LIE GROUPS EXERCISES
Gerard 't Hooft
Spinoza Instituut
Postbus 80.195
3508 TD Utrecht
e-mail: g.thooft@phys.uu.nl
internet: http://www.phys.uu.nl/~thooft/
Exercises with the lecture course Topics in Theoretical Physics, part II, Lie Groups 2007.
June 18, 2007
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1. Lecture April 23
Consider matrices A with two rows and columns(2 × 2 -matrices), and only real coefficients.
We demand thatA be orthogonal(A ~A = I), and that det(A) = 1.
a. Show that these matrices form a group, and that one parameter (de rotation angle)
suffices to describe the matrix. This group is called SO(2).
b. Show that these matrices commute: two matrices of this type, A and B, obey
[ A, B ] = 0. Such groups are called Abelian.
2. Lecture April 23
Consider the definition of a group as in the notes. A group obeys the 4 axioms listed
there.
a. Derive, only using these axioms, that the "left-inverse" R-1
defined at point 4, also
serves as a "right-inverse": R R-1
= I
(Hint: Consider R-1
R R-1
and make use of the fact that also R-1
has an inverse).
b. Prove that the unit element is also the "right-unit-element": R = R I for every
R
(Hint: multiply at the right with R-1
R)
c. Prove that there is exactly one unit element.
3. Lecture April 23
We give a few elements of a representation of the three-dimensional rotation group SO(3)
that consists of 2 dimensional matrices:
R3 =



0 1 0
-1 0 0
0 0 1


 
1

2
1 + i 0
0 1 - i
(3.1)
R1 =



1 0 0
0 0 1
0 -1 0


 
1

2
1 i
i 1
, (3.2)
Compute the matrices belonging to some other elements of this group, such as
R2 =



0 0 -1
0 1 0
1 0 0


 . (3.3)
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4. Lecture May 7
Let R(1) be a rotation over 90
around the z-axis and R(2) a rotation over 45
around
the x-axis.
a. Write the corresponding matrices.
b. Compute the product matrix R(3) = R(2) R(1) .
c. The matrix R(3) may also be viewed as a rotation over an angle  around an axis
(a1, a2, a3). Show that
Tr (R(3)) = 1 + 2 cos  , (4.1)
and compute the angle .
d. The coordinates of the axis can be deduced from the antisymmetric part of R(3) ,
see equation (3.10) of the notes. Find the orientation of this axis.
e. Check that the length of the vector indeed produces the sine of the rotation angle.
5. Lecture May 7
Consider the discussion of page 16--20 of the notes, but now applied to rotations in a four
dimensional space. Thus, we have vectors (x, y, z, u) that we rotate using orthogonal
4 × 4 matrices.
a. Show that there are 6 linearly independent generators. We may call them L1,    , L6 .
Thus, a 4 × 4 matrix R can be written as
R = exp i
6
k=1
k Lk (5.1)
b. Make a list of all structure constants c k
ij .
6. Lecture May 7.
a. Derive equation (3.32) (see the notes) for the structure constants c m
ij .
b. Show that, if we assume Li to be hermitean, the structure constants are purely
imaginary. Show furthermore that these constants are antisymmetric in the indices
i en j: c m
ij = -c m
ji .
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c. Now introduce the so-called metric tensor gij (using sommation convention):
gij
def
= c m
ik c k
mj . (6.1)
Show that gij is symmetric: gij = gji .
d. Show (with the use of (3.32)) that the `normalized structure constants',
~ckij = gkmc m
ij are completely antisymmetric:
~ckij = ~cijk = -~ckji . (6.2)
Often, we normealize the Li in such a way that gij = ij .
7. Lecture May 7.
Study Appendix D: The Campbell-Baker-Hausdorff formula. It implies that, given two
matrices A and B, a matrix C exists, such that
eA
eB
= eC
, (7.1)
and that C can be written as a power series in A and B consisting exclusively of commutators,
and no direct products.
a. Admitting complex numbers, show that there are several matrices Ci such that (7.1)
holds.
Hint: diagonalize C and add 2i to one of its eigenvalues,
b. There are several ways to prove the theorem. Instead of equation (D.2) we can also
define
eC(x)
= exA
eB
. (7.2)
Use the same method as in the Appendix to derive that
dC(x)
dx
= {A,
C
1 - e-C
} = A + 1
2
[A, C] + 1
12
[[A, C], C] +    . (7.3)
c. Use this result to compute the coefficients in (D.29).
This exercise is a bit more difficult than what will be required at the test of this course.
The proof of the CBH formula will not have to be reproduced in the test.
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8. Lecture May 14
Consider again the group of orthogonal rotations in 4 dimensions. The elementary representation
is formed by 4-vectors themselves. The matrices R en D then remain the same.
This is the 4 representation. Write the components of the 4 vector as x
,  = 1, 2, 3 or
4.
a. Now consider the 16-dimensional space spanned by the product of two such vectors.
Write A
, ,  = 1,    , 4. Now give the matrix D if under a rotation
A
 R
R
A
. (8.1)
b. This 16 representation however is not irreducible. Show that the two subspaces
spanned by the symmetric tensors A
, for which A
= A
, and those of the
antisymmetric tensors, A
= -A
, are invariant subspaces, with dimensions 10
and 6, respectively.
c. Show that the 10-dimensional representation can be split up further (by looking at
the trace A
: 10 = 9  1.
d. But also the antisymmetric representation 6 can be split up. Consider two kinds of
tensors:
A
L = 1
2
A
L , (8.2)
A
R = -1
2
A
R , (8.3)
to establish that there are two invariant subspaces: 6 = 3L 3R . Later we will find
out that the 3L en 3R are not unitarily equivalent.
9. Lecture May 14
Consider the generators Li of the rotations as given in equation (3.14) of the notes.
a. They generate what we call the `adjoint representation' of the group SO(3). Find
the eigenvalues m = -1, 0, +1 of L3 , and the associated eigenvectors (in the basis
where (3.14) holds), which we will call |m .
b. Construct the matrices L+ and L- , and find out how they act on the states |m .
c. Check whether you indeed get |m  1 , with the norm factors 2 - m(m  1) as
expected according to chapter 5.
d. Compute the Casimir operator i L2
i and check whether indeed the quantum number
is 1 .
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10. Lecture May 14 and May 21
This exercise is more difficult. It exhibits procedures that will be further elaborated on
in the lectures.
A representation A of SO(3) has basis elements A
 ,  = 1,    , NA , and a representation
B has basis elements B
 ,  = 1,    , NB . Consider the product representation
A  B having the products A
 B
 as its basis elements. The dimension of this representation
is NA NB .
a. Show that the generators of this representation can be written as IAB
i = IA
i + IB
i ,
where IA
i only act on the indices  of representation A and IB
i only on the indices
 of representation B.
b. Show that [IA
i , IB
j ] = 0. We still have [IA
i , IA
j ] = iijkIA
k and [IB
i , IB
j ] = iijkIB
k .
c. Show that the operators IAB
i obey the correct commutation rules. Omitting superscript
A  B: [Ii, Ij] = iijkIk .
d. Write the basis elements of A  B as |mA
, mB
. How do the operators I act on
this basis?
e. The new representation A  B is not irreducible. Consider the values that mAB
can have and use that to derive that the maximal value of AB
must be equal to
A
+ B
.
f. Take the case A
= 5
2
and B
= 3
2
. Show that there is only one state with mAB
= 4.
It hust have AB
= 4 (convince yourself of this by letting I+ act on this state). It is
indicated as | = 4, m = 4 . How many basis elements do we have with mAB
= 3,
with mAB
= 2, etc. ?
g. Now let I- act on them. Construct the state | = 4, m = 3 . Show that it is equal
to 5
8
|3
2
, 3
2
+ 3
8
|5
2
, 1
2
. Why do we still have = 4?
h. There is an other state with mAB
= 3 orthogonal to the one found above. Construct
this state and show that it is | = 3, m = 3 .
i. This way we can find all states | , m . Indicating the representations by their
dimension N = 2 + 1, show that 4  6 = 3  5  7  9.
j. In general: AB
= | A
- B
|, | A
- B
| + 1,    , A
+ B
. Show that the sums of the
dimensions of these irreducible parts match:
A+ B
=| A- B|
(2 + 1) = (2 A
+ 1)(2 B
+ 1) . (10.1)
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11. Lecture May 21.
a. Show that 2 × 2-matrices X of the form
a b
-b
a , where a an b are arbitrary
complex numbers, form a group because they obey the axioms of page 8.
b. Show that this remains to be true if we impose on a en b the condition that |a|2
+
|b|2
= 1.
c. Show that this condition corresponds to det(X) = 1.
12. Lecture May 21.
Consider the four-dimensinal vector space spanned by the 4 matrix elements of the 2 × 2
matrices.
a. Show that the three  matrices and the identity matrix, which we could write as
(0)
 = 

form orthonormal basis elements in this vector space, apart from a normalizing
coefficient:
(i) 
 (j)
 = Nij , i, j = 0,    , 3 .
Compute the coefficient N and then normalize these basis elements.
b. In this basis, therefore,
3
i=0
(i)
 (i) = N , ,  = 1,    , 4 .
This must hold in any basis. Show now that
3
i=0
(i) 
 (i)
 = N .
Use this to prove Eq. (6.41) in the lecture notes (6.42 in the new English version):
3
i=1
(i)
(i)
 = -
 
 + 2
 
 .
This argument may sound complicated at first sight, but it is in fact the fastest way to
prove this equation.
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13. Lecture May 21.
We consider analytical functions f of two complex variables 1
, 2
. The operators Li, i =
1, 2, 3, are defined as in eq. (6.38), or (6.39) in the English version:
Li = -1
2
(i)
 

. (13.1)
Show that they obey the commutation rules of SU(2) (and SO(3)):
[Li, Lj] = i ijkLk . (13.2)
Note the minus sign in the definition!
14. Lecture June 4.
Show, using the arguments on the last pages of section 6 of the Notes, that all representations
of SO(3) with = integer +1
2
have the property that the rotation over 2 is
mapped onto the matrix -I. Give an argument why, consequently, orbital wave functions
of quantum particles always have integral angular momentum.
15. Lectures until June 4
Consider again the rotation group in 4 dimensions, SO(4). Consider the 6 generators
found in exercise 5. Instead of writing them as L1,    , L6 , we write them as L
(1)
k =
(L23, L31, L12), L
(2)
k = (L14, L24, L34), with k = 1, 2, 3. Here, Lij generates a rotation in
the (ij) plane.
a) Compute the commutators [L
(a)
i , L
(b)
j ]. Hint: first discover the general rule for the
commutators [Lij, Lkl] by looking at some examples.
Consider two sets of vectors, L
(+)
k and L
(-)
k , defined by: L
()
k = 1
2
(L
(1)
k  L
(2)
k ).
b) Show that [L
(+)
i , L
(+)
j ] = iijkL
(+)
k and similarly for L
(-)
i .
c) Show that [L
(+)
i , L
(-)
j ] = 0.
d) This is the algebra for SU(2)L  SU(2)R , where L, R refer to the + and the components
of the generators. Now argue that the 4-vector x
is in the 2L  2R
representation of this group. Hint: compute how L
(1)
k acts on 2L2R , and note that
this representation indeed contains a 3-vector and a scalar (the 4th component).
e) Now take the skew symmetric (=antisymmetric) product of two of these: A =
-A , as in exercise 8. Explain that this representation decomposes into two
SU(2)L  SU(2)R representations: (2L  2R)  (2L  2R)antisymm = 3L  3R .
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f) Similarly, show that the symmetric part of this product, A = A , obeys
(2L  2R)  (2L  2R)symm = 3L  3R  1. Indeed, the 1 is the trace, and the 9
represents the rest.
16. Lecture June 11.
Derive Equations (7.18)--(7.20) from Eq. (7.17) in the notes.
17. Lecture June 11.
Consider an experiment as in Chapter 7, but now the scattering of a particle with spin
1 against a spherically symmetric target. We write the representation in the basis where
Lz is diagonal, so
+1
= f(r)



1
0
0


 , 0
= f(r)



0
1
0


 , -1
= f(r)



0
0
1


 . (17.1)
Now define the scattering function F() as the 3 × 3 matrix



f++() f+0() f+-()
f0+() f00() f0-()
f-+() f-0() f--()


 . (17.2)
a. Construct the matrix that generates infinitesimal rotations around the x axis in
this basis: Lx = 1
2
(L+ + L-) .
b. We find the rotation over a finite angle , or exp(iLx), by applying (3.10):
exp(iLx) = I + (cos  - 1)L2
x + i sin  Lx . (17.3)
Show that this follows from L2n+1
x = Lx and compute exp(iLx).
c. Find out how F() transforms under this rotation.
d. What is the most general angle dependence to be expected for f++, f00 en f-- ?
e. Now compute the angular distribution for scattering experiments with particles
having Sz = 1, 0, en -1. Arguments using parity symmetry may be used to
find further restrictions for the parameters in this result, which otherwise would be
allowed to be anything.
f. So, how do we distinguish particles with spin 1
2
from particles with spin 1?
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18. Lecture June 11.
Just like the ordinary nucleons N = (p, n), the N
has isospin 1
2
. Its mass is sufficient
to decay, just like the  particles, into a nucleon and a pion. The resulting states |N, 
now must have total isospin 1
2
. Of all states occurring in formula (8.6), we now search
for those combinations that have Itot
= 1
2
and I3 = +1
2
.
a. Find the two basis elements with I3 = +1
2
. One combination must have Itot
= 3
2
and one combination, orthogonal to that, has Itot
= 1
2
. Why are these combinations
orthogonal to one another?
b. Show that the state Itot
= 1
2
must obey I+| = 0. Now find this state.
c. The state Itot
= 1
2
, I3 = -1
2
is now obtained from the result of (b) by letting Iact
on it. Compute this state.
d. An alternative method is to solve the equation I-| = 0. Show that this produces
the same result, apart from a possible phase factor (or minus sign).
e. Now compute the decay probabilities for
N+
 p + 0
, N+
 n + +
,
N 0
 p + ,
N 0
 n + 0
. (18.1)
Observe the difference with the  decay, Eq. (8.14).
19. Lecture June 18
It is not difficult to show that
[Li, rj] = ihijkrk
and [Li, pj] = ihijkpk . (19.1)
Now show that, if two arbitrary vectors A and B both obey the commutation rules
with Li as in (19.1), and if an other operator R commutes with all Li , then the same
commutation relations are obeyed by vectors C and D defined according to C = R A,
and D = A × B, so
[Li, Cj] = ihijkCk
and [Li, Dj] = ihijkDk . (19.2)
This is the so-called Wigner-Eckhart theorem. With this, we prove both (9.18) and (9.21)
in one stroke.
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20. Lecture June 18
a. Check the calculations that lead to the equations [K, H] = 0 (9.24) and K  L = 0
(9.29) for the quantum mechanical Runge-Lenz vector (9.16).
b. Show that
(L × p)  (p × L) = -p2
L2
. (20.1)
Make sure that no extra terms arise by correctly manipulating the commutation
rules
c. We now calculate K2
by writing K as
K =
1
e2
(L × p - hip) +
r
r
=
1
e2
(-p × L + hip) +
r
r
, (20.2)
after which we multiply one expression with the other, making use of b. Now prove
(9.30).
21. Lecture June 18
Consider the quantum states of the hydrogen atom at given n.
a. Show that both L+
3 and L-
3 take the values -1
2
(n-1), -1
2
(n-1)+1,    , 1
2
(n-1).
b. What are now the possible values of L3 = L+
3 + L-
3 , and what is the degree of
degeneracy at these values?
c. Use this to derive that takes the integral values 0, 1,    n - 1 all exactly once,
according to equation (9.45).
(This was explained in the lecture; make sure that the argument is well understood)
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