The Dimensional-Reduction Anomaly

TL;DR

This paper investigates the breakdown of the expected dimensional reduction of the effective action in certain spacetimes after renormalization, revealing anomalous terms that violate the classical mode sum property.

Contribution

It explicitly calculates the anomalous terms responsible for the violation of dimensional reduction after renormalization in quantum field theory on curved spacetimes.

Findings

Renormalization introduces anomalous terms that break the mode sum property.

Explicit calculations demonstrate the effect in simple spacetime models.

The anomaly affects the effective action's dimensional reduction.

Abstract

In a wide class of D-dimensional spacetimes which are direct or semi-direct sums of a (D-n)-dimensional space and an n-dimensional homogeneous ``internal'' space, a field can be decomposed into modes. As a result of this mode decomposition, the main objects which characterize the free quantum field, such as Green functions and heat kernels, can effectively be reduced to objects in a (D-n)-dimensional spacetime with an external dilaton field. We study the problem of the dimensional reduction of the effective action for such spacetimes. While before renormalization the original D-dimensional effective action can be presented as a ``sum over modes'' of (D-n)-dimensional effective actions, this property is violated after renormalization. We calculate the corresponding anomalous terms explicitly, illustrating the effect with some simple examples.