Gauge transformations and gauge-fixing conditions in constraint systems

Gauge transformations and gauge-fixing conditions in the total Hamiltonian (HT) and extended Hamiltonian (HE) formalisms are investigated. For gauge-fixing conditions χα, only the condition det({ϕα, χβ}) ≠ 0 is usually imposed, where ϕα are first class constraints. This condition is not sufficient and one should (i) employ HT and (ii) choose the gauge-fixing conditions χα to be stationary under HT. Gauge degrees of freedom in the Lagrangian formalism are equal in number to the primary first class constraints . Hence the number of arbitrarily chosen primary gauge conditions is the same as that of . Secondary gauge-fixing conditions associated with secondary first class constraints should be determined by the stationarity conditions of . If a canonical Hamiltonian (weakly) vanishes, χα must be explicitly time-dependent, otherwise we have the trivial result. Further, it is pointed out that the HE formalism has discrepancies with the HT formalism in many aspects. As illustrations of these properties, a few typical models are examined.