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An algebraic QFT approach to the Wetterich equation on Lorentzian manifolds

Edoardo D'Angelo, Nicolò Drago, Nicola Pinamonti, Kasia Rejzner

TL;DR

The paper develops a covariant functional renormalization group framework for interacting quantum fields on generic Lorentzian spacetimes using perturbative algebraic QFT (pAQFT). By employing a local regulator, it derives a Wetterich-like flow equation that remains meaningful in curved backgrounds and generic quantum states, with the Hadamard parametrix providing the necessary covariant counterterms. The formalism is illustrated through Minkowski vacuum, thermal states, and the Bunch-Davies state on de Sitter space, where non-trivial fixed points are found, demonstrating the method's ability to capture state- and curvature-dependent RG dynamics. This Lorentzian FRG approach offers a principled route to study non-perturbative effects in curved spacetimes and could illuminate issues in gauge theories and quantum gravity, including cosmological contexts and asymptotic safety scenarios. The state-dependence of the flow and the Hadamard-based regularization ensure covariance and rigorous control over ultraviolet and infrared behavior in a framework that extends beyond Euclidean formulations.

Abstract

We discuss the scaling of the effective action for the interacting scalar quantum field theory on generic spacetimes with Lorentzian signature and in a generic state (including vacuum and thermal states, if they exist). This is done constructing a flow equation, which is very close to the renown Wetterich equation, by means of techniques recently developed in the realm of perturbative Algebraic Quantum Field theory (pAQFT). The key ingredient that allows one to obtain an equation which is meaningful on generic Lorentzian backgrounds is the use of a local regulator, which keeps the theory covariant. As a proof of concept, the developed methods are used to show that non-trivial fixed points arise in quantum field theories in a thermal state and in the case of quantum fields in the Bunch-Davies state on the de Sitter spacetime.

An algebraic QFT approach to the Wetterich equation on Lorentzian manifolds

TL;DR

The paper develops a covariant functional renormalization group framework for interacting quantum fields on generic Lorentzian spacetimes using perturbative algebraic QFT (pAQFT). By employing a local regulator, it derives a Wetterich-like flow equation that remains meaningful in curved backgrounds and generic quantum states, with the Hadamard parametrix providing the necessary covariant counterterms. The formalism is illustrated through Minkowski vacuum, thermal states, and the Bunch-Davies state on de Sitter space, where non-trivial fixed points are found, demonstrating the method's ability to capture state- and curvature-dependent RG dynamics. This Lorentzian FRG approach offers a principled route to study non-perturbative effects in curved spacetimes and could illuminate issues in gauge theories and quantum gravity, including cosmological contexts and asymptotic safety scenarios. The state-dependence of the flow and the Hadamard-based regularization ensure covariance and rigorous control over ultraviolet and infrared behavior in a framework that extends beyond Euclidean formulations.

Abstract

We discuss the scaling of the effective action for the interacting scalar quantum field theory on generic spacetimes with Lorentzian signature and in a generic state (including vacuum and thermal states, if they exist). This is done constructing a flow equation, which is very close to the renown Wetterich equation, by means of techniques recently developed in the realm of perturbative Algebraic Quantum Field theory (pAQFT). The key ingredient that allows one to obtain an equation which is meaningful on generic Lorentzian backgrounds is the use of a local regulator, which keeps the theory covariant. As a proof of concept, the developed methods are used to show that non-trivial fixed points arise in quantum field theories in a thermal state and in the case of quantum fields in the Bunch-Davies state on the de Sitter spacetime.
Paper Structure (31 sections, 10 theorems, 224 equations)

This paper contains 31 sections, 10 theorems, 224 equations.

Table of Contents

  1. Introduction
  2. Perturbative Algebraic Quantum Field Theory
  3. Classical field theory
  4. Deformation quantization
  5. States
  6. Normal ordering
  7. Interacting theories
  8. Functional renormalization
  9. Generating functionals
  10. Effective action
  11. Regularised generating functionals
  12. Properties of the average effective action
  13. Quantum equations of motion
  14. Classical limit
  15. Parity of $V$
  16. ...and 16 more sections

Key Result

Lemma 3.1

If $\omega$ does not fulfil the Gell-Mann-Low formula given in eq:gellmann-low, there is no functional $\zeta(j)$ satisfying Equation Eq: j not zero defining property of generating functional.

Theorems & Definitions (29)

  • Remark 2.1
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Remark 3.4
  • Proposition 3.5
  • proof
  • Remark 3.6
  • ...and 19 more