Gauge invariance of systems with first-class constraints

Abstract

The infinitesimal canonical transformations which map solutions of the total Hamiltonian equations of motion into each other are investigated. For that, the generating function Ψ of such transformations should satisfy certain conditions. In general, Ψ is a function which depends on the coordinates, the momenta, and the Lagrange multipliers λ. However, it is shown that the requirement of independence of the function Ψ on the Lagrange multipliers is sufficient for the existence of gauge invariant transformations in the Lagrange formalism. It is shown that the condition that the Poisson brackets between Ψ and all the primary first-class constraints are a linear combination of the latter ones provides the λ independence of the function Ψ. The existence of such a λ-independent function Ψ is proven for some systems. In particular, this is proven for the relevant case of systems having primary and secondary first-class constraints. The authors suggest the possibility that for some specific systems a λ-independent generating function Ψ cannot be constructed. This conclusion concerns the systems with more than primary and secondary constraints.