Matsubara Frequency Sums

TL;DR

This paper discusses the evaluation of Matsubara frequency sums in finite-temperature quantum field theory, introducing a method based on the infinite series expansion of coth to handle sums involving chemical potential.

Contribution

It presents a novel operational approach using the series expansion of coth to evaluate Matsubara sums with chemical potential in finite-temperature quantum field theory.

Findings

Provides a new method for summing Matsubara frequencies with chemical potential.

Simplifies calculations involving finite-temperature sums in quantum field theory.

Enhances analytical techniques for thermal quantum systems.

Abstract

We cannot use directly the results of zero-temperature at finite temperature, for at finite temperature the average is to be carried over all highly degenerate excited states unlike zero-temperature average is only on unique ground state. One of the formal way to take into account the finite temperature into quantum field theory is due to Matsubara, to replace temporal component of eigenvalues $k_{4}$ by $\omega_{n}=\frac{2\pi n}{\beta}$ $(\frac{2\pi (n+{1/2})}{\beta})$ with summation over all integer values of $n$. The summation is done with the infinite series expansion of $\coth (\pi y)$. With the chemical potential $\mu$, $\omega_{n}$ will be replaced by $\omega_{n} - \mu$ in the eigenvalues and the summation over $n$ cannot be done easily. Various methods exist to evaluate it. We use the infinite series expansion of $\coth (\pi y)$ to work operationally for such Matsubara frequency sums.