dorsal/arxiv
View SchemaQuantizing Constrained Systems: New Perspectives
| Authors | L. Kaplan, N. T. Maitra, E. J. Heller |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9810037 |
| URL | https://arxiv.org/abs/quant-ph/9810037 |
| DOI | 10.1103/PhysRevA.56.2592 |
| Journal | Phys.Rev. A56 (1997) 2592 |
Abstract
We consider quantum mechanics on constrained surfaces which have non-Euclidean metrics and variable Gaussian curvature. The old controversy about the ambiguities involving terms in the Hamiltonian of order hbar^2 multiplying the Gaussian curvature is addressed. We set out to clarify the matter by considering constraints to be the limits of large restoring forces as the constraint coordinates deviate from their constrained values. We find additional ambiguous terms of order hbar^2 involving freedom in the constraining potentials, demonstrating that the classical constrained Hamiltonian or Lagrangian cannot uniquely specify the quantization: the ambiguity of directly quantizing a constrained system is inherently unresolvable. However, there is never any problem with a physical quantum system, which cannot have infinite constraint forces and always fluctuates around the mean constraint values. The issue is addressed from the perspectives of adiabatic approximations in quantum mechanics, Feynman path integrals, and semiclassically in terms of adiabatic actions.
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"abstract": "We consider quantum mechanics on constrained surfaces which have\nnon-Euclidean metrics and variable Gaussian curvature. The old controversy\nabout the ambiguities involving terms in the Hamiltonian of order hbar^2\nmultiplying the Gaussian curvature is addressed. We set out to clarify the\nmatter by considering constraints to be the limits of large restoring forces as\nthe constraint coordinates deviate from their constrained values. We find\nadditional ambiguous terms of order hbar^2 involving freedom in the\nconstraining potentials, demonstrating that the classical constrained\nHamiltonian or Lagrangian cannot uniquely specify the quantization: the\nambiguity of directly quantizing a constrained system is inherently\nunresolvable. However, there is never any problem with a physical quantum\nsystem, which cannot have infinite constraint forces and always fluctuates\naround the mean constraint values. The issue is addressed from the perspectives\nof adiabatic approximations in quantum mechanics, Feynman path integrals, and\nsemiclassically in terms of adiabatic actions.",
"arxiv_id": "quant-ph/9810037",
"authors": [
"L. Kaplan",
"N. T. Maitra",
"E. J. Heller"
],
"categories": [
"quant-ph",
"chao-dyn",
"nlin.CD"
],
"doi": "10.1103/PhysRevA.56.2592",
"journal_ref": "Phys.Rev. A56 (1997) 2592",
"title": "Quantizing Constrained Systems: New Perspectives",
"url": "https://arxiv.org/abs/quant-ph/9810037"
},
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