Covariance systems

TL;DR

This paper develops new theoretical frameworks for covariance systems, extending the GNS construction, and applies these to quantum particles and spacetime models, advancing the mathematical foundations of quantum physics.

Contribution

It introduces novel definitions of states and representations for covariance systems and generalizes the GNS construction within this context.

Findings

Extended the GNS construction to covariance systems

Provided a new description of a nonrelativistic quantum particle

Presented a simple model of quantum spacetime by Doplicher et al.

Abstract

We introduce new definitions of states and of representations of covariance systems. The GNS-construction is generalized to this context. It associates a representation with each state of the covariance system. Next, states are extended to states of an appropriate covariance algebra. Two applications are given. We describe a nonrelativistic quantum particle, and we give a simple description of the quantum spacetime model introduced by Doplicher et al.