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

The Hilbert space is the fermionic Fock spaceIn a polarization where ${\hat{\eta }}^a$ acts by exterior multiplication and ${\hat{\bar{\eta }}}_a$ by contraction, a general state isThe fermion number operator $N={\hat{\eta }}^a{\hat{\bar{\eta }}}_a$ returns the exterior degree, and therefore