Euclidean Scalar Green Function in a Higher Dimensional Global Spacetime

TL;DR

This paper derives an explicit Euclidean scalar Green function for a massless field in higher-dimensional global monopole spacetimes, enabling calculation of vacuum expectation values of key operators.

Contribution

It provides a new explicit Green function expression in higher dimensions with solid angle deficit, facilitating quantum field calculations in such spacetimes.

Findings

Calculated the renormalized vacuum expectation value of $\

<T_{\mu\nu}\

Abstract

We construct the explicit Euclidean scalar Green function associated with a massless field in a higher dimensional global monopole spacetime, i.e., a $(1+d)$-spacetime with $d\geq3$ which presents a solid angle deficit. Our result is expressed in terms of a infinite sum of products of Legendre functions with Gegenbauer polynomials. Although this Green function cannot be expressed in a closed form, for the specific case where the solid angle deficit is very small, it is possible to develop the sum and obtain the Green function in a more workable expression. Having this expression it is possible to calculate the vacuum expectation value of some relevant operators. As an application of this formalism, we calculate the renormalized vacuum expectation value of the square of the scalar field, $<\Phi^2(x)>_{Ren.}$, and the energy-momentum tensor, $<T_{\mu\nu}(x)>_{Ren.}$, for the global monopole spacetime with spatial dimensions $d=4$ and $d=5$.