A method for calculating the imaginary part of the Hadamard Elementary function $G^{(1)}$ in static, spherically symmetric spacetimes

TL;DR

This paper introduces a method to calculate the imaginary part of the Hadamard Elementary function for a charged scalar field in static, spherically symmetric spacetimes using Euclidean space techniques and a WKB approximation, providing new insights into particle production.

Contribution

A novel method employing Euclidean space and WKB approximation to compute the imaginary part of $G^{(1)}$ in curved spacetime, extending the DeWitt-Schwinger approximation.

Findings

Derived the DeWitt-Schwinger approximation for the imaginary part of $G^{(1)}$.

Established the relation between the imaginary part and particle production.

Provided analytical expressions for vacuum expectation values in curved spacetime.

Abstract

Whenever real particle production occurs in quantum field theory, the imaginary part of the Hadamard Elementary function $G^{(1)}$ is non-vanishing. A method is presented whereby the imaginary part of $G^{(1)}$ may be calculated for a charged scalar field in a static spherically symmetric spacetime with arbitrary curvature coupling and a classical electromagnetic field $A^{\mu}$. The calculations are performed in Euclidean space where the Hadamard Elementary function and the Euclidean Green function are related by $(1/2)G^{(1)}=G_{E}$. This method uses a $4^{th}$ order WKB approximation for the Euclideanized mode functions for the quantum field. The mode sums and integrals that appear in the vacuum expectation values may be evaluated analytically by taking the large mass limit of the quantum field. This results in an asymptotic expansion for $G^{(1)}$ in inverse powers of the mass $m$ of the quantum field. Renormalization is achieved by subtracting off the terms in the expansion proportional to nonnegative powers of $m$, leaving a finite remainder known as the ``DeWitt-Schwinger approximation.'' The DeWitt-Schwinger approximation for $G^{(1)}$ presented here has terms proportional to both $m^{-1}$ and $m^{-2}$. The term proportional to $m^{-2}$ will be shown to be identical to the expression obtained from the $m^{-2}$ term in the generalized DeWitt-Schwinger point-splitting expansion for $G^{(1)}$. The new information obtained with the present method is the DeWitt-Schwinger approximation for the imaginary part of $G^{(1)}$, which is proportional to $m^{-1}$ in the DeWitt-Schwinger approximation for $G^{(1)}$ derived in this paper.