Conservation Law and Trace Anomaly for the Stress Energy Tensor of a Self-Interacting Scalar Field

TL;DR

This paper extends the construction of a divergence-free quantum stress-energy tensor for a self-interacting scalar field to curved spacetimes, using algebraic quantum field theory techniques and perturbation theory up to third order.

Contribution

It generalizes a method for defining a divergence-free quantum stress-energy tensor from free to interacting scalar fields on Minkowski and curved spacetimes.

Findings

Construction valid up to third order in coupling constant

Method applicable to arbitrary globally hyperbolic spacetimes

Extension of previous free field results to interacting case

Abstract

We consider a self-interacting, massive, real scalar field on a four-dimensional globally hyperbolic spacetime and the associated stress-energy tensor. Using techniques proper of the algebraic approach to perturbative quantum field theory, we study the associated, Wick-ordered, quantum observable. In particular we generalize a construction, first developed in the free field theory scenario by Moretti in [Mor03], aimed at exploiting the existing freedoms in the definition of the classical stress-energy tensor, in order to define a quantum counterpart which is divergence free. We focus on Minkowski spacetime proving that this procedure can be adapted also to cubic or quartic self-interactions, at least up to order $\mathcal{O}(\lambda^3)$ in perturbation theory. We remark that this result can be extended to arbitrary globally hyperbolic spacetimes, although, in this case one needs to exploit the existing regularization freedom in the construction of the Wick ordered stress-energy tensor.