An Essay on the Completion of Quantum Theory. I: General Setting

TL;DR

This paper introduces a geometric framework that completes the mathematical structures of quantum theory, providing a more comprehensive understanding similar to the completion of geometric spaces, with future work planned on dynamics.

Contribution

It proposes a novel geometric completion of quantum theory's axiomatic formalism, extending the spaces of observables and states akin to classical geometric completions.

Findings

Provides a geometric completion of quantum state and observable spaces

Frames usual quantum theory as a special case within a broader structure

Sets the stage for future analysis of quantum dynamics

Abstract

We propose a geometric setting of the axiomatic mathematical formalism of quantum theory. Guided by the idea that understanding the mathematical structures of these axioms is of similar importance as was historically the process of understanding the axioms of geometry, we complete the spaces of observables and of states in a similar way as in classical geometry linear or affine spaces are completed by projective spaces. In this sense, our theory can be considered as a "completion of usual linear quantum theory" , such that the usual theory appears as the special case where a reference frame is fixed once and for all. In the present first part, this general setting is explained. Dynamics (time evolution) will be discussed in subsequent work.