Iy ¼
1
2
s2; Iz ¼
1
2
s3 [88]
such that [86] takes the final form of the Zeeman
Hamiltonian
Hint;spin ¼ gB Á I [89]
The Lande´ factor g introduced above in Eq. [83]
appears to be slightly larger than 2 for electrons as
genuine Dirac particles. As discussed earlier, for
nuclei, g cannot be determined this way—Dirac’s
theory is not capable to predict nuclear g factors.
However, the principal expression [88] and [89]
remain valid with the nuclear gyromagnetic ratio
[87] obtained either from experiment or from a
theory of strong nuclear interactions.
Let us return briefly to the classical picture of Larmor
precession of the magnetic dipolar moment vector
g"hI in a static magnetic field. For the sake of connecting
this classical picture with the QED view,
consider the following heuristic argument. Take a
particle with the intrinsic angular momentum
L ¼ "hI [90]
with the spin vector operator I ¼ ðIx; Iy; IzÞ according
to Eq. [88]. The particle is situated in an external
static field aligned along the z axis. According to
classical physics the angular momentum vector L
will perform a Larmor precession with angular frequency
o around the z axis, as illustrated in Fig. 7A.
Although during the precession motion the length of
L remains constant, the direction of L changes with
time. Consider the angular momentum at two successive
instants of time separated by the interval Dt. Let
L1 ¼ Lðt1Þ and L2 ¼ Lðt1 þ DtÞ and denote the difference
vector that arises from this discretized Larmor
precession by M ¼ L1 À L2. We just have
expressed the conservation of angular momentum for
the system particle þ field: the change of angular
momentum L2 À L1 during the precession motion of
the vector L in the static field is compensated by the
angular momentum M conveyed from the particle to
the static magnetic field, or vice versa, from the field
to the particle, such that M þ ðL2 À L1Þ ¼ 0 [Fig.
7(B,C)]. The interaction energy [89] of the particle
does not change during Larmor precession. If we
declare a virtual photon as the arbiter of the interaction
between the source of the static magnetic field
and the spin particle, then this photon carries the
angular momentum M ¼ L1 À L2. In addition, this
photon possesses the linear momentum K and a nonnegative
energy K0. Yet we have to conclude that the
energy of this virtual photon is equal to zero, because
absorption or emission of the photon just identified
by the spin particle with angular momentum L while
performing Larmor precession does not change the
energy of the spin particle in the static magnetic
field. This is also a statement about the certainty of
energy in this specific interaction process—certainty
of the energy of the spin particle as well as certainty
of the photon energy. In other words, because of the
condition that the spin particle before and after the
emission or absorption event of the virtual photon
has the same energy, the virtual photon cannot carry
finite energy. The emission or absorption event in
Figs. 6 and 7(C) occurs, symbolically, at the vertex
where the lines meet, the photon line is associated
with K0 ¼ 0 and with the finite angular momentum
M as well as linear momentum K (both may be
expressed in combination with each other as the helicity
operator MÁK/(|M||K|) of the virtual photon.
Note, in the argument for the zero energy of the
virtual photon involved in Larmor precession we
have not included yet any appeal that this photon is
off-shell. We just stated that at the interaction vertex
[Figs. 6 and 7(c)] the balance of energy, momentum,
Figure 7 Conservation of angular momentum during the
Larmor precession of an magnetic dipole associated with
the spin angular momentum L of a particle in an external
static field B. Although the vector L has constant length,
its direction changes over time. The resulting angular momentum
M is taken from, or conveyed to a virtual photon
which mediates the interaction.
298 ENGELKE
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and angular momentum between incoming particles
(here the spin particle) and outgoing particles (again
the spin particle and the virtual photon) has to be
maintained. This vertex property is universally
valid—we will elaborate that in a slightly more general
way in Section ‘‘Outlook and Conclusion’’ when
discussing Feynman rules and Feynman diagrams.
The statement that the virtual photon carries zero
energy means that we consider the energy to be certain,
i.e., for the uncertainty we have DE ? 0. In
Section ‘‘A QED NMR Probe Model: Pulsed NMR
as a Scattering Process’’ we have already spoken
about the signature of virtual photons.
Ascribing the energy K0 ¼ 0 to the photon creates
an extreme case for a virtual photon: the zero-mass
shell (with radius K0) degenerates to a point in k
space and the three-momentum vector K is uncertain,
so the photon is really far off-shell. As the uncertainty
relation dictates, according to Eq. [35] and as
discussed in Section ‘‘A QED NMR Probe Model:
Pulsed NMR as a Scattering Process,’’ in the present
case we consider the two complementary (i.e., noncommuting)
observables energy E and three-momentum
K (or energy E and angular momentum M). The
characteristic time interval associated to E and K is
given by DtK ¼ DK/FK with the force FK equal to
the total time derivative of hKi. This leads to the
uncertainty relation DEDtK ! "h=2 and for DE ? 0
we have to conclude DtK ? 1, the latter either
because DK ? 1 (or because FK ? 0). The situation
that the momentum uncertainty DK ? 1 refers
to the fact that for virtual photons in static interactions,
the sharp energy (zero radius of the zero mass
shell) is associated with an completely uncertain momentum
K, an extreme case for virtual photons.
The heuristic argument for the exchange of virtual
photons with zero energy between the spin particle
and the surrounding magnet current generating the
static induction field is not only appropriate when
starting from the classical view with Larmor precession,
we may also take the spin-1/2 particle quantummechanically.
The spin angular momentum I is quantized
along the axis of the magnetic induction field.
Let this be defined as the z axis, so that eigenstates of
the spin component Iz are simultaneously energy
eigenstates of the spin particle in the static field B.
There are two such eigenstates for spin-1/2 particles,
both states differ in energy (Zeeman levels) by "ho
with o being equal to the Larmor frequency. However,
the spin particle thus characterized does not
perform transitions from one Zeeman state to the
other Zeeman state as long as there is no perturbation
acting on the spin particle or as long as spontaneous
emission does not occur, hence the spin particle does
not change its energy. The external perturbation on
the spin could, e.g., arise from a time-harmonic field
superimposed on the static field or it could originate
from relaxation processes. Except for such external
perturbations, the spin remains in its Zeeman energy
eigenstate, hence its energy does not change.
The diagram in Fig. 7(C) has been drawn in such
a way to show the similarity to the diagram in Fig. 6,
or to the general Feynman diagram in Figs. 1 and 2.
In contrast to our discussion in Section ‘‘A QED
NMR Probe Model: Pulsed NMR as a Scattering Process’’
where we modeled the interaction between an
rf or microwave pulse of short duration with a spin
particle as a scattering process, the QED picture of
Larmor precession confronts us with a different situation.
First, it is not appropriate to view Larmor precession
as a short-time scattering process, i.e., assuming
that the spin particle is submitted to the static
magnetic field only for a limited short time interval.
It is rather the opposite, we face a situation where the
spin particle is permanently interacting with the external
static field, at least for time periods many orders
of magnitude larger than the duration of any rf or
microwave pulse. Second, it becomes thus clear that
the diagram in Fig. 6 can only represent the elementary
process, a building block showing the principle
of photon exchange during Larmor precession such as
in Fig. 7(A) where it becomes obvious that only a
small time interval is considered. A more appropriate
diagrammatic representation covering the spirit of the
whole process of permanent interaction of a spin particle
in a constant magnetic field is shown in Fig. 8. It
illustrates the repeated exchange of virtual photons
between the spin current density [69,75] and the current
density of the magnet coil generating the constant
magnetic field in which the spin particle is embedded.
The Feynman diagram of Fig. 6 can be seen as an element
of the extended diagram in Fig. 8. The QED formal
treatment of multiple exchange processes as illustrated
in Fig. 8 would require to go into the full theory
of bound states, i.e., a theory that takes into account
not just short scattering events but time-like extended,
permanent interactions. Having in mind the introductory
character of the present article, this clearly would
exceed its scope.
INTERACTION OF A SPIN-1/2 PARTICLE
WITH EXTERNAL TIME-HARMONIC
FIELDS
In the model for a magnetic resonance probe introduced
in Section ‘‘A QED NMR Probe Model:
Pulsed NMR as a Scattering Process’’ we supposedly
VIRTUAL PHOTONS IN MAGNETIC RESONANCE 299
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dealt with current density functions. We will reexamine
this assumption more closely now. In spacetime
we do easily distinguish two current densities,
j(x) and j(y). The first one, j(x), is related to the rf
current where electrons in the conduction band in a
metal are the carriers of the current. These conduction-band
electrons could be characterized by Bloch
wave functions in the periodic potential of a metal
lattice or simply approximated as nearly free electrons
with the spatial boundary condition that
restricts their position distribution to inside the macroscopic
metal parts that constitute, e.g., the resonator
or coil. The second current density, j(y), relates to
the spins in the sample in a different space region.
With this distinction, j(x) and j(y) certainly commute,
because even when both current densities would be
operators, they refer to different particles or different
variables—conduction electrons in the coil or circuit
on one side, spin particles in the sample on the other
side—we consider the two currents as distinguishable.
The situation would change if we have a more
general situation: we may regard all nearly free electrons
in the metal conduction band as indistinguishable
among themselves, or likewise, an assembly of
identical spin particles represents a statistical ensemble
of indistinguishable particles. Therefore, within
one family of particles for one specific current density
when treating the particles as fermions characterized
by wave or field functions and then performing
quantization of the fermion field, it is not justified to
treat the current density in each family as noncommuting
for different space-time points. However, this
is not the case that we treat in the present article.
In the sequel we will study the interaction between
the rf current density j(x) and spin particles under the
following circumstances: we are concerned with j(x)
as a classical, macroscopic current density and look at
the coupling with a single spin. Hence, even with j(x)
being a classical function, it generates the quantized
electromagnetic field in some coherent state |ai
whose time evolution governed by Glauber’s displacement
operator Da is given by Eq. [61]. The spin
current density j(y), as given by the first term on the
right-hand side in Eqs. [69], has vector character with
vector components being functions. Hence the condition
that the current densities are functions (thus they
commute) is satisfied and all the previous theoretical
implications for the quantized electromagnetic field
apply, as discussed in Section ‘‘A QED NMR Probe
Model: Pulsed NMR as a Scattering Process.’’ Turning
back to Eq. [52], which expresses the time evolution
operator for the electromagnetic field, we look at
the exponential expression containing the real part
d((xÀy)2
) of the photon propagator DF(xÀy), and take
j(y) as a spin current density function according to
Eq. [69]. The electromagnetic field (one mode) is
transformed by Da. The exponential expression in
[52] depends on the spin current density, it does not
affect coherent states |ai. There are spin states |si,
these in turn are not affected by Da. As we recognize
now, we have achieved a factorization of the timeevolution
operator U(t) such that for any matrix element
of U(t) formed with product states |a,si ¼ |ai |si
it holds
ha; sjUðtÞja0
; s0
i ¼ hajDaðtÞja0
i
 hsj exp
i
8p
Zt
À1
dx0
Z
d3
x
Zt
À1
dy0
0
@
Â
Z
d3
yjn
ðxÞdððx À yÞ2
ÞjnðyÞ

js0
i ½91Š
In the derivation of the unitary operator U performed
in Section ‘‘A QED NMR Probe Model:
Pulsed NMR as a Scattering Process,’’ we relied on
Figure 8 A special Feynman diagram illustrating the
repeated photon exchange between the magnet static current
density j0(x) and the spin current density jspin(y). The
space-time points y1, y2, . . . denote time intervals and
refer to orientations of the spin angular momentum during
Larmor precession.
300 ENGELKE
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the assumption that, formally, the current densities
j(x) and j(y) represent functions. Furthermore, we
have shown, beginning with the general Dirac current
density as given in Eq. [62], that the spin current density
[69] is, indeed, a vector with spin wave functions
as components. Furthermore in Section ‘‘Spin Current
Density, Zeeman Hamiltonian, and Larmor Precession’’
we have seen that the spin current density
function, jspin, Eq. [69], has an associated operator
counterpart, ^jspin, Eq. [75]. If we want to include in
our arguments not only the time evolution of the
electromagnetic field but also the quantum time evolution
of the spin-1/2 particle when discussing the
unitary operator U, we have to perform the subtle
switch from taking j(y) as a function to taking ^jðyÞ
(note the caret symbol) as an operator depending on
spin operators. We observe that this is an a posteriori
construction like in Section ‘‘Spin Current Density,
Zeeman Hamiltonian, and Larmor Precession,’’ with
the purpose to incorporate the more familiar spin operator
formalism. We could live without this construct,
in that case the exponential in Eq. [91] inside
the Dirac bra hs| and the Dirac ket |s0
i would become
simply a complex number and U would describe the
time evolution of the electromagnetic field only.
With the operator ^jðyÞ for the spin current density the
unitary operator for the time evolution of the spin
reads
VðtÞ ¼ exp
i
8p
Zt
À1
dx0
Z
d3
x
Zt
À1
dy0
0
@
Â
Z
d3
yjn
ðxÞdððx À yÞ2
Þ^jnðyÞ
1
A ½92Š
such that the complete time evolution operator for
the electromagnetic field and the spin can be
written as
UðtÞ ¼ DaðtÞ  VðtÞ [93]
The two unitary operators Da and V act on different
kinds of state functions, either states of the electromagnetic
field or spin states, thus their product in
[93] is rather to be understood as direct product, not
as an ordinary matrix product.
We may summarize the situation as follows:
(i) the single-mode electromagnetic field Am
evolves according to the unitary transformation
[61] with Glauber’s displacement operator
Da, where the field parameter a depends
on the current density in k space. This part
describes the action on the field originating
from all the current densities (arising from
the rf current as well as the spin current).
The photons involved are asymptotically free
photons, the ‘‘constituents’’ of the interacting
and outgoing rf pulse.
(ii) the states |si of the spin particle evolve
according to the unitary time evolution
V(t)|si with the unitary operator V given by
Eq. [92]. Here we have built-in the direct
interaction between spin current and rf current
density. The photons appearing in V(t)
are virtual photons being exchanged between
the classical current j(x) and the spin particle
current density ^jðyÞ as formally indicated by
the d function in the real part of DF(xÀy)
inside the integral in Eq. [92].
For the purpose of illustration, let us investigate
coherent states |ai with a small number of photons
(Fig. 9). First, the ground state of the electromagnetic
field is the vacuum state |0i, this is a coherent state
and simultaneously it is also a Fock state (with the
Figure 9 Poisson probability distributions over numbers
n of asymptotically free photons with the electromagnetic
field in various coherent states |ai according to Eq. [57]
with associated average photon numbers "n ¼ aaÃ
.
VIRTUAL PHOTONS IN MAGNETIC RESONANCE 301
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lowest possible energy). Only the state |0i can claim that
double role. With the field in state |0i the average number
"n ¼ 0 and the probability to find exactly zero photons
in the field, i.e., n ¼ 0, is equal to unity (Fig. 9,
top histogram), hence the probability that there are
n . 0 photons is identical to zero. That means that in
the vacuum state |0i there are no asymptotically free
photons, but virtual photons do occur. Now consider
the coherent state |ai with an average photon number
"n ¼ aaÃ
¼ 1. According to Eq. [56], this state is
equal to the superposition of Fock states
jaðtÞi ¼ h0jaij0i þ h1jaij1i þ h2jaij2i þ . . .
so due to the histogram in the second row in Fig. 9,
there is a finite probability to find n ¼ 1 or n ¼ 2 or
. . . asymptotically free photons, but also there is still
a significant probability for n ¼ 0 (the case where
only virtual photons, but no asymptotically free photons
occur). In that specific coherent state with "n ¼ 1
the probability to have zero asymptotically free (i.e.,
to find only virtual) photons in the field is equal to the
probability to find exactly n ¼ 1 photons in the field.
The same argument can be iterated now for coherent
states |ai with higher average numbers "n ¼ aaÃ
¼
2; 3; . . . where from inspection of the histograms in
Fig. 9 for these cases we recognize that the maximum
of the probability distribution occurs always for
n ¼ "n and the width of the distribution (not shown
here, but being equal to the standard deviation) is
equal to
ffiffiffi
"n
p
. Having thus illustrated the uncertainty
of the photon number n in electromagnetic fields
being in coherent states (we only know the average
and the standard deviation), it becomes obvious that
this uncertainty directly relates to the energy of the
field—here not to be confused with the energy of a
single photon: every asymptotically free photon for a
given field mode k carries an energy "ho and is onshell,
but the photon number is uncertain, and so is
the total electromagnetic field energy being the sum
over all photon energies. Specifically the uncertainty
of the photon number could be understood as photon
emission to the field or photon absorption from the
field, hence a varying number of photons over time
requiring that the average "n ¼ aaÃ
, now being taken
as time average, is sustained and as long as the Poissonian
photon distribution is maintained for the field
mode considered. In particular, the time evolution operator
Da allows photon emission, either from the rf
current density or from the spin current density into
the rf field. Since in time-harmonic fields oscillating
with the angular frequency o each asymptotically free
photon carries energy equal to "ho, for a given field
mode, each act of emission or absorption of these
asymptotically free photons changes the energy of the
participating particle whose motion or spin constitutes
the respective current density and changes the
energy of the field, accordingly. The only exception
from this general picture is the vacuum state |0i;
insofar here specifically all emissions lead to subsequent
complete absorption with the result of net number
zero of asymptotically free photons in the field.
We want to corroborate the fact that the unitary
operator V(t) characterizes the evolution of the spin
state as claimed above. For that purpose, let us derive
a more familiar form for V(t) explicitly exhibiting the
spin operators in the spin current density as defined
in Eq. [75]. We introduce the four-potential function
as the convolution integral of the classical rf current
density j(x) in the coil or resonator and the real part
<e(DF(x)) of the photon propagator:
~An
ðyÞ¼À
1
4p
Zt
À1
dx0
Z
d3
xjn
ðxÞdððxÀyÞ2
޼1
4p
Z
d3
x
Â
Zt
À1
dx0
jn
ðxÞ
dððx0
Ày0
ÞÀrÞþdððx0
Ày0
ÞþrÞ
2r
½94Š
where in the second equation in [94] we have
inserted the first identity as given in Eqs. [18]. We
note again the two d functions indicating the two different
time orderings as in our discussion of Eq. [12]
in Section ‘‘The Feynman Propagator.’’ Performing
the integration over the time variable x0
in [94] yields
the four-potential generated by the coil current arbitrated
by <e(DF(x)) as
~An
ðyÞ ¼ À
1
8p
Z
d3
x
jn
ðx;y0
þ rÞ þ jn
ðx;y0
À rÞ
r
[95]
Having obtained [95], we may write the time-evolution
operator [92] acting on spin states as
VðtÞ ¼ exp i
Zt
À1
dy0
Z
d3
y ~An
ðyÞ^jnðyÞ
0
@
1
A [96]
According to Eq. [64] the integral over ~An
ðyÞ^jnðyÞ
is equal to the interaction Hamiltonian density for the
spin particle interacting with the time-harmonic field,
and as we have derived with Eq. [89], it holds
Z
d3
y ~An
ðyÞ^jnðyÞ ¼ Hint;spin ¼ gI Á Bðy0
Þ [97]
Just slightly changing the notation by setting y0
¼
t0
and assuming a finite time interval (t1, t) during
which the electromagnetic field of the incident pulse
302 ENGELKE
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interacts with the spin, Eq. [97] inserted into [96]
finally leads to
Vðt; t1Þ ¼ exp ig
Zt
t1
dt0
I Á Bðt0
Þ
0
@
1
A; [98]
an expression for the time evolution operator of the
spin particle that is familiar from the formalism used
in magnetic resonance. Supposing that the magnetic
induction field B(t) represents a linearly polarized
time-harmonic field,
BðtÞ ¼ B1ðtÞ ¼ B10 cosðotÞ [99]
with amplitude B10, Eq. [98] characterizes exactly
the time evolution operator in the laboratory frame
for a spin-1/2 angular momentum interacting with
this time-harmonic field oscillating at the angular frequency
o. Expressing the time evolution operator as
in Eq. [98], we have formulated the semi-classical
limit for the interaction of a spin-1/2 with the radiofrequency
field. The term semi-classical refers to the
approximation in which the time-harmonic electromagnetic
field B(t) in Eq. [98] is taken as a classical
field, obtained from the vector potential ~AðtÞ via
BðtÞ ¼ r  ~AðtÞ, where ~AðtÞ represents the spacelike
part of the four-potential conveyed by Eqs. [94,
95]. The time evolution operator V(t) according to
Eq. [98], which characterizes the action of the electromagnetic
field on the spin-1/2 particle during an rf
pulse of duration t ¼ tÀt1, is associated with the
exchange of virtual photons.
The interaction Hamiltonian Hint;spin ¼ gI Á B in
Eq. [97] may contain the static Zeeman interaction
and the interaction with the time-harmonic field. Let
us briefly discuss how this interaction Hamiltonian is
related to the total Hamiltonian (free-field Hamiltonian
Hem plus Hint,spin). The Hamiltonian for a singlemode
free electromagnetic field reads
Hem ¼ oaþ
a þ o=2 [100]
where the term o/2 represents the zero-point or vacuum
energy term. Note, Hem is an operator independent
of time. Taking the expectation value with the
electromagnetic field in a coherent state |ai yields
the field energy in that state:
hajHemjai ¼ hajoaþ
ajai þ o=2
¼ oaÃ
a þ o=2 ¼ "no þ o=2 ½101Š
which is, apart from the zero-point energy, equal to
the average number, "n, of photons in the field multiplied
by the energy of a single photon, o (or rather
"ho). Let us take a look at the effect of performing
the unitary transformation of the free-field Hamiltonian
Hem with Glauber’s displacement operator Da,
which gives
"Hem ¼DaHemDþ
a ¼oðaþ
aÀaþ
aÀaaÃ
þaaÃ
Þþo=2;
[102]
and calculate again the expectation value in coherent
state |ai:
haj "Hemjai¼hajDaHemDþ
a jai¼h0jHemj0i¼o=2
[103]
where we have applied Eq. [55]. Eq. [103] means
that the average free field electromagnetic energy
with the field in coherent state |ai in the frame corotating
with or rotated by Da is equal to the vacuum
energy, i.e., in that frame there are no free photons,
or more precisely, no asymptotically free photons.
Let us refer to the frame co-rotating with Da as the
a-rotating frame. We consider the complete Hamiltonian
H, first in the laboratory frame (without unitary
transformation with Da),
H ¼Hem þHint;spinðtÞ¼oaþ
aþo=2þgIÁðB0 þB1ðtÞÞ
[104]
where Hint,spin(t) incorporates the interaction of the
spin with the static field B0 (Zeeman interaction) as
well as with the (e.g., linearly polarized) rf field B1(t)
(i.e., for example, [99]). In the free-field Hamiltonian
Hem we have electromagnetic field operators,
i.e., photon creation and annihilation operators only.
Hint,spin depends on spin operators and the electromagnetic
field appears as functions of classical
space-time variables, as exhibited by Eq. [94].
According to Eq. [93], which shows the factorization
of the time development operator for the entire system,
electromagnetic field plus spin, we have
achieved that states depending on field and spin variables
can be expressed as direct products. While the
time evolution of the field operators is given by Eq.
[54] as Da(t)Am
(x)Da
þ
(t), the time variation of the
coherent state |ai can be expressed as
q
qt
jai ¼
q
qt
DaðtÞj0i [105]
where we took into account Eq. [55], i.e., Da acts as
a creation operator of the coherent state |ai from the
vacuum state |0i. Because the vacuum state is a state
which does not dependent on time, we only need to
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calculate the time derivative of Glauber’s displacement
operator following from Eq. [51],
qDa
qt
¼
qa
qt
aþ
Da À
qaÃ
qt
Daa À
1
2
qaÃ
qt
aDa
À
1
2
qa
qt
aÃ
Da ¼¼ Àioðaaþ
þ aÃ
a À aÃ
aÞDa ½106Š
so that
qjai
qt
¼ Àioðaaþ
þ aÃ
a À aÃ
aÞDaj0i
¼ Àioðaaþ
þ aÃ
a À aÃ
aÞjai
¼ Àioaaþ
jai ¼ Àioaþ
ajai ¼ Àioaþ
ajai ½107Š
That means, in the laboratory frame the time evolution
of jai occurs under the free-field Hamiltonian
(without vacuum energy term Hvac ¼ o/2),
qjai
qt
¼ ÀiðHem À HvacÞjai
with the solution
jaðtÞi ¼ expðÀioaþ
atÞjað0Þi [108]
For the equation of motion in the a-rotating
frame, let us look at the equation of motion analogous
to [105] for the state jai transformed to the arotating
frame. As a field state it transforms according
to
j¯ai ¼ Dþ
a jai [109]
According to Eq. [55] we observe that j¯ai ¼ j0i,
i.e., the coherent field state in the a-rotating frame
equals the vacuum state, which also follows from Eq.
[102],
haj "Hemjai ¼ hajDaHemDþ
a jai ¼ h¯ajHemj¯ai
¼ h0jHemj0i ¼ o=2
As a consequence, because the state |0i is time-independent,
we conclude qj¯ai=qt ¼ 0. Continuing to
work in the a-rotating frame, by first transforming
the complete Hamiltonian H (Eq. [104]), excluding
the vacuum energy part,
H0
¼ DaðHem À Hvac þ Hint;spinÞDþ
a ¼ "H À Hvac
¼ "Hint;spinðtÞ ¼ Hint;spinðtÞ ¼ gI Á BðtÞ ½110Š
shows that in this manner we have removed the freefield
part. The interaction Hamiltonian Hint,spin is not
affected by the transformation with Da, because it
does not contain field variables as electromagnetic
field operators anymore, here the field appears as
classical field ~AðtÞ (Eqs. [94, 95]), via B(t), derived
from the real part of Feynman’s photon propagator.
The semi-classical Hamiltonian H0
contains the timedependent
induction field function BðtÞ ¼ B0þ B1ðtÞ,
i.e., the rf field, and the static field function B0 leading
to the static Zeeman interaction. Nevertheless,
when seen from its origin, the way we have obtained
B1(t) or B0 it becomes clear that we deal with virtual
photons: we say B(t) is equal to the curl of the vector
potential ~AðtÞ, Eq. [95], the latter is equal to the convolution
integral [94] of the rf or static current density
(generating the induction fields) with the real
part of the photon propagator,
~An
ðyÞ¼
1
4p
Zt
À1
dx0
Z
d3
xjmðxÞ<eðDmn
F ðxÀyÞÞ [111]
This vector potential appears here as a field function,
not an operator anymore. According to Table 1,
<e(DF(xÀy)) signifies the presence of virtual photons,
concealed via Eq. [111] in the semi-classical
Hamiltonian H0
of Eq. [110].
SINGLE-SPIN FID AND NMR RADIATION
DAMPING
We return to Eq. [92] as a tool for explaining the free
induction decay (FID) as virtual photon exchange
between the spin current density j(y) and the current
density in the rf coil. This statement needs some
more clarification: specifically we may say that the
spin current density j(y) generates an electromagnetic
field that acts on the conducting elements of the coil
or resonator and causes a current density j(x)—which
we may observe as FID. This latter current density
j(x), now in turn being the source of the rf field, acts
back on the spin current density j(y)—a phenomenon
called NMR radiation damping (105–120). A more
proper term would be back-action or reaction, since
no radiation is involved here at all and radiation
damping in general electromagnetics is a different
phenomenon. Nevertheless, NMR radiation damping
as an established technical term means the second
step of the interaction between a spin dipolar moment
and the surrounding resonator: (1) the spin current
density generates an rf current j(x) that can be
detected as FID in a closed circuit (such that actually
we have a macroscopic current density j(x)), and (2)
this rf current j(x) acts back on the spin current den-
304 ENGELKE
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sity. Note, Eq. [92] does not allow us to actually
compute the FID, or more precisely, does not allow
to compute the effects imposed upon the FID by simultaneous
NMR radiation damping, because to be
able to do so would require knowledge of the magnitude
of rf current density in the coil or resonator, and
this depends on the details of our macroscopic device
contained in the probe circuitry. Thus here we meet
one border aspect to the engineering side of the
NMR probe.
We need to clarify the differences that appear
when on one hand we apply an rf pulse to our spin
system and on the other hand when after the pulse
the spin current density becomes the source of electromagnetic
fields. In Sections ‘‘A QED NMR Probe
Model: Pulsed NMR as a Scattering Process’’ and
‘‘Interaction of a Spin-1/2 Particle with External
Time-Harmonic Fields’’ we have described the
impact of an rf pulse on the spin as a kind of scattering
process with an incident pulse with the singlemode
field in a coherent state |ai characterized by an
unsharp, average photon number "n obeying a Poisson
distribution [57]. The four-potential of the interacting
and outgoing pulse obeys a unitary time evolution
according to Eq. [36], provided the losses of the
NMR probe circuitry and coil as well as the dielectric
losses possibly caused by the sample are negligibly
small. Assuming commuting current densities, the
overall time evolution operator U(t) turned out to be
decomposable into a direct product Da(t)  V(t) with
Da governing the propagation of the rf pulse and V(t)
characterizing the interaction between spin current
density and rf current. We could show that during a
time interval when the pulsed rf field interacts with
the spin we can perform a transformation into the arotating
frame, which leaves, apart from the vacuum
energy of the electromagnetic field, only the interacting
part Hint,spin of the total Hamiltonian including
the Zeeman coupling of the spin situated in a strong
external field and the interaction of the spin with the
time-harmonic electromagnetic field associated with
the rf pulse following V(t). This holds for each distinct
mode in the electromagnetic field. If more than
one mode is present, the state of the electromagnetic
field is equal to the product state |a1i |a2i . . . |aki
. . ., which is not a coherent state anymore, but the
property to be dispersion free (vanishing relative
uncertainties) in the classical limit is maintained (see
the discussion at the end of Section ‘‘A QED NMR
Probe Model: Pulsed NMR as a Scattering Process’’).
Suppose prior to the arrival of the incident rf pulse
that the spin state is equal to one of the two Zeeman
energy eigenstates |þi or |Ài while the electromagnetic
field is found to be in the vacuum state |0i. The
compound state for the composite system, spin plus
field, appears to be |0, 6i with |6i denoting one of
the two Zeeman spin states. The arrival of the rf
pulse suddenly changes the electromagnetic field
vacuum into a coherent state |ai, where the complex
field amplitude a = 0 depends on the three-dimensional
Fourier transform of the current density. As
becomes clear from Eqs. [D18, D19], in k space we
do not distinguish between rf and spin current as we
did in space-time based on the spatial separation of
currents j(x) and j(y) in regions x and y. So the emergence
of Glauber’s displacement operator Da contains
the joint contribution to the electromagnetic
field from both current densities, rf current and spin
current, the latter depending on the spin state at a
given instant in time. Due to Eqs. [D18, D19], the
photons in the field associated with the rf pulse are
on-shell, i.e., these are asymptotically free photons
either emitted by the rf current density or emitted by
the spin current density, or likewise, absorbed. The
overall electromagnetic field is characterized by an
average number of photons "n in the field with a
standard deviation
ffiffiffi
"n
p
of the Poisson distribution
[57]. The exchange of virtual photons between rf current
density and spin current density is hidden in the
unitary time evolution V1(t) during the pulse and in
V2(t) after the pulse (Fig. 10, vide infra). Thus in Eq.
[92] the role of the current densities is clear during
the time interval when the electromagnetic field of
the rf pulse interacts with the spin (time evolution
V1) and the rf current density j(x) associated with the
incident pulse is injected from outside the probe.
This role of j(x) and j(y) is different prior to the
arrival of the pulse in the spatial region of the spin
particle and after the pulse has passed that region. In
the latter case, the current density j(x) is no longer
externally impressed, it is either zero (when the cirFigure
10 Time evolution in pulsed magnetic resonance:
during the rf pulse, the field operators A(x) are governed
by the unitary operator Da whereas the spin operators follow
a time evolution due to V1(t). After the pulse, the
state of the electromagnetic field returns to the vacuum
state as new initial state at time t, associated with the
time evolution operator for the electromagnetic field equal
to the unity operator 1. The interaction of the spin is characterized
by the unitary evolution as given by the unitary
operator V2(t).
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cuit is an open circuit) or it is originating from the
effect of the spin current density j(y) generating an
electromagnetic field which in turn produces a current
j(x) when the circuit is a closed circuit and an
macroscopic rf current can exist. The details of the
field generated by the spin current density depend on
the spin state present at the end of the rf pulse at time
t. With the spin state known after the pulse, in complete
analogy to Eq. [111] we may claim that the
electromagnetic four-potential generated by the spin
particle is equal to
~An
ðxÞ ¼
1
4p
Zt
À1
dy0
Z
d3
y^jmðyÞ<eðDmn
F ðx À yÞÞ;
[112]
where, compared to Eq. [111], we have exchanged
the coordinates x and y—this is a form of reciprocity—because
now the spin current density represents
the source. Note, ~An
ðxÞ in [112] is a function as far
the electromagnetic field variables are concerned, but
it is an operator with respect to spin variables,
because we have inserted the spin operator version
^jðyÞ. It becomes a function in all variables when we
take expectation values over spin states (see below,
Eq. [113]). It is the field ~An
ðxÞ at space-time position
x generated by the magnetic dipole of the spin particle
in space-time position y, which in turn may generate
j(x) leading to an FID detected as a current.
At that point we need to emphasize again the two
possible macroscopic boundary conditions imposed
by the rf circuit. Either we look at an open circuit,
e.g., an isolated resonator or a simple nonresonant
Hertzian loop with open ports where there is no macroscopic
current j(x) propagating through the resonator
or the loop and as a consequence no NMR radiation
damping occurs. Alternatively we have a closed
circuit, which in practice means that our coil or resonator
or loop is embedded in some larger circuitry
connected to the outer world by cables to transmitters
and receivers outside the probe. In that latter case
NMR radiation damping appears because we allow a
macroscopic classical rf current density through the
coil—this is the case that occurs more often in practice.
The closed circuit case fits easily into our
scheme of two interacting current densities (Eq.
[92]).
However for the purpose of a crucial understanding,
we do elaborate the case with the open circuit,
because it allows us to separate the concept of a single-spin
FID from additional phenomena associated
with NMR radiation damping. Without a macroscopic
rf current j(x) we cannot straightforwardly rely
on Eq. [92] for an elucidation of the quantum electrodynamic
origin of the FID. Nonetheless we can turn
to Eqs. [94–95] which in our context can be interpreted
such that the single spin particle generates a
very weak, but in principle macroscopically detectable
electromagnetic field ~An
ðxÞ, according to Eq.
[112]. The nature of that field, being at the heart of
realizing the FID as arising from a single spin particle,
becomes more obvious when we write out Eq.
[112] in an explicit manner with all the constituent
quantities involved:
~An
ðxÞ ¼
1
4p
e
m
Zt
À1
dy0
Z
d3
yrðxÃ
xÞ Â IÞm<eðh0jiTðAm
ðxÞAn
ðyÞÞj0iÞ
where we took into account Eqs. [31, 75, 88] inserted
into [112]. In Eq. [75] for the spin current density,
we took only the field-independent part, because in
the case we treat here, the emergence of the FID,
there is no externally applied time-harmonic field.
We recognize that ~An
ðxÞ becomes a three-vector
~AðxÞ, because in the approximation applied for the
non-relativistic limit the spin current density is a
three-vector as well, the time-like component being
equal to zero. Hence the spin-dipolar momentum g"hI
generates a three-vector potential. The time-like
component of the four-potential, the scalar potential,
vanishes. Furthermore we see, although ~AðxÞ is a
field function as far as the variables of the electromagnetic
field are concerned, it still depends on the
spin vector operator I. Thus in order to complete our
discussion of virtual photons in the context to link
them to an entirely classical electromagnetic field,
we need to take the expectation value over the single-spin
state function |si:
AavðxÞ ¼ hsj~AðxÞjsi ¼
1
4p
e
m
Zt
À1
dy0
Â
Z
d3
yrððxÃ
ðyÞxðyÞÞ Â hsjIjsiÞm
<eðh0jiTðAm
ðxÞAn
ðyÞÞj0iÞ ½113Š
Equation [113] contains all the information we
need to know about the origin of the classical electromagnetic
field Aav(x) generated by a spin particle situated
in an electromagnetic field in its vacuum state
|0i, the spin particle is characterized by a spatial
wave function x(y), spin operator I, and spin wave
function |si. Note first that Aav(x) depends on the
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vacuum correlation function of the quantum fields
Am(x) and An(y) at positions x (resonator) and y
(spin). More specifically, the vector field Aav(x)
depends on the real part of these vacuum correlations,
which indicates the presence of virtual photons—either
photons being exchanged between the
spin and the surrounding resonator or photons being
emitted and reabsorbed by the spin particle itself,
indicating an effect called self interaction or selfenergy.
Furthermore Aav(x) depends on the expectation
value of the spin vector operator in spin state |si.
The spatial extension of the spin particle plays a role,
characterized by the spatial wave function x (y).
Given the classical vector potential Aav(x) produced
by the single spin particle, we may infer the
associated classical induction field B ¼ r  AavðxÞ.
Suppose, for the sake of simplicity, our coil surrounding
the spin particle is just an open Hertzian
loop circumscribing a surface area S with surface
area element dS and associated normal vector dS,
then the magnetic flux through the loop surface area
is equal to F ¼
R
B Á dS and the change of the magnetic
flux over time through a the loop (here it is an
immobile loop), qF=qt ¼
R
qB=qt Á dS, is equal to
the negative emf or voltage induced across the ports
of the open loop—Faraday’s law of induction.
Equation [113] also makes obvious why in the
semi-classical picture—spin particle as quantum particle
and rf field as classical—we are allowed to
express the FID, the observable signal here manifested
as an emf, as expectation value over spin variables,
more specifically, hsjIjsi. In Eq. [113] we are
allowed to move the expression hsjIjsi out of the integral
over space coordinates such that
qAðxÞ=qt ¼ R Â hsjIjsi with vector R representing
the integral of all the space-coordinate dependent
contributions over d3
y.
One crucial characteristics of the classical electromagnetic
field Aav(x) should not go unnoticed. The
field Aav(x) does not propagate through space in the
sense of classically propagating electromagnetic
waves. This becomes evident when we take Eq. [95]
and swap coordinates x and y so that it applies to the
spin particle as source j(y) of the field,
~An
ðxÞ ¼ À
1
8p
Z
d3
y
^jn
ðy; x0
þ rÞ þ ^jn
ðy; x0
À rÞ
r
[114]
The four-potential in Eq. [114] appears formally
as the sum of a time-retarded and time-advanced part
(arising from time ordering, Section ‘‘The Feynman
Propagator,’’ e.g., Eq. [14]). Such a composition of
two propagating fields, one travelling from the spin
particle outbound, the other travelling inbound
towards the spin particle, yields a standing wave pattern
in the space around the spin particle with an amplitude
proportional to the inverse distance 1/r. The
corresponding induction field would drop off with 1/
r2
, a characteristic that one finds for the region at distances
below one wavelength for the given field
mode—in classical electrodynamics also referred to
as the near field region. The standing wave pattern is
the classical signature of virtual photon exchange
between the spin particle and the electrons in the
loop wire—even if there is no macroscopic rf current
through the wire for an open loop—the electrons in
the metal experience a torque equal to the change in
angular momentum hsj"hIjsi of the spin particle—
while their spatial motion is negligibly small (57). In
the situation observed here, the exchange of a virtual
photon does not transmit energy to an electron in the
conduction band of the metal material of the open
loop where it would appear as kinetic energy (as it
would be the case if a current flows in a closed circuit).
Thus we conclude, for an open loop the virtual
photon exchange transmits three-momentum but zero
energy, the energy of the spin particle as well as the
energy of the electrons in the metal wire do not
change upon photon exchange.
The situation becomes different when the Hertzian
loop is closed and a macroscopic rf current can be
induced originating from the induced emf or Faraday
voltage. Equation [92] becomes directly applicable
again. Now the photon exchange transmits energy
from the single spin to a conduction band electron
current where the transmitted energy appears as kinetic
energy, the electron current performs an oscillatory
motion at a frequency equal to the Larmor frequency
o0 with an amplitude that depends on the
details of the macroscopic circuit (the load) in which
the current loop is embedded. The fact that the oscillation
frequency o0 is equal to the rf frequency of the
induced current, however, does not imply that now
each virtual photon involved in the exchange process
must have exactly the energy "ho0. In order to make
this evident, consider the following Gedanken experiment
with a line of arguments: (a) suppose initially
our single spin is in a definite Zeeman eigenstate,
say, state |þi such that Hz |þi ¼ þ (1/2) "ho0 with Hz
denoting the Zeeman Hamiltonian. Furthermore suppose
we know with certainty that the spin is in that
state, because, for example, we may have prepared
the spin in that state by some earlier experiment. As
a consequence, we do know the precise energy of the
spin in the static magnetic field B0 ¼ o0/g, the energy
being equal to þ"ho0/2. Now (b) we apply an rf pulse
of amplitude a and some duration t with circular
VIRTUAL PHOTONS IN MAGNETIC RESONANCE 307
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polarization transverse to the static magnetic field. It
follows (c) that after the rf pulse our spin is, in general,
not in a Zeeman eigenstate anymore, instead the
spin state appears to be a superposition |s(t)i ¼ c1(t)
|þi þ c2(t) |Ài of Zeeman eigenstates with timeperiodic
coefficients c1 and c2 depending on the
rf pulse parameters a and t as well as on the
time elapsed after the pulse. We note, |s(t)i is not
an energy eigenstate anymore, thus the energy of
the spin particle is uncertain—expressed in technical
terms: the spin energy given by its expectation
value hEspini ¼ hs(t) |Hz |s(t)i is associated with
an uncertainty, which is given by the variance or
standard deviation hDEspini ¼ (hs(t) |H2
z-hHzi2
|s(t)i)1/2
,
the latter being nonzero for states |s(t)i different
from energy eigenstates. We conclude, (d) the
energy of a virtual photon emitted or absorbed by
the spin particle might be well within the range,
or is on the order of the energy uncertainty
h2DEspini. Therefore (e) the energy of the spin
may change as well as the energy of the other
interaction partner (an electron in our current
loop): the photons being exchanged between spin
and closed loop may carry nonzero energy in the
range from 0 to h2DEspini and they transfer three-
momentum.
Having characterized the virtual photon exchange
between two nonvanishing current densities, spin
current and rf current, let us briefly return to the situation
of an open current loop where we found out
that for this specific situation the photons do not
carry energy, even if they transfer three-momentum.
The associated classical vector potential field is
given by Eq. [113] where we recognize that this
field does not depend on any quantities or parameters
of the open loop at all. So we may assert that,
for example, when we would choose a larger loop
diameter or even if we omit the loop leaving a solitary
spin particle sitting in the external static magnetic
field, the principal form of Aav(x) remains
unchanged and, more importantly, the existence of
Aav(x) is independent of whether the Hertzian loop
is present or not. Of course the question arises, if
there is only the single spin particle generating the
field Aav(x), how do we understand virtual photon
exchange for this case? Who does exchange photons
with whom? The simple sounding answer of
quantum electrodynamics (although it is not really
simple) is that the spin particle exchanges photons
with itself, i.e., the spin particle interacts with itself.
Although this kind of interaction seems to be rather
obscure on the first sight, in QED self-interaction
must be counted as a real process being part of
other interaction processes.
OUTLOOK AND CONCLUSION
The present article aimed to introduce the quantized
electromagnetic interaction field in magnetic resonance
and to elaborate the related concept of virtual
photons. For that purpose we have applied the Feynman
propagator formalism for photons. For the physical
interpretation of the photon propagator, which
mathematically represents a special Green function
associated with the inhomogeneous wave equation for
the electromagnetic field, we encountered two kinds
of photons—asymptotically free photons and virtual
photons, the former being always on the zero-mass
shell, the latter being allowed off-shell. When considering
a model for pulsed NMR as a scattering process,
we inferred that the incident rf pulse, considering a
single mode of that field, is characterized well by an
electromagnetic field in a coherent state, which also
provides an explanation of the features of the field in
the classical limit, where the relative uncertainties of
the field amplitude and phase become vanishingly
small. Relying on the assumption that the current densities
(rf current and spin current) interacting with
each other are functions, we succeeded in a separation
of electromagnetic field variables from the spin variables.
This separation enabled us to describe a scenario
where, during the rf pulse, both kinds of photons
appear—asymptotically free photons as well as virtual
photons—whereas after the rf pulse the emerging FID,
either with or without NMR radiation damping effects,
can be characterized by virtual photon exchange only.
The way we have presented QED here leaves one
principal aspect open—the incompleteness that we
took into account by focusing on the quantum electromagnetic
field while not taking care of the fermion
field as a quantum field, although we took spin-1/2
carrying particles (fermions) as quantum objects. We
remain aware of this limitation and take it as preliminary
conclusion: it might be well justified for educational
reasons, but it leaves open a series of crucial
questions. Formally seen, fermion field quantization
requires us to endow the former fermion wave function
c with operator character (in analogy to the
four-potential Am
of the electromagnetic field) and to
submit the c operators to anticommutation rules (as
opposed to commutation rules for the Am
). In the
present article, we have not taken this step of performing
fermion field quantization, thus some of the
statements made above rest on an incomplete basis
insofar that we have not shown in detail the evidence
for these statements. For example, we have claimed
that a solitary spin particle as described above undergoes
self-interaction as a consequence of the observation
that the classical field Aav(x), originating from
308 ENGELKE
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the real part of the photon propagator and hence signifying
virtual photons, does not depend on the circuitry
to detect either the emf or the rf current density. On
the first sight it may seem plausible that it plays no
role whether the detecting circuit is present at all,
however, the difference turns out to be that for the
virtual photon exchange in the case of an existing circuit
surrounding the spin particle the exchange occurs
between circuit and spin and in the case of a nonexisting
circuit the spin particle exchanges photons only
with itself. In this concluding section there is certainly
not enough space to complete a presentation of
QED suitably worked out for magnetic resonance, it
is nevertheless worthwhile to at least point out some
possible extensions that may enhance and deepen the
basic notions of QED discussed in the present article.
A sort of central position is taken by the time evolution
operator for the compound system, spin particle
plus electromagnetic field, whose general form we
have expressed in Eq. [39]. If we take the limit of the
time parameter t, being the upper integration boundary
in [39], towards þ1, we obtain a unitary operator
S ¼ U (1) that is referred to as scattering operator
and that plays an important general role in QED. The
scattering operator S completely characterizes a scattering
process of incident particles or fields, interacting
with each other and then reappearing in one or the
other form as outgoing particles or fields. In that
respect, U(t) as used in our probe model for a scattering
process represents a finite-time version of S. In
full QED the scattering operator S reads
S ¼ T exp Ài
Z
d4
yAn
inðyÞjnðyÞ
 
¼ T exp Àie
Z
d4
y"cðyÞgkAn
inðyÞcðyÞ
 
½115Š
where An
and jn are electromagnetic field and fermion
current density operators, not commuting among each
other. With fermion fields quantized, i.e., with Dirac
wave functions c becoming (non-Hermitian) operators,
the form of Eq. [62] for the fermion current density
is still generally valid. A large part of understanding
QED from a formal or technical point of view is
to learn (a) how to expand the time-ordered exponential
operator S, Eq. [115], in a perturbative power series,
(b) to find a meaningful physical interpretation
for the terms of increasing order in this series expansion
with each term seen as an elementary process,
and (c) to be able to actually calculate transition
amplitudes hf |S |ii between initial states |ii and final
states |fi, and following from them, scattering cross
sections for the various elementary processes as well
as the total scattering cross section as the sum of all
particular cross sections. The transition amplitude
hf|S(n)
|ii for a term S(n)
of nth order in the power series
expansion of S yields a probability measure associated
with the corresponding elementary process in
the overall scattering process. It is interesting to
observe that when, in the series expansion of S, we reintroduce
the assumption that current densities commute,
no terms higher than second order appear. This
becomes evident by reminding ourselves that (i) with
only noncommuting electromagnetic field operators
being present and (ii) the commutator of two electromagnetic
field operators yielding a nonoperator
(a complex-valued function), the nested commutator
[. , [ . , . ]] of electromagnetic field operators appearing
in third and higher order terms is equal to zero.
Thus, we might say that the assumption of commuting
current densities and current densities representing
functions led us to consider QED processes up to
second order in the perturbative series expansion of
the scattering matrix S given by [115].
In the present article we have tacitly introduced
Feynman diagrams such as Figs. 1–3 and 6–8 in a
rather informal way, often as a kind of illustration for
the related formal analytical expressions. Nevertheless,
Feynman diagrams are not just graphical representations
of some intuitive pictures what quantum
objects might do. They are concise and precise diagrammatic
tools corresponding to well defined analytical
terms appearing in the power series expansion of
the scattering operator S. However, despite their
abstract and rigorous meaning as a graphical tool and
even if one may not push the pictorial information too
far, Feynman diagrams certainly provide some kind of
intuitive insight—beside the rigorous meaning—that
is hardly experienced when just looking at the corresponding
analytical expressions. Nevertheless, as we
do not have the opportunity and space here to elaborate
this correspondence between Feynman diagrams
and analytical terms given by strict rules, we refer to
the textbook literature, e.g., (79, 81, 89, 90, 95), and
just discuss one aspect in some exemplary fashion that
appears relevant and interesting when studying nonrelativistic
spins (i.e., we will avoid encounters with
antiparticles) interacting with electromagnetic fields.
We first introduce some technical terminology.
Feynman diagrams as in Figs. 1 and 2 are composed
of fermion lines (spin-1/2 particles) and boson lines
(photons). A fermion line is drawn as a solid line
with an arrow, to symbolize fermion propagation.
Likewise, boson lines are drawn as wavy lines, symbolizing
photon propagation. Interaction between fermions
and photons is symbolized by points where two
fermion lines meet one boson line, such a point is
called vertex. Boson or fermion lines that connect two
VIRTUAL PHOTONS IN MAGNETIC RESONANCE 309
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vertices in a diagram are called internal lines—they
represent virtual particles. Fermion or boson lines that
either end in one vertex or start in one vertex but do
not connect to other vertices are called external
lines—they stand for asymptotically free particles.
Feynman diagrams can be drawn to represent processes
in space-time, like in Figs. 2, 3, and 8, where the
vertices are labeled with space-time coordinates, each
vertex representing an event in space-time (which is
not necessarily and not always literally point-like in
space or time) when some photon emission or absorption
by a fermion occurs. But also in four-dimensional
k space Feynman diagrams have their usage and benefit,
examples are shown in Figs. 6, 7, and 11. In k
space vertices also symbolize interaction events (emission,
absorption), but here we label the inner and outer
fermion and boson lines with their respective fourmomenta
or, in some cases, angular momenta (as in
Figs. 6 and 7) carried either by the photon or fermion.
A few strict rules, derived from the underlying analytical
formalism, apply in a universal manner to all kspace
Feynman diagrams in QED: (1) all fermions
symbolized by external fermion lines correspond to
asymptotically free fermions and obey the energy-momentum
relationship p2
¼ m2
, synonymously, these
fermions are on-shell (Appendix A, Eqs. [A8, A9]).
Likewise, all photons corresponding to external photon
lines satisfy the energy-momentum relationship
for photons, k2
¼ 0, (Appendix A, Eq. [A10]), hence
they are on-shell and asymptotically free. For internal
fermion lines p2
¼ m2
is not required, as well as for internal
photon lines k2
¼ 0 is not required. Internal fermions
or photons correspond to virtual particles, these
can be off-shell. Finally, in addition to these rules that
apply to lines, there is one general rule for vertices: in
every vertex, the meeting point of two fermion lines
and one photon line, the energy and the three-momentum
of all incident and outgoing particles is strictly
balanced—that means, as a vertex property managing
interacting fermions and photons, energy and threemomentum
are conserved quantities!
Let us consider a few examples compiled in Fig.
11. Each diagram is an example for a particular kind
of elementary interaction process. Starting with Fig.
11(B), the Feynman diagram drawn here is the k
space equivalent of the diagram in Fig. 2. In general,
it shows the elementary process of exchanging a virtual
photon (internal wavy line) between two fermions
(making up the current densities between
which the interaction occurs). There are two incident
fermions, one with four-momentum p, the other with
four-momentum q, the virtual photon being
exchanged carries the four-momentum k. After the
photon exchange one fermion carries four-momentum
p0
, the other q0
. Since all fermion lines are external, it
must hold p2
¼ p02
¼ m2
for the left fermion (with rest
mass m), likewise it must hold q2
¼ q02
¼ M2
for the
right fermion (rest mass M). For the internal photon
line there is no restriction on k, virtual photons can be
off-shell. However for the two vertices we have to
insist that p ¼ p0
þ k [left vertex in Fig. 11(B)] and q
þ k ¼ q0
[right vertex in Fig. 11(B)]—conservation of
energy and momentum in vertices. Diagram 11B has
been the elementary process discussed in Section
‘‘Single-Spin FID: NMR Radiation Damping’’ as far
as the FID and radiation damping is concerned. We
could continue to go through the other diagrams in
Fig. 11 in the same detailed way, but let us dwell for a
moment on diagram 11A. It shows a single vertex
with one incident external fermion line, one outgoing
external fermion line and one external photon line.
This is a diagram of the first order term in the series
expansion of the scattering operator S. As it turns out
generally, the number of vertices in a diagram indicates
the order in the series expansion—so first order,
one vertex. First formally, for the fermion and the
photon in Fig. 11(A) we have to require
p2
¼ p02
¼ m2
; k2
¼ 0; p ¼ p0
þ k [116]
Second, attempting to interpret diagram 11A, we
may say that it represents the spontaneous emission
of a photon with four-momentum k (on-shell) where
the fermion with initial momentum p after the spontaneous
emission event appears to have the final
four-momentum p0
. So, we could try to imagine our
solitary spin particle suddenly emitting a photon. Do
such first-order spontaneous emission processes
really occur? The answer is no, for the following reason.
At first we notice, although we have correctly
applied the rules for Feynman diagrams, that the
equations in [116] are incompatible with each other, or
Figure 11 Feynman diagrams in k space illustrating the
following elementary electromagnetic processes: (A)
spontaneous emission of a photon as a first order process,
(B) fermion-fermion interaction by virtual photon
exchange, (C) Compton scattering including photon
absorption and emission, (D) fermion self interaction.
Except for the process in (A), which is of first order, all
other processes shown are of second order.
310 ENGELKE
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even harder, we may prove that the equations p2
¼ p02
¼ m2
in conjunction with k2
¼ 0 are in contradiction
with the equation p ¼ p0
þ k. To see this, one could
write down these four-dimensional equations separately
for the time-like components (energy terms) and
the space-like components (three-momentum terms)
and one recognizes, that they are only noncontradictory
if the photon energy k0
¼ 0 as well as the photon
three-momentum k ¼ 0, in other words—there is no
photon. If we calculate the transition amplitude
hf|S(1a)
|ii for the first order process in diagram 11A
which is represented by the term S(1a)
in the expansion
of S, we obtain zero (a detailed derivation of this result
can be found, e.g., in Ref. 124). That means: spontaneous
emission of a photon as a first order elementary
process is a forbidden process. These findings do not
imply that spontaneous emission is impossible. They
just show that spontaneous emission as a first order
process does not occur. It might be possible that a diagram
with the topology like the one in Fig. 11(A)
appears as a subdiagram in higher order diagrams. This
is the case in diagram 11C which can be read as photon
absorption with subsequent photon emission—a process
that for optical, UV and X-ray photons we would
call Compton effect. In the second-order diagram 11C
we do not meet contradictory energy-momentum relationships
and vertex relations for energy and three-momentum
conservation because in that diagram there
appears a virtual fermion that can be off-shell. Calculating
the corresponding transition amplitude hf |S(2c)
|ii
gives a nonzero result. Processes like in Fig. 11(C)
occur during the interaction between the spin-1/2 particle
with the incident rf pulse.
In summary, the probability that a spin-1/2 particle
spontaneously emits an asymptotically free photon is
zero in first order. Higher order processes may lead to
a finite probability for such emission processes.
Finally, let us say a few words about the diagram in
Fig. 11(D). It depicts a process where a virtual photon is
emitted by a spin-1/2 particle and subsequently this photon
is absorbed by the same spin-1/2 particle. In the literature
this diagram is referred to as self-energy diagram.
The elementary process behind it and other processes
play a crucial role for the understanding of the anomalous
magnetic moment of the electron (g . 2) (21, 124).
Summing up, the elementary processes depicted
in Fig. 11 are relevant for electromagnetic interactions
of a spin-1/2 particle: (a) first-order spontaneous
emission of an asymptotically free photon does
not occur—it is a forbidden process. (b) the basic
interaction process, the exchange of a virtual photon
between two electromagnetic current distributions, is
of second order—this is the elemental process in FID
formation and NMR radiation damping as well as the
interaction between an externally generated rf current
density and a spin particle or a spin particle system. (c)
the absorption of a photon with subsequent emission of
an asymptotically free photon is an allowed process.
This process can be seen as elemental for the emission
and absorption of photons during the rf pulse interacting
with the spins where the rf field is in a coherent
state. (d) self energy generally plays a role in interaction
processes. It leads to a mass renormalization of
the electron and plays a partial role in the origin of the
anomalous magnetic moment of the electron.
In our discussion of QED processes we have met
other approximations. We have relied on the unitarity
of the time evolution for the electromagnetic field and
the spin particle by explicitly excluding dissipative
processes—loss mechanisms for conduction band
electrons in metals, dielectric losses in macroscopic
samples, or spin-lattice relaxation processes. In moving
beyond this approximation, it might be possible to
develop models for magnetic resonance that include
the thermal photon field—photons with an energy distribution
corresponding to thermal equilibrium—in
such a way to study noise processes using to the QED
view of magnetic resonance. Another possibility
would be to focus on the inclusion of spin-spin couplings
besides the coupling between spin and externally
applied rf field. As a result of such studies, an
NMR probe model more complex than the one discussed
in the present article as well as a more realistic
model of the spin system in the NMR sample, and
finally, e.g., the actual study of micro or nanosamples
and micro or nanocoils could provide a further application
for the QED framework as a tool for physical
interpretation and analysis of magnetic resonance.
ACKNOWLEDGMENTS
This article is dedicated to Dieter Michel, teacher
and friend, on occasion of his 70th birthday.
The author gratefully acknowledges the enlightening
discussions with Mark Butler, Lyndon Emsley, and
Alexander Krahn. Particular thanks to Mark Butler for
the critical reading of the manuscript. Further thanks to
Francis Taulelle for communicating Ref. 54, to Huub
de Groot for alerting the author to Ref. 62, and to Mark
Butler for pointing out and providing Ref. 99.
APPENDIX A
Covariant Notation, Minkowski Space,
Lorentz Transformation, Maxwell’s
Equations (78)
It is in the very spirit of special relativity to treat
space and time at an equal footing. Thus, threeVIRTUAL
PHOTONS IN MAGNETIC RESONANCE 311
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dimensional space and one-dimensional time are
united in four-dimensional space-time, providing a
vector space with pseudo-Euclidean metric called
Minkowski space. A position vector in Minkowski
space has four components with the first one representing
the time coordinate x0
¼ ct, and the remaining
three being equal to the common coordinates of
a point in three-dimensional space. Such a vector in
Minkowski space is called four-vector and it comes
in two flavors: one is called contravariant fourvector
indicated by a superscript index,
x ¼ xk
¼ ðct; x1
; x2
; x3
Þ ¼ ðct; rÞ [A1]
where time and space coordinates have the same
sign, the other is referred to as covariant four-vector
labeled by a subscript index
xk ¼ ðct; Àx1
; Àx2
; Àx3
Þ ¼ ðct; ÀrÞ [A2]
where the time and usual space coordinates have the
opposite sign. The superscripts or subscripts i, k, m,
. . . enumerate the component, e.g., k ¼ 0, 1, 2, 3.
The infinitesimal distance ds between two points in
Minkowski space is called line element and given as,
ds2
¼
X3
i¼0
X3
k¼0
gikdxi
dxk
 gikdxi
dxk
¼ c2
dt2
À dx1
À dx2
À dx3
½A3Š
with coordinate differentials dxi, which are also fourvectors.
In the notation of Eq. [A3] we have made
use of an important convention: any expression with
one or more pair(s) of equal covariant and contravariant
indices is summed up (Einstein summation convention).
In Eq. [A3] there are two such pairs. This
simplifies notation—in [A3], for example, one can
omit the two sum signs. Also, appearing in [A3] is
the metric tensor, a symmetric, second-rank tensor
(written as a diagonal 4 Â 4 matrix), which appears
to be Lorentz-invariant. In Cartesian coordinates, the
metric tensor reads
gik ¼ gik
¼
1 0 0 0
0 À1 0 0
0 0 À1 0
0 0 0 À1
0
B
B
@
1
C
C
A [A4]
Note that for the metric tensor, covariant and
contravariant components coincide. By means of the
metric tensor, we can raise or lower covariant or
contravariant indices of a vectors and tensors,
respectively. For a vector,
ui ¼ gikuk
; ui
¼ gik
uk [A5]
The scalar product of two four-vectors x and y
reads xm
ym ¼ x0
y0 þ ðx1
y1 þ x2
y2 þ x3
y3Þ. This does
not hold only for space-time position vectors allocating
points in space-time, it holds for arbitrary
four-vectors,
uiwi
¼ ui
wi ¼ u0
w0
À u Á w: [A6]
The norm square (equal to the squared magnitude)
of vector ui
is defined as
ui
ui ¼ uiui
¼ ðu0
Þ2
À juj2
[A7]
We recognize that the norm square ui
ui of a
four-vector ui
cannot be positive-definite. It might
be a positive or negative number or equal to zero. If
ui
ui . 0, vector ui
is called a time-like vector, if
ui
ui , 0, vector ui
is called a space-like vector, and
if ui
ui ¼ 0, vector ui
is called a null vector or a
light-like vector. All time-like coordinate vectors xk
correspond to points in four-space that are within
the lightcone whose surface is defined by coordinate
vectors that are light-like, i.e., xk xk
¼ 0 or c2
t2
¼
r2
, which means, the surface of the lightcone is
given by light rays that travel the distance r during
the time interval t with the speed of light c
(Fig. A1).
Now let us switch to four-dimensional k space
which is the Fourier domain of space-time. From
special relativity we know that for the total energy
E of a free particle (i.e., a particle without interaction)
with momentum p and rest mass m it holds
Figure A1 In four-dimensional Minkowski space-time
with pseudo-Euclidean metric the norm square of any
four-vector x is given by x2
¼ x2
0Àx2
1Àx2
2Àx2
3. Hence, it
can be positive, negative, or zero. In the latter case, it
follows x2
0 ¼ c2
t2
¼ x2
1þx2
2þx2
3 ¼ r2
. This represents a
(double) cone surface in space-time which defines the
light cone.
312 ENGELKE
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E2
¼ jpj2
c2
þ m2
c4
[A8]
For the particle at rest, p ¼ 0 we see that its
energy is equal to its rest energy mc2
. For particle
velocities v small compared to c, it follows
E ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jpj2
c2 þ m2c4
q
¼ mc2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
v2=c2 þ 1
p
% mc2
ð1 þ v2
=2c2
Þ ¼ mc2
þ
m
2
v2
i.e., we obtain the total energy E being equal to the
rest energy plus kinetic energy of a slowly moving
particle. Let us rewrite Eq. [A8] in four-dimensional
form with the total energy E equal to the time-like
component k0
of the four-momentum k and for the
three-momentum it holds p ¼ "hk. Then Eq. [A8]
reads (k0
)2
¼ |k|2
þ m2
, where we have set "h ¼ 1,c
¼ 1 to simplify notation. With the four-momentum
vector k ¼ kn
¼ (k0
, k), Eq. [A8] becomes
k2
¼ m2
[A9]
Equation [A8] or its four-dimensional variant
[A9] is called energy-momentum relationship for
the particle with rest mass m. We are allowed to
apply [A8] or [A9] also for the case of free photons.
In that case it holds m ¼ 0, photons have no rest
mass, and Eq. [A9] converts into
k2
¼ 0 [A10]
In classical electromagnetism Eq. [A10] has a well
familiar meaning. Writing k0
¼ o/c and |k|¼ 2p/l, we
see that [A10] becomes the well-known relationship
between angular frequency o and wavelength l of
freely propagating electromagnetic waves:
o ¼ 2pc=l [A11]
In three-dimensional momentum space we may
define the spherical shell with radius r given by
jkj2
¼ ðk1
Þ2
þ ðk2
Þ2
þ ðk3
Þ2
¼ ðk0
Þ2
À m2
¼ r2
! 0
[A12]
This spherical shell is referred to as the mass
shell with radius r ¼ þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðk0Þ2
À m2
q
and drawn in
Fig. A2. For photons with m ¼ 0 it is called zero
mass shell and its radius is equal to k0
.
Photons or electromagnetic waves that satisfy
Eq. [A10] or Eq. [A11] are called on-shell or onzero
mass shell.
We define the covariant partial derivative with
respect to the space-time coordinates as
qk 
q
qxk
¼
q
cqt
; r
 
[A13]
and d’Alembert’s operator (the operator for the
wave equation) in four-dimensional notation
q2
 qkqk
¼
1
c2
q2
qt2
À r2
[A14]
Any expression which can be written as
qiuk À qkui [A15]
is called four-rotation (a generalization of the curl
in three dimensions) of the vector ui
and is an antisymmetric
tensor of rank 2. Analogously, a scalar
quantity a given by
a ¼ qiui
¼ qi
ui ¼
1
c
qu0
qt
À r Á u
 
[A16]
is called four-divergence of the vector ui. A vector
wi that results from a scalar quantity b such that
wi ¼ qib ¼
1
c
qb
qt
; rb
 
; [A17]
is called four-gradient of the scalar b.
The principle of special relativity states that the
physical phenomena observed by observers in
Figure A2 Definition of the zero-mass shell for photons
in three-dimensional momentum space.
VIRTUAL PHOTONS IN MAGNETIC RESONANCE 313
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different inertial frames are independent from these
frames, i.e., that all inertial frames are equivalent.
Inertial frames are frames that are moving with constant
velocity relative to each other or that transform
into each other by spatial translations and rotations.
The transition from one frame to another is
accomplished by the Lorentz transformation. As an
example, a Lorentz transformation may describe the
transformation from one inertial system with coordinate
frame xk
to a second one, coordinate frame "xk
,
moving with the velocity v along the x1
axis:
"x0
¼
x0
À bx1
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À b2
p ; "x1
¼
x1
À bx0
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À b2
p ;
"x2
¼ x2
; "x3
¼ x3
; b ¼ v=c ½A18Š
The specific transformation in [A18] is referred
to as a Lorentz boost along the x1
axis. Setting
tanhV ¼ b, then Eq. [A18] can be rewritten as
"x0
"x1
"x2
"x3
0
B
B
@
1
C
C
A ¼
cosh O À sinh O 0 0
À sinh O cosh O 0 0
0 0 1 0
0 0 0 1
0
B
B
@
1
C
C
A
x0
x1
x2
x3
0
B
B
@
1
C
C
A
[A19]
which can be understood as a rotation in the x0
x1
plane with the (imaginary) rotation angle V. Generally,
Lorentz transformations can be viewed as either
hyperbolic rotations (like the one in [A19]) or
as real rotations in three-space (without the time
dimension being involved). They belong to the class
of linear coordinate transformations in Minkowski
space and can be expressed as
"xk
¼ Ãk
i xi
; Ãk
i ¼
q"xk
qxi
; Ãk
i Ãi
m ¼ dk
m ¼ gmngnk
[A20]
with the transformation matrices Lk
i (an example of
such a matrix is given in [A19]), forming a group,
called the (homogeneous) Lorentz group.
The common three-vector of the electromagnetic
vector potential A ¼ (A1
, A2
, A3
) and the scalar
potential f form a four-vector
Ak
¼ ðf; A1
; A2
; A2
Þ ¼ ðf; AÞ;
Ak ¼ ðf; ÀA1
; ÀA2
; ÀA3
Þ ¼ ðf; ÀAÞ ½A21Š
The electromagnetic field strength tensor (a completely
antisymmetric tensor of rank 2) is defined as
the four-rotation (see Eq. [A15]) of the four-potential:
Fik ¼
qAk
qxi
À
qAi
qxk
 qiAk À qkAi [A22]
i.e., Fii ¼ 0 and Fik ¼ ÀFki. For example, we have
F01 ¼
qA1
qx0
À
qA0
qx1
¼
1
c
qA1
qt
À
qf
qx1
¼ E1
¼ Ex;
F02 ¼
qA2
qx0
À
qA0
qx2
¼
1
c
qA2
qt
À
qf
qx2
¼ E2
¼ Ey; etc.
and
F12 ¼
qA2
qx1
À
qA1
qx2
¼ ÀB3
¼ ÀBz;
F13 ¼
qA3
qx1
À
qA1
qx3
¼ B2
¼ By; etc.
hence
Fik ¼
0 E1
E2
E3
ÀE1
0 ÀB3
B2
ÀE2
B3
0 ÀB1
ÀE3
ÀB2
B1
0
0
B
B
B
@
1
C
C
C
A
;
Fik
¼
0 ÀE1
ÀE2
ÀE3
E1
0 B3
ÀB2
E2
ÀB3
0 B1
E3
B2
ÀB1
0
0
B
B
B
@
1
C
C
C
A
; ½A23Š
and Eqs. [A22] correspond to the well-known threespace
equations
E ¼ À
1
c
qA
qt
À rf; B ¼ r  A [A24]
defining the electric field strength E and the magnetic
induction B in terms of the electromagnetic potentials.
Note, the present definitions suppose that electric
field strength and magnetic induction have identical
units. In order to convert the units for E and B into
conventional SI units (with V/m for the electric and
Vs/m2
for the induction field), E needs to be divided
by c (with B unchanged), or alternatively, B needs to
be multiplied by c (with E unchanged). Since we are
treating time and space as equivalent as well as unifying
the scalar and the vector potential in a four-vector,
it appears natural to also treat E and B unified in the
field strength tensor in a system of identical units.
The electromagnetic field tensor Fik satisfies the
cyclic equation
Fhikjmi  qmFik þ qiFkm þ qkFmi ¼ 0 [A25]
Identities for antisymmetric second-rank tensors
Fik of the form of Eq. [A25] are called Bianchi
314 ENGELKE
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identities. The validity of [A25] can be easily
proved by inserting [A22] into [A25].
Fhik|mi is a tensor of rank 3 with 64 components
being nonzero only if i = k = m. Fhik|mi is completely
antisymmetric, i.e., Fhik|mi ¼ À Fhki|mi, Fhik|mi
¼ À Fhim|ki, etc. This condition leaves only 24 nonzero,
thus 12 nonzero and different components.
Since for every index triple i, k, m with if i = k = m
there are three permutations yielding the same equation,
we conclude that Eq. [A25] corresponds to
actually four distinct equations:
Fh12j3i ¼ q3F12 þ q1F23 þ q2F31
¼ À
qBz
qz
À
qBx
qx
À
qBy
qy
¼ Àr Á B ¼ 0
and
Fh23j0i ¼ q0F23 þ q2F30 þ q3F02
¼ À
qBx
cqt
À
qEz
qy
þ
qEy
qz
¼ À
qBx
cqt
À ðr  ~EÞx ¼ 0
Fh01j3i ¼ q3F01 þ q0F13 þ q1F30
¼
qEx
qz
þ
qBy
cqt
À
qEz
qx
¼ þ
qBy
cqt
þ ðr  ~EÞy ¼ 0
Fh02j1i ¼ q1F02 þ q0F21 þ q2F10
¼ þ
qEy
qx
þ
qBz
cqt
À
qEx
qy
¼ þ
qBz
cqt
þ ðr  ~EÞz ¼ 0
These four equations, and therefore Eq. [A25],
constitute the first two of Maxwell’s differential
equations (one vector equation and one scalar equa-
tion)
r  E ¼ À
1
c
qB
qt
; r Á B ¼ 0 [A26]
Note, these two equations are the strict consequence
of the definition of the field strength tensor
[A22].
We turn our attention to the second set of Maxwell’s
equations. The differential electric charge of a
particle, dq, is a scalar quantity. Let us denote by r
the three-dimensional electric charge density, with dq
¼ r dV and dV ¼ dx1
dx2
dx3
equal to the volume element
in three-space that contains the charge dq. Multiplying
dq ¼ r dV on both sides with dxi
, we obtain
dqdxi
¼ rdVdxi
¼ rdVdt
dxi
dt
[A27]
and we observe, since dq dxi
is a four-vector (dq is a
scalar, dxi
is a four-vector), that the right-hand side of
Eq. [A27] must be a four-vector. Because dVdt is
scalar, we conclude that rdxi
/dt is a four-vector defining
the electromagnetic current density
jk
¼ r
dxk
dt
¼ ðcr; jÞ [A28]
j ¼ rdr/dt ¼ rv is equal to the three-dimensional current
density. The third and fourth of Maxwell’s equations
read in four-dimensional notation
qiFik
¼ Àjk
; [A29]
In three-dimensional notation it follows from Eq.
[A29]
r Á E ¼ r; r  B ¼
qE
cqt
þ j
 
[A30]
Furthermore from Eq. [A29] we obtain
qkqiFki
¼ Àqkjk
Because qkqi is a symmetric tensor (the partial
derivatives commute) and the electromagnetic field
tensor Fki
is an antisymmetric tensor, we find that
qkqiFki
: 0. Therefore the four-divergence of the
current density vanishes,
qkjk
¼ 0 [A31]
expressing the continuity equation
qr
qt
þ r Á j ¼ 0 [A32]
APPENDIX B
Gauge Transformation, Lorenz
Gauge, Wave Equation,
and Green Functions
The electromagnetic potentials Ak
do not uniquely
determine the field strength tensor Fik, Eq. [A22].
To each Ak
one might add or subtract the four-gradient
of a scalar function f without changing the
electromagnetic field tensor Fik
:
~Ak ¼ Ak À qkf [B1]
Equation [B1] is called gauge transformation and
the arbitrary scalar function f is referred to as gauge
function. The invariance of Fik under the transformation
[B1] can easily be demonstrated,
VIRTUAL PHOTONS IN MAGNETIC RESONANCE 315
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~Fik ¼ qi
~Ak À qk
~Ai ¼ qiAk À qiqkf À qkAi þ qkqif
¼ qiAk À qkAi ¼ Fik ½B2Š
For that reason, Maxwell’s equations are also
invariant under gauge transformations. In threespace,
Eq. [B1] reads
~A ¼ A À rf; ~f ¼ f À
qf
cqt
[B3]
The four-potentials ~Ak obtained by Eq. [B1] and
the four-potentials Ak describe electromagnetic fields
which are equivalent in the sense that both lead to the
same field strength tensor [B2], i.e., the same electric
and magnetic fields. From Maxwell’s equations
[A28] we obtain by inserting the definition [A22] of
the field tensor Fik
by the four-potentials ~Ak
:
qiðqi ~Ak
À qk ~Ai
Þ ¼ qiqi ~Ak
À qiqk ~Ai
¼ Àjk
i.e.,
qiqi ~Ak
À qk
ðqi
~Ai
Þ ¼ Àjk
[B4]
That means, Maxwell’s equations allow to derive
Eq. [B4] for the four-potential containing the mixed
term qk
ðqi
~Ai
Þ. The ambiguity of the potentials
allows to impose one arbitrary additional condition
for Ak, thus gauging the electromagnetic field. Let
us therefore choose the condition
qiqi
f ¼ qiAi
[B5]
which leads to
qi
~Ai
¼ 0 ¼ qiAi
À qiqi
f [B6]
Eqs. [B6] is called Lorenz gauge condition (not
Lorentz gauge, see Ref. 121). Inserting Eq. [B6]
into Eq. [B4] yields the inhomogeneous wave equation
(d’Alembert’s equation)
q2 ~Ak
¼ qiqi ~Ak
¼ Àjk
[B7]
which reads in three-space
1
c2
q2 ~f
qt2
À r2 ~f ¼ Àr
1
c2
q2 ~A
qt2
À r2 ~A ¼ Àj [B8]
Details on electromagnetic gauge invariance
linked to local phase invariance in quantum theory
can be found in Ref. 10.
Taking the wave equation [B7] (omitting the
tilde over A),
q2
Am
ðxÞ ¼ Àjm
ðxÞ [B9]
we ask how to solve [B9] for certain given boundary
conditions. Formally, we can Fourier transform
the partial differential equation [B9] to obtain the
algebraic equation in four-dimensional k space
À k2
Am
ðkÞ ¼ Àjm
ðkÞ [B10]
Equation [B10] could be formally solved for
Am(k), however, this solution would diverge for
the case
k2
¼ 0 ¼ ðk0
Þ2
À j~kj2
¼ ðo=cÞ2
À ð2p=lÞ2
[B11]
Equation [B11] represent the energy-momentum
or, equivalently, the frequency-wavelength relationship
for free electromagnetic waves, see Appendix A,
Eqs. [A10–A12]. The theory of linear partial differential
equations suggests the following particular solution
for the inhomogeneous differential equation [B9]:
Am
ðxÞ ¼
Z
d4
yGðx À yÞjm
ðyÞ [B12]
The function G(x) is referred to as a Green function
associated with the differential operator q2
:
qn
qn. A Green function is a fundamental solution of
the associated differential equation,
q2
GðxÞ ¼ Àdð4Þ
ðxÞ; [B13]
which is, in our case here, the inhomogeneous wave
equation with the general source current density
jm
(x) replaced by the Dirac d function in space-time
representing a point source in space as well as in
time. When we take into account the definition
[A13] for the wave operator q2
, Eq. [B13] reads, in
ordinary vector notation,
1
c2
q2
Gðx0; xÞ
qt2
À ~r2
Gðx0; xÞ ¼ Àdðx0Þdð3Þ
ðxÞ
Applying the wave operator q2
to Eq. [B12] and taking
into account Eq. [B13], we see immediately, that
q2
Am
ðxÞ ¼ q2
Z
d4
yGðx À yÞjm
ðyÞ
¼
Z
d4
yq2
xGðx À yÞjm
ðyÞ ¼
¼ À
Z
d4
ydð4Þ
ðx À yÞjm
ðyÞ ¼ Àjm
ðxÞ
316 ENGELKE
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
i.e., Am
(x) given by [B12] is a particular solution to
the wave equation [B9]. There exists a variety of
Green functions G(x), each representing specific
boundary conditions. Examples are the retarded
Green function or the Feynman Green function
treated in more detail in Appendix C.
Equations [10] and [D10] for the retarded
propagator Dret(xÀy) are equivalent.
We want to show that the retarded Green function
defined by Eq. [10] (with m0 ¼ 1, c ¼ 1),
Dretðx À yÞ ¼
1
4p
dððx0
À y0
Þ À rÞ
r
; r ¼ jx À yj [10]
is equivalent to the retarded Green function appearing
in Eq. [D10] (see Appendix D)
Dretðx À yÞ ¼ yðx0
À y0
ÞDðx À yÞ [D10]
with D(xÀy) denoting the Pauli-Jordan function
defined by
À igmn
Dðx À yÞ ¼ ½Am
inðxÞ;An
inðyފ [B14]
We start with Eq. [24] for the Fourier expansion
of the four-potential
AmðxÞ ¼
Z
d3
k
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2pð Þ3
2ok
q
Â
X3
l¼0
eðlÞ
m ðkÞ aðlÞ
ðkÞeÀikx
þ aðlÞþ
ðkÞeikx
 
½24Š
to calculate the commutator of four- potentials, taking
into account the CCR’s [25],
½aðlÞ
ðkÞ;aðrÞþ
ðk0
ފ ¼ Àglr
d3
ðk À k0
Þ; l;r ¼ 0;1;2;3
[25]
Therefore we write down at first
Àigmn
DðxÀyÞ¼½Am
inðxÞ;An
inðyފ¼
¼
Z
d3
k0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2pð Þ3
2ok0
q
Z
d3
k
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2pð Þ3
2ok
q
X3
l0
¼0
X3
l¼0
eðl0
Þ
m ðk0
ÞeðlÞ
n ðkÞ
 aðl0
Þ
ðk0
ÞeÀik0
x
þaðl0
Þþ
ðk0
Þeik0
x
 
;
h
aðlÞ
ðkÞeÀiky
þaðlÞþ
ðkÞeiky
 i
which gives, after disposing of commutator terms
that yield zero by definition,
À igmn
Dðx À yÞ ¼ ½Am
inðxÞ; An
inðyފ ¼
¼
Z
d3
k0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2pð Þ3
2ok0
q
Z
d3
k
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2pð Þ3
2ok
q
X3
l0
;l¼0
eðl0
Þ
m ðk0
ÞeðlÞ
n ðkÞ
 aðl0
Þ
ðk0
ÞeÀik0
x
; aðlÞþ
ðkÞeiky
h i
þ aðl0
Þþ
ðk0
Þeik0
x
; aðlÞ
ðkÞeÀiky
h i
Applying the commutator relationships [25]
yields
Àigmn
DðxÀyÞ¼½Am
inðxÞ;An
inðyފ¼
¼
Z
d3
k0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2pð Þ3
2ok0
q
Z
d3
k
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2pð Þ3
2ok
q
X3
l0
;l¼0
eðl0
Þ
m ðk0
ÞeðlÞ
n ðkÞ
 Àgll0
d3
ðk0
ÀkÞeÀiðk0
xÀkyÞ
þgll0
d3
ðk0
ÀkÞeiðk0
xÀkyÞ
 
Because the metric tensor is diagonal, the double
sum over l, l0
reduces to a single sum over l ¼l0
,
À igmn
Dðx À yÞ ¼ ½Am
inðxÞ; An
inðyފ ¼
¼
Z
d3
k0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2pð Þ3
2ok0
q
Z
d3
k
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2pð Þ3
2ok
q
X3
l¼0
ðÀgll
ÞeðlÞ
m ðk0
Þ
ÂeðlÞ
n ðkÞ d3
ðk0
À kÞeÀiðk0
xÀkyÞ
À d3
ðk0
À kÞeiðk0
xÀkyÞ
 
Performing the integration over k0
yields
À igmn
Dðx À yÞ ¼ ½Am
inðxÞ; An
inðyފ ¼
¼
Z
d3
k
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2pð Þ3
2ok
q
X3
l¼0
ðÀgll
ÞeðlÞ
m ðkÞeðlÞ
n ðkÞ
 expðÀikðx À yÞÞ À expðikðx À yÞÞð Þ ½B15Š
We observe that for the sum over the polarization
dependent terms the completeness relation in
four dimensions holds:
X3
l¼0
ðÀgllÞeðlÞ
m ðkÞeðlÞ
n ðkÞ ¼ Àgmn [B16]
Inserting [B16] into [B15] yields
À igmn
Dðx À yÞ ¼ ½Am
inðxÞ; An
inðyފ ¼ Àgmn
Â
Z
d3
k
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2pð Þ3
2ok
q expðÀikðx À yÞÞ À expðikðx À yÞÞð Þ
[B17]
VIRTUAL PHOTONS IN MAGNETIC RESONANCE 317
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In order to calculate the integral over threedimensional
k space, we introduce spherical coordinates
ok ¼ x ¼ jkj; # ¼ ffðk; ðx À yÞ ¼ rÞ, j, such
that Eq. [B17] reads
igmn
Dðx À yÞ ¼ gnm
Z1
0
x2
dx
2pð Þ3
2x
Zp
0
sin #d#
Â
Z2p
0
dj eÀik0
ðx0Ày0Þ
eixr cos #
À eik0
ðx0Ày0Þ
eÀixr cos #
 
Integrating over dj and observing that Àd cos W
¼ sin WdW allows us to write
igmn
Dðx À yÞ ¼ À
gnm
8p2
Z1
0
xdx
Â
ZÀ1
1
d cos # eÀik0
ðx0Ày0Þ
eixr cos #
À eik0
ðx0Ày0Þ
eÀixr cos #
 
After integrating over d cos W, we arrive at
igmn
Dðx À yÞ ¼ À
gnm
8p2
Z1
0
xdx
Â
eÀik0
ðx0Ày0Þ
ixr
ðeÀixr
À eixr
Þ À
eik0
ðx0Ày0Þ
Àixr
ðeixr
À eÀixr
Þ
!
which after rearrangement of terms gives
igmn
Dðx À yÞ ¼
ignm
8p2r
Â
Z1
0
dx eÀik0
ðx0Ày0Þ
À eik0
ðx0Ày0Þ
 
ðeÀixr
À eixr
Þ
Multiplying out the parentheses with the exponentials,
we get
Dðx À yÞ ¼
1
8p2r
Z1
0
dx eÀiðk0
ðx0Ày0ÞþxrÞ

ÀeÀiðk0
ðx0Ày0ÞÀxrÞ
À eiðk0
ðx0Ày0ÞÀxrÞ
þ eiðk0
ðx0Ày0ÞþxrÞ

Suppose that for the energy it holds x ¼ k0
. Then
the exponentials simplify,
Dðx À yÞ ¼
1
8p2r
Z1
0
dx eÀixððx0Ày0ÞþrÞ
À eÀixððx0Ày0ÞÀrÞ

Àeixððx0Ày0ÞÀrÞ
þ eixððx0Ày0ÞþrÞ

The third and fourth exponential term can be
omitted when we extend the range of integration to
the x interval from À1 to þ1. Hence
DðxÀyÞ¼
1
8p2r
Z1
À1
dx eÀixððx0Ày0ÞþrÞ
ÀeÀixððx0Ày0ÞÀrÞ
 
Performing finally the integration over x yields
DðxÀyÞ¼À
1
4pr
dððx0 Ày0ÞÀrÞÀdððx0 Ày0ÞþrÞð Þ
[B18]
Then with Eq. [D10] we get
Dretðx À yÞ ¼ yðx0 À y0ÞDðx À yÞ ¼ À
dððx0 À y0Þ À rÞ
4pr
[B19]
The term in Eq. [B18] with d((x0Ày0) þ r) does
not contribute because d((x0Ày0) þ r) only contributes
for (x0Ày0) þ r ¼ 0, hence (x0Ày0) ¼ Àr , 0,
but for (x0Ày0) , 0 the Heaviside step function
yields y(x0Ày0) ¼ 0. This concludes the proof for
the equivalence or equality of Eqs. [10] and [D10].
APPENDIX C
Generalized Functions and How to
Work with Them
Relying with our description of quantum electrodynamics
on propagators (i.e., Green functions as fundamental
solutions to the wave equation) and using
their analytical properties, we have to admit and
manage the fact that these propagators are not
‘‘well-behaved’’ ordinary functions. In this Appendix
we aim at some more careful explanation of some
of the mathematics of propagators. In Section ‘‘The
Feynman Propagator’’ we have introduced the mathematical
objects d(x) (Dirac’s d function) and dþ(x)
(a modification of d) where we have tacitly assumed
that they may behave like ordinary functions. We
even have written down integral expressions that we
have attempted to consider as definitions for d and
dþ, like Eqs. [15] and [16],
dðx0
Þ ¼
1
2p
Zþ1
À1
expðÀiox0
Þdo; [15]
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dþðx0
Þ ¼
1
4p2
Zþ1
0
expðÀiox0
Þdo: [16]
However, if we take these two integral expressions
literally, in the ordinary classical sense of integral
calculus, the integrals in Eqs. [15, 16] are not
existing at all, because periodic functions like
eÀiox0
, or likewise, cos(ox0
) or sin(ox0
) have no
definite value for o ¼ 61. Furthermore, d(x) is
singular, it holds d(x) ¼ 0 for x = 0 and d(x)
diverges for x ¼ 0. The function dþ(x0
) possesses’
similar features. Apart from this singular behavior,
all the features often applicable to ordinary ‘‘wellbehaved’’
functions like, e.g., continuity, differentiability,
and integrability, do not seem to work in a
straightforward way with objects like d and dþ. One
way to handle them in a mathematical appropriate
way is to understand them as the result of limit
processes, i.e., we start with ordinary functions
being finite or regular versions of the singular
objects, containing a parameter e specifying their
functional form and let this parameter go towards
zero such that during this limit process the function
approaches more and more the behavior associated
with the singular objects. So we may understand
integrals like [15, 16] as synonymous to expressions
with limits of the following kind,
dðx0
Þ ¼
1
2p
lim
e!0
Z0
À1
expðÀoðÀe þ ix0
ÞÞdo
þ
1
2p
lim
e!0
Z1
0
expðÀoðe þ ix0
ÞÞdo; [C1]
dþðx0
Þ ¼
1
4p2
lim
e!0
Zþ1
0
expðÀoðe þ ix0
ÞÞdo [C2]
with the cutoff or attenuation functions exp(Àeo) for
o . 0, with exp(eo) for o , 0, and with e being a
positive, real number. Now with the integrands in
[C1, C2] containing regular functions being finiteintegrable,
we may perform the integration while
keeping the limit operation outside the integral,
dðx0
Þ ¼
1
2p
lim
e!0
eÀoðÀeþix0
Þ
ÀðÀe þ ix0Þ
" #0
À1
þ
1
2p
lim
e!0
eÀoðeþix0
Þ
Àðe þ ix0Þ
" #1
0
¼
1
2p
lim
e!0
1
ÀðÀe þ ix0Þ
!
þ
1
2p
lim
e!0
1
ðe þ ix0Þ
!
4p2
dþðx0
Þ ¼ lim
e!0
eÀoðeþix0
Þ
Àðe þ ix0Þ
" #þ1
0
¼ lim
e!0
1
ðe þ ix0Þ
!
¼ Ài lim
e!0
1
ðx0 À ieÞ
!
[C3]
Separating real and imaginary parts, we obtain
dðx0
Þ ¼
1
2p
lim
e!0
ðÀe À ix0
Þ
ÀðÀe þ ix0ÞðÀe À ix0Þ
!
þ
1
2p
lim
e!0
ðe À ix0
Þ
ðe þ ix0Þðe À ix0Þ
!
¼
1
2p
lim
e!0
ÀðÀe À ix0
Þ þ ðe À ix0
Þ
e2 þ ðx0Þ2
" #
4p2
dþðx0
Þ ¼ lim
e!0
ðe À ix0
Þ
ðe þ ix0Þðe À ix0Þ
!
¼ lim
e!0
ðe À ix0
Þ
e2 þ ðx0Þ2
" #
¼ lim
e!0
e
e2 þ ðx0Þ2
À
ix0
e2 þ ðx0Þ2
" #
dðx0
Þ ¼
1
p
lim
e!0
e
e2 þ ðx0Þ2
" #
[C4]
The expressions inside the square brackets of
Eqs. [C3, C4] are regularized versions of the singular
functions on the left-hand sides of Eqs. [C3, C4] with
e as the regularization parameter. We recognize
e/(e2
þ(x0
)2
) as the function with a Lorentzian shape.
For e ? 0 its width becomes infinitesimally narrow
while its amplitude diverges for x0
¼ 0, thus becoming
the Dirac d generalized function or distribution as
we know it, as infinitely narrow and infinitely high
‘‘spike’’. The expression lime!0ðx0
=ðe2
þ ðx0
Þ2
ÞÞ has
a singularity at x0
¼ 0, everywhere else it behaves
like the function 1/(x0
). We denote it by
}
1
x0
 
¼ lim
e!0
x0
e2 þ ðx0Þ2
" #
[C5]
and refer to it as the prinicipal value distribution
with respect to 1/x0
.
An equivalent way of introducing the distributions
d(x0
) and <(1/x0
) can be found by considering
them as functionals, i.e.,
dðjÞ ¼
Z1
À1
dðx0
Þjðx0
Þdx0
¼ jð0Þ
}
1
x0
 
ðjÞ ¼ lim
e!0
ZÀe
À1
dx0 jðx0
Þ
x0
þ
Z1
e
dx0 jðx0
Þ
x0
8
<
:
9
=
;
[C6]
VIRTUAL PHOTONS IN MAGNETIC RESONANCE 319
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
which means for a given function j (called test
function, provided it fulfills certain special properties,
not discussed here), the distribution d(x0
) or
}(1/x0
) taken as functionals defined in [C6] associates
a numerical value (d(j) or }(1/x0
)(j)) to the
function j. Note, Eqs. [C4, C5] and Eqs. [C6] are
equivalent to each other. For example, d(x0
) associates
to each test function j the numerical value
j(0). This interpretation via functionals using integral
expressions like [C6] is particularly interesting
when we take a look at the action functional W in
Eq. [19] or the time evolution operator US(t) in Eq.
[50]. There we were tacitly introducing them in
Sections ‘‘The Feynman Propagator’’ and ‘‘A QED
NMR Probe Model: Pulsed NMR as a Scattering
Process’’ and they appeared naturally as functionals
via integrals like in [C6]. The role of the test function
j in Eq. [C6] has its counterpart with a physical
meaning, namely the current density function
j(x).
With [C4, C5] and [C3] we obtain Sokhotski’s
formula, i.e., the decomposition of dþ(x0
) into the
distribution d (x0
) and the principal value distribution
}(1/x0
),
4p2
dþðx0
Þ ¼ pdðx0
Þ À i}
1
x0
 
¼ Ài lim
e!þ0
1
x0 À ie
 
[C7]
which plays a central role in the interpretation of
the photon propagator in terms of virtual photons.
Making use of the identities [18] (see also below,
Eqs. [C18, C19]),
dððx0
À y0
Þ À rÞ þ dððx0
À y0
Þ þ rÞ
2r
¼ dððx0
À y0
Þ2
À r2
Þ ¼ dððx À yÞ2
Þ
dþððx0
À y0
Þ À rÞ þ dþððx0
À y0
Þ þ rÞ
2r
¼ dþððx0
À y0
Þ2
À r2
Þ ¼ dþððx À yÞ2
Þ [18]
we are also allowed to write
4p2
dþðx2
Þ ¼ pdðx2
Þ À i}
1
x2
 
¼ Ài lim
e!þ0
1
x2 À ie
 
[C8]
DERIVATION OF EQ. [31]
Inserting Eq. [24] for the field operators appearing
in h0|T (Am(x)An(y)) |0i we find
h0jTðAm
ðxÞAn
ðyÞÞj0i ¼ h0jT
Z
d3
k0
ð2pÞ3
2ok0
Â
Z
d3
k
ð2pÞ3
2ok
X
l0
X
l
eðl0
Þm
ðk0
ÞeðlÞn
ðkÞ
 aðl0
Þ
ðk0
ÞaðlÞ
ðkÞeÀiðk0
xþkyÞ

þ aðl0
Þ
ðk0
ÞaðlÞþ
ðkÞeÀiðk0
xÀkyÞ
þ aðl0
Þþ
ðk0
ÞaðlÞ
ðkÞeiðk0
xÀkyÞ
þaðl0
Þþ
ðk0
ÞaðlÞþ
ðkÞeiðk0
xþkyÞ

j0i
and taking into account Eqs. [26], we get
h0jTðAm
ðxÞAn
ðyÞÞj0i ¼ h0jT
Z
d3
k0
ð2pÞ3
2ok0
Z
d3
k
ð2pÞ3
2ok
Â
X
l0
X
l
eðl0
Þm
ðk0
ÞeðlÞn
ðkÞaðl0
Þ
ðk0
ÞaðlÞþ
ðkÞeÀiðk0
xÀkyÞ
j0i
According to [30], applying the time-ordering
operator T yields
h0jTðAm
ðxÞAn
ðyÞÞj0i ¼ h0j
Z
d3
k0
ð2pÞ3
2ok0
Â
Z
d3
k
ð2pÞ3
2ok
X
l0
X
l
eðl0
Þm
ðk0
ÞeðlÞn
ðkÞ
yðx0 À y0Þaðl0
Þ
ðk0
ÞaðlÞþ
ðkÞeÀiðk0
xÀkyÞ

þyðy0 À x0ÞaðlÞ
ðkÞaðl0
Þþ
ðk0
ÞeÀiðkyÀk0
xÞ

j0i
From the CCR’s [25] we conclude
aðl0
Þ
ðk0
ÞaðlÞþ
ðkÞeÀiðk0
xÀkyÞ
j0i
¼ Àgl0
l
dðk0
À kÞeÀiðk0
xÀkyÞ
j0i
aðlÞ
ðkÞaðl0
Þþ
ðk0
ÞeÀiðkyÀk0
xÞ
j0i
¼ Àgll0
dðk À k0
Þeiðk0
xÀkyÞ
j0i
such that after integration over k0
,
h0jTðAm
ðxÞAn
ðyÞÞj0i ¼ h0j
Z
d3
k
ð2pÞ3
2ok
Â
X
l0
;l
ðÀgll0
Þeðl0
Þm
ðkÞeðlÞn
ðkÞ
yðx0 À y0ÞeÀikðxÀyÞ
þ yðy0 À x0ÞeikðxÀyÞ
 
j0i
Because the metric tensor is diagonal and
because for the polarization sum holds (see [B16]),
X
l
ðÀgll
ÞeðlÞm
ðkÞeðlÞn
ðkÞ ¼ Àgmn
320 ENGELKE
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
we arrive at
h0jTðAm
ðxÞAn
ðyÞÞj0i ¼ Àgmn
Z
d3
k
2okð2pÞ3
 yðx0 À y0ÞeÀikðxÀyÞ
þ yðy0 À x0ÞeikðxÀyÞ
 
[C9]
We define the energy ok of one photon as a
strictly positive quantity or ok ¼ 0. We observe with
kðx À yÞ ¼ k0ðx0
À y0
Þ À k Á ðx À yÞ that, depending
on the time order, we obtain from Eq. [C9],
yðx0 À y0ÞeÀikðxÀyÞ
¼ yðx0 À y0ÞeÀiokðx0
Ày0
Þ
eikðxÀyÞ
yðy0 Àx0ÞeikðxÀyÞ
¼yðy0 Àx0Þeiokðx0
Ày0
Þ
eÀikðxÀyÞ
[C10]
Physically, the energy variable k0 being the timelike
component of the momentum four-vector km ¼
(k0, Àk) is a real-valued quantity. For the further
calculation the following trick turns out to be helpful:
we extend k0 to a complex quantity with a fictitious
imaginary part. Then we can submit this to
complex functional analysis to evaluate Eqs. [C9,
C10], and in a final step we perform again the limit
of vanishing imaginary part to re-obtain real values
of k0. Towards this goal, we first apply Cauchy’s
formula (88),
1
2pi
Z
C0
fðzÞ
z À zS
dz ¼ f ðzSÞ [C11]
that relates the value f (zS) at the specific point z ¼
zS in the complex plane of a function f (z) being analytic
inside the closed contour C0 and on the contour
C0, to the integral over f (z)/(zÀzS) taken along the
closed contour C0 of arbitrary shape surrounding the
point zS that represents a singularity (i.e., here a
pole) for f (z)/(zÀzS). Let us introduce the following
substitutions: z ¼ k0, zS ¼ ok, and fðzÞ ¼ f ðk0Þ ¼
eÀik0ðx0
Ày0
Þ
: Then it follows from [C11]:
eÀiokðx0
Ày0
Þ
¼ À
1
2pi
Z
Cþ
dk0
eÀik0ðx0
Ày0
Þ
k0 À ok
for the singularity at k0 ¼ þok
eiokðx0
Ày0
Þ
¼ þ
1
2pi
Z
CÀ
dk0
eÀik0ðx0
Ày0
Þ
k0 þ ok
for the singularity at k0 ¼ Àok
The first equation holds for x0
. y0
, i.e., y(x0
À
y0
) ¼ 1, where the closed contour Cþ
surrounding
the singularity at k0 ¼ þ ok runs clockwise (minus
sign in front of the integral) and the second equation
holds for x0
, y0
, i.e., y(y0
À x0
) ¼ 1, where the
closed contour CÀ
runs counterclockwise (plus sign
in front of the integral) enclosing the singularity at k0
¼ Àok. The integration contours Cþ
and CÀ
are
drawn in Fig. C1. For both cases, the integration
along the real axis is identical, only the infinite semicircle
is different when closing the contour (once in
the lower, once in the upper half plane) and thus
defining the different outcomes of the integral. The
integral vanishes when taken only over the semicircle
with infinite radius, however this semi-circle
matters as the element closing the contour, because it
defines which of the pole is inside and which is outside
the closed contour. Nevertheless, because the
integration along the real axis are identical in both
cases, we may write the sum of the two integrals as
one integral over the contour CF with the sum of the
two integrands such that for [C9] we obtain
h0jTðAm
ðxÞAn
ðyÞÞj0i ¼ À
gmn
2pi
Z
d3
k
2pð Þ3
2ok
Â
Z
CF
dk0
eikðxÀyÞ
eÀik0ðx0
Ày0
Þ
k0 À ok
À
eikðxÀyÞ
eÀik0ðx0
Ày0
Þ
k0 þ ok
!
¼
gmn
2pi
Z
d3
k
2pð Þ3
2ok
Z
CF
dk0eikðxÀyÞ
eÀik0ðx0
Ày0
Þ
Â
Àðk0 þ okÞ þ ðk0 À okÞ
ðk0 À okÞðk0 þ okÞ
 
¼
gmn
2pi
Z
d3
k
2pð Þ3
2ok
Â
Z
CF
dk0eikðxÀyÞ
eÀik0ðx0
Ày0
Þ À2ok
ðk0 À okÞðk0 þ okÞ
 
¼
¼ À
gmn
2pi
Z
d3
k
2pð Þ3
Z
CF
dk0
eikðxÀyÞ
eÀik0ðx0
Ày0
Þ
ðk0 À okÞðk0 þ okÞ
¼ À
gmn
2pi
Z
d3
k
2pð Þ3
eikðxÀyÞ
Z
CF
dk0eÀik0ðx0
Ày0
Þ
k2
0 À o2
k
[C12]
where on the right-hand side of the first equation in
the sum term with (k0 þok) in the denominator, we
have replaced the three-momentum vector k by Àk in
the exponent, i.e., as before the reversal of time order
is understood in conjunction with a reversal of the
motion. The contour CF in Eqs. [C12] above encloses
a singularity either at k0 ¼ þok or at k0 ¼ Àok, as
shown in Fig. C1. For these singularities holds k2
0 À
o2
k ¼ k2
¼ 0. Introducing the regularization parameter
e (analogous to Eq. [C7]), Eq. [C12] finally reads
VIRTUAL PHOTONS IN MAGNETIC RESONANCE 321
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h0jTðAm
ðxÞAn
ðyÞÞj0i ¼ lim
e!0
igmn
2p
Z
d3
k
2pð Þ3
eikðxÀyÞ
Â
Z
Reðk0Þ
dk0eÀik0ðx0
Ày0
Þ
k2
0 À o2
k þ ie
¼ igmn
lim
e!0
Z
d4
k
ð2pÞ4
eÀikðxÀyÞ
k2 þ ie
where in the last equation we have united the two
integrals into one four-dimensional momentum integral.
Thus
h0jTðAm
ðxÞAn
ðyÞÞj0i ¼ lim
e!0
igmn
Z
d4
k
ð2pÞ4
eÀikðxÀyÞ
DFðk; eÞ [C13]
where Dmn
F ðkÞ ¼ gmn
lime!0 DFðk; eÞ denotes the fourdimensional
Fourier transform of the two-point correlation
function h0jTðAm
ðxÞAn
ðyÞÞj0i,
Dmn
F ðkÞ ¼ gmn
DFðkÞ ¼ gmn
lim
e!0
DFðk; eÞ ¼ lim
e!0
gmn
k2 þ ie
[C14]
So far we have not yet obtained an analytical
expression for h0jTðAm
ðxÞAn
ðyÞÞj0i, instead we
derived its Fourier transform in four-dimensional k
space. On the first glance it seems straightforward to
compute the correlation function h0jTðAm
ðxÞAn
ðyÞÞj0i
in space-time by inverse Fourier transformation.
However we recognize that Dmn
F (k) has singularities
for k2
¼ ðk0Þ2
À j~kj2
¼ 0, thus the integral which
appears as the inverse Fourier transformation is not
existing in the classical sense. Nevertheless, following
the derivation given by Castellani, et al. (94), we
can explicitly calculate the four-dimensional Fourier
integral DFðxÞ ¼ lime!0þ
R d4
k
ð2pÞ4
eÀikx
k2þie as follows. At
first we observe that
i
Z1
0
eisk2
ds ¼ lim
e!0þ
i
Z1
0
eÀes
eisk2
ds ¼ lim
e!0þ
i
Z1
0
eÀðeÀik2
Þs
ds
¼ À lim
e!0þ
i
eÀðeÀik2
Þs
ðeÀik2Þ





1
0
¼ lim
e!0þ
i
ðeÀik2Þ
¼ lim
e!0þ
1
ðÀk2 ÀieÞ
¼ À lim
e!0þ
1
k2 þie
[C15]
From Eq. [C15] we infer
DFðxÞ ¼ lim
e!0þ
Z
d4
k
ð2pÞ4
eÀikx
k2 þ ie
¼ À
i
ð2pÞ4
Z
d4
keÀikx
Z1
0
dseisk2
¼ À
i
ð2pÞ4
Z1
0
ds
Z
d4
keÀikx
eisk2
¼ À
i
ð2pÞ4
Z1
0
ds
Z
d4
keiðk2
sÀkxÞ
Figure C1 Integration contour CF in the complex plane for the two-point correlation function
h0jTðAm
ðxÞAn
ðyÞÞj0i: (A) for time order x0
. y0
the contour extends to infinity on both
sides and encloses the point þok by a semi-circle with infinite radius in the lower half
plane. (B) For the opposite time order, x0
, y0
, the contour extends to infinity to both
sides and encloses the point Àok by a semi-circle with infinite radius in the upper half
plane. The two respective half circles do not contribute directly to the contour integrals,
nevertheless they are necessary to close the contour around the associated singularities
thus defining the value of the integral. The two small semi-circles at the poles
have infinitesimally small radius e.
322 ENGELKE
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
Introduce the new momentum variable q by the
substitution k ¼ q þ x
2s, then we arrive at
DFðxÞ ¼ lim
e!0þ
Z
d4
k
ð2pÞ4
eÀikx
k2 þ ie
¼ À
i
ð2pÞ4
Z1
0
ds
Z
d4
qeiðq2
sÀx2
=ð4sÞÞ
¼ À
i
ð2pÞ4
Z1
0
ds
Z
d4
qeiq2
s
eÀix2
=ð4sÞ
i.e., we can separate the variables such that the two
integrals appear as a product:
DFðxÞ ¼ lim
e!0þ
Z
d4
k
ð2pÞ4
eÀikx
k2 þ ie
¼ À
i
ð2pÞ4
Z1
0
dseÀix2
=ð4sÞÞ
Z
d4
qeiq2
s
We need to calculate the Gaussian integral over
d4
q:
Z
d4
qeiq2
s
¼
Z
dq0
Z
dq1
Z
dq2
Z
dq3eisðq2
0Àq2
1Àq2
2Àq2
3Þ
¼
Z
dq0eisq2
0
Z
dq1eÀisq2
1
Z
dq2eÀisq2
2
Z
dq3eÀisq2
3
¼
Z1
0
dq0eÀaq2
0
Z1
0
dq1eÀbq2
1
Z1
0
dq2eÀbq2
2
Z1
0
dq3eÀbq2
3
where we have used Cartesian coordinates for the
four-vector q and substituted a ¼ Àis and b ¼ þis.
For one particular integral holds (122)
Z1
0
dq0eÀaq2
0 ¼
ffiffiffi
p
a
r
Hence
Z
d4
qeiq2
s
¼
p2
ffiffiffiffiffiffiffi
ab3
p ¼
p2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ÀisðisÞ3
q ¼
p2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðÀiÞðÀiÞss3
p
¼
p2
ffiffiffiffiffiffiffiffi
Às4
p ¼
p2
is2
¼ À
ip2
s2
such that we obtain
DFðxÞ¼ lim
e!0þ
Z
d4
k
ð2pÞ4
eÀikx
k2 þie
¼
ðÀiÞðÀiÞp2
ð2pÞ4
Z1
0
ds
s2
eÀix2
=ð4sÞÞ
¼À
1
4ð2pÞ2
Z1
0
ds
s2
eÀix2
=ð4sÞÞ
Now substitute u¼ 1
4s; du
ds ¼À 1
4s2 to arrive at
DFðxÞ ¼ lim
e!0þ
Z
d4
k
ð2pÞ4
eÀikx
k2 þ ie
¼
1
4ð2pÞ2
Z0
1
4s2
du
s2
eÀiux2
¼ À
1
ð2pÞ2
Z1
0
dueÀiux2
Analogous to the derivation given in [C15] we
observe
Z1
0
eÀiux2
du¼ lim
e!0þ
Z1
0
eÀeu
eÀiux2
du¼ lim
e!0þ
Z1
0
eÀðeþix2
Þu
du
¼À lim
e!0þ
eÀðeþix2
Þu
ðeþix2Þ





1
0
¼ lim
e!0þ
1
ðeþix2Þ
¼À lim
e!0þ
i
x2 Àie
such that finally
DFðxÞ ¼ lim
e!0þ
Z
d4
k
ð2pÞ4
eÀikx
k2 þ ie
¼
i
4p2
lim
e!0þ
1
x2 À ie
[C16]
According to [18], taking into account Sokhotski’s
formula [C7, C8], we arrive at
DFðxÞ ¼
i
4p2
lim
e!þ0
1
x2 À ie
 
¼ Àdþðx2
Þ
such that finally it follows
iDmn
F ðx À yÞ ¼ h0jTðAm
ðxÞAn
ðyÞÞj0i
¼ À
igmn
4p2
dþððx À yÞ2
Þ [31]
or
iDmn
F ðx À yÞ ¼ h0jTðAm
ðxÞAn
ðyÞÞj0i
¼ Àigmn 1
4p
dððx À yÞ2
Þ À
i
4p2
}
1
ðx À yÞ2
!" #
[C17]
Proof of Eq. [18]
The first equation in [18] involving the d function is
a special case of the general feature
dðjðxÞÞ ¼
X
n
1
jj0ðxnÞj
dðx À xnÞ [C18]
VIRTUAL PHOTONS IN MAGNETIC RESONANCE 323
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
where j0
(x) denotes the derivative of the function
j(x) and xn denotes the nth zero of the function
j(x). Thus j0
(xn) is equal to the derivative of j(x)
for x ¼ xn. A proof for the general relation [C18]
can be found, for example, in the appendix of Ref.
98.
Setting j(x) ¼ x2
¼ (x0
)2
Àr2
and letting x2
¼ 0,
we have two zeroes (x0
) ¼ þ r and (x0
) ¼ Àr. That
means,
dðx2
Þ ¼ dððx0
Þ2
À r2
Þ ¼
1
j2rj
ðdðx0
À rÞ þ dðx0
þ rÞÞ
¼
dðx0
À rÞ þ dðx0
þ rÞ
2r
[C19]
The proof of Eq. [18] for dþ would involve the
principal value distribution }(1/x2
). In order to
achieve that, the Fourier transform relations that
exist between [33] and [34] can be used—i.e., we
prove the analogous relation [C19] for d(k2
) which
then, transformed to the space-time domain, shows
the validity for }(1/x2
), and both together, d(x2
) and
}(1/x2
), the validity for the second equation in [18].
APPENDIX D
Derivation of Eq. [40]
Let us consider some finite time interval (ti, tf)
between some initial and final instant in time and
divide that interval into N subintervals of duration
Dt ¼ (tfÀti)/N being sufficiently small such that during
each short interval Dt we may assume that the
interaction-Hamiltonian Hint(tm) (with m ¼ 1, 2, . . .,
N) does not change in time, i.e., we assume a piecewise
constant Hamiltonian over time. The time- ordered
exponential appearing in Eq. [39] can then be
approximated by the following expression
T exp Ài
Ztf
ti
dt0
Hintðt0
Þ
2
4
3
5
% expðÀiDtHintðtNÞÞ Á Á Á expðÀiDtHintðt2ÞÞ
 expðÀiDtHintðt1ÞÞ [D1]
with the time instants tm ¼ ti þ ((2mÀ1)/2)Dt such
that they appear time-ordered: tN ! tNÀ1 ! . . .! t2
! t1. We remind ourselves that the commutator of
the Hamiltonian with itself at different times does
not vanish. However this commutator is equal to a
complex number whenever the current densities j(x)
and j(y) are functions. In order to see this, remember
that the Hamiltonian density contains the product
jm Am
in of current density and field operator. Let
us focus upon the commutator expression
½Am
inðxÞjmðxÞ; An
inðyÞjnðyފ ¼ Am
inðxÞjmðxÞAn
inðyÞjnðyÞ
À An
inðyÞjnðyÞAm
inðxÞjmðxÞ ¼ Am
inðxÞAn
inðyÞjmðxÞjnðyÞ
À An
inðyÞAm
inðxÞjnðyÞjmðxÞ ¼ ½Am
inðxÞ; An
inðyފjmðxÞjnðyÞ
þ Cmnðx; yÞAn
inðyÞAm
inðxÞ; Cmnðx; yÞ¼½jmðxÞ; jnðyފ
[D2]
where we have assumed that the A field commutes
with the j field. Thus it becomes clear that for commuting
current densities, i.e., Cmn(x, y) ¼ 0, the
commutator of Hamiltonian densities is equal to the
commutator [Am
in(x),An
in(y)] (which is a complex
number) times the product jm(x) jn(y) of current densities.
If the current densities are functions (then
they obviously commute), not operators, then the
commutator of the Hamiltonian densities is itself a
complex number.
We take into account the operator identity
expðBÞ expðCÞ ¼ expðB þ C þ ½B; CŠ=2Þ ½D3Š
which holds if [B,[B,C]] ¼ [C,[B,C] ¼ 0. The operator
identity [D3] is a special case of the general
Baker-Campbell-Hausdorff (BCH) formula (95, 96).
Applying Eq. [D3] piece by piece to Eq. [D1], we
arrive at
T exp Ài
Ztf
ti
dt0
Hintðt0
Þ
2
4
3
5 % exp ÀiDt
XN
m¼1
HintðtmÞ
À
ðDtÞ2
2
XN
m¼1
XN
n¼1
1 m n N
½HintðtmÞ; Hintðtnފ
1
C
C
A
Now we let go N to infinity, which means that
the artificial time intervals Dt become infinitesimally
small again. We replace the sums by integrals again
and obtain
T exp Ài
Ztf
ti
dt0
Hintðt0
Þ
2
4
3
5 ¼ exp Ài
Ztf
ti
dt0
Hintðt0
Þ
0
@
þ
1
2
Ztf
ti
dt0
Ztf
ti
dt00
yðt0
À t00
Þ½Hintðt0
Þ; Hintðt00
ފ
1
A [D4]
and because on the right-hand side of Eq. [D4] the
second term in the exponential (the one with the
324 ENGELKE
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
double integral) is equal to a complex number, we
may write it as a separate exponential factor:
T exp Ài
Ztf
ti
dt0
Hintðt0
Þ
2
4
3
5 ¼ exp Ài
Ztf
ti
dt0
Hintðt0
Þ
0
@
1
A
 exp þ
1
2
Ztf
ti
dt0
Ztf
ti
dt00
yðt0
À t00
Þ½Hintðt0
Þ; Hintðt00
ފ
0
@
1
A
Taking, as initially in Eq. [39], for the initial
time ti ? À1 and set tf ¼ t, we may rewrite the
time evolution operator [39] as
UðtÞ ¼ exp Ài
Zt
À1
dt0
Hintðt0
Þ
0
@
1
A exp þ
1
2
Zt
À1
dt0
0
@
Â
Zt
À1
dt00
yðt0
À t00
Þ½Hintðt0
Þ; Hintðt00
ފ
1
A ½D5Š
With the time-dependent interaction Hamiltonian
as given in Eq. [38], we may express the integrals
in Eq. [D5] as
UðtÞ ¼ exp Ài
Zt
À1
dt0
Z
d3
xAn
inðxÞjnðxÞ
0
@
1
A
 exp þ
1
2
Zt
À1
dx0
Z
d3
x
Zt
À1
dy0
0
@
Â
Z
d3
yyðx0
À y0
Þ½Am
inðxÞjmðxÞ; An
inðyÞjnðyފ
1
A [D6]
Because the field operators commute with the
current densities and the current densities are supposed
to commute with themselves, we have
½Am
inðxÞjmðxÞ; An
inðyÞjnðyފ ¼ Àigmn
Dðx À yÞjmðxÞjnðyÞ
[D7]
where for the commutator of the electromagnetic
field operators we have introduced the Pauli-Jordan
function:
½Am
inðxÞ; An
inðyފ ¼ Àigmn
Dðx À yÞ [D8]
defined by the commutator in [D8], see also Eq.
[B14], Appendix B. Now Eq. [D6] may be reexpressed
as
UðtÞ ¼ exp Ài
Zt
À1
dx0
Z
d3
xAn
inðxÞjnðxÞ
0
@
1
A
 exp i
2
Zt
À1
dx0
Z
d3
x
Zt
À1
dy0
0
@
Â
Z
d3
yyðx0
À y0
Þgmn
jmðxÞDðx À yÞjnðyÞ
1
A [D9]
The retarded propagator Dret can be expressed by
the function D as
Dretðx À yÞ ¼ yðx0
À y0
ÞDðx À yÞ [D10]
y denotes Heaviside’s step function as defined for
Eq. [30], it encodes now the time ordering. The retarded
propagator in Eq. [D10] corresponds exactly
to the retarded Green function as introduced in Eq.
[10]. The equivalence of Eqs. [10] and [D10] is
demonstrated in Appendix B. Eq. [D10] inserted
into [D9] leads us further to
UðtÞ ¼ exp Ài
Zt
À1
dx0
Z
d3
xAn
inðxÞjnðxÞ
0
@
1
A
 exp i
2
gmn
Zt
À1
dx0
Z
d3
x
Zt
À1
dy0
0
@
Â
Z
d3
yjmðxÞDretðx À yÞjnðyÞ
1
A [40]
Derivation of Eq. [46]
Starting with Eq. [D9],
UðtÞ ¼ exp Ài
Zt
À1
dx0
Z
d3
xAn
inðxÞjnðxÞ
0
@
1
A
 exp i
2
Zt
À1
dx0
Z
d3
x
Zt
À1
dy0
0
@
Â
Z
d3
yyðx0
À y0
Þgmn
jmðxÞDðx À yÞjnðyÞ
!
we decompose the four-potential operator, according
to Eq. [24], into a positive frequency part containing
the annihilation operator and negative frequency
VIRTUAL PHOTONS IN MAGNETIC RESONANCE 325
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
part containing the creation operator (see Eqs.
[44, 45]:
Am
inðxÞ ¼ A
mðþÞ
in ðxÞ þ A
mðÀÞ
in ðxÞ [D11]
such that we can write
À igmn
jnðxÞDðx À yÞjmðyÞ ¼ jmðxÞ½A
mðþÞ
in ðxÞ
þ A
mðÀÞ
in ðxÞ; A
nðþÞ
in ðyÞ þ A
nðÀÞ
in ðyފjnðyÞ
¼ jmðxÞðA
mðþÞ
in ðxÞA
nðþÞ
in ðyÞ þ A
mðþÞ
in ðxÞA
nðÀÞ
in ðyÞ
þ A
mðÀÞ
in ðxÞA
nðþÞ
in ðyÞ þ A
mðÀÞ
in ðxÞA
nðÀÞ
in ðyÞÀ
À A
nðþÞ
in ðyÞA
mðþÞ
in ðxÞ À A
nðÀÞ
in ðyÞA
mðþÞ
in ðxÞ
À A
nðþÞ
in ðyÞA
mðÀÞ
in ðxÞ À A
nðÀÞ
in ðyÞA
mðÀÞ
in ðxÞÞjnðyÞ ¼
¼ jmðxÞð½A
mðþÞ
in ðxÞ; A
nðþÞ
in ðyފ þ ½A
mðþÞ
in ðxÞ; A
nðÀÞ
in ðyފþ
þ½A
mðÀÞ
in ðxÞ; A
nðþÞ
in ðyފ þ ½A
mðÀÞ
in ðxÞ; A
nðÀÞ
in ðyފÞjnðyÞ
We observe that when we take the vacuum expectation
value h0 |. . . |0i of this last result, we do
not change it, because the result represents already
a complex number. We may even write
À igmn
jnðxÞDðx À yÞjmðyÞ
¼ Àigmn
h0jjnðxÞDðx À yÞjmðyÞj0i
¼ h0jjmðxÞð½A
mðþÞ
in ðxÞ; A
nðþÞ
in ðyފ
þ ½A
mðÀÞ
in ðxÞ; A
nðÀÞ
in ðyފ þ ½A
mðþÞ
in ðxÞ; A
nðÀÞ
in ðyފ
þ ½A
mðÀÞ
in ðxÞ; A
nðþÞ
in ðyފÞjnðyÞj0i ¼
¼ h0jjmðxÞð½A
mðþÞ
in ðxÞ; A
nðÀÞ
in ðyފ
þ ½A
mðÀÞ
in ðxÞ; A
nðþÞ
in ðyފÞjnðyÞj0i
¼ jmðxÞh0jð½A
mðþÞ
in ðxÞ; A
nðÀÞ
in ðyފ
þ½A
mðÀÞ
in ðxÞ; A
nðþÞ
in ðyފÞj10ijnðyÞ
and Eq. [D9] becomes
UðtÞ ¼ exp Ài
Zt
À1
dx0
Z
d3
xAn
inðxÞjnðxÞ
0
@
1
A
 exp þ
1
2
Zt
À1
dx0
Z
d3
x
Zt
À1
dy0
0
@
Â
Z
d3
yyðx0
À y0
ÞjmðxÞh0jð½A
mðþÞ
in ðxÞ; A
nðÀÞ
in ðyފ
þ ½A
mðÀÞ
in ðxÞ; A
nðþÞ
in ðyފÞj0 i jnðyފ
1
A [D12]
According to the operator identity Eq. [D2], we
are allowed to transform the first exponential term
in [D12] into
exp Ài
Zt
À1
dx0
Z
d3
xAn
inðxÞjnðxÞ
0
@
1
A
¼ exp Ài
Zt
À1
dx0
Z
d3
xðAnÀ
in ðxÞ þ Anþ
in ðxÞÞjnðyÞ
0
@
1
A
¼ exp Ài
Zt
À1
dx0
Z
d3
xAnÀ
in ðxÞjnðxÞ
0
@
1
A
 exp Ài
Zt
À1
dy0
Z
d3
yAnþ
in ðyÞjnðyÞ
0
@
1
A
 exp 1
2
Zt
À1
dx0
Z
d3
x
Zt
À1
dy0
0
@
Â
Z
d3
y½AnÀ
in ðxÞjnðxÞ; Amþ
in ðyÞjmðyފ
1
A
Introducing this into [D12] yields
UðtÞ ¼ exp Ài
Zt
À1
dx0
Z
d3
xAnÀ
in ðxÞjnðxÞ
0
@
1
A
 exp Ài
Zt
À1
dy0
Z
d3
yAnþ
in ðyÞjnðyÞ
0
@
1
A
 exp 1
2
Zt
À1
dx0
Z
d3
x
Zt
À1
dy0
0
@
Â
Z
d3
y½AmÀ
in ðxÞjnðxÞ; Anþ
in ðyÞjrðyފ
1
A
 exp þ
1
2
Zt
À1
dx0
Z
d3
x
Zt
À1
dy0
0
@
Â
Z
d3
yyðx0
À y0
ÞjmðxÞh0jð½A
mðþÞ
in ðxÞ; A
nðÀÞ
in ðyފ
þ ½A
mðÀÞ
in ðxÞ; A
nðþÞ
in ðyފÞj0ijnðyފ
1
A
The third and fourth exponential term are usual
exponential functions, not operator exponentials.
Therefore we can collect them into one common
exponent which gives
326 ENGELKE
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
UðtÞ ¼ exp Ài
Zt
À1
dx0
Z
d3
xAnÀ
in ðxÞjnðxÞ
0
@
1
A
 exp Ài
Zt
À1
dy0
Z
d3
yAnþ
in ðyÞjnðyÞ
0
@
1
A
 exp þ
1
2
Zt
À1
dx0
Z
d3
x
Zt
À1
dy0
0
@
Â
Z
d3
yjmðxÞh0jðyðx0
À y0
Þ½A
mðþÞ
in ðxÞ; A
nðÀÞ
in ðyފ
þ yðx0
À y0
Þ½A
mðÀÞ
in ðxÞ; A
nðþÞ
in ðyފ
À½AmÀ
in ðxÞ; Anþ
in ðyފj0ijnðyÞÞ [D13]
In Eq. [D13] let us focus on the two terms
that contain specifically the commutator [Ain
m(À)
(x),
Ain
n(þ)
(y)] and we find
yðx0
À y0
Þ½A
mðÀÞ
in ðxÞ; A
nðþÞ
in ðyފ À ½AmÀ
in ðxÞ; Anþ
in ðyފ
¼
0 for x0
.y0
À½AmÀ
in ðxÞjnðxÞ; Anþ
in ðyފ for x0
< y0
;
(
hence it holds
yðx0
À y0
Þ½A
mðÀÞ
in ðxÞ; A
nðþÞ
in ðyފ À ½AmÀ
in ðxÞ; Anþ
in ðyފ
¼ Àyðy0
À x0
Þ½A
mðÀÞ
in ðxÞ; A
nðþÞ
in ðyފ
such that Eq. [D13] can be rewritten as
UðtÞ ¼ exp Ài
Zt
À1
dx0
Z
d3
xAnÀ
in ðxÞjnðxÞ
0
@
1
A
 exp Ài
Zt
À1
dy0
Z
d3
yAnþ
in ðxÞjnðxÞ
0
@
1
A
 exp þ
1
2
Zt
À1
dx0
Z
d3
x
Zt
À1
dy0
Z
d3
yjmðxÞ
h0jðyðx0
À y0
Þ½A
mðþÞ
in ðxÞ; A
nðÀÞ
in ðyފ
À yðy0
À x0
Þ½AmÀ
in ðxÞ; Anþ
in ðyފj0ijnðyÞÞ
We observe that
½A
mðþÞ
in ðxÞ;A
nðÀÞ
in ðyފ¼h0j½A
mðþÞ
in ðxÞ;A
nðÀÞ
in ðyފj0i
¼h0jA
mðþÞ
in ðxÞA
nðÀÞ
in ðyÞj0i
½A
mðÀÞ
in ðxÞ;A
nðþÞ
in ðyފ
¼h0j½A
mðÀÞ
in ðxÞ;A
nðþÞ
in ðyފj0i¼Àh0jA
nðþÞ
in ðyÞA
mðÀÞ
in ðxÞj0i
because, due to Eqs. [24, 26], it holds
A
mðþÞ
in ðxÞj0i¼h0jA
nðÀÞ
in ðyÞ¼0. In addition,
h0j½A
mðþÞ
in ðxÞ;A
nðÀÞ
in ðyފj0i
¼h0jA
mðþÞ
in ðxÞA
nðÀÞ
in ðyÞj0i¼h0jAm
inðxÞAn
inðyÞj0i
h0j½A
mðÀÞ
in ðxÞ;A
nðþÞ
in ðyފj0i¼Àh0jA
nðþÞ
in ðyÞA
mðÀÞ
in ðxÞj0i
¼Àh0jAn
inðyÞAm
inðxÞj0i ½D14Š
Therefore
UðtÞ ¼ exp Ài
Zt
À1
dx0
Z
d3
xAnÀ
in ðxÞjnðxÞ
0
@
1
A
 exp Ài
Zt
À1
dy0
Z
d3
yAnþ
in ðxÞjnðxÞ
0
@
1
A
 exp þ
1
2
Zt
À1
dx0
Z
d3
x
Zt
À1
dy0
0
@
Z
d3
yjmðxÞh0jðyðx0
À y0
ÞAm
inðxÞAn
inðyÞ
þ yðy0
À x0
ÞAn
inðyÞAm
inðxÞÞj0ijnðyÞ
1
A [D15]
The term inside the expectation value h0 |. . . |0i
in the third exponential term of Eq. [D15] is equal
to the time-ordered product of An
in(x)Am
in(y) as given
by Dyson’s chronological operator T according to
Eqs. [29]:
UðtÞ ¼ exp Ài
Zt
À1
dx0
Z
d3
xAnÀ
in ðxÞjnðxÞ
0
@
1
A
 exp Ài
Zt
À1
dy0
Z
d3
yAnþ
in ðxÞjnðxÞ
0
@
1
A
 exp þ
1
2
Zt
À1
dx0
Z
d3
x
Zt
À1
dy0
0
@
Â
Z
d3
yjmðxÞh0jTðAm
inðxÞAn
inðyÞÞj0ijnðyÞ
1
A [D16]
The first two exponentials in [D16] define the
normal ordering form indicated by the enclosure in
colons, : . . . : (see Eq. [43]),
VIRTUAL PHOTONS IN MAGNETIC RESONANCE 327
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
: exp Ài
Zt
À1
dx0
Z
d3
xAn
inðxÞjnðxÞ
0
@
1
A :
¼ exp Ài
Zt
À1
dx0
Z
d3
xAnÀ
in ðxÞjnðxÞ
0
@
1
A
 exp Ài
Zt
À1
dx0
Z
d3
yAnþ
in ðxÞjnðxÞ
0
@
1
A [43]
where in the product of the series expansion of the
exponentials all creation operators appear to the left
from all annihilation operators. With Eqs. [D16, 43]
the notation of Eq. [D15] simplifies to
UðtÞ ¼: exp Ài
Zt
À1
dx0
Z
d3
xAn
inðxÞjnðxÞ
0
@
1
A :
exp þ
1
2
Zt
À1
dx0
Z
d3
x
Zt
À1
dy0
0
@
Â
Z
d3
yjmðxÞh0jTðAm
inðxÞAn
inðyÞÞj0ijnðyÞ
1
A
The photon propagator is defined as
À igmn
DFðx À yÞ ¼ h0jTðAm
inðxÞAn
inðyÞÞj0i [D17]
thus
UðtÞ ¼: exp Ài
Zt
À1
dx0
Z
d3
xAn
inðxÞjnðxÞ
0
@
1
A :
exp À
i
2
Zt
À1
dx0
Z
d3
x
Zt
À1
dy0
0
@
Â
Z
d3
yjmðxÞgmn
DFðx À yÞjnðyÞ
1
A
and finally
UðtÞ ¼: exp Ài
Zt
À1
dx0
Z
d3
xAn
inðxÞjnðxÞ
0
@
1
A :
exp À
i
2
Zt
À1
dx0
Z
d3
x
Zt
À1
dy0
0
@
Â
Z
d3
yjmðxÞDmn
F ðx À yÞjnðyÞ
1
A [46]
Derivation of Eq. [50]
We begin with
UðtÞ ¼ exp Ài
Zt
À1
dx0
Z
d3
xAnÀ
in ðxÞjnðxÞ
0
@
1
A
 exp Ài
Zt
À1
dx0
Z
d3
yAnþ
in ðxÞjnðxÞ
0
@
1
A
 exp þ
1
2
Zt
À1
dx0
Z
d3
x
Zt
À1
dy0
0
@
Â
Z
d3
yjmðxÞ
gmn
4p2
}
1
ðx À yÞ2
!
jnðyÞ
1
A
 exp þ
i
2
Zt
À1
dx0
Z
d3
x
Zt
À1
dy0
0
@
Â
Z
d3
yjmðxÞ
gmn
4p
dððx À yÞ2
ÞjnðyÞ
1
A [47]
From [33, 34] it follows
Àp
Z
d4
k
ð2pÞ4
dðk2
ÞeÀikðxÀyÞ
¼
1
4p2
}
1
ðxÀyÞ2
!
[D18]
which inserted into [47] gives
UðtÞ¼exp Ài
Zt
À1
dx0
Z
d3
xAnÀ
in ðxÞjnðxÞ
0
@
1
A
Âexp Ài
Zt
À1
dx0
Z
d3
yAnþ
in ðxÞjnðxÞ
0
@
1
A
Âexp À
pgmn
2
Zt
À1
dx0
Z
d3
x
Zt
À1
dy0
0
@
Â
Z
d3
yjmðxÞ
Z
d4
k
ð2pÞ4
dðk2
ÞeÀikðxÀyÞ
jnðyÞ
1
A
Âexp
igmn
8p
Zt
À1
dx0
Z
d3
x
Zt
À1
dy0
0
@
Â
Z
d3
yjmðxÞdððxÀyÞ2
ÞjnðyÞ
1
A
Taking the integrals of the three-dimensional
space volumes, the third exponential (the one containing
the on-shell contribution d(k2
)) can be
rewritten in terms of the three-dimensional k space
328 ENGELKE
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
Fourier transform Jmðk;x0
Þ (Eq. [49]) of the current
density jm(x) as
UðtÞ¼exp Ài
Zt
À1
dx0
Z
d3
xAnÀ
in ðxÞjnðxÞ
0
@
1
A
Âexp Ài
Zt
À1
dx0
Z
d3
yAnþ
in ðxÞjnðxÞ
0
@
1
A
Âexp À
pgmn
2
Zt
À1
dx0
eÀik0x0
Zt
À1
dy0
eik0y0
0
@
Â
Z
d4
k
ð2pÞ4
dðk2
ÞJÃ
mðk;x0
ÞJnðk;y0
Þ
1
Aexp
igmn
8p
Zt
À1
dx0
0
@
Â
Z
d3
x
Zt
À1
dy0
Z
d3
yjmðxÞdððxÀyÞ2
ÞjnðyÞ
1
A [D19]
Taking into account the Fourier expansion [24]
for the four-potentials, let us define the complex
valued amplitudes
aðlÞ
ðk; x0
Þ ¼ ÀiemðlÞ
ðkÞJmðk; x0
Þeikx0
[D20]
where
a
Ã
ðl0
Þ
ðk; x0
ÞaðlÞ
ðk; y0
Þ
¼ gl0
l
JnÃ
ðk; x0
ÞJnðk; y0
ÞeÀik0x0
eik0y0
[D21]
in view of the fact that the polarization vectors form
an orthogonal frame of vectors in space-time. Performing
the integral over x0
, we may further define
aðk; tÞ ¼
Zt
À1
dx0
aðk; x0
Š [D22]
such that [D19] appears as
UðtÞ ¼ exp
Z
d3
k
ð2pÞ3
2ok
X
l
aðlÞ
ðk; tÞaðlÞþ
ðkÞ
!
 exp Z
d3
k
ð2pÞ3
2ok
X
l
aðlÞÃ
ðk; tÞaðlÞ
ðkÞ
!
 exp p
2
Z
d4
k
ð2pÞ4
dðk2
Þ
X
l
aðlÞÃ
ðk; tÞaðlÞ
ðk; tÞ
!
 exp
i
8p
Zt
À1
dx0
Z
d3
x
Zt
À1
dy0
0
@
Â
Z
d3
yjn
ðxÞdððx À yÞ2
ÞjnðyÞ
1
A
Decomposing the four-dimensional integral over
k into one over three-dimensional k space and one
over k0 leads us to
UðtÞ ¼ exp
Z
d3
k
ð2pÞ3
2ok
X
l
aðlÞ
ðk; tÞaðlÞþ
ðkÞ
!
 exp Z
d3
k
ð2pÞ3
2ok
X
l
aðlÞÃ
ðk; tÞaðlÞ
ðkÞ
!
 exp p
2
Z
d3
k
ð2pÞ3
Z
dk0
2p
dðk2
0 À jkj2
Þ
Â
X
l
aðlÞÃ
ðk; tÞaðlÞ
ðk; tÞ
!
exp
i
8p
Zt
À1
dx0
Z
d3
x
0
@
Â
Zt
À1
dy0
Z
d3
yjn
ðxÞdððx À yÞ2
ÞjnðyÞ
1
A
In analogy to the identities [18], it holds in k
space,
dðk2
Þ ¼ dðk2
0 À jkj2
Þ ¼
dðk0 À jkjÞ þ dðk0 þ jkjÞ
2ok
;
ok ¼ jkj [D23]
such that
UðtÞ ¼ exp Ài
Z
d3
k
ð2pÞ3
2ok
X
l
aðlÞ
ðk; tÞaðlÞþ
ðkÞ
!
 exp Ài
Z
d3
k
ð2pÞ3
2ok
X
l
aðlÞÃ
ðk; tÞaðlÞ
ðkÞ
!
 exp 1
4
Z
d3
k
ð2pÞ3
2
2ok
X
l
aðlÞÃ
ðk; tÞaðlÞ
ðk; tÞ
!
 exp
i
8p
Zt
À1
dx0
Z
d3
x
Zt
À1
dy0
0
@
Â
Z
d3
yjn
ðxÞdððx À yÞ2
ÞjnðyÞ
1
A [D24]
For the sake to simplify notation, not giving
up too much in generality, let us consider only
one discrete field mode with polarization l and
four-momentum k such that we can skip the inte-
grals
R d3
k
ð2pÞ3
2ok
and sums
P
l over polarizations. In
that way k becomes fixed for that particular
mode:
VIRTUAL PHOTONS IN MAGNETIC RESONANCE 329
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
UðtÞ ¼ exp aðk; tÞaþ
ðkÞð Þ exp ÀaÃ
ðk; tÞaðkÞð Þ
 exp 1
2
aÃ
ðk; tÞaðk; tÞ
 
exp
i
8p
Zt
À1
dx0
0
@
Â
Z
d3
x
Zt
À1
dy0
Z
d3
yjn
ðxÞdððx À yÞ2
ÞjnðyÞ
1
A [D25]
Applying the identity [D2] to factorize exponential
operators, exp(B)exp(C) exp(À[B,C]/2)¼
exp(BþC), with exp(B) being the first and exp(C)
being the second exponential in [D24], we arrive at
UðtÞ ¼ exp aðk; tÞaþ
ðkÞ À aÃ
ðk; tÞaðkÞð Þ
 exp
i
8p
Zt
À1
dx0
Z
d3
x
Zt
À1
dy0
0
@
Â
Z
d3
yjn
ðxÞdððx À yÞ2
ÞjnðyÞ

[50]
APPENDIX E
Dirac Equation
Dirac postulated that the equation of motion sought
as a relativistic generalization of the Schro¨dinger
equation has to be of first order, both in space and
time derivatives (covariance) and he assumed it to
be in the form (123)
i"h
qc
qt
¼
"hc
i
a Á r þ bmc2
 
c [E1]
with certain algebraic quantities a ¼ (ax, ay, az)
and b yet to be determined by means of the following
conditions:
(i) Equation [E1] has to admit plane-waves
being solutions to the Klein-Gordon equation—the
latter follows as second-order relativistic
wave equation from Eq. [A8], Appendix
A, and reads (q2
þ m2
)c ¼ 0 (in Heaviside-Lorentz
units) for particles with rest
mass m,
(ii) the Hamiltonian operator resulting from Eq.
[E1] has to be a Hermitian operator,
(iii) there exists a four-current density whose
time component is positive-definite such that
it allows an interpretation as a probability
density, and
(iv) Equation [E1] can brought into a form that
is covariant.
Equation [E1] is called Dirac equation in non-covariant
form. We first focus on condition (i) by showing that
from Eq. [E1] and the requirement that plane waves of
the Dirac field c have to satisfy the Klein-Gordon equation
several constraints follow that allow us to determine
the quantities ~a ¼ (ax, ay, az) and b. Taking the
square of the operators in Eq. [E1] and taking into
account the Klein Gordon equation, we have
"hc
i
a Á r þ bmc2
 2
¼ À"h2
c2
a2
r2
þ b2
m2
c4
þ
"hc
i
ab Á r þ
"hc
i
ba Á r
¼ À"h2
c2
r2
þ m2
c4
¼ À"h2 q2
qt2
and we obtain
a2
¼ a2
x þ a2
y þ a2
z ¼ b2
¼ 1;
axb þ bax ¼ ayb þ bay ¼ azb þ baz ¼ 0
i.e.,
a2
¼ 1; b2
¼ 1; ab þ ba ¼ 0 [E2]
Introducing the quantities
gk ¼ ðb; baÞ ¼ ðb; gÞ ¼ ðb; bax; bay; bazÞ [E3]
we may rewrite Eq. [E1] in four-dimensional notation
as follows
i"h
q
qt
þ ca Á r
 
c À bmc2
c ¼ 0 ¼
À
1
i"hc
b i"h
q
qt
þ i"hca Á r À bmc2
 
c; b2
¼ 1
Àb
q
cqt
À ba Á r þ b2 mc
i"h
 
c
¼ Àg0
q
qx0
À g1
q
qx1
À g2
q
qx2
À g3
q
qx3
þ
mc
i"h
 
c ¼ 0
i.e., with the notational conventions
qk ¼ q
qxk ¼ q0
; r
À Á
and qk
¼ gkm
qm ¼ ðq0
; ÀrÞ and
in Heaviside-Lorentz units, we arrive at
Àigkqk
þ m
À Á
c ¼ 0 [E4]
In contrast to Eq. [E1], Eq. [E4] is the Dirac
equation in covariant notation. Now, if we multiply
Eq. [E4] from the left with igmqm
þ mð Þ, we get
igmqm
þmð Þ Àigmqm
þmð Þc ¼ ðgmgkqm
qk
þm2
Þc ¼ 0
330 ENGELKE
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
Because qm
qk
¼ qk
qm
, it holds gmgkqm
qk
¼
1
2 gmgk þgkgmð Þqm
qk
, and we obtain
1
2
fgm;gkgqm
qk
þm2
 
c ¼ 0 [E5]
with the anticommutator
fgm;gkg ¼ gmgk þgkgm [E6]
Because we require Eq. [E5] to be identical to
the Klein-Gordon equation (q2
þ m2
)c ¼ 0, we
may write
1
2
fgm; gkgqm
qk
þ m2
 
c ¼ ðq2
þ m2
Þc ¼ 0
such that we have to conclude that
fgm; gkg ¼ 2gmk [E7]
For m ¼ k this implies the first two equations in
[E2], for m ¼ 1,2,3 and k ¼ 0 the third equation in
[E2]. For m = k and m = 0, k = 0, we obtain in
addition to Eqs. [E2]:
axay þayax ¼ 0; ayaz þazay ¼ 0; axaz þazax ¼ 0
[E8]
With Eq. [E7] (or Eqs. [E2, E8]) we have found
all constraints for gk
(or a ¼ ðax; ay; azÞ and b).
The conditions [E2, E8] for the quantities
b; ax; ay; az, or equivalently Eq. [E7] for the quantities
gk defined by Eq. [E3], are a consequence that
the ‘‘square of the Dirac equation should be equal
to the Klein-Gordon equation’’—they cannot be fulfilled
by real or complex numbers, but by matrices
b; ax; ay; az being at least of size 4-by-4. One possible
set of matrices satisfying Eqs. [E2, E8], the so
called standard or Dirac representation, reads
b ¼
1 0 0 0
0 1 0 0
0 0 À1 0
0 0 0 À1
0
B
B
@
1
C
C
A; ax ¼
0 0 0 1
0 0 1 0
0 1 0 0
1 0 0 0
0
B
B
@
1
C
C
A;
ay ¼
0 0 0 Ài
0 0 i 0
0 Ài 0 0
i 0 0 0
0
B
B
@
1
C
C
A; az ¼
0 0 1 0
0 0 0 À1
1 0 0 0
0 À1 0 0
0
B
B
@
1
C
C
A
[E9]
or the corresponding set satisfying Eq. [E7],
g0 ¼
1 0 0 0
0 1 0 0
0 0 À1 0
0 0 0 À1
0
B
B
@
1
C
C
A; g1 ¼
0 0 0 1
0 0 1 0
0 À1 0 0
À1 0 0 0
0
B
B
@
1
C
C
A;
g2 ¼
0 0 0 Ài
0 0 i 0
0 i 0 0
Ài 0 0 0
0
B
B
@
1
C
C
A; g3 ¼
0 0 1 0
0 0 0 À1
À1 0 0 0
0 1 0 0
0
B
B
@
1
C
C
A
[E10]
The matrices gk are called Dirac matrices. We use
the indices k, m, . . . to enumerate the matrices, not the
elements within one particular matrix, thus the notation
for the Dirac matrices is complete when we write
ðgabÞk with a, b indicating the matrix elements in matrix
gk. In distinction to the index k indicating the
dimensions in 4D space, the inner indices a, b in conjunction
with g matrices are called spinor indices.
The fact that gk are 4-by-4 matrices implies that the
wave functions c satisfying Dirac’s equation [E4]
have to have four components,
c ¼
c1
c2
c3
c4
0
B
B
@
1
C
C
A [E11]
These wave functions are also referred to as
bispinors, defined in Hilbert space and satisfying
specific transformation properties in that space (see,
for example, 78–81). Here again, the indices 1, 2, 3,
4 are spinor indices, not space-time indices. Each
component is a complex function, i.e., there exists a
complex conjugate wave function cþ
given as
cþ
¼ ðcÃ
1; cÃ
2; cÃ
3; cÃ
4Þ [E12]
with ca* being the complex conjugate of the component
ca. As it turns out, bispinors c describe spin-1/
2 particles and their associated antiparticles. If we
define the following 2-by-2 matrices (Pauli matrices)
as well as the 2-by-2 unit matrix 1 and the 2by-2
zero matrix 0,
s1 ¼
0 1
1 0
 
; s2 ¼
0 Ài
i 0
 
;
s3 ¼
1 0
0 À1
 
; 1 ¼
1 0
0 1
 
; 0 ¼
0 0
0 0
 
[E13]
the Dirac matrices gk in the standard representation
can be written as
VIRTUAL PHOTONS IN MAGNETIC RESONANCE 331
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
g0 ¼
1 0
0 À1
 
; g1 ¼
0 s1
Às1 0
 
;
g2 ¼
0 s2
Às2 0
 
; g3 ¼
0 s3
Às3 0
 
: ½E14Š
Introducing the ‘‘matrix vector’’ s ¼
ðs1; s2; s3Þ, we arrive at the convenient and compact
expression
gk ¼ ðg0;gÞ ¼
1 0
0 À1
 
;
0 s
Às 0
  
[E15]
Besides the Dirac standard representation in Eqs.
[E10] or [E14, E15], there are other matrix representations
possible that are obtainable by unitary
transformations g0
k ¼ UgkUÀ1
in the Hilbert space
of the bispinor wave functions [E11, E12], i.e., by
forming linear combinations of Dirac wave functions.
Each wave function c transforms into the
new representation c0
¼ Uc satisfying again Dirac’s
equation, if the latter is simultaneously submitted
to a Lorentz transformation of space-time coordinates
(Refs. 78–81), as expressed by condition
(iv) above.
Turning to condition (ii), in order to verify that
the Dirac-Hamiltonian is Hermitian, we first observe
that all Pauli matrices [E13], r ¼ ðs1; s2; s3Þ, are
Hermitian, therefore it follows
g0 ¼ gþ
0 ; gþ
¼ Àg; gþ
k ¼ g0gkg0 [E16]
(because forming the adjoint matrix requires taking
the conjugate complex of its elements and the transpose
of the matrix). This leads to
g0gð Þþ
¼ g0g [E17]
From Eq. [E1] we read that the Hamiltonian is
equal to
HD ¼
"hc
i
a Á ~r þ bmc2
¼ ca Á ~p þ bmc2
[E18]
Multiplying both sides with b2
¼ 1 and taking
into account Eq. [E3] with g ¼ ðg1; g2; g3Þ, we
arrive at
HD ¼ cg0g Á p þ g0mc2
[E19]
Since the momentum operator p is Hermitian,
and, according to Eqs. [E16, E17], g0g as well as g0
are Hermitian, we conclude that the Dirac Hamiltonian
HD is also Hermitian. Thus, Dirac’s equation
can also be written in the form i"hqc=qt ¼ HDc,
being in the form of Schro¨dinger’s equation, however,
containing the Dirac Hamiltonian HD linear in
momentum.
For the sake of brevity, in the sequel we call particles
characterized by Dirac’s field equation [E4]
free Dirac particles. We wish to derive the combined
field equations when a electromagnetic field
is present. In order to achieve this, we have to
replace the linear momentum pk
¼ iqk
for the free
particle by the momentum pk
À eAk
, i.e., we replace
the partial differential operator iqk
in Eq. [E4] by
iqk
À eAk
, in the literature sometimes the latter is
referred to as the covariant derivative. The replacement
of momentum pk
by pk
À eAk
is called the
minimum coupling condition and its physical relevance
can be derived from requiring invariance of
the Dirac equation under local phase transformations
of the field function c, which leads, in conjunction
with gauge invariance of the electromagnetic
wave equation, to that condition (see for
example, Ref. 10). We arrive at the Dirac equation
in the presence of an electromagnetic field:
Àgk iqk
À eAk
À Á
þ m
À Á
c ¼ 0 [E20]
which is the relevant equation of motion for Dirac
particles with electromagnetic interactions.
APPENDIX F
Current Density for Particles Obeying
the Dirac Equation
We want to show that condition (iii) (Appendix E)
is satisfied by constructing the four-current density
for Dirac particles. We start with the free particle
Dirac equation [E4] and take the Hermitian conjugate
of it,
cþ
iqk
gþ
k þ m
À Á
¼ 0
where it is understood that the differential operator
qk
now acts on those quantities standing on the left
side of it and cþ
is given by Eq. [E12]. To determine
the Hermitian conjugate gþ
k of g, we notice
that according to Eqs. [E16], gþ
k ¼ g0gkg0, and that
g2
0 ¼ 1, hence
cþ
iqk
g0gkg0 þ m
À Á
¼ cþ
iq0
g0g2
0 À iqm
g0gmg0 þ m
À Á
¼ 0; m ¼ 1; 2; 3
332 ENGELKE
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
Defining the adjoint of c to be
"c ¼ cþ
g0 [F1]
we arrive at
cþ
iq0
g0g2
0 À iqm
g0gmg0 þ m
À Á
¼ "c iq0
À iqm
gmg0
À Á
þ cþ
m ¼ 0
Multiplying from the right with g0 yields
"c iq0
g0 À iqm
gmg2
0
À Á
þ cþ
g0m
¼ "c iq0
g0 À iqm
gm þ m
À Á
¼ 0
simplifying to
"c iqk
gk þ m
À Á
¼ 0 [F2]
Now we consider the bilinear expression "cgkc
which we will identify with the current density of
Dirac particles. By means of Eqs. [E4] and [F2], we
calculate the derivative
qk
ð"cgkcÞ¼ðqk "cÞgkcþ "cgkðqk
cÞ¼im"ccÀim"cc¼0
i.e., the quantity
jk ¼e"cgkc [62; F3]
has vanishing four-divergence, qk
jk ¼0. The timelike
component "cg0c¼cþ
c is equal to the squared
magnitude of c, hence positive-definite, and can be
interpreted as the probability density for the Dirac
particle. Therefore jk as defined in Eq. [F3] satisfies
the needs for a four-current density (condition (iii)):
it is a genuine four-vector, its four-divergence vanishes,
and its time-like component is positive-
definite.
Just for the sake of demonstration, Eqs. [62,F3]
written in conventional form would result in the following
matrix equations for the time-like component
j0 and the space-like components j1:
j0 ¼ e"cg0c ¼ ecþ
g2
0c ¼ ecþ
c
¼ eðcÃ
1; cÃ
2; cÃ
3; cÃ
4Þ
c1
c2
c3
c4
0
B
B
@
1
C
C
A
¼ eðcÃ
1c1 þ cÃ
2c2 þ cÃ
3c3 þ cÃ
4c4Þ
j1 ¼ e"cgkc ¼ cþ
g0g1c ¼ eðcÃ
1;cÃ
2;cÃ
3; cÃ
4Þ
Â
1 0 0 0
0 1 0 0
0 0 À1 0
0 0 0 À1
0
B
B
B
@
1
C
C
C
A
0 0 0 1
0 0 1 0
0 À1 0 0
À1 0 0 0
0
B
B
B
@
1
C
C
C
A
c1
c2
c3
c4
0
B
B
B
@
1
C
C
C
A
¼
¼ eðcÃ
1; cÃ
2; cÃ
3;cÃ
4Þ
0 0 0 1
0 0 1 0
0 1 0 0
1 0 0 0
0
B
B
B
@
1
C
C
C
A
c1
c2
c3
c4
0
B
B
B
@
1
C
C
C
A
¼ eðcÃ
1; cÃ
2; cÃ
3;cÃ
4Þ
c4
c3
c2
c1
0
B
B
B
@
1
C
C
C
A
¼ eðcÃ
1c4 þ cÃ
2c3 þ cÃ
3c2 þ cÃ
4c1Þ
Considering these and similar lengthy expressions,
the concise covariant form [F3] can only be
appreciated.
Gordon Decomposition (91, 92) of
Dirac Current Density
The four-current density [F3] associated with the
Dirac equation [E20] can be decomposed into two
parts—one related to the spatial motion of particles,
the other related to the spin of particles. We begin
with the Dirac equation including the electromagnetic
field,
Àgkðiqk
À eAk
Þ þ m
À Á
c ¼ 0 [E20; F4]
We introduce the notation with the momentum
operator Pk that takes into account the presence of
the electromagnetic field,
Pk
¼ pk
À eAk
; pk
¼ iqk
[F5]
Two four-vectors am and bn always satisfy the
relationship
gmam
gnbn
¼ gm
amgn
bn ¼ ambngm
gn
¼
1
2
ambnðgm
gn
þ gn
gm
þ gm
gn
À gn
gm
Þ ¼
¼ ambnðgmn
À ismn
Þ ¼ am
bm À iambmsmn
½F6Š
where we have used the fact that the anticommutator
of Dirac matrices yields the metric tensor and
the spin tensor smn
is defined by the commutator of
Dirac matrices. Eqs. [F6] indicate that a certain
VIRTUAL PHOTONS IN MAGNETIC RESONANCE 333
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
product of four-vectors and Dirac matrices may be
decomposed into a ‘‘metric’’ part and a ‘‘spinor’’
part, the latter containing the spin tensor smn
. We
will now apply the above relationships to arrive at
analogous decomposition of the Dirac current density.
For that purpose we consider two particular
vector solutions ca and cb of the Dirac equation,
ðgkPk
À mÞca ¼ 0; ðgkPk
À mÞcb ¼ 0 [F7]
For the second solution cb let us take the adjoint
equation
"cbððgkPk
Þ þ mÞ ¼ 0 [F8]
with the notations
"cbðgkPk
Þ ¼ ðPm
"cbÞgm
; ðgkPk
Þ!ca ¼ ðPmcaÞgm
[F9]
and
"cb ¼ cþ
b g0 [F10]
Multiplying the first equation in [F7] by "cbðgkak
Þ
and multiplying the second equation in [F7] by
Àðgkak
Þca, then subtracting the two resulting equations
yields
0 ¼ "cbðÀðgkPk
Þ À mÞðgkak
Þca
þ"cbðgkak
ÞððgkPk
Þ! À mÞca
From this equation it follows
"cbðgkak
Þca¼
1
m
"cbðgkak
ÞðgkPk
Þ!ÀðgkPk
Þ ðgkak
ÞÞca
and taking into account the relation [F6] this gives
"cbðgkak
Þca ¼
1
m
ðPm
am
"cbÞca À "cbðPm
amcaÞ
Â
ÀiðPmansmn "cbÞca þ i"cbðPmansmn
caÞ
Ã
½F11Š
The tensor smn
is, by definition, an antisymmetric
tensor,
smn
¼
i
2
½gm
; gn
Š [F12]
hence
Pmansmn "cb ¼ ÀPnamsnm "cb
which yields
"cbðgkak
Þca ¼
1
m
ðPm
am
"cbÞca À "cbðPm
amcaÞ
Â
þiððPnamsnm "cbÞca þ "cbðPmansmn
caÞÞ
Ã
½F13Š
Pn is a differential operator such that the product
rule of differentiation applies,
Pnamð"cbsmn
caÞ¼ðPnamsnm "cbÞca þ "cbðPmansmn
caÞ
which inserted into [F13] gives
"cbðgkak
Þca ¼
1
m
ðPm
am
"cbÞca À "cbðPm
amcaÞ
Â
þiPnamð"cbsmn
caÞ
Ã
For the special case that am is a spatially constant
vector, we may write
"cbgm
caam ¼ À
1
m
"cbðPm
caÞ À ðPm "cbÞca
Â
ÀiPnð"cbsmn
caÞ
Ã
am ½F14Š
and in this case we may omit am on both sides of
the equation. Furthermore setting ca ¼ cb and ca ¼
c and introducing a factor 1/2 for the purpose of
normalization, we arrive at
jm
ðxÞ ¼ e"cðxÞgm
cðxÞ ¼ À
e
2m
"cðxÞðPm
cðxÞÞ
Â
ÀðPm "cðxÞÞcðxÞ À iPnð"cðxÞsmn
cðxÞÞ
Ã
½F15Š
and the spin-dependent part, i.e., the spin current
density reads
jm
spin ¼
þie
2m
Pnð"csmn
cÞ [F16]
Nonrelativistic First-Order Limit
for the Spin Current Density
(See ref, 92 and 93)
We remind ourselves that the Dirac wave function
c is a four-component vector in Hilbert space. We
divide this vector into two 2-component vectors
(spinors) and write the Dirac equation [F4] as
Àgniqn
þ egnAn
þ mð Þ
j
w
 
¼ 0 [F17]
With iqn
¼ pn
Eq. [F17] becomes
Àgnpn
þ egnAn
þ mð Þ
j
w
 
¼ 0
334 ENGELKE
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
and if we separate this into timelike (n ¼ 0) and
spacelike (n ¼ 1,2,3) components we obtain
Àðg0p0
À g Á pÞ þ eðg0A0
À g Á AÞ þ m
À Á j
w
 
¼ 0
Àg0ðp0
À eA0
Þ þ g Á ðp À eAÞ þ m
À Á j
w
 
¼ 0
[F18]
With the definitions [E13-E15] for the Dirac matrices
this becomes
À
1 0
0 À1
 
ðp0
À eA0
Þ
j
w
 
þ
0 s
Às 0
 
Áðp À eAÞ
j
w
 
þ m
j
w
 
¼ 0 ½F19Š
hence we obtain two equations:
À ðp0
À eA0
Þj þ s Á ðp À eAÞw þ mj ¼ 0
ðp0
À eA0
Þw À s Á ðp À eAÞj þ mw ¼ 0 ½F20Š
The constant term m in both equations leads to a
time dependence of spinor wave functions j and w
expressed by the factor exp(Àimt). We may easily
remove the mj term in [F20] by introducing the
new spinor wave functions j0
, w0
:
j0
ðtÞ
w0
ðtÞ
 
¼
jðtÞ
wðtÞ
 
expðþimtÞ [F21]
Taking into account that p0
¼ iq=qt, inserting Eq.
[F21] into Eqs. [F20] yields
À ðp0
À eA0
Þj0
þ s Á ðp À eAÞw0
¼ 0
ðp0
À eA0
þ 2mÞw0
À s Á ðp À eAÞj0
¼ 0 ½F22Š
For a nonrelativistic particle, as mentioned
above, we take now into account that its energy
p0
À eA0
in the field is small compared to its rest
energy m, i.e.,
p0
<< m; eA0
<< m [F23]
An analogous relation holds for the magnitude of
the three-momentum p and the magnitude of the
vector potential A. For a free particle, p0
is equal to
its kinetic energy, so in the nonrelativistic regime,
the kinetic energy is small compared to the rest
energy. Likewise, eA0
, representing the energy of
the particle in the electromagnetic field with scalar
potential A0
, is supposed to be small compared to
the rest energy.
With the assumption in [F23], we keep only the
term 2m in the second equation of [F22] in the first
pair of parentheses,
À ðp0
À eA0
Þj0
þ s Á ðp À eAÞw0
¼ 0
2mw0
À s Á ðp À eAÞj0
¼ 0 ½F24Š
The second equation in [F24] allows us to
express the wave function w0
in terms of the wave
function j0
:
w0
¼
1
2m
s Á ðp À eAÞj0
[F25]
For a nonrelativistic particle the components of
the spinor wave function w0
thus become small compared
to the components of the 2-spinor wave function
j0
. Consequently, the particle can be thus
approximately described by the spinor j0
only.
Derivation of Eq. [78] From Eq. [76]
Hint;spin ¼
e
2m
A Á rðxÃ
xÞ Â sÞ þ iexÃ
xðA Â sð Þ
¼
e
2m
A Á rðxÃ
xÞ Â sð Þ ½76Š
First, with the vector-algebraic identity
a Á ðb  cÞ ¼ ða  bÞ Á c [F26]
and setting
a ¼ A; b ¼ ðrxÃ
xÞ; c ¼ s
we may write
A Á rðxÃ
xÞ Â sð Þ ¼ A Â rðxÃ
xÞð Þ Á s [F27]
Secondly, we have
r  ðfaÞ ¼ fðr  aÞ À a  rf; [F28]
i.e.,
A Â rðxÃ
xÞð Þ ¼ xÃ
xðr  AÞ À r  ðxÃ
xAÞð Þ
[F29]
so that [F28] becomes
VIRTUAL PHOTONS IN MAGNETIC RESONANCE 335
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
A Á ðrðxÃ
xÞ Â sÞ ¼ xÃ
xðr  AÞ À ðxÃ
xAÞð Þ Á s
[F30]
According to the chain rule for the vector prod-
uct,
r Á ða  bÞ ¼ b Á ðr  aÞ À a Á ðr  bÞ [F31]
with a ¼ s; b ¼ xÃ
xA we obtain for the second
term on the right-hand side of [F30]
À r  ðxÃ
xAÞð Þ Á s ¼ ÀðxÃ
xAÞ Á ðr  sÞ
þr Á ðs  ðxÃ
xAÞÞ ½F32Š
Because the spin operators are not depending on
space variables, we have
r  s ¼ 0; [F33]
thus
À ðr  ðxÃ
xAÞÞ Á s ¼ r Á ðs  ðxÃ
xAÞÞ
and from [F30, F32] we arrive at
A Á rðxÃ
xÞ Â sð Þ ¼ xÃ
xðr  AÞ Á s
þr Á ðs  ðxÃ
xAÞÞ ½F34Š
Introducing the magnetic induction field associated
with the vector potential, i.e.,
B ¼ r  A [F35]
[F34] simplifies to
A Á rðxÃ
xÞ Â sð Þ ¼ xÃ
xB Á s þ r Á ðs  ðxÃ
xAÞÞ
and the interaction Hamiltonian density [77] now
reads
Hint;spin ¼
e
2m
A Á rðxÃ
xÞ Â sð Þ
¼
e
2m
xÃ
xB Á s þ r Á ðs  ðxÃ
xAÞÞð Þ [78]
REFERENCES
1. Vaara J, Jokisaari J, Wasylishen RE, Bryce DL.
2002. Spin-spin coupling tensors as determined by
experiment and computational chemistry. Prog
NMR Spectrosc 41:233–304.
2. Romero RH, Aucar GA. 2002. Self-energy effects
on nuclear magnetic resonance parameters within
quantum electrodynamics perturbation theory. Int J
Mol Sci 3:914–930.
3. Slichter CP. 1978. Principles of Magnetic Resonance,
2nd ed. Berlin: Springer. pp 100–106.
4. Walls DF, Milburn GJ. 1994. Quantum Optics. Berlin:
Springer.
5. Glauber R. 1963. Coherent and incoherent states of
the radiation field. Phys Rev 131:2766.
6. Ehrenburg W, Siday RE. 1949. The refractive index
in electron optics and the principles of dynamics.
Proc Phys Soc (London) B 62:8–21.
7. Aharonov Y, Bohm D. 1959. Significance of electromagnetic
potentials in the quantum theory. Phys
Rev 115:485–491.
8. Aharonov Y, Bohm D. 1961. Further considerations
on electromagnetic potentials in the quantum theory.
Phys Rev 123:1511–1524.
9. Chambers RG. 1960. Shift of an electron interference
pattern by enclosed magnetic flux. Phys Rev
Lett 5:3–5.
10. Moriyasu K. 1983. An Elementary Primer for
Gauge Theory. Singapore: World Scientific.
11. Berry MV. 1984. Quantal phase factors accompanying
adiabatic changes. Proc R Soc (London) A 392:45–57.
12. Aharonov Y, Anandan J. 1987. Phase change during
cyclic quantum evolution. Phys Rev Lett 58:1593–
1596.
13. Tycko R. 1987. Adiabatic rotational splittings and
Berry’s phase in nuclear quadrupole resonance. Phys
Rev Lett 58:2281–2284.
14. Suter D, Chingas GC, Harris RA, Pines A. 1987.
Berry’s phase in magnetic resonance. Mol Phys
61:1327–1340.
15. Suter D, Mueller KT, Pines A. 1988. Study of the
Aharonov-Anandan quantum phase by NMR interferometry.
Phys Rev Lett 60:1218–1220.
16. Zwanziger JW, Koenig M, Pines A. 1990. Non-Abelian
effects in a quadrupole system rotating around
two axes. Phys Rev A 42:3107–3110.
17. Furman GB, Kadzhaya IM. 1993. Bloch-Siegert
shift and Berry phase effect in nuclear magnetic resonance.
J Magn Reson A 105:7–9.
18. Steffen M, Vandersypen LMK, Chuang IL. 2000.
Simultaneous soft pulses applied at nearby frequencies.
J Magn Reson 146:369–374.
19. Kayanuma Y. 1997. Stokes phase and geometrical
phase in a driven two-level system. Phys Rev A
55:R2495–R2498.
20. Schwinger J. 1948. On quantum electrodynamics
and the magnetic moment of the electron. Phys Rev
73:416–417.
21. Weinberg S. 1995. The Quantum Theory of Fields.
Vol. I—Foundations. Cambridge: Cambridge University
Press. pp 452–457 and 472–498.
22. Zee A. 2003. Quantum Field Theory in a Nutshell.
Princeton and Oxford: Princeton University Press.
23. Greiner W, Schramm S, Stein E. 2002. Quantum
Chromodynamics, 2nd ed. Berlin: Springer.
336 ENGELKE
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
24. Pich A. 1995. Quantum Chromodynamics. Lecture
Notes. arXiv:hep-ph/9505231v1. Available at: http://
arxiv.org/abs/hep-ph/9505231.
25. The web pages of the German Electron Synchrotron
(DESY), Hamburg: www-hermes.desy.de and
www.desy.de/f/jb/desy2000-066-077.pdf and the
web pages at CERN, Geneva, wwwcompass.cern.ch.
26. Landau LD, Lifschitz EM. 1975. Theoretische
Physik Kurzgefasst. Band II: Quantentheorie. Berlin:
Akademieverlag. pp 181–183 and 134–135.
27. Bowers CR, Weitekamp DP. 1986. Transformation
of symmetrization order to nuclear-spin magnetization
by chemical reaction and nuclear magnetic resonance.
Phys Rev Lett 57:2645–2648.
28. Natterer J, Bargon J. 1997. Parahydrogen induced
polarization. Prog Nucl Magn Reson Spectrosc
31:293–315.
29. Adams RW, Aguilar JA, Atkinson KD, Cowley MJ,
Elliott PIP, Duckett SB, Green GGR, Khazal IG,
Lopez-Serrano J, Williamson DC. 2009. Reversible
interactions with para-hydrogen enhance NMR sensitivity
by polarization transfer. Science 323:1708–
1711.
30. Rugar D, Yannoni CS, Sidles JA. 1992. Mechanical
detection of magnetic resonance. Nature 360:563–
566.
31. Sidles JA. 1991. Noninductive detection of singleproton
magnetic resonance. Appl Phys Lett 58:
2854–2856.
32. Sidles JA, Rugar D. 1993. Signal-to-noise ratios in
inductive and mechanical detection of magnetic resonance.
Phys Rev Lett 70:3506–3509.
33. Verhagen R, Wittin A, Hilbers CW, van Kempen H,
Kentgens AP. 2002. Spatially resolved spectroscopy
and structurally encoded imaging by magnetic resonance
force microscopy of quadrupolar spin systems.
J Am Chem Soc 124:1588–1589.
34. Lin Q, Degen CL, Tomaselli M, Hunkeler A, Meier
U, Meier BH. 2006. Magnetic double resonance in
force microscopy. Phys Rev Lett 96:137604.
35. Eberhardt KW, Degen CL, Hunkeler A, Meier BH.
2008. One- and two-dimensional NMR spectroscopy
with a magnetic-resonance force microscope.
Angew Chem Int Ed Eng 47:8961–8963.
36. Butler M. 2008. Novel Methods for Force-Detected
Nuclear Magnetic Resonance, PhD thesis. Pasadena:
California Institute of Technology.
37. Olson L, Peck TL, Webb AG, Magin R, Sweedler
JV. 1995. Radiofrequency microcoils in magnetic
resonance. Prog Nucl Magn Reson Spectrosc 31:1–
42.
38. Zhou LX, Potter CS, Lauterbur PC, Both BW.
1989. NMR imaging with (6.37 mm)3
isotropic resolution.
Abstr Soc Magn Reson Med, 8th Annual
Meeting, Amsterdam. p 128.
39. Ciobanu L, Seeber DA, Pennington CH. 2002. 3D
MR microscopy with resolution 3.7 mm by 3.7 mm
by 3.3 mm. J Magn Reson 158:178,182.
40. Minard KR, Wind RA. 2002. Picoliter 1
H NMR
spectroscopy. J Magn Reson 154:336–343.
41. Yusa G, Muraki K, Takashina K, Hashimoto K, Hirayama
Y. 2005. Controlled multiple quantum coherences
of nuclear spins in a nanometre-scale device.
Nature 434:1001.
42. van Bentum PJM, Janssen JWG, Kentgens APM,
Bart J, Gardeniers JGE. 2007. Stripline probes for
nuclear magnetic resonance. J Magn Reson
189:104–113.
43. Maguire Y, Chuang IL, Zhang S, Gershenfeld N.
2007. Ultra-small-sample molecular structure detection
using microslot waveguide nuclear spin resonance.
Proc Natl Acad Sci USA 104:9198–9203.
44. Busch P, Lahti PJ, Mittelstaedt P. 1996. The Theory
of Quantum Measurement, 2nd ed. Berlin:
Springer.
45. Braginsky VB, Khalili FY. 1992. Quantum Measurement.
Cambridge: Cambridge University Press.
46. von Neumann J, Grundlagen der Quantenmechanik.
1955. Mathematical Foundations of Quantum
Mechanics. Princeton, NJ: Princeton University Press.
47. Birkhoff G, von Neumann J. 1936. The logics of
quantum mechanics. Ann Math 37:823.
48. Feynman RP. 1982. Simulating physics with computers.
Int J Theor Phys 21:467–488.
49. Deutsch D. 1987. Quantum theory, the ChurchTuring
principle and the universal quantum computer.
Proc R Soc Lond A 400:97–117.
50. Deutsch D. 1989. Quantum computational networks.
Proc R Soc Lond A 425:73–90.
51. Preskill J. 1997. Quantum Information and Computation.
Lecture Notes. Pasadena: California Institute
of Technology.
52. Stolze J, Suter D. 2004. Quantum Computing.
Weinheim: Wiley-VCH.
53. Zeilinger A. 2000. Quantum computing: quantum
entangled bits step closer to IT. Science 289:405–
406.
54. Jeener J, Henin F. 1986. Fixed bound atom interacting
with a coherent pulse of quantized radiation:
Excitation and short time behavior. Phys Rev A
34:4897–4928.
55. Jeener J, Henin F. 2002. A presentation of pulsed
nuclear magnetic resonance with full quantization of
the radio frequency magnetic field. J Chem Phys
116:8036–8047.
56. Hoult DI. 1989. The magnetic resonance myth of
radio waves. Concepts Magn Reson 1:1–5.
57. Hoult DI, Bhakar B. 1997. NMR signal reception:
virtual photons and coherent spontaneous emission.
Concepts Magn Reson 9:277–297.
58. Hoult DI, Ginsberg NS. 2001. The quantum origins
of the free induction decay signal and spin noise.
J Magn Reson 148:182–199.
59. Hoult DI. 2004. The Quantum Origins of the NMR
Signal: A Paradigm for Faraday Induction. Physics
in Canada Conference.
VIRTUAL PHOTONS IN MAGNETIC RESONANCE 337
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
60. Hoult DI. 2009. The origins and the present status
of the radio wave controversy in NMR. Concepts
Magn Reson 34A:193–216.
61. Boender GC, Vega S, de Groot HJM. 2000.
Quantized field description of rotor-frequency
driven dipolar recoupling. J Chem Phys 112:1096–
1106.
62. Blok H, Orlinski SB, Schmidt J, Baranov PG. 2004.
Overhauser effect of 67
Zn nuclear spins in ZnO via
cross relaxation induced by the zero-point fluctuations
of the phonon field. Phys Rev Lett 92:047602-
1–047602-4.
63. Hahn EL. 1997. Concepts of NMR in quantum
optics. Concepts Magn Reson 9:69–81.
64. Zhao W, Wright JC. 2000. Doubly vibrationally
enhanced four wave mixing: the optical analog to
2D NMR. Phys Rev Lett 84:1411–1414.
65. Ka¨lin M, Fedin M, Gromov I, Schweiger A. 2006.
Multiphoton transitions in EPR spectroscopy. Lect
Notes Phys 684:143–183.
66. Ka¨lin M, Gromov I, Schweiger A. 2003. The
continuous wave electron paramagnetic resonance
experiment revisited. J Magn Reson 160:166–182.
67. Ka¨lin MA. 2003. Multiple Photon Processes in
Electron Paramagnetic Resonance Spectroscopy.
PhD thesis. ETH Zu¨rich.
68. Gromov I, Schweiger A. 2000. Multiphoton resonances
in pulse EPR. J Magn Reson 146:110–121.
69. Hatanaka H, Tabuchi N. 2002. New line-narrowing
effect in triple-quantum resonance in a two-level
NMR system. J Magn Reson 155:119.
70. Zur Y, Levitt MH, Vega S. 1983. Multiphoton
NMR spectroscopy on a spin system with I ¼ 1/2. J
Chem Phys 78:5293.
71. Goelman G, Zax DB, Vega S. 1987. Observation of
three-photon resonances in NMR under four-field
irradiation. J Chem Phys 87:31.
72. Goelman G, Vega S, Zax DB. 1989. Squared amplitude-modulated
composite pulses. J Magn Reson
81:423–429.
73. Sternin E, Goelman G, Bendahan Y, Vega S. 1991.
Squared multiphoton pulses for spectral selectivity.
J Magn Reson 92:538–549.
74. Michal CA. 2003. Nuclear magnetic resonance noise
spectroscopy using two-photon excitation. J Chem
Phys 118:3451–3454.
75. Eles PT, Michal CA. 2003. Two-photon excitation
in nuclear quadrupole resonance. Chem Phys Lett
376:268–273.
76. Eles PT, Michal CA. 2004. Two-photon two-color
nuclear magnetic resonance. J Chem Phys 121:
10167–10173.
77. Feynman RP. 1949. Space-time approach to quantum
electrodynamics. Phys Rev 76:769–821.
78. Jackson JD. 1998. Classical Electrodynamics, 3rd
ed. New York: Wiley. p 775.
79. Itzykson C, Zuber JB. 1980. Quantum Field Theory.
New York: McGraw Hill.
80. Cohen-Tannoudji C, Dupont-Roc J, Grynberg G.
1989. Photons and Atoms—Introduction to Quantum
Electrodynamics. New York: Wiley.
81. Greiner W, Reinhardt J. 1995. Quantenelektrodynamik,
2nd ed. Thun: Harri Deutsch.
82. Greiner W, Reinhardt J. 1993. Feldquantisierung.
Thun: Harri Deutsch.
83. Schwartz L. 1951. Theorie des Distributiones. Paris:
Hermann.
84. Kolmogorov IN, Fomin SV. 1975. Introductory Real
Analysis. New York: Dover Publications.
85. Zeidler E. 2006. Quantum Field Theory I. Basics in
Mathematics and Physics—A Bridge Between
Mathematicians and Physicists. Berlin: Springer.
pp 607–625.
86. Szantay C Jr. 2008. NMR and the uncertainty principle:
how to and how not to interpret homogeneous
line broadening and pulse nonselectivity. Part III.
Uncertainty? Concepts Magn Reson 32A:302–325
Part IV. (Un)certainty. Concepts Magn Reson
32A:373–404.
87. Engelke F. 2002. Electromagnetic wave compression
and radio frequency homogeneity in NMR solenoidal
coils: computational approach. Concepts
Magn Reson 15:129–155.
88. Byron FW Jr, Fuller RW. 1992. Mathematics of
Classical and Quantum Physics. New York: Dover
Publications. Chapter 6 (Theory of analytic functions)
and 7 (Green functions).
89. Ziman JM. 1969. Elements of Advanced Quantum
Theory. Cambridge: Cambridge University Press.
90. Kaempffer FA. 1965. Concepts in Quantum
Mechanics. New York: Academic Press.
91. Gordon W. 1928. Der Strom der Diracschen Elektronentheorie.
Z Phys A 50:630.
92. Bjorken JD, Drell SD. 1998. Relativistische Quantenmechanik.
Heidelberg: Spektrum—Akademischer
Verlag. pp 48–49.
93. Landau LD, Lifshitz EM. 1962. Theoretical Physics.
Vol. III. Quantum Mechanics. Oxford: Pergamon.
94. Castellani G, Sivasubramanian S, Widom A, Srivastava
YN. 2003. Radiating and non-radiating current
distributions in quantum electrodynamics. arXiv:
quant-ph/0306185 v1. Available at: http://arxiv.org/
abs/quant-ph/0306185.
95. Greiner W, Reinhardt J. 1993. Feldquantisierung.
Thun: Harri Deutsch. pp 32–34.
96. Untidt TS, Nielsen NC. 2002. Closed solution to the
Baker-Campbell-Hausdorff problem: exact effective
Hamiltonian theory for analysis of nuclear-magneticresonance
experiments. Phys Rev E 65:021108–1-17.
97. Louisell WH. 1973. Quantum Statistical Properties
of Radiation. New York: Wiley.
98. Fick E. 1981. Einfu¨hrung in die Grundlagen der
Quantentheorie, 2nd ed. Leipzig: Geest & Portig.
99. Wigner E. 1939. On the unitary representations of
the inhomogeneous Lorentz group. Ann Math
40:149–204.
338 ENGELKE
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
100. Stoll ME, Vega AJ, Vaughn RW. 1977. Explicit demonstration
of spinor character for a spin-1/2 nucleus
via NMR interferometry. Phys Rev A 16:1521–1524.
101. Stoll ME, Wolff EK, Mehring M. 1978. Demonstration
of 4p and 2p rotational symmetry in a spin-1
system (deuterons). Phys Rev A 17:1561–1567.
102. Wolff EK, Mehring M. 1979. Spinor behavior of a
double quantum transition in three-level systems.
Phys Lett 70A:125–126.
103. Mehring M, Wolff EK, Stoll ME. 1980. Exploration
of the eight-dimensional spin space of a spin-1 particle.
J Magn Reson 37:475–495.
104. Pauli W. 2000. Wave mechanics. In: Enz CP, ed.
Pauli Lectures on Physics, Vol. 5. Cambridge (Mass.):
MIT Press.
105. Bloembergen N, Pound RV. 1954. Radiation damping
in magnetic resonance experiments. Phys Rev 95:8–12.
106. Bruce CR, Norberg RE, Pake GE. 1956. Radiation
damping and resonance shapes in high resolution
nuclear magnetic resonance. Phys Rev 104:419–420.
107. Bloom S. 1957. Effects of radiation damping on
spin dynamics. J Appl Phys 28:800–805.
108. Szoeke A, Meiboom S. 1959. Radiation damping in
nuclear magnetic resonance. Phys Rev 113:585–586.
109. Leigh JS Jr. 1968. New technique for radio frequency
magnetic field measurement. Rev Sci Instrum 39:
1594–1595.
110. Gueron M, Plateau P, Decorps M. 1991. Solvent signal
suppression in NMR. Prog NMR Spectrosc 23:135–209.
111. Abergel D, Lallemand JY. 1994. Some microscopic
aspects of radiation damping in NMR. J Magn
Reson A 110:45–51.
112. Gueron M, Leroy JL. 1989. NMR of water protons—
the detection of their nuclear spin noise and a simple
determination of absolute probe sensitivity based on
radiation damping. J Magn Reson 85:209–215.
113. Gueron M. 1991. A coupled resonator model of the
detection of nuclear magnetic resonance: radiation
damping, frequency pushing, spin noise, and the signal-to-noise
ratio. Magn Reson Med 19:31.
114. Warren WS, Hames SL, JBates JL. 1989. Dynamics
of radiation damping in nuclear magnetic resonance.
J Chem Phys 91:5895–5904.
115. Vlassenbroek A, Jeener J, Broekart P. 1995. Radiation
damping in high resolution liquid NMR, a simulation
study. J Chem Phys 103:5886–5897.
116. Mao XA, Ye CH. 1997. Understanding radiation damping
in a simple way. Concepts Magn Reson 9:173–187.
117. Hahn EL. 1997. Radiation damping of an inhomogeneously
broadened spin ensemble. Concepts Magn
Reson 9:65–67.
118. Barjat H, Matiello DL, Freeman R. 1998. Suppression
of radiation damping in high-resolution NMR.
J Magn Reson 136:114–117.
119. Augustine MP, Bush SD, Hahn EL. 2000. Noise
triggering of radiation damping from the inverted
state. Chem Phys Lett 322:111–118.
120. Marion DJY, Desvaux H. 2008. An alternative tuning
approach to enhance NMR signals. J Magn
Reson 193:153–157.
121. According to Nevels and Crowell, IEE Proc.-H,
137, 6 (1990), p 384, and various other authors, this
condition is due to L.V. Lorenz and not to H.A.
Lorentz. See: L. V. Lorenz, U¨ ber die Identita¨t der
Schwingungen des Lichts mit den elektrischen Stro¨men,
Poggendorfs Annalen der Physik, Bd. 131,
1867, and B. Riemann, Schwere, Elektrizita¨t und
Magnetismus, Carl Ru¨mpler, Hannover, 1861.
122. Bronstein IN, Semendjajew KA. 1965. Taschenbuch
der Mathematik. Zu¨rich: Harry Deutsch.
123. Dirac PAM. 1958. Quantum Mechanics, 4th ed.
London: Oxford University Press.
124. Schwabl F. 2004. Quantenmechanik fu¨r Fortgeschrittene,
3rd ed. Berlin: Springer. pp 345–246 and
364–379.
BIOGRAPHIES
Frank Engelke received his diploma in
physics and his doctorate in natural sciences
from Leipzig University, Germany,
working under Dieter Michel on NMR of
solid-state phase transitions. As a postoctoral
fellow he focused on phase transitions
of plastic crystals at the Physics Department
of Saarbru¨cken University in the
group of Jo¨rn Petersson and on the field of
in situ NMR of heterogeneous catalysis at
Ames Laboratory with Marek Pruski and Bernie Gerstein. In
1994, he joined the solid-state probe development team of Bruker
BioSpin. In 1996, he became head of the probe development
department at Bruker BioSpin in Rheinstetten, Germany. His current
research interests focus on rf and microwave technology for
NMR and DNP probes, MAS technology, and quantum electrodynamics.
He is author and/or coauthor of about 40 research articles
in the field of NMR spectroscopy.
VIRTUAL PHOTONS IN MAGNETIC RESONANCE 339
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a