Reduction of Quantum Systems on Riemannian Manifolds with Symmetry and Application to Molecular Mechanics

TL;DR

This paper presents a general method for reducing quantum systems with symmetry on Riemannian manifolds, including stratified cases, and applies it to molecular mechanics, enabling analysis of molecules with constrained motions.

Contribution

It introduces a reduction framework for quantum systems on stratified Riemannian manifolds with symmetry, extending principal bundle concepts and applying the Peter-Weyl theorem.

Findings

Reduction method works for free and non-free group actions.

Reduced Laplacian is well-defined as a self-adjoint operator.

Application demonstrated in molecular mechanics with constrained molecular motion.

Abstract

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 Q = M/G 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 Q = M/G. 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.