Discussion of the loop formula for the fermionic determinant

TL;DR

This paper derives and discusses a formula that expresses the fermionic determinant as an infinite product of fixed-size determinants indexed by loops, simplifying the analysis of fermionic systems on lattices.

Contribution

It introduces a novel loop formula for the fermionic determinant, enabling a new way to factorize and analyze fermionic systems in lattice theories.

Findings

The fermionic determinant can be expressed as an infinite product of smaller determinants.

These smaller determinants are of fixed size, independent of the lattice.

The formula is indexed by loops of increasing length.

Abstract

A formula expressing the fermionic determinant (a large order polynomial) as an infinite product of smaller determinants is derived and discussed. These smaller determinants are of a fixed size, independent of the size of the lattice and are indexed by loops of increasing length.