Hilbert space structure in quantum gravity: an algebraic perspective

TL;DR

This paper explores the algebraic structure of quantum gravity, emphasizing the importance of observable algebras over Hilbert space factorization, and highlights fundamental differences from local quantum field theory.

Contribution

It proposes an algebraic perspective on quantum gravity, revealing significant deviations from local quantum field theory and challenging the role of entanglement in the fundamental structure.

Findings

Hilbert space factorization is problematic in quantum gravity.

Algebraic structures differ significantly from local quantum field theory.

Obstacles exist to defining localized commuting algebras in gravity.

Abstract

If quantum gravity respects the principles of quantum mechanics, suitably generalized, it may be that a more viable approach to the theory is through identifying the relevant quantum structures rather than by quantizing classical spacetime. This viewpoint is supported by difficulties of such quantization, and by the apparent lack of a fundamental role for locality. In finite or discrete quantum systems, important structure is provided by tensor factorizations of the Hilbert space. However, even in local quantum field theory properties of the generic type III von Neumann algebras and of long range gauge fields indicate that factorization of the Hilbert space is problematic. Instead it is better to focus on the structure of the algebra of observables, and in particular on its subalgebras corresponding to regions. This paper suggests that study of analogous algebraic structure in gravity gives an important perspective on the nature of the quantum theory. Significant departures from the subalgebra structure of local quantum field theory are found, working in the correspondence limit of long-distances/low-energies. Particularly, there are obstacles to identifying commuting algebras of localized operators. In addition to suggesting important properties of the algebraic structure, this and related observations pose challenges to proposals of a fundamental role for entanglement.