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Journal of Physics A Mathematical and Theoretical

Quantizing conditionally variational nonholonomic systems

Oscar E. Fernandez

Published Jul 15, 2014• 3 citations
Publisher

Abstract

In this paper we present a quantization procedure for a class of nonholonomic systems—briefly, mechanical systems subject to nonintegrable constraints on the velocities—whose reduced mechanics is Hamiltonian. We illustrate the theory developed through several examples, one of which demonstrates how the classical nonholonomic constraints are sometimes manifested quantum mechanically as a shift in the ground state energy of the system.

DOI
10.1088/1751-8113/47/30/305206

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In this paper we present a quantization procedure for a class of nonholonomic systems—briefly, mechanical systems subject to nonintegrable constraints on the velocities—whose reduced mechanics is Hamiltonian. We illustrate the theory developed through several examples, one of which demonstrates how the classical nonholonomic constraints are sometimes manifested quantum mechanically as a shift in the ground state energy of the system.
In this Letter, we consider the problem of quantizing a nonholonomic system. This is highly nontrivial since such a system, which is subject to nonholonomic constraints, is not variational (or Hamiltonian). Our approach is to couple the system to a field which enforces the constraint in a suitable limit. We consider a simple but representative nonholonomic system, the Chaplygin sleigh. We then quantize the full (Hamiltonian) system. This system exhibits a key complicating feature of some nonholonomic systems-internal dissipative dynamics.
Nonholonomic mechanics describes systems subject to non-integrable velocity constraints, such as rolling bodies and skating motion. These systems generally lack a canonical Hamiltonian formulation, obstructing standard quantization methods. Here we quantize nonholonomic systems as Markovian open quantum systems, with the nonholonomic constraint appearing in a large-dissipation limit. We find explicit Lindblad superoperators that reproduce the classical dynamics of the Chaplygin sleigh and the Suslov problem in the semiclassical limit. The master equation is numerically simulated, and the covariance is shown to satisfy a relation predicted by the theory of metastability in open quantum systems.
This paper discusses the Hamiltonian aspects of the quantum theory of constrained Hamiltonian systems; in particular, we extend results of [9, 6] to such systems. In this paper, we usually refer to these systems as Hamilton–Dirac systems, because it was Dirac who first defined them (under the name of generalized Hamiltonian systems). Such systems arise in the application of the Legendre transform to unconstrained Lagrangian systems with singular Lagrange function (for which the Legendre transform is noninvertible). It should be emphasized that constrained Hamiltonian systems bear no relation to Lagrangian systems with holonomic constraints; however, them can be regarded as objects dual to Lagrangian systems with nonholonomic constraints. Namely, the Legendre transform turns Lagrangian systems with nonholonomic constraints into unconstrained Hamiltonian systems with singular Hamilton function
Introduction The main theme of this survey is the quantization of classical mechanical systems in terms of algebraic geometry. The aim is thus to relate questions of mathematics and theoretical physics. We first recall briefly the main problems and methods which turn us to study the new subject. Quantization itself is the main problem of theoretical physics. The need to introduce and develop it was dictated by the creators of quantum theory. According to the ‘Copenhagen philosophy’, the physical predictions of a quantum theory must be formulated in terms of classical concepts (the first sentence of Woodhouse; here we quote the beginning of this survey). Thus in addition to the usual structures (Hilbert space, unitary transformations, selfadjoint operators …, see, for example, Landau and Lifschitz) any reasonable quantum theory has to admit an appropriate passage to a classical limit under which quantum observables are transferred to their classical analogues. However, as Dirac pointed out at the beginning of the quantum age, the correspondence between quantum theory and classical theory should be based not just on numerical coincidences taking place in the limit h → ∞, but on an analogy between their mathematical structures. Classical theory does approximate the quantum theory, but it does more – it supplies a framework for some interpretation of quantum theory. Using this idea, we can in general understand a quantization procedure as a correspondence between classical and quantum theories. In this sense, quantization of the classical mechanical systems is a movement in one direction, while taking the quasiclassical limit goes in the opposite direction.