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Journal of Mathematical Physics

Quantization of dynamical systems subject to second class constraints

Masayoshi Nakamura, Hiroaki Minowa

Published Jan 1, 1993• 5 citations
Publisher

Abstract

It is shown that the operator formulation of the quantization of dynamical systems subject to second class constraints by Batalin and Fradkin (BF method) is equivalent to the projection-operator method (POM), which is the alternative method of the quantization of such systems, at the commutator level. It is also shown that the commutators in the BF method as well as those in the POM contain a quantum effect, which is completely missed in the usual quantization scheme with the Dirac brackets.

DOI
10.1063/1.530386

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It is shown that the operator formulation of the quantization of dynamical systems subject to second class constraints by Batalin and Fradkin (BF method) is equivalent to the projection-operator method (POM), which is the alternative method of the quantization of such systems, at the commutator level. It is also shown that the commutators in the BF method as well as those in the POM contain a quantum effect, which is completely missed in the usual quantization scheme with the Dirac brackets.
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.
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 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.
Using the projection operator method, the quantization of a dynamical system constrained to a submanifold MN−1 embedded in the Euclidean space RN is investigated. The resultant Hamiltonian contains the quantum correction terms which are missed in the usual approach with Dirac's formalism. Such correction terms are investigated from a geometrical point of view in the case that the system is constrained to a 2-dimensional orientatable compact curved spaces M2 embedded in R3. It is shown that the curvatures of these submanifolds are closely related to the quantum correction terms.
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.