Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 307 / 2 / b

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- Solution b

Solution

The Hilbert space is the fermionic Fock space

H ≅ Λ^{∙} C^{n} .

In a polarization where

η^{a}

acts by exterior multiplication and

\overset{η}{ˉ}_{a}

by contraction, a general state is

Ψ (η) = \sum_{p = 0}^{n} \frac{1}{p !} Ψ_{a_{1} \dots a_{p}} η^{a_{1}} \dots η^{a_{p}} .

The fermion number operator

N = η^{a} \overset{η}{ˉ}_{a}

returns the exterior degree, and therefore

(- 1)^{N} Ψ (η) = Ψ (- η) = p \sum (- 1)^{p} Ψ_{(p)} (η) .

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