Temperature in Fermion Systems and the Chiral Fermion Determinant

TL;DR

This paper interprets the chiral fermion determinant using a heat-kernel approach by relating the extra dimension to temperature, deriving the Dirac equation via Wick rotation, and analyzing regularizations and anomalies in fermion systems.

Contribution

It introduces a novel interpretation of the extra dimension as temperature in the heat-kernel approach and derives the Dirac equation and anomaly considerations within this framework.

Findings

The extra dimension is interpreted as inverse temperature.

A natural relation between fermion momentum, mass, and cutoff is established.

Two different regularizations for the fermion determinant are identified.

Abstract

We give an interpretation to the issue of the chiral determinant in the heat-kernel approach. The extra dimension (5-th dimension) is interpreted as (inverse) temperature. The 1+4 dim Dirac equation is naturally derived by the Wick rotation for the temperature. In order to define a ``good'' temperature, we choose those solutions of the Dirac equation which propagate in a fixed direction in the extra coordinate. This choice fixes the regularization of the fermion determinant. The 1+4 dimensional Dirac mass ($M$) is naturally introduced and the relation: $|$4 dim electron momentum$|$ $\ll$ $|M|$ $\ll$ ultraviolet cut-off, naturally appears. The chiral anomaly is explicitly derived for the 2 dim Abelian model. Typically two different regularizations appear depending on the choice of propagators. One corresponds to the chiral theory, the other to the non-chiral (hermitian) theory.