The Fermion Determinant, its Modulus and Phase

TL;DR

This paper derives a closed-form expression for the fermion determinant's dependence on boundary conditions, challenging the conventional view that this dependence diminishes over long times, with implications for supersymmetric quantum mechanics.

Contribution

It provides a novel analytical expression for the fermion determinant's boundary condition dependence using self-adjoint extensions, highlighting its significance in quantum models.

Findings

Fermion determinant depends crucially on boundary conditions

Dependence persists over long time evolution, contrary to previous assumptions

Application to supersymmetric quantum mechanics illustrates the impact

Abstract

We consider path integration of a fermionic oscillator with a one-parameter family of boundary conditions with respect to the time coordinate. The dependence of the fermion determinant on these boundary conditions is derived in a closed form with the help of the self-adjoint extension of differential operators. The result reveals its crucial dependence on them, contrary to the conventional understanding that this dependence becomes negligible over sufficiently long time evolution. An example in which such dependence plays a significant role is discussed in a model of supersymmetric quantum mechanics.