Crossed Products, Conditional Expectations and Constraint Quantization

Understand

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.