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1 Can we look at the quantisation rules as constraints?

E. Gozzi

Published Jan 1, 1997• 1 citation
OpenAlex

Abstract

In this paper we explore the idea of looking at the Dirac quantisation conditions as ¯h-dependent constraints on the tangent bundle to phase-space. Starting from the path-integral version of classical mechanics and using the natural Poisson brackets structure present in the cotangent bundle to the tangent bundle of phase- space, we handle the above constraints using the standard theory of Dirac for constrained systems. The hope is to obtain, as total Hamiltonian, the Moyal operator of time-evolution and as Dirac brackets the Moyal ones. Unfortunately the program fails indicating that something is missing. We put forward at the end some ideas for future work which may overcome this failure. 1.

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On Dirac Quantisation rules and the trace anomaly

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A tutorial on Dirac quantisation by analysing the problem of a ball on an inclined plane as a Hamiltonian system with constraints

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Dirac versus reduced phase space quantization

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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.

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Bozhidar Zakhariev Iliev

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Weyl-Wigner-Moyal formulation of a Dirac quantized constrained system

Louis-Martinez, D

An extension of the Weyl-Wigner-Moyal formulation of quantum mechanics suitable for a Dirac quantized constrained system is proposed. In this formulation, quantum observables are described by equivalent classes of Weyl symbols. The Weyl product of these equivalent classes is defined. The new Moyal bracket is shown to be compatible with the Dirac bracket for constrained systems.

•Jan 1, 2000•5 citations
In this paper we explore the idea of looking at the Dirac quantisation conditions as ¯h-dependent constraints on the tangent bundle to phase-space. Starting from the path-integral version of classical mechanics and using the natural Poisson brackets structure present in the cotangent bundle to the tangent bundle of phase- space, we handle the above constraints using the standard theory of Dirac for constrained systems. The hope is to obtain, as total Hamiltonian, the Moyal operator of time-evolution and as Dirac brackets the Moyal ones. Unfortunately the program fails indicating that something is missing. We put forward at the end some ideas for future work which may overcome this failure. 1.
In this article I shall clarify various aspects of the Dirac quantisation rules of 1930\cite{Dirac}, namely (i) the choice of antisymmetric Poisson brackets, (ii) the first quantisation Rule 1 (iii) the second quantisation Rule 2, and their relations to the trace anomaly. In fact in 1925 Dirac already had a preliminarily formulation of these rules \cite{Dirac3}. Using them, he had independently rediscovered the Born-Jordan quantisation rule \cite{BornJordan1925} and called it the quantum condition. This is the best known and undoubtedly most significant of the canonical quantisation rules of quantum mechanics. We shall discuss several violations of the Poisson-Lie algebra (assumed by Dirac), starting from antisymmetry, which is the first criterion for defining a Lie algebra. Similar violations also occur for the Leibniz's rule and the Jacobi identity, the latter we shall also prove for all our quantum Poisson brackets. That none of these violations jeopardised Dirac's ingenious original derivation \cite{Dirac} of his first quantisation Rule 1, is quite remarkable. This is because the violations are all of higher orders in ℏ\hbarℏ. We shall further show that (ii) does not automatically lead to a trace anomaly for certain bounded integrable operators. Several issues that are both pedagogical and foundational arising from this study show that quantum mechanics is still not a finished product. I shall briefly mention some attempts and options to complete its development.
In this paper, we present a detailed review/analysis of the Dirac quantisation of Hamiltonian systems with constraints. To this end, we use, as a guide, the physical example provided by the dynamics of a solid ball rolling, without slipping, down an inclined plane under the action of gravity. After all, however simple this physical system may be, it provides a rich framework for this analysis since, in addition to allowing us to discuss scenarios involving holonomic and non-holonomic constraints, it is also a gauge system. Indeed, due to this latter fact, we have carefully detailed how the transition, from classical to quantum mechanics, must be guided by the Dirac-Bergmann algorithm and by the consequent replacement of Dirac brackets with commutators. As a central result, we demonstrate that the restriction of the Hamiltonian operator of this system with constraints to the physical Hilbert subspace (which is identified with the quantisation of these constraints) reproduces the same Schrödinger equation that can be originally obtained in intrinsic terms, a fact that only reinforces the consistency of the Dirac quantisation method.
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.
We propose a new fibre bundle formulation of the mathematical base of relativistic quantum mechanics. At the present stage the bundle form of the theory is equivalent to its conventional one, but it admits new types of generalizations in different directions. In the present first part of our investigation we consider the time-dependent or Hamiltonian approach to bundle description of relativistic quantum mechanics. In it the wavefunctions are replaced by (state) liftings of paths or sections along paths of a suitably chosen vector bundle over space-time whose (standard) fibre is the space of the wavefunctions. Now the quantum evolution is described as a linear transportation (by means of the evolution transport along paths in the space-time) of the state liftings/sections in the (total) bundle space. The equations of these transportations turn to be the bundle versions of the corresponding relativistic wave equations.
The quantization of a dynamical system with time-dependent constraints or time-dependent Hamiltonians presents particular difficulties not only from a technical, but also from a theoretical point of view. We give here an approach to the evolution of such a quantum system in terms of star products on symplectic or canonical manifolds. These star products are particular deformations of the usual product of functions, deformations that start with the Poisson bracket. The introduction of the star products corresponds to a generalization of the Weyl–Wigner quantization on a flat phase space. This approach leads to the introduction of ‘‘two times’’ denoted t and τ, playing different roles; the time t plays the role of a geometrical time corresponding to the control, while the other, τ, which parametrizes the evolution, plays the role of a dynamical time. The classical as well as the quantum results point out the role played by the variable (t+τ). This reflects the coherence of the formalism.