Hamiltonian constraints and Dirac observables

Summary

Presymplectic manifolds underlie all relevant physical theories, since all of them are, or may be, described by singular Lagrangians [1] and therefore by Dirac-Bergmann constraints [2] in their Hamiltonian description. In Galilean physics both Newtonian mechanics [3] and gravity [4] have been reformulated in this framework. In particular one obtains a multitime formulation of non-relativistic particle systems, which generalizes the non-relativistic limit of predictive mechanics [5] and helps one to understand features unavoidable at the relativistic level, where each particle, due to manifest Lorentz covariance, has its own time variable. Instead, all both special and general relativistic theories are always described by singular Lagrangians. See the review in Ref.[6] and Ref.[7] for the so-called multitemporal method for studying systems with first class constraints (second class constraints are not considered here).

The basic idea relies on Shanmugadhasan canonical transformations [8], namely one tries to find a new canonical basis in which all first class constraints are replaced by a subset of the new momenta [when the Poisson algebra of the original first class constraints is not Abelian, one speaks of Abelianization of the constraints]; then the conjugate canonical variables are Abelianized gauge variables and the remaining canonical pairs are special Dirac observables in strong involution with both Abelian constraints and gauge variables. These Dirac observables, together with the Abelian gauge variables, form a local Darboux basis for the presymplectic manifold [9] defined by the first class constraints (maybe it has singularities) and coisotropically embedded in the ambient phase space when there is no mathematical pathology.