•
Γ =
∏d
j=1
(1− 2 a†FjaFj)
•
Γ =cγ 1 γ 2 ···γ 2 d
for some constantc. Computec.
- γjΓ + Γγj= 0
for allj.
Γ^2 = 1
P±=
1
2
( 1 ±Γ)
are projection operators onto subspacesH+andH−ofHF.
- Show thatH+andH−are each separately representations ofspin(n) (i.e.,
the representation operators commute withP±).
Problem 4:
Using the fermionic analog of Bargmann-Fock to construct spinors, and the
inner product 31.3, show that the operatorsaFj andaF†j are adjoints with
respect to this inner product.
B.15 Chapters 33 and 34
Problem 1:
Consider a two dimensional version of the Pauli equation that includes a
coupling to an electromagnetic field, with Hamiltonian
H=
1
2 m
((P 1 −eA 1 )^2 + (P 2 −eA 2 )^2 )−
e
2 m
Bσ 3
whereA 1 andA 2 are functions ofq 1 ,q 2 and
B=
∂A 2
∂q 1
−
∂A 1
∂q 2
Show that this is a supersymmetric quantum mechanics system, by finding
operatorsQ 1 ,Q 2 that satisfy the relations 33.1.
Problem 2:
For the three choices of inner product given in section 34.3, show that the
inner product is invariant under the action of the groupE ̃(3) on the space of
solutions.