1 Functional Integrals for Fermions

This approach can be extended to Fermions in two formally similar but conceptually inequivalent ways :–

In the first we define coherent states of Fermions by allowing to take linear combination of states with coefficients valued in a Grassmann algebra, i.e., the algebra generated by 2N anticommuting objects {ξ_{i}ξ_{i}^{∗}; i = 1, ⋯N} . Any member of this algebra is called a Grassmann number. ∫ dξ_{i} ≡ \frac{∂}{∂ξ_{i}} anticommutes with ξ_{j}, j ≠ i

We define a fermion coherent state as

|ξ⟩ ≡ exp (− ∑_{α} ξ_{α}a_{α}^{†} |0⟩) = ∏_{α} (1 − ξ_{α}a_{α}^{†}) |0⟩

We require that {ξ, a} = 0 and (ξa)^{†} = a^{†}ξ^{∗}

Then it is clear that

a_{α} |ξ⟩ = ξ_{α} |ξ⟩
⟨ξ| a_{α}^{†} = ⟨ξ| ξ_{α}^{∗}

To understand the properties of the Fermionic coherent states consider first a single Fermion degree of freedom {a, a^{†}}. Define

|ξ⟩ ≡ e^{−ξa^{†}} |0⟩

{ξ, a} = 0 and (ξa)^{†} = a^{†}ξ^{∗}. The overcompleteness relations are easily worked out

∫ dξ^{∗}dξ |ξ⟩ ⟨ξ| = \frac{∂}{∂ξ^{∗}} \frac{∂}{∂ξ} (|0⟩ − ξa^{†} |0⟩)(⟨0| − ⟨0| aξ^{∗})(1 − ξ^{∗}ξ)
= |0⟩ ⟨0| + a^{†} |0⟩ ⟨0| a = I

Notice that ξ anticommutes with vectors in the Hilbert space containing an odd number of Fermions in which case

ξ |ψ⟩ = − |ψ⟩ ξ
⟨η|ξ⟩ = (⟨0| − ⟨0| aη^{∗})(|0⟩ − ξa^{†} |0⟩)
= 1 + η^{∗}ξ = e^{η^{∗}ξ}

Finally, notice that if |ψ_{i}⟩ are vectors in the Fock space with a definite number of Fermions then ⟨ξ|ψ_{i}⟩ is a grassmann variable which contains an odd number of η_{α} if |ψ_{i}⟩ has an odd number of Fermions and contains an even number of η_{α} if |ψ_{i}⟩ has an even number of Fermions. Hence:

⟨ξ|ψ_{i}⟩ ⟨ψ_{j}|ξ⟩ = ⟨ψ_{j}|ξ⟩ ⟨ξ|ψ_{i}⟩ (−1)^{F}
= ⟨ψ_{j}|ξ⟩ ⟨ψ_{i}| − ξ⟩ = ⟨ψ_{j}| − ξ⟩ ⟨ψ_{i}|ξ⟩

This gives an expression for the trace of an operator A which acts on the Fock space and preserves the number of particles.

TrA = ∑_{n} ⟨n|A|n⟩ = ∑_{n} ∫ ∏_{α} dξ_{α}^{∗}dξ_{α}e^{− ∑_{α} ξ_{α}^{∗}ξ_{α}} ⟨n|ξ⟩ ⟨ξ|A|n⟩

= ∫ ∏_{α} dξ_{α}^{∗}dξ_{α}e^{− ∑_{α} ξ_{α}^{∗}ξ_{α}} ∑_{n} ⟨−ξ|A|n⟩ ⟨n|ξ⟩

= ∫ ∏_{α} dξ_{α}^{∗}dξ_{α}e^{− ∑_{α} ξ_{α}^{∗}ξ_{α}} ⟨−ξ|A|ξ⟩

Equipped with these techniques we can write functional integral expression for Fermi systems in complete analogy with the Bosonic theory.

U(τ, τ′) = ∏_{i=0}^{N−1} U(τ_{i+1}, τ_{i})

τ_{0} = τ′, τ_{N} = τ . We express

U(τ_{i+1,}τ_{i}) = exp −H (\frac{τ_{i+1} + τ_{i}}{2} ) Δτ

and insert at each point a resolution of the identity to find

⟨η_{N} |U(τ, τ′)| η_{0}⟩ = ∏_{i=1}^{N−1} ∫ dη_{i}^{∗}dη_{i}e^{−η_{i}^{∗}η_{i}} ⟨η_{i+1} |U(τ_{i+1,}τ_{i})| η_{i}⟩ ⟨η_{1} |U(τ_{1,}τ_{0})| η_{0}⟩

Using the expression for the trace of an operator derived earlier one finds

TrU(τ, τ′) = ∫ dη_{0}^{∗}dη_{0}e^{−η_{0}^{∗}η_{0}} ⟨η_{0} |U(τ_{,}τ′)| η_{0}⟩

If U ≃ e^{−H(\frac{τ_{i+1}+τ_{i}}{2} )Δτ} and H_{c}(η^{∗}, η) with η^{∗} and η and replaced by a^{†} and a give the normal ordered form of H we find

TrU(τ, τ′) = ∏_{i=0}^{N−1} ∫ dη_{i}^{∗}dη_{i}e^{− ∑_{i=0}^{N−1} η_{i}^{∗}η_{i}} ⟨η_{i+1}|η_{i}⟩ e^{−H_{c}(η_{i+1}^{∗},η_{i})Δτ}

= ∏_{i=0}^{N−1} ∫ dη_{i}^{∗}dη_{i}e^{− ∑_{i=0}^{N−1} η_{i}^{∗}η_{i}}e^{∑_{i=0}^{N−1} η_{i}^{∗}η_{i}}e^{−H_{c} ∑_{i=0}^{N−1}(η_{i+1}^{∗},η_{i})Δτ}

(η_{N}^{∗} ≡ −η_{0}^{∗}). Notice that if H_{c} contained some explicit time dependence due to the source or an auxillary field, it should be evaluated at time τ = \frac{τ_{i+1}+τ_{i}}{2} . Proceeding heuristically we could define

Tre^{−βH} = ∫ Dη^{∗}Dηe^{−(η^{∗} \frac{∂η}{∂τ} +H_{c}(η^{∗},η))}

where the operator \frac{∂}{∂τ} is defined on the space of functions obeying η(β) ≡ −η(0).

2 The Standard Gaussian Integral:

∫ ∏_{i} dη_{i}^{∗}dη_{i}e^{−η^{∗}Aη+η^{∗}j+j^{∗}η} = ∫ ∏_{i} dη_{i}^{∗}dη_{i}e^{−η^{∗}Aη}e^{j^{∗}A^{−1}j}

using the transformation

η → η + A^{−1}j
η^{∗} → η^{∗} + j^{∗}A^{−1}

Thus

∫ ∏_{i} dη_{i}^{∗}dη_{i}e^{−η^{∗}Aη+η^{∗}j+j^{∗}η} ≃ (det A)e^{j^{∗}A^{−1}j}

expanding to second order.

η_{i}^{∗}j_{i}j_{k}^{∗}η_{k} = j_{k}^{∗}A_{kl}^{−1}j_{l}
⟨η_{k}|η_{l}^{∗}⟩ = A_{kl}^{−1}