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

Beatrice Costeri, Claudio Dappiaggi, Michele Goi

TL;DR

This paper extends Moretti’s approach for a divergence-free quantum stress-energy tensor to self-interacting real scalar fields within the perturbative algebraic QFT framework on four-dimensional globally hyperbolic spacetimes. By carefully selecting on-shell-improvement parameters (notably $\eta=\tfrac{1}{4}$ for interactions) and employing the Bogoliubov map, it shows that the quantum EMT can be made divergence-free up to $O(\lambda^3)$ in Minkowski space for cubic or quartic potentials, at the expense of a modified trace. The trace acquires interaction-dependent, state-independent contributions, with explicit expressions for quartic and cubic cases that relate to Hadamard coefficients like $v_1(z,z)$ and to the contracted kernels of Hadamard/Feynman propagators. The work highlights the role of renormalization freedoms in preserving conservation laws in interacting QFTs and discusses the extension to arbitrary globally hyperbolic spacetimes, where geometric counterterms can cancel residual divergences.

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 $\mathcal{O}(λ^3)$ 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.

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

TL;DR

This paper extends Moretti’s approach for a divergence-free quantum stress-energy tensor to self-interacting real scalar fields within the perturbative algebraic QFT framework on four-dimensional globally hyperbolic spacetimes. By carefully selecting on-shell-improvement parameters (notably for interactions) and employing the Bogoliubov map, it shows that the quantum EMT can be made divergence-free up to in Minkowski space for cubic or quartic potentials, at the expense of a modified trace. The trace acquires interaction-dependent, state-independent contributions, with explicit expressions for quartic and cubic cases that relate to Hadamard coefficients like and to the contracted kernels of Hadamard/Feynman propagators. The work highlights the role of renormalization freedoms in preserving conservation laws in interacting QFTs and discusses the extension to arbitrary globally hyperbolic spacetimes, where geometric counterterms can cancel residual divergences.

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 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.

Paper Structure

This paper contains 11 sections, 11 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Functional spaces
  3. Hadamard states
  4. Introduction to perturbative Algebraic Quantum Field Theory
  5. Deformation quantization
  6. Interacting Theories in pAQFT
  7. The S-matrix and the Bogoliubov map.
  8. Examples
  9. On the stress-Energy Tensor for a Free Theory
  10. Stress-Energy tensor for a self-interacting theory
  11. Microlocal analysis: Wavefront set and scaling degree

Key Result

Theorem 1.12

Let $\mathcal{M}$ be a globally hyperbolic spacetime and let $\Delta_+\in\mathcal{D}^\prime(\mathcal{M}\times\mathcal{M})$. It abides by Definition Def: Hadamard 2-pt if and only if it also satisfies the hypotheses of Definition Def: local Hadamard 2-pt.

Theorems & Definitions (61)

  • Definition 1.1: Functional derivative
  • Definition 1.2: Microcausal functionals
  • Definition 1.3: Regular functionals
  • Definition 1.4: Local functionals
  • Example 1.5
  • Remark 1.6
  • Definition 1.7: Classical Microcausal Algebra
  • Remark 1.8
  • Definition 1.9
  • Remark 1.10
  • ...and 51 more