Spectral Functions at finite temperature and chemical potential

TL;DR

This paper clarifies the differences between two formulations of spectral functions at finite temperature and chemical potential, proposing a modification to Fourier transforms to reconcile discrepancies in calculations.

Contribution

It introduces a spectral function-based method to unify the two formulations and resolves the discrepancy caused by frequency differences.

Findings

Discrepancies between formulations are due to frequency differences.

Modifying the Fourier transform eliminates the discrepancies.

The method simplifies calculations at finite temperature and chemical potential.

Abstract

There are two formulations at non-zero chemical potential; one is the formulation that a Lagrangian includes a chemical potential, the other is the formulation that a Lagrangian does not include a chemical potential. The existence of two formulations makes a calculation complicated. The results from those formulations are not corresponding directly. This discrepancy exists in the imaginary time formalism and the real time formalism. However, since this is essentially caused by a difference of a frequency, the discrepancy vanishes by modifying the Fourier transform. We show a calculational procedure with a spectral function to understand this.