Download Problem Set 11

Introduction to Quantum Mechanics, Spring 2015
Problem Set 11
Due Tuesday, March 3
Problem 1: Show that for vectors v ∈ Rn and jk the basis element of so(n)
corresponding to an infinitesimal rotation in the j − k plane, one has
θ
θ
e− 2 γj γk γ(v)e 2 γj γk = γ(eθjk v)
and
1
[− γj γk , γ(v)] = γ(jk v)
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Problem 2: Prove the following change of variables formula for the fermionic
integral
Z
1
F (ξ)dξ1 dξ2 · · · dξn =
det A
Z
F (Aξ 0 )dξ10 dξ20 · · · dξn0
where ξ = Aξ 0 , i.e.
ξj =
n
X
Ajk ξk0
k=1
for any invertible matrix A with entries Ajk .
For a skew-symmetric matrix A, and n = 2d even, show that one can evaluate
the fermionic version of the Gaussian integral as
Z
Pn
1
e 2 j,k=1 Ajk ξj ξk dξ1 dξ2 · · · dξn = P f (A)
where
P f (A) =
1 X
(−1)|σ| Aσ(1)σ(2) Aσ(3)σ(4) · · · Aσ(n−1)σ(n)
d!2d σ
Here the sum is over all permutations σ of the n indices. P f (A) is called the
Pfaffian of the matrix A.
Problem 3: For the fermionic oscillator construction of the spinor representaPd
tion in dimension n = 2d, with number operator NF = j=1 a†F j aF j , define
Γ = eiπNF
Show that
•
Γ=
d
Y
(1 − 2a†F j aF j )
j=1
1
•
Γ = cγ1 γ2 · · · γ2d
for some constant c. Compute c.
•
γj Γ + Γγj = 0
for all j.
•
Γ2 = 1
•
1
(1 ± Γ)
2
are projection operators onto subspaces H+ and H− of HF .
P± =
• Show that H+ and H− are each separately representations of spin(n) (i.e.
the representation operators commute with P± ).
Problem 4: Use the fermionic analog of Bargmann-Fock to 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.
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