Hamiltonian dynamics of gauge systems

Gauge systems are approached by deemphasizing the role of the gauge group and replacing it by a subgroup of point transformations on the phase space scrS=$T^{\mathrm{*}}$scrM which is a cotangent bundle over a ‘‘big’’ configuration manifold scrM. These transformations are generated by a proper subalgebra scrV of the Poisson algebra of dynamical variables linear in the momenta. The orbits of scrV which lie on the constraint surface scrC, on which all υ from scrV vanish, form the physical phase space scrs. Observables are identified with dynamical variables on scrS which are constant along the orbits in scrC, and physical variables are identified with equivalence classes of observables. Special observables (including the Hamiltonian of the system) are at most quadratic in the momenta. The kinetic part of the Hamiltonian endows scrM with a metric which, together with the gauge algebra scrV, leads to a unique splitting of all special observables into standard physical parts and gauge parts. The splitting also leads to observables which represent conjugate canonical variables in the physical space. The Poisson brackets of all special observables can be explicitly evaluated, and the gauge theory can be explicitly reduced to a physical theory. The canonical formalism is manifestly covariant under point transformations in scrS and in scrs, and under changes of the basis in scrV. It enables us to construct a covariant factor ordering leading to a consistent canonical quantization of gauge systems.