PROBLEM SETS, 2012-3
1. Problem Set 1
Problem 1:
Consider the group S3 of permutations of 3 objects. This group acts on the set
of 3 elements. Consider the representation (π, C3 ) this gives on the vector space C3
of complex valued functions on the set of 3 elements (as defined in class). Choose
a basis of this set of functions, and find the matrices π(g) for each element g ∈ S 3 .
Is this representation irreducible? If not, can you give its decomposition into irreducibles, and find a basis in which the representation matrices are block diagonal?
Problem 2:
Use a similar argument to the one given in class for G = U (1) to classify the
irreducible representations of the group R under the group law of addition. Which
of these are unitary?
Problem 3:
Consider the group SO(2) of 2 by 2 real orthogonal matrices of determinant one.
What are the complex representations of this group?
There is an obvious representation of SO(2) on R2 given by matrix multiplication
on real 2-vectors. If I replace the real 2-vectors by complex 2-vectors I get a 2complex dimensional representation. How does this decompose as a direct sum of
irreducibles?
Problem 4:
Consider a quantum mechanical system with state space H = C3 and Hamiltonian operator


0 1 0
H = 1 0 0
0 0 2
Solve the Schrödinger equation for this system to find its state vector |Ψ(t)i at
any time t > 0, given that the state vector at t = 0 was
 
ψ1
ψ2 
ψ3
for ψi ∈ C.
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PROBLEM SETS, 2012-3
2. Problem Set 2
Problem 1: Calculate the exponential etM for


0 π 0
−π 0 0
0 0 0
by two different methods:
• Diagonalize the matrix M (i.e. write as P DP −1 , for D diagonal), then
show that
−1
etP DP = P etD P −1
and use this to compute etM .
• Calculate etM using the Taylor series expansion for the exponential, as well
as the series expansions for the sine and cosine.
Problem 2: Consider a two-state quantum system, with Hamiltonian
H = −Bx σ1
(this is the sort of thing that occurs for a spin-1/2 system subjected to a magnetic
field in the x-direction).
• Find the eigenvectors and eigenvalues of H. What are the possible energies
that can occur in this quantum system?
• If the system starts out at time t = 0 in the state
1
|ψ(0)i =
0
(i.e. spin “up”) find the state at later times.
Problem 3: By using the fact that any unitary matrix can be diagonalized by
conjugation by a unitary matrix, show that all unitary matrices can be written as
eX , for X a skew-adjoint matrix in u(n).
By contrast, show that
−1 1
A=
0 −1
is in the group SL(2, C), but is not of the form eX for any X ∈ sl(2, C) (this Lie
algebra is all 2 by 2 matrices with trace zero.
Hint: For 2 by 2 matrices X, one can show (this is the Cayley-Hamilton theorem:
matrices X satisfy their own characteristic equation det(λ1 − X) = 0, and for 2 by
2 matrices, this equation is λ2 − tr(X)λ + det(X) = 0)
X 2 − tr(X)X + det(X)1 = 0
For X ∈ sl(2, C), tr(X) = 0, so here X 2 = −det(X)1. Use this to show that
p
p
sin( det(X))
eX = cos( det(X))1 + p
X
det(X)
Try to use this for eXp= A and derive a contradiction (taking the trace of the
equation, what is cos( det(X))?)
Problem 4: Show that the Lie algebra u(n) is not a complex vector space.
PROBLEM SETS, 2012-3
3
3. Problem Set 3
Problem 1: On the Lie algebras g = su(2) and g = so(3) one can define the
Killing form K(·, ·) by
(X, Y ) ∈ g × g → K(X, Y ) = tr(XY )
(1) For both Lie algebras, show that this gives a bilinear, symmetric form,
negative definite, with the basis vectors sa in one case and La in the other
providing an orthogonal basis if one uses −K(·, ·) as an inner product.
(2) Another possible way to define the Killing form is as
K 0 (X, Y ) = tr(ad(X) ◦ ad(Y ))
Here the Lie algebra adjoint representation (ad, g) gives for each X ∈ g a
linear map
ad(X) : R3 → R3
and thus a 3 by 3 real matrix. This K 0 is determined by taking the trace
of the product of two such matrices. How are K and K 0 related?
Problem 2: Under the homomorphism Φ given in class, what elements of SO(3)
do the quaternions i, j, k (unit length, so elements of Sp(1)) correspond to?
Problem 3: In special relativity, we consider space and time together as R4 ,
with an inner product such that < v, v >= v02 −v12 −v22 −v32 , where v = (v0 , v1 , v2 , v3 , v4 ) ∈
R4 . The group of linear transformations of determinant one preserving this inner
product is written SO(1, 3) and known as the Lorentz group. Show that, just as
SO(4) has a double-cover Spin(4) = Sp(1) × Sp(1), the Lorentz group has a double cover SL(2, C), with action on vectors given by identifying R4 with 2 by 2
Hermitian matrices according to
v0 + v3 v1 − iv2
(v0 , v1 , v2 , v3 , v4 ) ↔
v1 + iv2 v0 − v3
and using the conjugation action of SL(2, C) on these matrices. (Hint: use determinants).
Note that the Lorentz group has a spinor representation, but it is not unitary.
Problem 4: Fix the mistaken formula in my notes for
α β
Φ
−β α
deriving a correct one. The notes refer to a place this is done with another convention. To do this calculation, note that your result will be a 3 by 3 real matrix,
with the first column given by the result of applying this linear transformation to
 
1
0
0
then doing the same for the next two columns. Each of these three calculations is
done by conjugating the corresponding basis element in su(2) and expanding the
result in terms of these basis elements.
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PROBLEM SETS, 2012-3
4. Problem Set 4
Problem 1: Using the definition
1
< f, g >= 2
π
Z
2
f (z1 , z2 )g(z1 , z2 )e−(|z1 |
+|z2 |2 )
dx1 dy1 dx2 dy2
C2
for an inner product on polynomials on homogeneous polynomials on C2
• Show that the representation π on such polynomials given in class (induced
from the SU (2) representation on C2 ) is a unitary representation with
respect to this inner product.
• Show that the
zj zk
√1 2
j!k!
are orthonormal with respect to this inner product (break up the integrals
into integrals over the two complex planes, use polar coordinates).
• Show that the differential operator π(S3 ) is self-adjoint, the π(S− ) and
π(S+ ) are adjoint of each other.
Problem 2: Using the formulas for the Y1m (θ, φ) and the inner product given
in the notes, show that
• The Y11 , Y10 , Y1−1 are orthonormal.
• Y11 is a highest weight vector.
• Y10 and Y1−1 can be found by repeatedly applying ρ(L− ) to a highest weight
vector.
Problem 3: Recall that the Casimir operator L2 of so(3) is the operator that
in any representation ρ is given by
L2 = ρ0 (L1 )ρ0 (L1 ) + ρ0 (L2 )ρ0 (L2 ) + ρ0 (L3 )ρ0 (L3 )
Show that this operator commutes with the ρ0 (X) for all X ∈ so(3). Use this to
show that L2 has the same eigenvalue on all vectors in an irreducible representation
of so(3).
Problem 4: For the case of the SU (2) representation π on polynomials on C2
given in the notes, find the Casimir operator
L2 = π 0 (S1 )π 0 (S1 ) + π 0 (S2 )π 0 (S2 ) + π 0 (S3 )π 0 (S3 )
as a explicit differential operator. Show that homogeneous polynomials are eigenfunctions, and calculate the eigenvalue.
5. Problem Set 5
Problem 1: Consider the action of SU (2) on the tensor product V 1 ⊗ V 1 of
two spin one-half representations. According to the Clebsch-Gordan decomposition,
this breaks up into irreducibles as V 0 ⊕ V 2 .
PROBLEM SETS, 2012-3
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(1) Show that
1
1
0
0
1
√ (
⊗
−
⊗
)
0
1
1
0
2
is a basis of the V 0 component of the tensor product, by computing first the
action of SU (2) on this vector, and then the action of su(2) on the vector
(i.e. compute the action of π 0 (X) on this vector, for π the tensor produ ct
representation, and X basis elements of su(2)).
(2) Show that
1
1
0
0
1
0
0
1
1
⊗
+
⊗
),
⊗
⊗
,√ (
0
1
1
0
1
1
0
0
2
give a basis for the irreducible representation V 2 , by showing that they are
eigenvectors of π 0 (s3 ) with the right eigenvalues (weights), and computing
the action of the raising and lowering operators for su(2) on these vectors.
Problem 2: In class we described the quantum system of a free non-relativistic
particle of mass m in R3 . Using tensor products, how would you describe a system
of two identical such particles? Find the Hamiltonian and momentum operators.
Find a basis for the energy and momentum eigenstates for such a system, first under
the assumption that the particles are bosons, then under the assumption that the
particles are fermions.
Problem 3: Consider a quantum system describing a free particle in one spatial
dimension, of size R (the wavefunction satisfies ψ(x, t) = ψ(x + R, t)). If the wavefunction at time t = 0 is given by
4π
6π
ψ(x, 0) = C(sin( x) + cos( x + φ))
R
R
where C is a constant and φ is an angle, find the wave-function for all t. For what
values of C is this a normalized wave-function (|ψ(x, t)|2 = 1)?
6. Problem Set 6
Problem 1: Fill in the details of the proof of the Groenewold-van Hove theorem
following the outline given in Chapter 5.4 of Rolf Berndt’s An Introduction to
Symplectic Geometry.
Problem 2: If a quantum harmonic oscillator is in a state
1
√ (|0i + |1i)
2
at time t = 0, find its position-space wavefunction ψ(q, t) for all t.
Problem 3: For the one-dimensional quantum harmonic oscillator, compute
the expectation values in the energy eigen-state |ni of the following operators
Q, P, Q2 , P 2
and
Q4
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PROBLEM SETS, 2012-3
Use these to find the standard deviations in the statistical distributions of observed
values of q and p in these states. These are
p
p
∆Q = hn|Q2 |ni − hn|Q|ni2 , ∆P = hn|P 2 |ni − hn|P |ni2
For two energy eigenstates |ni and |n0 i, find
hn0 |Q|ni and hn0 |P |ni
7. Problem Set 7
Problem 1: Show that the Lie algebras of Spin(n) and SO(n) are the same by
showing that the quadratic elements
1
γj γk
2
for j < k of the Clifford algebra Cliff(n, R) satisfy the same commutation relations
as the Ljk (elementary antisymmetric matrices).
Problem 2: Show that conjugation by an exponential of the quadratic Clifford
algebra element of the previous problem gives a rotation in the j − k plane.
Problem 3: Using the construction of spinors given in class, consider the cases
of the Clifford algebra in 4 or 6 dimensions, corresponding to the fermionic oscillator
in 2 or 3 variables.
• The Hamiltonian operator generates a U (1) action on the spinors. What
is it explicitly? This U (1) is a subgroup of the Spin group (in 4 or 6
dimensions respectively), and so acts not just on spinors, but on vectors as
a rotation. What is the rotation on vectors?
• For a rotation by an angle θ in the j − k plane in 4 or 6 dimensions, what
are the elements of the Spin group that correspond to this, and how do
they act on the fermionic oscillator states? Do this by expressing things in
terms of annihilation and creation operators and their action on the spinors,
thought of as a fermionic oscillator state space.
Problem 4: Use the anti-commuting variable analog of the Bargmann-Fock
construction construct spinors in even dimensions as spaces of functions of anticommuting variables. Find the inner product on such spinors that is the analog of
the one constructed using an integral in the bosonic case. Show that the operators
aF j and aF †j are adjoints with respect to this inner product.
PROBLEM SETS, 2012-3
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8. Problem Set 8
Problem 1: Compute the propagator
G(x0 , t0 , x, t) = h0|ψ̂(x0 , t0 )ψ̂ † (x, t)|0i
for the free non-relativistic particle of mass m (for t0 > t). First do this in momentum space, showing that
2
k
(t0 −t)
e 0 , t0 , k, t) = e−i 2m
G(k
δ(k 0 − k)
then Fourier transform to find the position space result
m
3
m
(x0 −x)2
G(x0 , t0 , x, t) = (
) 2 e i2π(t0 −t)
0
i2π(t − t)
Show also that
lim0 G(x0 , t0 , x, t) = δ(x0 − x)
t→t
Problem 2: For non-relativistic quantum field theory of a free particle, including interaction with a potential, show that there is a particle number operator N̂
that is conserved (commutes with the Hamiltonian) and generates a U (1) symmetry.
Problem 3: For non-relativistic quantum field theory of a free particle in three
dimensions, find the “classical” angular momentum functions Lj on the phase space
of complex valued functions on R3 . Show that these functions Poisson-commute
with the Hamiltonian function. Find the corresponding quantized operators Lˆj ,
show that these commute with the Hamiltonian operator, and satisfy the Lie algebra
commutation relation
[L̂1 , L̂2 ] = L̂3
Problem 4: Show that the Lie algebra so(4, C) is sl(2, C) × sl(2, C). Within
this Lie algebra, identify the sub-Lie algebras of the groups Spin(4), Spin(1, 3) and
Spin(2, 2).
9. Problem Set 9
Problem 1: Compute the following commutators of elements of the Lie algebra
of the Poincaré group
[Kl , Pm ]
where the Kl (l = 1, 2, 3) generate boosts in the Lorentz subgroup, and Pm (m =
0, 1, 2, 3) generate translations.
Problem 2:
• For the real scalar field theory, find the momentum operator acting on the
quantum field theory state space, in terms of the scalar quantum field. Show
that this gives the expected expression in terms of a sum of the number
operators for each momentum mode and the corresponding momentum.
• Repeat the same calculation for the complex scalar field theory.
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PROBLEM SETS, 2012-3
Problem 3: In class we showed that taking two real free scalar fields, one
could make a theory with SO(2) symmetry, and we found the charge operator Q
that gives the action of the Lie algebra of SO(2) on the state space of this theory.
Instead, consider two complex free scalar fields, and show that this theory has a
U (2) symmetry. Find the four operators that give the Lie algebra action for this
symmetry on the state space, in terms of a basis for the Lie algebra of U (2).
Note that this is the field content and symmetry of the Higgs sector of the
standard model (where the difference is that the theory is not free, but interacting,
and has a lowest energy state not invariant under the symmetry).