On Dirac's incomplete analysis of gauge transformations

Dirac's approach to gauge symmetries is discussed. We follow closely the steps that led him from his conjecture concerning the generators of gauge transformations at a given time-to be contrasted with the common view of gauge transformations as maps from solutions of the equations of motion into other solutions-to his decision to artificially modify the dynamics, substituting the extended Hamiltonian (including all first-class constraints) for the total Hamiltonian (including only the primary first-class constraints). We show in detail that Dirac's analysis was incomplete and, in completing it, we prove that the fulfilment of Dirac's conjecture-in the "non-pathological" cases-does not imply any need to modify the dynamics. We give a couple of simple but significant examples. (c) 2005 Elsevier Ltd. All rights reserved.