On Dirac's incomplete analysis of gauge transformations

Abstract

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. © 2005 Elsevier Ltd. All rights reserved.