Unified Description and Canonical Reduction to Dirac’s Observables of the Four Interactions

The standard SU(3) × SU(2) × U(1) model of elementary particles, all its extensions with or without supersymmatry, all variants of string theory, all formulations of general relativity are described by singular Lagrangians. Therefore, their Hamiltonian formulation needs Dirac’s theory of 1st and 2nd class constraints [1] determining the submanifold of phase space relevant for dynamics: this means that the basic mathematical structure behind our description of the four interactions is presymplectic geometry (namely the theory of submanifolds of phase space with a closed degenerate 2-form; strictly speaking only 1st class constraints are associated with presymplectic manifolds: the 2nd class ones complicate the structure). For a system with 1st and 2nd class constraints the physical description becomes clear in coordinates adapted to the presymplectic submanifold. Locally in phase space, an adapted Darboux chart can always be found by means of Shanmugadhasan’s canonical transformations [2] [strictly speaking their existence is proved only for finite-dimensional systems, but they underlie the existence of the Faddeev-Popov measure for the path integral]. The new canonical basis has: i) as many new momenta as 1st class constraints (Abelianization of 1st class constraints); ii) their conjugate canonical variables (Abelianized gauge variables);