Quantum Mechanics for Mathematicians

In the bosonic case (see equation 26.7) extending the Poisson bracket from
MtoM⊗Cby complex linearity gave an indefinite Hermitian form onM⊗C

〈·,·〉=i{·,·}=iΩ(·,·)

positive definite onM+J for positiveJ. In the fermionic case we can extend
the fermionic Poisson bracket fromVtoV ⊗Cby complex linearity, getting a
Hermitian form onV ⊗C

〈·,·〉={·,·}+= (·,·)

This is positive definite onV+J (and also onVJ−) if the initial symmetric bilinear
form was positive.
To quantize this system we need to find operators Γ+(θj) and Γ+(θj) that
satisfy
[Γ+(θj),Γ+(θk)]+= [Γ+(θj),Γ+(θk)]+= 0
[Γ+(θj),Γ+(θk)]+=δjk 1

but these are just the CAR satisfied by fermionic annihilation and creation
operators. We can choose

Γ+(θj) =aF†j, Γ+(θj) =aFj

and realize these operators as

aFj=

∂

∂θj

, aF†j= multiplication byθj

on the state space Λ∗Cdof polynomials in the anticommuting variablesθj. This
is a complex vector space of dimension 2d, isomorphic with the state spaceHF
of the fermionic oscillator inddegrees of freedom, with the isomorphism given
by

1 ↔ | 0 〉F

θj↔aF†j| 0 〉F

θjθk↔aF†jaF†k| 0 〉

···

θ 1 ...θd↔aF† 1 aF† 2 ···aF†d| 0 〉F

where the indicesj,k,...take values 1, 2 ,...,dand satisfyj < k <···.
If one defines a Hermitian inner product〈·,·〉onHFby taking these basis
elements to be orthonormal, the operatorsaFjanda†Fjwill be adjoints with
respect to this inner product. This same inner product can also be defined
using fermionic integration by analogy with the Bargmann-Fock definition in
the bosonic case as

〈f 1 (θ 1 ,···,θd),f 2 (θ 1 ,···,θd)〉=

∫

e−

∑d

j=1θjθjf 1 f 2 dθdθ 1 ···dθddθd (31.3)