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arXiv:2605.28878

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

M. F. Araujo de Resende, Thales Machado F

quant-phhep-thphysics.class-phSubmitted May 26, 2026• Announced May 27, 2026• 0 citations
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Abstract

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.

Comments
63 pages, 9 figures, submitted to publication

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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.
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.
The authors discuss the Dirac-bracket quantisation procedure for the nonrelativistic rigid rotator. In cartesian coordinates this system exhibits two important features which are also characteristic of the nonabelian gauge theories, it is constrained, and nonlinear. By comparing to the known solution in polar coordinates the author demonstrate that simple hermiticity conditions are sufficient to allow the unambiguous application of the Dirac quantisation method to such nonlinear dynamical systems.
The Dirac-Bergmann algorithm for the Hamiltonian analysis of constrained systems is a nice and powerful tool, widely used for quantization and non-perturbative counting of degrees of freedom. However, certain aspects of its application to systems with first-class constraints are often overlooked in the literature, which is unfortunate, as a naive treatment leads to incorrect results. In particular, when transitioning from the total to the extended Hamiltonian, the physical information encoded in the constrained modes is lost unless a suitable redefinition of gauge invariant quantities is made. An example of this is electrodynamics, in which the electric field gets an additional contribution to its longitudinal component in the form of the gradient of an arbitrary Lagrange multiplier. Moreover, Dirac's conjecture, the common claim that all first-class constraints are independent generators of gauge transformations, is somewhat misleading in the standard notion of gauge symmetry used in field theories. At the level of the total Hamiltonian, the true gauge generator is a specific combination of primary and secondary first-class constraints; in general, Dirac's conjecture holds only in the case of the extended Hamiltonian. The aim of the paper is primarily pedagogical. We review these issues, providing examples and general arguments. Also, we show that the aforementioned redefinition of gauge invariants within the extended Hamiltonian approach is equivalent to a form of the Stueckelberg trick applied to variables that are second-class with respect to the primary constraints.
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.
A framework for deriving equations of motion for constrained quantum systems is introduced and a procedure for its implementation is outlined. In special cases, the proposed new method, which takes advantage of the fact that the space of pure states in quantum mechanics has both a symplectic structure and a metric structure, reduces to a quantum analogue of the Dirac theory of constraints in classical mechanics. Explicit examples involving spin- particles are worked out in detail: in the first example, our approach coincides with a quantum version of the Dirac formalism, while the second example illustrates how a situation that cannot be treated by Dirac's approach can nevertheless be dealt with in the present scheme.
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.