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Secular growths and their relation to equilibrium states in perturbative QFT

Stefano Galanda, Nicola Pinamonti, Leonardo Sangaletti

TL;DR

Secular growths arise in truncated perturbative expansions when time-dependent interactions are adiabatically switched on. The authors develop a perturbative algebraic QFT (pAQFT) framework and prove that, for spatially compact interaction Lagrangians $V$ and background states with clustering or time-translation invariance (e.g., KMS states), secular terms are absent at all perturbative orders; the return-to-equilibrium property under the interacting dynamics plays a central role. They extend these results from free to interacting equilibrium states, and further to general secularly bounded states, providing corollaries that relate two-point functions to the absence of secular growth. The paper then applies the theorems to a complex scalar and a Dirac field on a background KMS state with external electromagnetic potentials, analyzing first-order corrections and loop diagrams to illustrate when adiabatic and large-time limits may be interchanged safely. Overall, the work clarifies when perturbative predictions remain reliable in non-equilibrium QFT by exploiting spatial localization of interactions and appropriate background states.

Abstract

In the perturbative treatment of interacting quantum field theories, if the interaction Lagrangian changes adiabatically in a finite interval of time, secular growths may appear in the truncated perturbative series also when the interaction Lagrangian density is returned to be constant. If this happens, the perturbative approach does not furnish reliable results in the evaluation of scattering amplitudes or expectation values. In this paper we show that these effects can be avoided for adiabatically switched-on interactions, if the spatial support of the interaction is compact and if the background state is suitably chosen. We start considering equilibrium background states and show that, when thermalisation occurs (interaction Lagrangian of spatial compact support), secular effects are avoided. Furthermore, no secular effects pop up if the limit where the Lagrangian is supported everywhere in space is taken after thermalisation (large time limit), in contrast to the reversed order. This result is generalized showing that if the interaction Lagrangian is spatially compact, secular growths are avoided for generic background states which are only invariant under time translation and to states whose explicit dependence of time is not too strong. Finally, as an application, the presented theorems are used to study a complex scalar and a Dirac field, on a background KMS state, in a classical external electromagnetic potential and the contribution to the two point-function given by a generic loop diagram arising from a second order perturbative expansion.

Secular growths and their relation to equilibrium states in perturbative QFT

TL;DR

Secular growths arise in truncated perturbative expansions when time-dependent interactions are adiabatically switched on. The authors develop a perturbative algebraic QFT (pAQFT) framework and prove that, for spatially compact interaction Lagrangians and background states with clustering or time-translation invariance (e.g., KMS states), secular terms are absent at all perturbative orders; the return-to-equilibrium property under the interacting dynamics plays a central role. They extend these results from free to interacting equilibrium states, and further to general secularly bounded states, providing corollaries that relate two-point functions to the absence of secular growth. The paper then applies the theorems to a complex scalar and a Dirac field on a background KMS state with external electromagnetic potentials, analyzing first-order corrections and loop diagrams to illustrate when adiabatic and large-time limits may be interchanged safely. Overall, the work clarifies when perturbative predictions remain reliable in non-equilibrium QFT by exploiting spatial localization of interactions and appropriate background states.

Abstract

In the perturbative treatment of interacting quantum field theories, if the interaction Lagrangian changes adiabatically in a finite interval of time, secular growths may appear in the truncated perturbative series also when the interaction Lagrangian density is returned to be constant. If this happens, the perturbative approach does not furnish reliable results in the evaluation of scattering amplitudes or expectation values. In this paper we show that these effects can be avoided for adiabatically switched-on interactions, if the spatial support of the interaction is compact and if the background state is suitably chosen. We start considering equilibrium background states and show that, when thermalisation occurs (interaction Lagrangian of spatial compact support), secular effects are avoided. Furthermore, no secular effects pop up if the limit where the Lagrangian is supported everywhere in space is taken after thermalisation (large time limit), in contrast to the reversed order. This result is generalized showing that if the interaction Lagrangian is spatially compact, secular growths are avoided for generic background states which are only invariant under time translation and to states whose explicit dependence of time is not too strong. Finally, as an application, the presented theorems are used to study a complex scalar and a Dirac field, on a background KMS state, in a classical external electromagnetic potential and the contribution to the two point-function given by a generic loop diagram arising from a second order perturbative expansion.
Paper Structure (28 sections, 14 theorems, 237 equations, 2 figures)

This paper contains 28 sections, 14 theorems, 237 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Simple toy model - adiabatic mass changes in Klein--Gordon fields
  3. Perturbative approach to interacting quantum field theories in the algebraic setting
  4. Interacting quantum field theory, S matrices and Bogoliubov maps
  5. Equilibrium states for interacting theories
  6. Equilibrium states for the free dynamics
  7. Equilibrium states for the interacting dynamics
  8. Secular effects and return to equilibrium
  9. Return to equilibrium for KMS states and absence of secular growths
  10. Dirac field
  11. Complex scalar field
  12. Absence of secular growths for general states
  13. Applications and examples
  14. Scalar electrodynamics
  15. General introduction
  16. ...and 13 more sections

Key Result

Theorem 3.1

Let $\omega^{\beta}$ be an equilibrium state with respect to the free dynamics $\tau_t$. Given an interaction $V$ (Equation eq:formaV) and the corresponding interacting dynamics $\tau_t^V$, if return to equilibrium holds at all orders in perturbation theory: then secular effects are absent. Here, $\omega^{\beta,V}$ is the equilibrium state for the interacting dynamics.

Figures (2)

  • Figure 1: The sum of the two diagrams on the left corresponds to $Z^{\mathfrak{A}}$ while the two on the right to $Z^{\mathfrak{B}}$. See Appendix \ref{['appendix_scalar_el']} for their analytic expression.
  • Figure 2: Generic diagram

Theorems & Definitions (32)

  • Definition 2.1
  • Definition 2.2
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Lemma 3.3: Clustering for $\tau_t$
  • proof
  • Definition 3.1
  • Theorem 3.4
  • proof
  • ...and 22 more