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Modern Physics Letters A

GEOMETRY-INDUCED YANG-MILLS FIELDS IN CONSTRAINED QUANTUM MECHANICS

Paolo Maraner, C. Destri

Published Mar 21, 1993• 32 citations
Publisher

Abstract

We derive the effective Hamiltonian for a quantomechanical system constrained to move on a submanifold M of its configuration space R n by a confining potential V. Besides potential terms proportional to the intrinsic and mean curvature of M the restriction to the constraint produce the minimal interaction with a geometry-induced Yang-Mills field.

DOI
10.1142/s0217732393000891

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We derive the effective Hamiltonian for a quantomechanical system constrained to move on a submanifold M of its configuration space R n by a confining potential V. Besides potential terms proportional to the intrinsic and mean curvature of M the restriction to the constraint produce the minimal interaction with a geometry-induced Yang-Mills field.
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.
We discuss the motion of a quantum-mechanical system constrained to move on an arbitrary submanifold M of its configuration space R n by a real confining potential. A complete perturbative expansion for the Hamiltonian describing the dynamics of the system is obtained in terms of the quantities characterizing the intrinsic and extrinsic geometric properties of the constraint's surface M. In accordance with Heisenberg's principle the zeroth-order term of the expansion represents strong fluctuations of the system in directions normal to M, whereas the first-order term is naturally interpreted as the Hamiltonian describing the effective dynamics along the constraint's surface. The effective motion on M results in coupling with Abelian/non-Abelian gauge fields and quantum potentials. The rest of the perturbative expansion allows one to take into account further interactions between normal and effective degrees of freedom. As a concrete example we consider the dynamics of an electron constrained on a circle by a hypothetical semiconductor device. Dunham's expansion for the rotovibrational energy of a rigid diatom is also obtained.
A realistic analysis shows that constraining a quantomechanical system produces the effective dynamics to be coupled with abelian/nonabelian gauge fields and quantum potentials induced by the intrinsic and extrinsic geometrical properties of the constraint’s surface. This phenomenon is observable in the effective rotational motion of some simple polyatomic molecules. By considering specific examples it is shown that the effective Hamiltonians for the nuclear rotation of linear and symmetric top molecules are equivalent to that of a charged system moving in a background magnetic-monopole field. For spherical top molecules an explicit analytical expression of a non-abelian monopole-like field is found. Quantum potentials are also relevant for One of the first questions addressed by researchers who first studied the properties of surfaces embedded in the three-dimensional euclidean space, at
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.
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.