Algebras and their Covariant representations in quantum gravity

TL;DR

This paper explores covariant representations of operator algebras in quantum gravity, highlighting their connection to crossed product algebras and their physical interpretation in gravitational theories.

Contribution

It establishes a detailed link between covariant representations and crossed product algebras, providing new insights into algebraic structures in quantum gravity.

Findings

Covariant representations implement symmetries unitarily.

Crossed product algebras correspond to covariant representations.

The work clarifies the physical meaning of algebraic constructions in quantum gravity.

Abstract

We study a physically motivated representation of an algebra of operators in gravitational and non gravitational theories called the covariant representation of an algebra. This is a representation where the symmetries of the operator algebra are implemented unitarily on the Hilbert space. We emphasize the very close similarity of this representation to the crossed product of an algebra. In fact, as an example of (and sometimes identified with) a covariance algebra, the crossed product of an algebra is in one to one correspondence with the covariant representation of the algebra. This will in turn illuminate physically what the crossed product algebra is in the context of quantum gravity.