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A novel class of functionals for perturbative algebraic quantum field theory

Eli Hawkins, Kasia Rejzner, Berend Visser

Abstract

Perturbative Algebraic Quantum Field Theory (pAQFT) is based upon formal power series valued in spaces of functionals. This is usually done with microcausal functionals, which are defined using microlocal analysis and motivated by propagation of singularities. In this paper, we prove that the class of microcausal functionals is not closed under the Peierls (Poisson) bracket by showing that a Peierls bracket of regular functionals can fail to be smooth. Consequently, microcausal functionals are not a suitable basis for pAQFT. To remedy these issues, we introduce the class of equicausal functionals. We show that this class contains the local functionals and that it closes under the star-product and Peierls bracket. Furthermore, we prove the time-slice axiom for equicausal functionals, using a chain homotopy. The class of microcausal functionals is not closed under this chain homotopy, which strongly suggests that the class of microcausal functionals does not fulfill the time slice axiom.

A novel class of functionals for perturbative algebraic quantum field theory

Abstract

Perturbative Algebraic Quantum Field Theory (pAQFT) is based upon formal power series valued in spaces of functionals. This is usually done with microcausal functionals, which are defined using microlocal analysis and motivated by propagation of singularities. In this paper, we prove that the class of microcausal functionals is not closed under the Peierls (Poisson) bracket by showing that a Peierls bracket of regular functionals can fail to be smooth. Consequently, microcausal functionals are not a suitable basis for pAQFT. To remedy these issues, we introduce the class of equicausal functionals. We show that this class contains the local functionals and that it closes under the star-product and Peierls bracket. Furthermore, we prove the time-slice axiom for equicausal functionals, using a chain homotopy. The class of microcausal functionals is not closed under this chain homotopy, which strongly suggests that the class of microcausal functionals does not fulfill the time slice axiom.
Paper Structure (19 sections, 34 theorems, 236 equations, 2 figures)

This paper contains 19 sections, 34 theorems, 236 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Equicontinuous sets
  4. Smooth functionals
  5. Some Microlocal Analysis
  6. Functionals on spacetime
  7. Dynamics
  8. The Koszul complex for smooth functionals
  9. Time-slice axiom
  10. Undesirable features of microcausal functionals
  11. Equicausal multivector fields
  12. Examples of equicausal functionals
  13. Local Functionals
  14. Wick polynomials
  15. Time-slice axiom for equicausal functionals
  16. ...and 4 more sections

Key Result

Proposition 2.2

The topology of a topological vector space $A$ is equivalent to the topology of uniform convergence on equicontinuous subsets of $A'$, the continuous linear dual of $A$.

Figures (2)

  • Figure 1: Graphical depiction of $\sigma_{n,m,k}$.
  • Figure 2: The covectors in Lemma \ref{['coneinclusion']}.

Theorems & Definitions (71)

  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3: Leibniz integral rule
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8
  • Definition 2.9
  • ...and 61 more