Dirac\'s penumbra: constraints and gauge transformations in reparametrization invariant theories

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A simple theorem of Dirac identifies primary first-class constraints as generators of transformations, \'that do not affect the physical state\'. This result has profound implications for the definition of physical states and observables in the quantization of constrained systems, and leads to one aspect of the infamous \'problem of time\' in quantum gravity. As I will discuss, a close look at the theorem reveals that it depends crucially on the assumption of an absolute time. This assumption does not hold for reparametrization invariant theories, such as parametrized particle mechanics, and in these theories, the primary Hamiltonian constraint does generate physical change. I will also look at just what Dirac did and did not say about this case, and what has been said by reviewers since.