Quantum Mechanics for Mathematicians

Problem 2:
Prove that, as algebras overC,

• Cliff(2d,C) is isomorphic toM(2d,C)


• Cliff(2d+ 1,C) is isomorphic toM(2d,C)⊕M(2d,C)

B.14 Chapters 29 to 31

Problem 1:
Show that for vectorsv∈Rnandjkthe basis element ofso(n) correspond-
ing to an infinitesimal rotation in thejkplane, one has

e−

θ 2 γjγk

γ(v)e

θ 2 γjγk

=γ(eθjkv)

and [

−

1

2

γjγk,γ(v)

]

=γ(jkv)

Problem 2:
Prove the following change of variables formula for the fermionic integral
∫
F(ξ)dξ 1 dξ 2 ···dξn=

1

detA

∫

F(Aξ′)dξ′ 1 dξ′ 2 ···dξ′n

whereξ=Aξ′, i.e.,

ξj=

∑n

k=1

Ajkξk′

for any invertible matrixAwith entriesAjk.
For a skew-symmetric matrixA, andn= 2deven, show that one can evaluate
the fermionic version of the Gaussian integral as
∫
e

12 ∑nj,k=1Ajkξjξk

dξ 1 dξ 2 ···dξn=Pf(A)

where

Pf(A) =

1

d!2d

∑

σ

(−1)|σ|Aσ(1)σ(2)Aσ(3)σ(4)···Aσ(n−1)σ(n)

Here the sum is over all permutationsσof thenindices. Pf(A) is called the
Pfaffian of the matrixA.

Problem 3:
For the fermionic oscillator construction of the spinor representation in di-
mensionn= 2d, with number operatorNF=

∑d

j=1a

†

FjaFj, define

Γ =eiπNF

Show that