A Fermionic Hodge Star Operator

TL;DR

This paper introduces a fermionic analogue of the Hodge star operator, providing an explicit operator representation in fermionic models across various spacetime dimensions, and explores its mathematical properties and implications.

Contribution

It presents a novel fermionic Hodge star operator with explicit representation, linking it to conjugation and metric structures in fermionic quantum models.

Findings

Defines a fermionic Hodge star operator with explicit form.

Shows the operator induces a metric similar to the standard one.

Explores the properties and potential applications of the operator.

Abstract

A fermionic analogue of the Hodge star operation is shown to have an explicit operator representation in models with fermions, in spacetimes of any dimension. This operator realizes a conjugation (pairing) not used explicitly in field-theory, and induces a metric in the space of wave-function(al)s just as in exterior calculus. If made real (Hermitian), this induced metric turns out to be identical to the standard one constructed using Hermitian conjugation; the utility of the induced complex bilinear form remains unclear.