A Theory of Backgrounds and Background Independence

TL;DR

This paper develops a framework using $C^*$ algebra representation theory to understand backgrounds and background independence in quantum systems, illustrated through superconductivity and quantum gravity, linking symmetries, inequivalent Hilbert spaces, and observable extensions.

Contribution

It introduces a unified algebraic approach to backgrounds in quantum systems, connecting symmetry breaking, inequivalent representations, and background independence with physical processes and recent algebraic symmetry work.

Findings

Backgrounds correspond to inequivalent Hilbert space representations.

Gauging broken symmetries achieves background independence.

Intertwining operators relate different backgrounds and physical phenomena.

Abstract

In this note, we describe how the study of backgrounds for general quantum systems can be formulated in terms of the representation theory of abstract $C^*$ algebras. We illustrate our general framework through two example systems: superconductivity and perturbative quantum gravity. In both cases, spontaneously broken symmetries imply the existence of unitarily inequivalent Hilbert spaces that play the role of distinct backgrounds relative to which observables are measured. Background independence can be realized by gauging the broken symmetry; extending the algebra of observables for the theory to include new physical processes that intertwine between these disjoint representations. From the point of view of the background independent theory, different backgrounds have an interpretation as different vacuum expectation values of these intertwining operators. In superconductivity, the intertwiners are intimately related to the Josephson effect. In gravity, they are related to geometric fluctuations. We explain how this framework is connected to recent work on generalized symmetries and algebraic extensions. To this end, we close with some remarks about how the operator algebra of a closed universe may arise from a generalized symmetry associated with an algebraic extension of the causal wedge by appealing to subregion-subalgebra duality.