Cartan algorithm and Dirac constraints for Griffiths variational problems

Dirac algorithm allows to construct Hamiltonian systems for singular systems, and so contributing to its successful quantization. A drawback of this method is that the resulting quantized theory does not have manifest Lorentz invariance. This motivated the quest of alternatives to the usual Hamiltonian theory on the space of sections; a particular instance of this search is the so called restricted Hamiltonian theory, where the equations of motion of a field theory are formulated by means of a multisymplectic structure, a kind of generalization of the symplectic form to the multidimensional context, and even a constraint algorithm working in this context has been proposed. In the present article we will try to provide partial aswers to two questions intimately related with these issues: First, to assign multisymplectic spaces to variational problems in the Griffiths formalism in such a way that the equations of motion can be written as restricted Hamiltonian systems, and second, to propose a covariant Dirac-like algorithm suitable to work with them; it must be recalled that given the Griffiths formalism contains the classical variational problems as particular instances, it yields to a novel covariant algorithm for deal with constraints in field theory. Moreover, in this formulation the constraint algorithm becomes simply the Cartan algorithm designed for deal with Pfaffian systems.

Cartan algorithm and Dirac constraints for Griffiths variational problems
Hernán Cendra, Santiago Capriotti
Preprint Arxiv