Crossed Products, Conditional Expectations and Constraint Quantization

TL;DR

This work develops an operator-algebraic framework in which the extended phase space and its associated crossed products provide a unifying foundation for constraint quantization in gauge theories and gravity. By treating gauge constraints and corner charges via a pair of codimension-one and codimension-two extensions, the authors show that refined algebraic quantization, BRST, path integral methods, and the commutation theorem are facets of a single projection/dressing mechanism using conditional expectations. The framework yields a semi-finite von Neumann algebra, enabling density operators and a meaningful entanglement entropy notion, and is illustrated through examples including a symmetric harmonic oscillator, Yang-Mills theory, and gravity on null hypersurfaces. The results offer a geometric, observer-centered view of dressing and constraints, with implications for edge modes, modular structure, and potentially holographic contexts.

Abstract

Recent work has highlighted the importance of crossed products in correctly elucidating the operator algebraic approach to quantum field theories. In the gravitational context, the crossed product simultaneously promotes von Neumann algebras associated with subregions in diffeomorphism covariant quantum field theories from type III to type II, and provides the necessary ingredients to gravitationally dress operators, thereby enforcing the constraints of the theory. In this note we enhance the crossed product construction to the context of general gauge theories with arbitrary combinations of internal and spacetime local symmetries. This is done by leveraging the correspondence between the crossed product and the extended phase space. We then undertake a detailed study of constraint quantization from the perspective of a generic crossed product algebra. We study and compare four distinct approaches to constraint quantization from this point of view: refined algebraic quantization, BRST quantization, path integral quantization, and the commutation theorem for crossed products. Far from simply reproducing existing analyses, the operator algebraic viewpoint sheds new light on old problems by reformulating the dressing of operators in terms of conditional expectations and other closely related projection maps. We conclude by applying our approach to the constraint quantization of three distinct gauge theories including a discussion of gravity on null hypersurfaces.