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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Evolution, symmetry, and canonical structure in dynamics

Narasimhaiengar Mukunda, A. P. Balachandran, Jan S. Nilsson, E. C. G. Sudarshan, F. Zaccaria

Published May 15, 1981• 9 citations
Publisher

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.

DOI
10.1103/physrevd.23.2189

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The behavior of symmetries of classical equations of motion under quantization is studied from a new point of view. GL(3, RRR), 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, RRR) are all discussed. The possible relevance of this manner of symmetry breakdown for solid state and particle physics is pointed out.
A formalism describing the dynamics of classical and quantum systems from a group theoretical point of view is presented. We apply it to the simple example of the classical free particle. The Galileo group G is the symmetry group of the free equations of motion. Consideration of the free particle Lagrangian semi-invariance under G leads to a larger symmetry group, which is a central extension of the Galileo group by the real numbers. We study the dynamics associated with this group, and characterize quantities like Noether invariants and evolution equations in terms of group geometric objects. An extension of the Galileo group by U(1) leads to quantum mechanics.
It is known that the equations of motion of a classical system do not in general determine its canonical description uniquely. Thus the corresponding quantum system obtained by canonical quantization is also ambiguous. We use this freedom to quantize a free nonrelativistic particle so that the corresponding quantum particle has half-integral or integral helicity. Methods are developed for finding inequivalent canonical descriptions of a given classical system. It is emphasized that classical symmetries can be broken at the quantum level by a suitable choice of the canonical formalism prior to quantization. It is suggested that this may provide a new mechanism for breaking internal symmetries at the quantum level.
One classical theory, as determined by an equation of motion or set of classical trajectories, can correspond to many unitarily {\em in}equivalent quantum theories upon canonical quantization. This arises from a remarkable ambiguity, not previously investigated, in the construction of the classical (and hence the quantized) Hamiltonian or Lagrangian. This ambiguity is illustrated for systems with one degree of freedom: An arbitrary function of the constants of motion can be introduced into this construction. For example, the nonrelativistic and relativistic free particles follow identical classical trajectories, but the Hamiltonians or Lagrangians, and the canonically quantized versions of these descriptions, are inequivalent. Inequivalent descriptions of other systems, such as the harmonic oscillator, are also readily obtained.
Relations between free motion on the GL + ( n , ℝ) group manifold and the dynamics of an n -particle system with spin degrees of freedom on a line interacting with a pairwise 1/sinh 2 x “potential” (Euler-Calogero-Sutherland model) are discussed within a Hamiltonian reduction. Two kinds of reductions of the degrees of freedom are considered: that which is due to continuous invariance and that which is due to discrete symmetry. It is shown that, upon projecting onto the corresponding invariant manifolds, the resulting Hamiltonian system represents the Euler-Calogero-Sutherland model in both cases.
We propose a new method of quantization of a wide class of dynamical systems that originates directly from the equations of motion. The method is based on the correspondence between the classical and the quantum Poisson brackets, postulated by Dirac. This correspondence applied to open (non-Hamiltonian) systems allows one to point out the way of transition from the quantum description based on the Lindblad equation to the dynamical description of their classical analogs by the equations of motion and vice versa. As the examples of using of the method we describe the procedure of the quantization of three widely considered dynamical systems: (1) the harmonic oscillator with friction, (2) the oscillator with a nonlinear damping that simulates the process of the emergence of the limit cycle, and (3) the system of two periodic rotators with a weak interaction that synchronizes their oscillations. We discuss a possible application of the method for a description of quantum fluctuations in Josephson junctions with a strong damping and for the quantization of open magnetic systems with a dissipation and a pumping.
Apart from the total energy, the two-dimensional isotropic harmonic oscillator possesses three independent constants of motion which, with the standard symplectic structure, generates a dynamical symmetry group isomorphic to SU(2). We show that, by suitably redefining the symplectic structure, any of these three constants of motion can be used as a Hamiltonian, and that the remaining two, together with the total energy, generate a dynamical symmetry group isomorphic to SU(1,1). We also show that the standard energy levels of the quantum two-dimensional isotropic harmonic oscillator and their degeneracies are obtained making use of the appropriate representations of SU(1,1), provided that the canonical commutation relations are modified according to the new symplectic structure. Whereas in classical mechanics the different symplectic structures lead to equivalent formulations of the equations of motion, in quantum mechanics the modifications of the commutation relations should be accompanied by modifications in the interpretation of the formalism in order to obtain results equivalent to those found with the common relations.