Heisenberg-picture quantum field theory

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.