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Progress of Theoretical Physics

Quantum Corrections in Constrained Systems

N. Okamoto, Masayoshi Nakamura

Published Jul 1, 1996• 4 citations
Publisher

Abstract

Using the projection operator method, the quantization of a dynamical system constrained to a submanifold MN−1 embedded in the Euclidean space RN is investigated. The resultant Hamiltonian contains the quantum correction terms which are missed in the usual approach with Dirac's formalism. Such correction terms are investigated from a geometrical point of view in the case that the system is constrained to a 2-dimensional orientatable compact curved spaces M2 embedded in R3. It is shown that the curvatures of these submanifolds are closely related to the quantum correction terms.

DOI
10.1143/ptp.96.235

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•Oct 20, 2008•15 citations
Using the projection operator method, the quantization of a dynamical system constrained to a submanifold MN−1 embedded in the Euclidean space RN is investigated. The resultant Hamiltonian contains the quantum correction terms which are missed in the usual approach with Dirac's formalism. Such correction terms are investigated from a geometrical point of view in the case that the system is constrained to a 2-dimensional orientatable compact curved spaces M2 embedded in R3. It is shown that the curvatures of these submanifolds are closely related to the quantum correction terms.
We analyze constrained quantum systems where the dynamics do not preserve the constraints. This is done, in particular, for the restriction of a quantum particle in Rn to a curved submanifold, and we propose a method of constraining and dynamics adjustment that produces the right Hamiltonian on the submanifold when tested on known examples. This method will be the germ of a “Dirac algorithm for quantum constraints.” We generalize it to the situation where the constraint is a general self-adjoint operator with some additional structures.
We derive an effective Hamiltonian for a quantum system constrained to a submanifold (the constraint manifold) of configuration space (the ambient space) by an infinite restoring force. We pay special attention to how this Hamiltonian depends on quantities which are external to the constraint manifold, such as the extrinsic curvature of the constraint manifold, the curvature of the ambient space, and the constraining potential. In particular, we find the remarkable fact that the twisting of the constraining potential appears as a gauge potential in the constrained Hamiltonian. This gauge potential is closely related to the geometric phase originally discussed by Berry. The constrained Hamiltonian also contains an effective potential depending on the extrinsic curvature of the constraint manifold, the curvature of the ambient space, and the twisting of the constraining potential. The general nature of our analysis allows applications to a wide variety of problems, such as rigid molecules, the evolution of molecular systems along reaction paths, and quantum strip waveguides.
A general prescription for the treatment of constrained quantum motion is outlined. We consider in particular constraints defined by algebraic submanifolds of the quantum state space. The resulting formalism is applied to obtain solutions to the constrained dynamics of systems of multiple spin particles. When the motion is constrained to a certain product space containing all of the energy eigenstates, the dynamics thus obtained are quasi-unitary in the sense that the equations of motion take a form identical to that of unitary motion, but with different boundary conditions. When the constrained subspace is a product space of disentangled states, the associated motion is more intricate. Nevertheless, the equations of motion satisfied by the dynamical variables are obtained in closed form.
A general prescription for the treatment of constrained quantum motion is outlined. We consider in particular constraints defined by algebraic submanifolds of the quantum state space. The resulting formalism is applied to obtain solutions to the constrained dynamics of systems of multiple spin particles. When the motion is constrained to a certain product space containing all of the energy eigenstates, the dynamics thus obtained are quasi-unitary in the sense that the equations of motion take a form identical to that of unitary motion, but with different boundary conditions. When the constrained subspace is a product space of disentangled states, the associated motion is more intricate. Nevertheless, the equations of motion satisfied by the dynamical variables are obtained in closed form.