Heisenberg-picture quantum field theory

TL;DR

This paper proposes a categorical framework for Heisenberg-picture quantum field theories, extending the axiomatic approach used for Schrödinger-picture theories by using pointed vector spaces and algebraic structures.

Contribution

It introduces a new axiomatic formulation for Heisenberg-picture quantum field theories using symmetric monoidal functors targeting pointed vector spaces and algebraic categories.

Findings

Outlines a categorical definition for Heisenberg-picture QFTs.

Suggests factorization algebras and skein theory as sources of such theories.

Abstract

What we should mean by "Heisenberg-picture quantum field theory"? Atiyah--Segal-type axioms do a good job of capturing the "Schr\"odinger picture": these axioms define a "$d$-dimensional quantum field theory" to be a symmetric monoidal functor from an $(\infty,d)$-category of "spacetimes" to an $(\infty,d)$-category which at the second-from-top level consists of vector spaces, so at the top level consists of numbers. This paper argues that the appropriate parallel notion "Heisenberg picture" should also be defined in terms of symmetric monoidal functors from the category of spacetimes, but the target should be an $(\infty,d)$-category that in top dimension consists of pointed vector spaces instead of numbers; the second-from-top level can be taken to consist of associative algebras or of pointed categories. The paper ends by outlining two sources of such Heisenberg-picture field theories: factorization algebras and skein theory.