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arXiv:gr-qc/9601042 · Progress of Theoretical Physics

Dirac versus Reduced Quantization and Operator Ordering

K. Shimizu

Published Jan 1, 1997• 2 citations
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Abstract

We demonstrate an equivalence between Dirac quantization and reduced phase space quantization. The equivalence of the two quantization methods determines the operator ordering of the Hamiltonian. Some examples of the operator ordering are shown in simple models.
DOI
10.1143/ptp.97.153

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The relationship between the Dirac and reduced phase space quantizations is investigated for spin models belonging to the class of Hamiltonian systems having no gauge conditions. It is traced out that the two quantization methods may give similar, or essentially different physical results, and, moreover, it is shown that there is a class of constrained systems, which can be quantized only by the Dirac method. A possible interpretation of the gauge degrees of freedom is given. Introduction There are two main methods to quantize the Hamiltonian systems with first class constraints: the Dirac quantization [1] and the reduced phase space quantization [2], whereas two other methods, the path integral method [3, 2] and the BRST quantization [4] being the most popular method for the covariant quantization of gauge-invariant systems, are based on and proceed from them [2, 5]. The basic idea of the Dirac method consists in imposing quantum mechanically the first class constraints as operator conditions on the states for singling out the physical ones [1]. The reduced phase space quantization first identifies the physical degrees of freedom at the classical level by the factorization of the constraint surface with respect to the action of the gauge group, generated by the constraints. Then the resulting Hamiltonian system is quantized as a usual unconstrained system [2]. Naturally, the problem of the relationship of these two methods arises. It was discussed in different contexts in literature [6], and there is an opinion that the differences between the two quantization methods can be traced out to a choice of factor ordering in the construction of various physical operators.
The constrained Hamiltonian systems admitting no gauge conditions are considered. The methods for dealing with such systems are discussed and developed. As a concrete application, the relationship between the Dirac and reduced phase space quantizations is investigated for spin models belonging to the class of systems under consideration. It is found that the two quantization methods may give similar, or essentially different, physical results, and moreover a class of constrained systems, which can be quantized only by the Dirac method, is discussed. A possible interpretation of the gauge degrees of freedom is given.
The constrained Hamiltonian systems admitting no gauge conditions are considered. The methods to deal with such systems are discussed and developed. As a concrete application, the relationship between the Dirac and reduced phase space quantizations is investigated for spin models belonging to the class of systems under consideration. It is traced out that the two quantization methods may give similar, or essentially different physical results, and, moreover, a class of constrained systems, which can be quantized only by the Dirac method, is discussed. A possible interpretation of the gauge degrees of freedom is given.
The formalism of the second quantization is developed entirely within the framework of the Dirac-Jordan representation theory. The equivalence of this formalism and the methods of ordinary coordinate-spin space is shown, and the equations of "connection" are fully developed. It is shown that the "creation" and "annihilation" operators, and indeed all operators which do not commute with the quantized Hamiltonian and hence whose representatives cannot be explicitly given in a configuration space of a fixed number of dimensions are easily treated in terms of the general representation theory and arise quite naturally. Both types of statistics are considered, and the Fock-Dirac density matrix and self-consistent field are treated as a simple illustration.