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arXiv:quant-ph/0511142 · The Journal of Chemical Physics

Statistical mechanics of quantum-classical systems with holonomic constraints

Alessandro Sergi

Published Jan 10, 2006• 14 citations
arXivHTMLPDF

Abstract

The statistical mechanics of quantum-classical systems with holonomic constraints is formulated rigorously by unifying the classical Dirac bracket and the quantum-classical bracket in matrix form. The resulting Dirac quantum-classical theory, which conserves the holonomic constraints exactly, is then used to formulate time evolution and statistical mechanics. The correct momentum-jump approximation for constrained systems arises naturally from this formalism. Finally, in analogy with what was found in the classical case, it is shown that the rigorous linear-response function of constrained quantum-classical systems contains nontrivial additional terms which are absent in the response of unconstrained systems.

DOI
10.1063/1.2159477

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The statistical mechanics of quantum-classical systems with holonomic constraints is formulated rigorously by unifying the classical Dirac bracket and the quantum-classical bracket in matrix form. The resulting Dirac quantum-classical theory, which conserves the holonomic constraints exactly, is then used to formulate time evolution and statistical mechanics. The correct momentum-jump approximation for constrained systems arises naturally from this formalism. Finally, in analogy with what was found in the classical case, it is shown that the rigorous linear-response function of constrained quantum-classical systems contains nontrivial additional terms which are absent in the response of unconstrained 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
The statistical mechanics of systems whose evolution is governed by mixed quantum-classical dynamics is investigated. The algebraic properties of the quantum-classical time evolution of operators and of the density matrix are examined and compared to those of full quantum mechanics. The equilibrium density matrix that appears in this formulation is stationary under the dynamics and a method for its calculation is presented. The response of a quantum-classical system to an external force which is applied from the distant past when the system is in equilibrium is determined. The structure of the resulting equilibrium time correlation function is examined and the quantum-classical limits of equivalent quantum time correlation functions are derived. The results provide a framework for the computation of equilibrium time correlation functions for mixed quantum-classical systems.
Abstract Interactions of a non-holonomic type are fundamentally different from interactions which can be treated as part of the Hamiltonian of a system. They usually lead to constraints which do not commute with the Hamiltonian, and cause important alterations in the development of a state vector. This paper deals with the Heisenberg equations of motion by analogy with classical dynamics using the Poisson bracket formalism of a previous paper (Eden 1951). The Schrödinger equation is investigated in co-ordinate representation, and it is shown that the wave function will have a non-integrabie phase factor or quasi phase. The quasi phase leads to an indefiniteness in the wave function, but does not violate the fundamental laws of quantum mechanics nor lead to any ambiguity in the physical interpretation of the theory. The relation between the Schrödinger and the Heisenberg equations shows that the Schrödinger treatment is also consistent with the classical analogue. If there is a given initial probability that the non-holonomic system has co-ordinates q(0)r, then there will be the same probability that the wave function in the subsequent motion will be zero except in a certain region of co-ordinate space. This region is the part of co-ordinate space which is accessible in the classical theory from the point q(0)r.
Abstract We construct an operational formulation of classical mechanics without presupposing previous results from analytical mechanics. In doing so, we rediscover several results from analytical mechanics from an entirely new perspective. We start by expressing the position and velocity of point particles as the eigenvalues of self-adjoint operators acting on a suitable Hilbert space. The concept of holonomic constraint is shown to be equivalent to a restriction to a linear subspace of the free Hilbert space. The principal results we obtain are: (1) the Lagrange equations of motion are derived without the use of D’Alembert or Hamilton principles, (2) the constraining forces are obtained without the use of Lagrange multipliers, (3) the passage from a position–velocity to a position–momentum description of the movement is done without the use of a Legendre transformation, (4) the Koopman–von Neumann theory is obtained as a result of our ab initio operational approach, (5) previous work on the Schwinger action principle for classical systems is generalized to include holonomic constraints.
In this paper a new formulation of quantum dynamics of totally constrained systems is developed, in which physical quantities representing time are included as observables. In this formulation the hamiltonian constraints are imposed on a relative probability amplitude functional Ψ which determines the relative probability for each state to be observed, instead of on the state vectors as in the conventional Dirac quantization. This leads to a foliation of the state space by linear manifolds on each of which Ψ is constant, and dynamics is described as linear mappings among acausal subspaces which are transversal to these linear manifolds. This is a quantum analogue of the classical statistical dynamics of totally constrained systems developed in the previous paper. It is shown that if the von Neumann algebra C generated by constants of motion is of type I, Ψ can be consistently normalizable on the acausal subspaces on which a factor subalgebra of C is represented irreducibly, and the mappings among these acausal subspaces are conformal. How the formulation works is illustrated by simple totally constrained systems with a single constraint such as the parametrized quantum mechanics, a relativistic free particle in Minkowski and curved spacetimes, and a simple minisuperspace model. It is pointed out that the inner product of the relative probability amplitudes induced from the original Hilbert space picks up a special decomposition of the wave functions to the positive and the negative frequency modes.
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.