Classical Dirac Observables: the Emergence of Rest-Frame Particle and Field Theories

Talk at the “Pacific Conference on Gravitation and Cosmology”, 1-6 February 1996, Seoul, Korea The second Noether theorem [1] and Dirac-Bergmann constraint theory [2,3] are the basis respectively of the Lagrangian and Hamiltonian formulations of all relativistic physical systems [4]. The need of redundant variables, to be reduced due to the presence of either first and/or second class constraints, is connected with requirements like manifest covariance and minimal coupling (the gauge principle). Also Newton mechanics [5] and Newton gravity with Galilean general covariance [6] can be reformulated in this language at the nonrelativistic level. Therefore, at the Hamiltonian level the fundamental geometric structure behind our description of physics is presymplectic geometry [7,8], the theory of a closed degenerate two-form (no definition of Poisson Brackets). Curiously, it has been much less studied than its dual structure, Poisson geometry, namely the theory of a closed (i.e. with vanishing Schouten-Nijenhuis bracket with itself) degenerate bivector (existence of degenerate Poisson brackets) [9]; in absence of degeneracy, Poisson manifolds coincide with symplectic manifolds. It is important to understand the properties of presymplectic manifolds embedded into ambient phase spaces, so to utilize their natural Poisson brackets as in the Dirac-Bergmann theory. In particular, for physical applications, Darboux charts of presymplectic manifolds are needed, for instance in the definition of the Faddeev-Popov measure in the path integral.