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