Involutions on the Algebra of Physical Observables From Reality Conditions

TL;DR

This paper examines the limitations of imposing reality conditions in algebraic quantization, showing that involutions on quantum operators cannot be unambiguously projected onto physical observables, leading to inherent ambiguities.

Contribution

It demonstrates the impossibility of unambiguous involution projection in algebraic quantization for systems with constraints and discusses the resulting ambiguities in Ashtekar's quantization approach.

Findings

Involutions on quantum operators cannot be unambiguously projected onto physical observables.

A method to induce involutions on physical observables under certain assumptions is proposed.

The induced involution depends on the choice of representatives, causing quantization ambiguities.

Abstract

Some aspects of the algebraic quantization programme proposed by Ashtekar are revisited in this article. It is proved that, for systems with first-class constraints, the involution introduced on the algebra of quantum operators via reality conditions can never be projected unambiguously to the algebra of physical observables, ie, of quantum observables modulo constraints. It is nevertheless shown that, under sufficiently general assumptions, one can still induce an involution on the algebra of physical observables from reality conditions, though the involution obtained depends on the choice of particular representatives for the equivalence classes of quantum observables and this implies an additional ambiguity in the quantization procedure suggested by Ashtekar.