Poisson-Lie Structures and Quantisation with Constraints

We develop here a simple quantisation formalism that make use of Lie algebra properties of the Poisson bracket. When the brackets {H,φi} and {φi, φj}, where H is the Hamiltonian and φi are primary and secondary constraints, can be expressed as functions of H and φi themselves, the Poisson bracket defines a Poisson-Lie structure. When this algebra has a finite dimension a system of first order partial differential equations is established whose solutions are the observables of the theory. The method is illustrated with a few examples. 1. The quantisation of systems with constraints is old as the quantum mechanics itself. The first such problem brilliantly solved was the finding of the hydrogen atom spectrum by Pauli in 1926 [1]. Enforcing the constraints in classical mechanics has a satisfactory solution [2, 3], but this is no more true in quantum mechanics. The constraints, i.e. a set of functions φi(q, p) = 0 , i = 1, 2, . . . , m (1.1) restrict the motion of the classical system to a manifold embedded in the initial Euclidean phase space and in consequence the canonical quantisation rules [qi, pj] = ih̄δij are no more sufficient for the quantum description of the physical system. 1 email: [email protected]