An Extension of the Chowla-Selberg Formula Useful in Quantizing with the Wheeler-De Witt Equation

TL;DR

This paper extends the Chowla-Selberg formula to inhomogeneous Epstein zeta-functions and applies it to compute determinants in quantum gravity models using the Wheeler-De Witt equation in 2+1 dimensions.

Contribution

It introduces a new formula extending the Chowla-Selberg formula for inhomogeneous Epstein zeta-functions and demonstrates its application in quantum gravity quantization.

Findings

Derived a simple extension of the Chowla-Selberg formula.

Applied the formula to compute determinants in 2+1 dimensional quantum gravity.

Provided a new analytical tool for zeta-function regularization in gravitational models.

Abstract

The two-dimensional inhomogeneous zeta-function series (with homogeneous part of the most general Epstein type): \[ \sum_{m,n \in \mbox{\bf Z}} (am^2+bmn+cn^2+q)^{-s}, \] is analytically continued in the variable $s$ by using zeta-function techniques. A simple formula is obtained, which extends the Chowla-Selberg formula to inhomogeneous Epstein zeta-functions. The new expression is then applied to solve the problem of computing the determinant of the basic differential operator that appears in an attempt at quantizing gravity by using the Wheeler-De Witt equation in 2+1 dimensional spacetime with the torus topology.