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arXiv:hep-th/9306144

Quantum Mechanics on Manifolds

Shogo Tanimura

Published Jun 28, 1993• 5 citations
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Abstract

A definition of quantum mechanics on a manifold $ M $ is proposed and a method to realize the definition is presented. This scheme is applicable to a homogeneous space $ M = G / H $. The realization is a unitary representation of the transformation group $ G $ on the space of vector bundle-valued functions. When $ H \ne \{ e \} $, there exist a number of inequivalent realizations. As examples, quantum mechanics on a sphere $ S^n $, a torus $ T^n $ and a projective space $ \RP $ are studied. In any case, it is shown that there are an infinite number of inequivalent realizations.

DOI
10.48550/arxiv.hep-th/9306144

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A definition of quantum mechanics on a manifold MMM is proposed and a method to realize the definition is presented. This scheme is applicable to a homogeneous space M=G/HM = G / HM=G/H. The realization is a unitary representation of the transformation group GGG on the space of vector bundle-valued functions. When H≠{e}H \ne \{ e \}H={e}, there exist a number of inequivalent realizations. As examples, quantum mechanics on a sphere SnS^nSn, a torus TnT^nTn and a projective space \RP\RP\RP are studied. In any case, it is shown that there are an infinite number of inequivalent realizations.
Extended Schwinger's quantization procedure is used for constructing quantum mechanics on a manifold with a group structure. The considered manifold MMM is a homogeneous Riemannian space with the given action of isometry transformation group. Using the identification of MMM with the quotient space G/HG/HG/H, where HHH is the isotropy group of an arbitrary fixed point of MMM, we show that quantum mechanics on G/HG/HG/H possesses a gauge structure, described by the gauge potential that is the connection 1-form of the principal fiber bundle G(G/H,H)G(G/H, H)G(G/H,H). The coordinate representation of quantum mechanics and the procedure for selecting the physical sector of states are developed.
Starting from the axioms of quantum mechanics as formalized by the systems of imprimitivity for homogeneous Riemannian manifolds, the classical theory is derived as a consequence, complete with its phase space realized as the space of pure classical states, a generalized version of the Wigner–Moyal correspondence rule, the Jordan and Lie algebra structures of functions on the cotangent bundle, given by point-wise multiplication and Poisson bracket, and the momentum map. A comparison is also given of the quantum and classical dynamics and equilibrium statistical mechanics of free particles on compact manifolds of constant negative curvature.
Group representations play a central role in theoretical physics. In particular, in quantum mechanics unitary — or, in general, projective unitary — representations implement the action of an abstract symmetry group on physical states and observables. More specifically, a major role is played by the so-called square integrable representations. Indeed, the properties of these representations are fundamental in the definition of certain families of generalized coherent states, in the phase-space formulation of quantum mechanics and the associated star product formalism, in the definition of an interesting notion of function of quantum positive type, and in some recent applications to the theory of open quantum systems and to quantum information.
Constructing a classical mechanical system associated with a given quantum-mechanical one entails construction of a classical phase space and a corresponding Hamiltonian function from the available quantum structures and a notion of coarser observations. The Hilbert space of any quantum-mechanical system naturally has the structure of an infinite-dimensional symplectic manifold ('quantum phase space'). There is also a systematic, quotienting procedure which imparts a bundle structure to the quantum phase space and extracts a classical phase space as the base space. This works straightforwardly when the Hilbert space carries weakly continuous representation of the Heisenberg group and one recovers the linear classical phase space . We report on how the procedure also allows extraction of nonlinear classical phase spaces and illustrate it for Hilbert spaces being finite dimensional (spin- j systems), infinite dimensional but separable (particle on a circle) and infinite dimensional but non-separable (polymer quantization). To construct a corresponding classical dynamics, one needs to choose a suitable section and identify an effective Hamiltonian. The effective dynamics mirrors the quantum dynamics provided the section satisfies conditions of semiclassicality and tangentiality.
Constructing a classical mechanical system associated with a given quantum mechanical one, entails construction of a classical phase space and a corresponding Hamiltonian function from the available quantum structures and a notion of coarser observations. The Hilbert space of any quantum mechanical system naturally has the structure of an infinite dimensional symplectic manifold (`quantum phase space'). There is also a systematic, quotienting procedure which imparts a bundle structure to the quantum phase space and extracts a classical phase space as the base space. This works straight forwardly when the Hilbert space carries weakly continuous representation of the Heisenberg group and recovers the linear classical phase space mathbbRmathrm2N\\mathbb{R}^{\\mathrm{2N}}mathbbRmathrm2N. We report on how the procedure also allows extraction of non-linear classical phase spaces and illustrate it for Hilbert spaces being finite dimensional (spin-j systems), infinite dimensional but separable (particle on a circle) and infinite dimensional but non-separable (Bohr quantization). To construct a corresponding classical dynamics, one needs to choose a suitable section and identify an effective Hamiltonian. The effective dynamics mirrors the quantum dynamics provided the section satisfies conditions of semiclassicality and tangentiality. There is also an alternative and a priori distinct notion of effective Hamiltonian used in the context of a WKB approximation. These two notions are compared for the example of loop quantum cosmology.
Within the generalized definition of coherent states as group orbits we study the orbit spaces and the orbit manifolds in the projective spaces constructed from linear representations. Invariant functions are suggested for arbitrary groups. The group SU (2) is studied in particular and the orbit spaces of its j = 1/2 and 1 representations completely determined. The orbits of SU (2) in CP N can be either two- or three-dimensional, the former being either isomorphic to S 2 or to RP 2 and the latter being isomorphic to quotient spaces of RP 3 . We end with a look from the same perspective to the quantum mechanical space of states in particle mechanics.