Poisson-Lie Structures and Quantisation with Constraints

Abstract

We develop here a simple quantisation formalism that make use of Lie algebra properties of the Poisson bracket. When the brackets $\{H,ϕ_i\}$ and $\{ϕ_i,ϕ_j\}$, where $H$ is the Hamiltonian and $ϕ_i$ are primary and secondary constraints, can be expressed as functions of $H$ and $ϕ_i$ 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.