Construction of spaces of kinematic quantum states for field theories via projective techniques

TL;DR

This paper introduces a general method for constructing quantum state spaces for field theories using projective techniques, extending previous linear-phase-space methods to more complex theories.

Contribution

It generalizes Kijowski's construction by removing the limitation to linear phase spaces, enabling the construction for a broader class of field theories.

Findings

Provides a systematic way to build quantum state spaces from phase space structures.

Extends the applicability of projective techniques beyond linear theories.

Lays groundwork for future inclusion of constraints in quantum state construction.

Abstract

We present a method of constructing a space of quantum states for a field theory: given phase space of a theory, we define a family of physical systems each possessing a finite number of degrees of freedom, next we define a space of quantum states for each finite system, finally using projective techniques we organize all these spaces into a space of quantum states which corresponds to the original phase space. This construction is kinematic in this sense that it bases merely on the structure of the phase space and does not take into account possible constraints on the space. The construction is a generalization of a construction by Kijowski - the latter one is limited to theories of linear phase spaces, while the former one is free of this limitation.