A note on the formal structure of quantum constrained systems

TL;DR

This paper discusses the proper way to construct the solution space of Dirac's quantum constraints, emphasizing the importance of 'complete' gauge transformations generated by the constraints on individual state components.

Contribution

It introduces the concept of 'complete' gauge transformations as a means to correctly factor the quantum state space for Dirac's constraints, addressing limitations of previous methods.

Findings

Simple gauge transformations are insufficient for constructing the solution space.

Complete gauge transformations are generated by the quantum constraints on state components.

This approach provides a more accurate framework for quantum constrained systems.

Abstract

The space of the solutions of Dirac's quantum constraints cannot be constructed factoring the quantum state space by the ``simple'' gauge transformations generated by the constraints. However, we show here that it can be constructed by factoring the state space by suitably defined ``complete'' gauge transformations. These are generated by the action of the quantum constraints on individual components of the quantum state.