Obtaining a closed-form representation for the dual bosonic thermal Green function by using methods of integration on the complex plane

TL;DR

This paper derives an exact closed-form expression for the Euclidean thermal Green function of a 2D massless scalar field, using complex analysis techniques to map the infinite strip to the upper-half-plane.

Contribution

It provides a novel closed-form representation of the dual bosonic thermal Green function using complex plane integration methods.

Findings

Exact expression for the thermal Green function obtained

Representation as the real part of a conformally mapped function

Identification of the dual Green function as the imaginary part

Abstract

We derive an exact closed-form representation for the Euclidean thermal Green function of the two-dimensional (2D) free massless scalar field in coordinate space. This can be interpreted as the real part of a complex analytic function of a variable that conformally maps the infinite strip $-\infty<x<\infty$ ($0<\tau<\beta$) of the $z=x+i\tau$ ($\tau$: imaginary time) plane into the upper-half-plane. Use of the Cauchy-Riemann conditions, then allows us to identify the dual thermal Green function as the imaginary part of that function.