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 {\it 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.Comment: 36 pages. Additional references and a new paragraph in the introduction. Version to be published in Studies in History and Philosophy of Modern Physic

Equivalence of Faddeev-Jackiw and Dirac approaches for gauge theories

The equivalence between the Dirac method and Faddeev-Jackiw analysis for gauge theories is proved. In particular we trace out, in a stage by stage procedure, the standard classification of first and second class constraints of Dirac's method in the F-J approach. We also find that the Darboux transformation implied in the F-J reduction process can be viewed as a canonical transformation in Dirac approach. Unlike Dirac's method the F-J analysis is a classical reduction procedure, then the quantization can be achieved only in the framework of reduce and then quantize approach with all the know problems that this type of procedures presents. Finally we illustrate the equivalence by means of a particular example.Comment: Latex v2.09, 15 pages, to appear in Int. J. Mod. Phys.