An approach to quanitzation of constrained systems

The dynamics of a classical system, specified by a HamiltonianH₀(q,p) and a number of constraint functions_k(q,p) in Euclidean phase space (q,p), is described by means of a new HamiltonianH(q,p), which is an invariant of the (closed) Poisson-bracket Lie algebra (H₀,_k). Fixed values of_k (not necessarily zero) are given by initial conditions, and they are conserved along the trajectories determined by the Hamilton equations. The quantization is performed by the standard Heisenberg commutation relations in the embedding phase space, while the constraint functions are put in correspondence with constraint operators which generate the Lie algebra of quantum commutators. A subset of commuting constraint operators may be chosen to have certain values in the initial state; and as soon as the Hamiltonian operator is an invariant of the Lie algebra, these conditions are maintained permanently. Simple examples are presented. Systems with both Bose and Fermi degrees of freedom (and constraints) can be treated universally.