Equations of motion, noncommutativity and quantization

Abstract

We study the relation between a given set of equations of motion in configuration space and a Poisson bracket. A Poisson structure is consistent with the equations of motion if the symplectic form satisfy some consistency conditions. When the symplectic structure is commutative these conditions are the Helmholtz integrability equations for the nonrestricted inverse problem of the calculus of variations [8]. We have found the corresponding consistency conditions for the symplectic noncommutative case.

A Variational Formulation of Symplectic Noncommutative Mechanics

“…where A and C are defined through the symplectic form (15). In summary we conclude that the θ-deformed Helmholtz conditions (17) are in fact the Helmholtz conditions associated to a θ-deformed Newtonian system (24) where N and M are related with the symplectic form σ by the identification (26).…”

“…The aim of the present note is to extend our previous investigation [14] where we have studied the dynamical compatibility between a Poisson bracket and a given set of second order equations of motion. We have found that if the Poisson bracket is noncommutative…”

“…Under these assumptions we will present in section 3 our central result that the NCC of our previous work [14] are indeed the Helmholtz conditions associated with a system of the form (2). We will find also the relation between the matrices N and M and the symplectic structure…”

“…As the ultimate goal is to understand the inherent space noncommutativity we propose a variational principle for noncommutative dynamical systems in configuration space, based on results of our previous work [14]. We hope that this variational formulation in configuration space can be of help to elucidate the definition of some global and dynamical properties of classical and quantum noncommutative space.…”

A Chiral Schwinger Model, Its Constraint Structure and Applications to Its Quantization

“…Now it is required that the primary constraint Ω 1 be preserved in time under the action of the Hamiltonian H,π 9) where these brackets denote the standard Poisson bracket defined as,…”