Constraints and gauge transformations: Dirac's theorem is not always valid

TL;DR

This paper challenges Dirac's theorem in the context of reparametrization invariant systems, demonstrating that the Hamiltonian constraint can generate physical motion contrary to traditional beliefs in quantum gravity.

Contribution

It identifies the limitations of Dirac's theorem for certain systems and explicitly shows how the primary Hamiltonian constraint can generate physical evolution.

Findings

Dirac's theorem does not hold for reparametrization invariant systems.

The primary Hamiltonian constraint can generate physical motion.

Implications for quantum gravity are discussed for specific systems.

Abstract

A standard tenet of canonical quantum gravity is that evolution generated by a Hamiltonian constraint is just a gauge transformation on the phase space and therefore does not change the physical state. The basis for this belief is a theorem of Dirac that identifies primary first-class constraints as generators of physically irrelevant motions. We point out that certain assumptions on which Dirac based his argument do not hold for reparametrization invariant systems, and show that the primary Hamiltonian constraint of these systems does generate physical motion. We show explicitly how the argument fails for systems described by Jacobi's principle, which has a structure closely resembling that of general relativity. We defer discussion of general relativity and the implications for quantum gravity to a later paper.