Fluctuations of quantum fields via zeta function regularization

TL;DR

This paper derives explicit formulas for expectation values and variances of quantum field observables on D-dimensional manifolds using zeta function regularization, revealing a universal regularized variance independent of the dimension.

Contribution

It introduces a method to compute variances of bilinear quantum field observables with a novel regularization approach, showing the variance's independence from the manifold's dimension.

Findings

Variance requires additional regularization

Regularized variance is 2/N, independent of D

Explicit examples demonstrate the method

Abstract

Explicit expressions for the expectation values and the variances of some observables, which are bilinear quantities in the quantum fields on a D-dimensional manifold, are derived making use of zeta function regularization. It is found that the variance, related to the second functional variation of the effective action, requires a further regularization and that the relative regularized variance turns out to be 2/N, where N is the number of the fields, thus being independent on the dimension D. Some illustrating examples are worked through.