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Progress of Theoretical Physics

On the Canonical Variables Derived from the Schwinger Boson Representation for the Quantized Rotator: Classical Theory

Masao Yamamura

Published Sep 1, 1979• 6 citations
Publisher

Abstract

With the aid of the correspondence between commutator and Poisson bracket, a set of canonical variables is derived from the Schwinger boson representation for the quantized rotator. By the use of these variables, Euler's equations of motion are given through Hamilton's equations of motion in the case of the rotator. Further, wobbling motion is described in the classical sense. Finally, for the purpose of applying the ATDHB theory to a microscopic description of collective roation, a basic idea is given.

DOI
10.1143/ptp.62.681

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With the aid of the correspondence between commutator and Poisson bracket, a set of canonical variables is derived from the Schwinger boson representation for the quantized rotator. By the use of these variables, Euler's equations of motion are given through Hamilton's equations of motion in the case of the rotator. Further, wobbling motion is described in the classical sense. Finally, for the purpose of applying the ATDHB theory to a microscopic description of collective roation, a basic idea is given.
Conventionally, when we construct a quantum model, we must first know the corresponding classical model. Applying the correspondence between the classical Poisson brackets and the canonical commutator, we can find the canonical quantization condition. Through the example given in this paper, we will find that it is not necessary to do this. In the example, a Lagrangian operator is operator gauge invariant. After localization, in order to keep the operator gauge invariance of the operator action, we must introduce a gauge potential. The Euler - Lagrange equation of motion of q gives the usual operator equation of motion, and the gauge potential gives a constraint. This constraint is just the usual canonical quantization condition.
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.
The position operator in the Heisenberg representation obeys the usual classical equation of motion. The similarity between the Heisenberg equations of motion and the classical Hamiltonian equations of motion arises from the fact that the commutator of the operators for the physical quantities plays a similar role in quantum mechanics, to that of the Poisson bracket of the classical quantities A and B, in classical mechanics. Generally, the Heisenberg equations of motion are more difficult to solve than the corresponding classical equations because of the lack of commutation of quantum mechanical operators. The chapter explains the notion of a time-ordered product of operators, and then can be written much more compactly.
In this chapter we set up a classical Lagrangian and Hamiltonian dynamics for matrix models. The fundamental idea is to set up an analog of classical dynamics in which the phase space variables are non-commutative, and the basic tool that allows one to accomplish this is cyclic invariance under a trace. Since no assumptions about commutativity of the phase space variables (such as canonical commutators/anticommutators) are made at this stage, the dynamics that we set up is not the same as standard quantum mechanics. Quantum mechanical behavior will be seen to emerge only when, in Chapters 4 and 5, we study the statistical mechanics of the classical matrix dynamics formulated here. In Section 1.1, we introduce our basic notation for bosonic and fermionic matrices, and give the cyclic identities that will be used repeatedly throughout the book. In Section 1.2, we define the derivative of a trace quantity with respect to an operator, and give the basic properties of this definition. In Section 1.3, we use the operator derivative to formulate a Lagrangian and Hamiltonian dynamics for matrix models. In Section 1.4, we introduce a generalized Poisson bracket appropriate to trace dynamics, constructed from the operator derivative defined in Section 1.2, and give its properties and some applications. Finally, in Section 1.5 we discuss the relation between the trace dynamics time evolution equations, and the usual unitary Heisenberg picture equations of motion obtained when one assumes standard canonical commutators/anticommutators. Bosonic and fermionic matrices and the cyclic trace identities We shall assume finite-dimensional matrices, although ultimately an extension to the infinite-dimensional case may be needed. The matrix elements of these matrices will be constructed from ordinary complex numbers, and from complex anticommuting Grassmann numbers.