On the geometry of quantum constrained systems

TL;DR

This paper explores the geometric structure of quantum constrained systems, comparing classical and quantum formalisms, and proposes a new condition for physical observables using the double commutator with the squared constraint.

Contribution

It introduces a geometric approach to quantum constrained systems and links the Dirac and Master Constraint procedures through a novel observable condition.

Findings

Identifies key differences between classical and quantum constraint geometries.

Proposes the double commutator with the squared constraint as a condition for physical observables.

Establishes a connection between Dirac and Master Constraint formalisms.

Abstract

The use of geometric methods has proved useful in the hamiltonian description of classical constrained systems. In this note we provide the first steps toward the description of the geometry of quantum constrained systems. We make use of the geometric formulation of quantum theory in which unitary transformations (including time evolution) can be seen, just as in the classical case, as finite canonical transformations on the quantum state space. We compare from this perspective the classical and quantum formalisms and argue that there is an important difference between them, that suggests that the condition on observables to become physical is through the double commutator with the square of the constraint operator. This provides a bridge between the standard Dirac procedure --through its geometric implementation-- and the Master Constraint program.