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Description
In this work, we present an overview of Dirac's classical approach for obtaining the Hamiltonian formulation of gauge systems, with a particular focus on systems where "gauge hits twice", resulting in the simultaneous appearance of two first-class Hamiltonian constraints for every given pure gauge mode. The validity of the Dirac conjecture, which states that all first-class constraints serve as independent generators of gauge transformations, is examined. It is demonstrated that under the standard notion of gauge symmetry in field theory this conjecture does not hold when using the Total Hamiltonian but is valid in the Extended Hamiltonian approach. We show that, without a proper redefinition of the model, transitioning from the Total to the Extended Hamiltonian modifies its contents by converting constrained modes into pure gauge modes. After performing the aforementioned redefinition, the constrained modes are restored. This redefinition essentially involves a change of zero-momentum variables, which is similar to the Stueckelberg's trick. The results are illustrated with examples of Yang–Mills theories and the ADM formalism of GR, and are then generalized to all mechanical gauge systems.
