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Progress of Theoretical Physics

Operator Ordering in Quantization on the Sphere

Masayoshi Nakamura, Hiroaki Minowa, Nobuhiko Mishima

Published Jan 1, 1988• 2 citations
Publisher

Abstract

The problem of the quantization of an elementary model with the second class constraints, O(n) rigid rotator model, is investigated by using the Dirac method, the projection operator method (POM) and the nonlinear quantum mechanics (NLQM). It is proved that the quantum effects, which are equivalent to those of NLQM, appear in the equation of motion of POM. Furthermore, it is shown that POM as well as NLQM preserves the isometric symmetry, which the system possesses in the classical level, through the quantization process.

DOI
10.1143/ptp.79.321

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The problem of the quantization of an elementary model with the second class constraints, O(n) rigid rotator model, is investigated by using the Dirac method, the projection operator method (POM) and the nonlinear quantum mechanics (NLQM). It is proved that the quantum effects, which are equivalent to those of NLQM, appear in the equation of motion of POM. Furthermore, it is shown that POM as well as NLQM preserves the isometric symmetry, which the system possesses in the classical level, through the quantization process.
The quantization of the second-class constraint systems is discussed within the projection operator method(POM) of constraint systems. Through the nonlocal representation of the constraint hyper-operators, new star-products are defined. Then, the projected operator-algebra of the quantized constraint systems is constructed with these star-products, and it is shown that the commutators and symmetrized products among the projected operators contain the quantum corrections due to the noncommutativity among operators.
A theory of quantization is given, for a model with symmetry, as the inverse of dequantization. Dequanrization is defined as tin� process of extracting the classical mechanics of a model from its quantal counterpart. It is shown that classical dynamics is embedded in quantum dynamics as a constrained subsystem. Thus, classical mechanics is obtained from quantum mechanics without taking limits of large-dimensional representations or letting h �? 0. The embedding shows that, under appropriate conditions, a quantal density can be identified with a classical state. The results are consistent with the conjecture that an algebraic model is quantized by constructing a unitary irrep of its spectrum generating algebra. Quantization of operators not in this algebra is determined by compatibility with dequantization: this method resolves the Dirac problem in a way that avoids the ambiguities of canonical quantization. Geometric quantization is used to express the vector coherent state inducing construction as a theory of quantization. The result provides physically useful quantum models for systems with an algebraic structure and intrinsic degrees of freedom. The results are illustrated by application to the S i (3) model of a rotor. An algebraic model of a rigid rotor is presented, anti the techniques of vector coherent state quantization are applied to give a quantization, including intrinsic degrees of freedom, that is compatible with the geometric structure of the corresponding classical model. An explicit dequantization of the asymmetric top, leading to Euler�s equations, illustrates the consistency of this quantization.
Systems with constraints pose problems when they are quantized. Moreover, the Dirac procedure of quantization prior to reduction is preferred. The projection operator method of quantization, which can be most conveniently described by coherent state path integrals, enables one to directly impose a regularized form of the quantum constraints. This procedure also overcomes conventional difficulties with normalization and second class constraints that invalidate conventional Dirac quantization procedures.
The problem of connecting systems with different numbers of degrees of freedom is discussed. Constraints appropriate for ``Bose'' and ``Fermi'' quantization are used to construct algebras of Dirac brackets associated with special solutions of the nonlinear complex oscillator. The constraints are shown to provide a basis for characterizing the elementary excitations of the oscillators. An alternative notion of quantization through a correspondence with an enveloping subalgebra of the Dirac brackets is introduced, a notion which simplifies the operator-ordering problem implied by the original Dirac brackets. The infinite- and the two-dimensional representations of the subalgebra are utilized to illustrate the quantization technique.
A quantum mechanical model with one bosonic degree of freedom is discussed in detail. Conventionally, when a quantum mechanical model is constructed, one must know the corresponding classical model. And by applying the correspondence between the classical Poisson brackets and the canonical commutator, the canonical quantization condition can be obtained. In the quantum model, study of the corresponding classical model is needed first. In this model, the Lagrangian is an operator gauge invariant. After localization, in order to keep gauge invariance, the operator gauge potential must be introduced. The Eular-Lagrange equation of motion of the dynamical argument gives the usual operator equation of motion. And the operator gauge potential just gjves a constraint. This constraint is just the usual canonical quantization condition.