Evolution, symmetry, and canonical structure in dynamics

Abstract

The behavior of symmetries of classical equations of motion under quantization is studied from a new point of view. GL(3, R), which is an invariance group of the linear equations of motion for the nonrelativistic free particle as well as the isotropic harmonic oscillator, is imposed as a group of automorphisms on acceptable Poisson brackets, and the consequences are examined in detail. The six independent variables of the classical system arrange themselves, in each acceptable bracket, into one canonical pair and four neutral elements. Consequences of this for the equations of motion, existence of a Hamiltonian, breakup of the states of motion into superselected sectors due to existence of neutral elements, and determination of the canonically realized subgroup of GL(3, R) are all discussed. The possible relevance of this manner of symmetry breakdown for solid state and particle physics is pointed out.