Quantization based on generalized Poisson brackets

Abstract

We regard classical mechanical equations of motion to be a set of first-order ordinary differential equations that include Hamiltonian systems as special cases. Generalized Poisson brackets are defined through the equivalent Jacobian form of equations of motion and the constants of the motion of the system. Fundamental quantum conditions are introduced through the generalized Poisson brackets. The classicalc-number dynamical variables are converted into quantumq-number dynamical operators by following the Dirac procedure. Quantum Heisenberg equations of motion are then obtained from the corresponding classical equations of motion. Some simple examples are studied in some detail. From the results of the simple examples there seems to exist a certain kind of relativity among the constants of the motion in the following sense: any one of the constants of the motion can be selected to be the motion generating Hamiltonian and the rest of the constants of the motion will all participate in defining the fundamental quantum conditions. The quantum Heisenberg equations of motion so obtained have all the same formal appearance as the corresponding classical equations.