Constraints in Covariant Field Theories

In this paper we have considered certain problems which arise when one attempts to cast a covariant field theory into a canonical form. Because of the invariance properties of the theory, certain identities exist between the canonical field variables. To insure that the canonical theory is equivalent to the underlying lagrangian formalism one must require that these identities, once satisfied, will remain satisfied through the course of time. In general, this will be true only if additional constraints are set between the canonical variables. We have shown that only a finite number of such constraints exist and that they form a function group. Our proof rests essentially on the possibility of constructing a generating function for an infinitesimal canonical transformation that is equivalent to an invariant infinitesimal transformation on the lagrangian formalism.

Once a hamiltonian is obtained by one of the procedures outlined in previous papers of this series, and the constraints have all been found, the consistent, invariant canonical formulation of the theory is completed. The main results of the paper have been formulated in such a manner as to make them applicable to a fairly general type of invariance. In the last sections we have applied these results to the cases of gauge and coordinate invariance. In the latter case a hamiltonian, corresponding to a quadratic lagrangian, has been constructed in a parameter-free form; and in both cases the constraints, together with the poisson bracket relations between them, have been obtained explicitly. As was to be expected, two constraints were found for a gauge-invariant theory and eight for a coordinate-invariant theory.