Quantum mechanics on spaces of nonconstant curvature: The oscillator problem and superintegrability

Strict deformation quantization of a particle in external gravitational and Yang-Mills fields

Let (M, g) be a pseudo-Riemannian manifold and 
$ \cal{F}_\lambda(M) $
 the space of densities of degree 
$ \lambda $
 on M. We study the space 
$ \cal{D}^2_{\lambda,\mu}(M) $
 of second-order differential operators from 
$ \cal{F}_\lambda(M) $
 to 
$ \cal{F}_\mu(M) $
 . If (M, g) is conformally flat with signature p - q, then 
$ \cal{D}^2_{\lambda,\mu}(M) $
 is viewed as a module over the group of conformal transformations of M. It turns out that, for almost all values of 
$ \mu-\lambda $
 , the O(p+1, q+1)-modules 
$ \cal{D}^2_{\lambda,\mu}(M) $
 and the space of symbols (i.e., of second-order polynomials on 
$ T^*M $
 ) are canonically isomorphic. This yields a conformally equivariant quantization for quadratic Hamiltonians. We furthermore show that this quantization map extends to arbitrary pseudo-Riemannian manifolds and depends only on the conformal class [g] of the metric. As an example, the quantization of the geodesic flow yields a novel conformally equivariant Laplace operator on half-densities, as well as the well-known Yamabe Laplacian. We also recover in this framework the multi-dimensional Schwarzian derivative of conformal transformations.

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Conformally equivariant quantum Hamiltonians

By using a sheaf-theoretical language, we introduce a notion of deformation quantization allowing not only for formal deformation parameters but also for real or complex ones as well. As a model for this approach to deformation quantization, we construct a quantization scheme for cotangent bundles of Riemannian manifolds. Here, we essentially use a complete symbol calculus for pseudodifferential operators on a Riemannian manifold. Depending on a scaling parameter, our quantization scheme corresponds to normally ordered, Weyl or antinormally ordered quantization. Finally, it is shown that our quantization scheme induces a family of pairwise isomorphic strongly closed star products on a cotangent bundle.

A Deformation-Theoretical Approach to Weyl Quantization on Riemannian Manifolds

A number of basic questions concerning quantization within the setting of operator algebras are stated, and in the process a brief survey of some of the recent developments is given. I collect here a number of basic questions concerning quantization within the setting of operator algebras, while at the same time surveying briefly some of the recent developments. For a less up-to-date but more leisurely survey see [R8]. Here I will only give reference to relatively recent papers. In particular, I will not repeat many important references which already appear in [R8]. An extensive treatment of many of the topics mentioned here, including historical notes and references, will appear in a forthcoming book by Landsman [La3]. See [Wr] for a recent comparison of various approaches to quantization, going beyond the operator algebra setting. For a fine survey of the approach using coherent states (which has significant interaction with operator algebras) see [AG]. For an interesting collection of questions about quantum groups in the operator algebra setting see [Wa]. The present manuscript is based on informal notes (based on earlier talks which I gave) which I distributed at the Dartmouth Workshop on E-theory, Quantization, and Deformations, September 1997. A number of people subsequently gave me suggestions for improvements, which are incorporated here. I am especially grateful to N. Landsman and A. Weinstein for their suggestions. I would, of course, very much like to be told about any progress on answering any of the questions I ask here. The kinematics of a classical mechanical system is given by its phase space, which is a manifold, say M , equipped with a Poisson bracket. It should be the “shadow” of a quantum mechanical system, whose kinematics is given by specifying a non-commutative C∗-algebra of operators and suitable “physical” states on it. There should be a length-scale, ~, Planck’s constant. As ~ → 0 the quantum system should converge in some sense to the classical system. We will not deal here with dynamics. One natural approach to finding quantum systems of which a classical system is the “shadow” is to deform the pointwise product on the algebra of functions on M into a family, parametrized by ~, of non-commutative products. There is an extensive literature dealing with the purely algebraic side of this, in which the 1991 Mathematics Subject Classification. Primary 46L60; secondary 46L87, 81S99. Supported in part by NSF grant DMS 96-13833.

http://cds.cern.ch/record/340075/files/9712009.pdf

Questions on quantization

For every pointwise polynomial function on each fiber of the cotangent bundle of a Riemannian manifold M, a family of diffential operators is given, which acts on the space of smooth sections of a vector bundle on M. Such a correspondence may be considered as a rule to quantize classical systems moving in a Riemannian manifold or in a gauge field. Some applications of our construction are also given in this paper

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Liu Zhang-Ju

Papers

Gauge invariant quantization on Riemannian manifolds