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An equivalence theorem for algebraic and functorial QFT

Severin Bunk, James MacManus, Alexander Schenkel

TL;DR

The paper develops a unifying framework linking AQFT and FQFT in globally hyperbolic Lorentzian spacetimes by introducing a globally hyperbolic Lorentzian bordism pseudo-operad $\mathscr{L}\!\mathsf{B}\mathsf{op}_m$ that encodes spatial locality via $n$-to-$1$ bordisms. It defines globally hyperbolic Lorentzian FQFTs as pseudo-multifunctors to the category of unital associative algebras, and shows that these admit a tractable description through the truncation $\tau(\mathscr{L}\!\mathsf{B}\mathsf{op}_m)$, enabling ordinary multifunctor formalisms. The central result is an equivalence $\mathbf{AQFT}_m^{W,\mathrm{add}} \simeq \mathbf{FQFT}_m^{W,\mathrm{add}}$ for all $m$, with explicit constructions between the two theories and reliance on the time-slice axiom and additivity. This advances the understanding of how AQFT and FQFT frameworks cohere in Lorentzian geometry, providing a robust basis for analyzing local observables and their time evolution in a unified, operadic setting.

Abstract

This paper develops a novel approach to functorial quantum field theories (FQFTs) in the context of Lorentzian geometry. The key challenge is that globally hyperbolic Lorentzian bordisms between two Cauchy surfaces cannot change the topology of the Cauchy surface. This is addressed and solved by introducing a more flexible concept of bordisms which provide morphisms from tuples of causally disjoint partial Cauchy surfaces to a later-in-time full Cauchy surface. They assemble into a globally hyperbolic Lorentzian bordism pseudo-operad, generalizing the geometric bordism pseudo-categories of Stolz and Teichner. The associated FQFTs are defined as pseudo-multifunctors into a symmetric monoidal category of unital associative algebras. The main result of this paper is an equivalence theorem between such globally hyperbolic Lorentzian FQFTs and algebraic quantum field theories (AQFTs), both subject to the time-slice axiom and a mild descent condition called additivity.

An equivalence theorem for algebraic and functorial QFT

TL;DR

The paper develops a unifying framework linking AQFT and FQFT in globally hyperbolic Lorentzian spacetimes by introducing a globally hyperbolic Lorentzian bordism pseudo-operad that encodes spatial locality via -to- bordisms. It defines globally hyperbolic Lorentzian FQFTs as pseudo-multifunctors to the category of unital associative algebras, and shows that these admit a tractable description through the truncation , enabling ordinary multifunctor formalisms. The central result is an equivalence for all , with explicit constructions between the two theories and reliance on the time-slice axiom and additivity. This advances the understanding of how AQFT and FQFT frameworks cohere in Lorentzian geometry, providing a robust basis for analyzing local observables and their time evolution in a unified, operadic setting.

Abstract

This paper develops a novel approach to functorial quantum field theories (FQFTs) in the context of Lorentzian geometry. The key challenge is that globally hyperbolic Lorentzian bordisms between two Cauchy surfaces cannot change the topology of the Cauchy surface. This is addressed and solved by introducing a more flexible concept of bordisms which provide morphisms from tuples of causally disjoint partial Cauchy surfaces to a later-in-time full Cauchy surface. They assemble into a globally hyperbolic Lorentzian bordism pseudo-operad, generalizing the geometric bordism pseudo-categories of Stolz and Teichner. The associated FQFTs are defined as pseudo-multifunctors into a symmetric monoidal category of unital associative algebras. The main result of this paper is an equivalence theorem between such globally hyperbolic Lorentzian FQFTs and algebraic quantum field theories (AQFTs), both subject to the time-slice axiom and a mild descent condition called additivity.

Paper Structure

This paper contains 14 sections, 13 theorems, 87 equations.

Table of Contents

  1. Introduction and summary
  2. Preliminaries on algebraic QFTs
  3. The globally hyperbolic Lorentzian bordism pseudo-operad
  4. The groupoid $(\mathscr{L}\!\mathscr{B}\mathsf{op}_m)_0$:
  5. The groupoids $(\mathscr{L}\!\mathscr{B}\mathsf{op}_m)_1^n$:
  6. Source and target functors:
  7. Operadic compositions:
  8. Operadic units:
  9. Permutation actions:
  10. Coherence isomorphisms:
  11. Globally hyperbolic Lorentzian functorial QFTs
  12. The equivalence theorem
  13. Basic theory of pseudo-operads in $\mathbf{Grpd}$
  14. Lorentzian geometric details

Key Result

Proposition 3.3

For each $m\in\mathbb{N}$, the $m$-dimensional globally hyperbolic Lorentzian bordism pseudo-operad defined above is fibrant in the sense of Definition def:fibrant, i.e. $\mathscr{L}\!\mathscr{B}\mathsf{op}_m\in\mathbf{PsOp}^{\mathrm{fib}}$. A choice of companion for a vertical morphism $[U,g]: (M,\ together with the $2$-cells

Theorems & Definitions (45)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Remark 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • ...and 35 more