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Compositional Quantum Field Theory: An axiomatic presentation

Robert Oeckl, Juan Orendain Almada

Abstract

We introduce Compositional Quantum Field Theory (CQFT) as an axiomatic model of Quantum Field Theory, based on the principles of locality and compositionality. Our model is a refinement of the axioms of General Boundary Quantum Field Theory, and is phrased in terms of correspondences between certain commuting diagrams of gluing identifications between manifolds and corresponding commuting diagrams of state-spaces and linear maps, thus making it amenable to formalization in terms of involutive symmetric monoidal functors and operad algebras. The underlying novel framework for gluing leads to equivariance of CQFT. We study CQFTs in dimension 2 and the algebraic structure they define on open and closed strings. We also consider, as a particular case, the compositional structure of 2-dimensional pure quantum Yang-Mills theory.

Compositional Quantum Field Theory: An axiomatic presentation

Abstract

We introduce Compositional Quantum Field Theory (CQFT) as an axiomatic model of Quantum Field Theory, based on the principles of locality and compositionality. Our model is a refinement of the axioms of General Boundary Quantum Field Theory, and is phrased in terms of correspondences between certain commuting diagrams of gluing identifications between manifolds and corresponding commuting diagrams of state-spaces and linear maps, thus making it amenable to formalization in terms of involutive symmetric monoidal functors and operad algebras. The underlying novel framework for gluing leads to equivariance of CQFT. We study CQFTs in dimension 2 and the algebraic structure they define on open and closed strings. We also consider, as a particular case, the compositional structure of 2-dimensional pure quantum Yang-Mills theory.
Paper Structure (24 sections, 30 theorems, 79 equations, 3 figures, 4 tables)

This paper contains 24 sections, 30 theorems, 79 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. General Boundary Quantum Field Theory
  3. Axioms
  4. Examples
  5. Plan for the paper
  6. Gluings: A new perspective
  7. Gluing functions
  8. Relative gluing diagrams
  9. Slice regions
  10. Implementing gluing functions: CQFT
  11. Axiomatic presentation
  12. First consequences and an inner product
  13. Infinite dimensions
  14. Comparison of axiomatic systems
  15. Admissibility
  16. ...and 9 more sections

Key Result

Lemma 2.22

Let $\Sigma$ be a hypersurface. Let $(S,s)$ be a slice region on $\Sigma$. $(S,s)$ is also a slice region on $\overline{\Sigma}$ and $(\overline{S},\overline{s})$ is also a slice region on $\Sigma$ and $\overline{\Sigma}$. Let $\Sigma_i$, $i=1,2$ be hypersurfaces. Let $(S_i,s_i)$ be a slice region f

Figures (3)

  • Figure 1: Compositional locality: With temporal locality spacetime regions are time-slices (left-hand side), in full spacetime locality, spacetime regions can be arbitrary (right-hand side). With the latter notion we can describe physics in an arbitrarily small spacetime neighborhood of an event $p$. This is crucial for implementing Wilsonian renormalization, as known from Lattice Gauge Theory. It would also be essential for quantum gravity.
  • Figure 2: Gluing of the disk to itself to obtain a sphere with $k$ holes. The blue intervals are glued pairwise as indicated by the dotted lines. The red intervals form the boundaries of the holes after gluing. The intervals marked $1$ and $k$ form the boundaries of holes $1$ and $k$ respectively. The intervals marked $2$ to $k-1$ appear in pairs and form the boundaries of the holes $2$ to $k-1$ with each interval contributing a half-circle.
  • Figure 3: Gluing of a sphere with $2g+k$ holes into a Riemann surface of genus $g$ with $k$ holes. Red ($2g$) and blue ($k$) circles are holes in the sphere. The $g$ pairs of red circles are glued together as indicated by the purple lines.

Theorems & Definitions (70)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Example 1.4
  • Example 1.5
  • Example 1.6
  • Remark 1.8
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • ...and 60 more