This paper deals with a general method for the reduction of quantum systems with symmetry. For a Riemannian manifold M admitting a compact Lie group G as an isometry group, the quotient space is not a smooth manifold in general but stratified into a collection of smooth manifolds of various dimensions. If the action of the compact group G is free, M is made into a principal fiber bundle with structure group G. In this case, reduced quantum systems are set up as quantum systems on the associated vector bundles over This idea of reduction fails, if the action of G on M is not free. However, the Peter–Weyl theorem works well for reducing quantum systems on M. When applied to the space of wave functions on M, the Peter–Weyl theorem provides the decomposition of the space of wave functions into spaces of equivariant functions on M, which are interpreted as Hilbert spaces for reduced quantum systems on Q. The concept of connection on a principal fiber bundle is generalized to be defined well on the stratified manifold M. Then the reduced Laplacian is well defined as a self-adjoint operator with the boundary conditions on singular sets of lower dimensions. Application to quantum molecular mechanics is also discussed in detail. In fact, the reduction of quantum systems studied in this paper stems from molecular mechanics. If one wishes to consider the molecule which is allowed to lie in a line when it is in motion, the reduction method presented in this paper works well.
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April 2000
Research Article|
April 01 2000
Reduction of quantum systems on Riemannian manifolds with symmetry and application to molecular mechanics
Shogo Tanimura;
Shogo Tanimura
Department of Engineering Physics and Mechanics, Kyoto University, Kyoto 606-8501, Japan
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Toshihiro Iwai
Toshihiro Iwai
Department of Applied Mathematics and Physics, Kyoto University, Kyoto 606-8501, Japan
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J. Math. Phys. 41, 1814–1842 (2000)
Article history
Received:
July 02 1999
Accepted:
November 19 1999
Citation
Shogo Tanimura, Toshihiro Iwai; Reduction of quantum systems on Riemannian manifolds with symmetry and application to molecular mechanics. J. Math. Phys. 1 April 2000; 41 (4): 1814–1842. https://doi.org/10.1063/1.533214
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