Physics 618
Homework #8
Due: April 2, 2012
1
Construct P and Q for the two standard tableaux of the Young Graph
. Construct Qi sij Pj , where s12 = s21 = (23), s11 = s22 = 1 Show that
I.
this generates a four-dimensional subspace of the group algebra whic
Chapter 2
Representations
In the last chapter we learned something about the structure of the groups
themselves, but we are really interested in how the symmetry groups act on
the eigenstates of a physical Hamiltonian. The set of states corresponding to
a
Chapter 3
Innite Groups
Most physical situations that have a symmetry group have an innite group.
Some examples:
Rotational invariance, SO (3). Here we can rotate through an arbitrary
angle specied by a continuous parameter , restricted to some nite
rang
Chapter 4
SU(2)
We will now work out in detail the properties of SU(2) and its representations.
We have already seen that the generators may be chosen to be
1
Li = i ,
2
Then cij k =
ijk
with i = the Pauli matrices.
are the structure constants, and
ij =
Chapter 5
Semisimple Compact Lie
Groups
We return now to considering a general nite dimensional semisimple compact Lie group and its Lie algebra.
In the algebra there are many abelian subalgebras, though not invariant.
For example, any one dimensional sub
Chapter 6
SU(3)
SU(3) rst hit the Physics world in 1961 through papers by Gell-Mann and
Neeman which applied it to what we now call the avor of hadrons, at a time
when particles involving charm, top, or bottom were unknown. In modern
language, these hadro
Chapter 7
Dynkin Diagrams
We now describe how to draw the Dynkin diagram for each semisimple compact nite-dimensional Lie group. We will see that many conceivable diagrams are forbidden, and therefore that the set of all such groups is quite
limited.
For
Chapter 8
Representations of Lie Groups
Let us label the simple roots i , i = 1, . . . , m. They and their conjugates generate (not linearly) any element of the Lie algebra. Thus any representation
can be determined by how it behaves under these roots1 .
Chapter 9
Sk and Tensor Representations
(Ref: Schensted Part II)
If we have an arbitrary tensor with k indices W i1 ,ik we can act on it
1 2 k
so
with a permutation P =
a b
(P w )i1 ,i2 ,ik = w ia ,ib ,i .
Consider the algebra A formed by taking arbitrar
Chapter 10
Tensor Products of Irreducible
Representations
Consider two representations with Young Graphs 1 and 2 , corresponding
to tensors of rank k1 and k2 . The tensor product is a tensor of rank k1 + k2 ,
and must be decomposed into irreducible repres
Physics 618
Homework #1
Due: Jan. 30, 2012
1 Find the matrix group generated by the matrices ix and iy , where
the s are the ordinary Pauli matrices
x =
01
,
10
y =
0 i
,
i0
z =
10
.
0 1
Show that the group has order 8 and has ve conjugacy classes, but th
Physics 618
Homework #2
Due: Feb. 6, 2012
1 If is a representation of a group G, show that the set of matrices
(A), which are the complex conjugates of (A), form a representation
of G. Note this is the complex congugate, not the hermitean conjugate,
( (A
Physics 618
Homework #3
Due: Feb. 13, 2012
1 Construct the character table and the irreducible representations of S3 ,
the permutation group on three objects.
2
Let i and j be two inequivalent irreducible representations of a
group G. Show that i j does n
Physics 618
Homework #4
Due: Feb. 20, 2012
1 Consider the 2 2 matrices with integer matrix elements and determinant 1. This set forms a group under ordinary matrix multiplication, called
the modular group or SL(2, Z).
a) Show it is a group.
11
01
b) Show
Physics 618
Homework #5
Due: Feb. 27, 2012
Reminder: There will be a midterm exam on Thursday,
March 8
You are allowed to use your notes, including my notes, and at most two
books.
1 [10 pts ] SU (1, 1) is the set of 2 2 complex matrices of determinant
1
Physics 618
Homework #6
Due: March 5, 2012
Reminder: There will be a midterm exam on Thursday,
March 8
You are allowed to use your notes, including my notes, and at most two
books.
1 [10 pts ] Consider a semisimple Lie algebra L of rank 2, with simple
roo
Physics 618
Homework #7
Due: March 26, 2012
1 [Note: this is Georgis problem IX.A] If | is the state of the highest
weight ( = 1 + 2 ) of the adjoint representation of SU(3), show that the
states
and
|A
|B
= E1 E2 |
= E2 E1 |
are linearly independent.
Hin
Physics 618
Homework #9
Due: April 9, 2012
1
Prove that, for any operator A(x),
A(x)
e
=
x
1
0
deA(x)
A(x) (1)A(x)
e
.
x
[Hints: Expand all exponentials as power series. Use the fact that the beta
function
B (z, w ) =
1
0
dz 1(1 )w1 =
(z )(w )
.
(z + w )
Chapter 1
Groups
1.1
Introduction
One of the fundamental problems in quantum physics is to solve the equation
H = Ei ,
(1.1)
where H is the Hamiltonian operator of the system, and is a wave function of the dynamical coordinates. Ei is an unknown eigenvalu