悬链曲面上的点粒子动力学及扩展空间约束系统量子化

扩张型正则量子化方案的核心内容是位置、动量以及哈密顿量同时量子化. 通过分析悬链面上粒子的扩张型正则量子化方案, 并且与薛定谔理论进行比较, 发现内禀几何中二维悬链面给不出与薛定谔理论相一致的结果, 而考虑将二维悬链面嵌入在三维欧氏空间之后, 还需要将正则量子化方案进行扩张, 可以得到体系的几何势能和几何动量, 并与薛定谔理论相一致.     

Symplectic structure of the moduli space of flat connection on a Riemann surface

We consider the canonical symplectic structure on the moduli space of flatg-connections on a Riemann surface of genusg withn marked points. Forg being a semisimple Lie algebra we obtain an explicit efficient formula for this symplectic form and prove that it may be represented as a sum ofn copies of Kirillov symplectic form on the orbit of dressing transformations in the Poisson-Lie groupG ^* andg copies of the symplectic structure on the Heisenberg double of the Poisson-Lie groupG (the pair (G, G ^*) corresponds to the Lie algebrag).Supported by Swedish Natural Science Research Council (NFR) under the contract F-FU 06821-304Supported in part by a Soros Foundation Grant awarded by the American Physical Society     

Real polarization of the moduli space of flat connections on a Riemann surface

We prove that the moduli space of flatSU(2) connections on a Riemann surface has a real polarization, that is, a foliation by lagrangian subvarieties. This polarization may provide an alternative quantization of the Chern-Simons gauge theory in higher genus, in line with the results of [11] for genus one.Supported by NSF Mathematical Sciences Postdoctoral Research Fellowship DMS 88-07291     

On the Quillen determinant

We explain the bundle structures of the Determinant line bundle and the Quillen determinant line bundle considered on the connected component of the space of Fredholm operators including the identity operator in an intrinsic way. Then we show that these two are isomorphic and that they are non-trivial line bundles and trivial on some subspaces. Also we remark a relation of the Quillen determinant line bundle and the Maslov line bundle.     

On geometric quantization of the Dirac magnetic monopole

We give a simple derivation of the spectrum of the Dirac magnetic monopole on a unit sphere S² based on geometric quantization and the Frobenius reciprocity formula. The starting point is the calculation by Novikov and Schmelzer of the canonical symplectic structure on the coadjoint orbits of the isometry group of 3-dimensional Euclidean space E(3), which showed the appearance of the Dirac magnetic term.     

Non-Abelian Symplectic Cuts and the Geometric Quantization of Noncompact Manifolds

Let (M, ) be a Hamiltonian U(n)-space with proper moment map. In the case where n = 1, Lerman constructed a one-parameter family of Hamiltonian U(1)-spaces M called the symplectic cuts of M. We generalize this construction to Hamiltonian U(n) spaces. Motivated by recent theorems that show that 'quantization commutes with reduction,' we next give a definition of geometric quantization for noncompact Hamiltonian G-spaces with proper moment map, and use our cutting technique to illustrate the proof of existence of such quantizations in the case of U(n) spaces. We then show (Theorem 1) that such quantizations exist in general.     

On a conjecture of Guillemin and Sternberg in geometric quantization of multiplicity-free symplectic spaces

We study the relation between multiplicity-free symplectic manifolds and the multiplicities of group representations obtained by geometric quantization. With the help of a general equivalence theorem we can prove a conjecture of Guillemin and Sternberg in the compact Kähler case.     

Pre-quantization of the moduli space of flat G-bundles over a surface

For a simply connected, compact, simple Lie group G$G$, the moduli space of flat G$G$-bundles over a closed surface Σ$\Sigma$ is known to be pre-quantizable at integer levels. For non-simply connected G$G$, however, integrality of the level is not sufficient for pre-quantization, and this paper determines the obstruction–namely a certain cohomology class in H³(G²;Z)$H^3(G^2;\mathbb{Z})$–that places further restrictions on the underlying level. The levels that admit a pre-quantization of the moduli space are determined explicitly for all non-simply connected, compact, simple Lie groups G$G$.     

The Selberg zeta function and a new Kähler metric on the moduli space of punctured Riemann surfaces

A local index theorem for families of -operators on Riemann surfaces with functures is proved. A new Kähler metric on the moduli space of punctured surfaces is described in terms of the Eisenstein-Maass series.     

Representations of the Heisenberg algebra on a Riemann surface

We formulate reflection positivity for meromorphic functions and for 1-forms on a Riemann surface. This construction yields representations of the Heisenberg algebra on a Riemann surface.Supported in part by the Department of Energy under Grant DE-FG02-88ER25065     

Rotational symmetry of classical orbits, arbitrary quantization of angular momentum and the role of the gauge field in two-dimensional space

We study quantum–classical correspondence in terms of the coherent wave functions of a charged particle in two- dimensional central-scalar potentials as well as the gauge field of a magnetic flux in the sense that the probability clouds of wave functions are well localized on classical orbits. For both closed and open classical orbits, the non-integer angular-momentum quantization with the level space of angular momentum being greater or less than is determined uniquely by the same rotational symmetry of classical orbits and probability clouds of coherent wave functions, which is not necessarily 2π-periodic. The gauge potential of a magnetic flux impenetrable to the particle cannot change the quantization rule but is able to shift the spectrum of canonical angular momentum by a flux-dependent value, which results in a common topological phase for all wave functions in the given model. The well-known quantum mechanical anyon model becomes a special case of the arbitrary quantization, where the classical orbits are 2π-periodic.     

A negative answer to a Rieffel’s question on the behavior of K-groups under strict deformation quantization

In this paper, we address one of the questions raised by Rieffel in his collection of questions on deformation quantization. The question is whether the K-theory groups remain the same under flabby strict deformation quantizations. By “deforming” the question slightly, we produce a negative answer to the question.     

Nonlinear integrable flow on the framed moduli space of instantons

A nonlinear integrable flow of the Lax-type on the framed moduli space of SU(_n) Yang-Mills instantons is found.     

Landau quantization and equatorial states on the surface of a nanosphere

The Landau quantization for the electron gas on the surface of a sphere is considered. It is shown that in the regime of strong fields the lowest energy states are those with magnetic quantum numbers m of order of Φ /Φ₀, the number of magnetic flux quanta piercing the sphere. For an electron gas of low density (semiconducting situation) it leads to the formation of an electronic stripe on the equator of the sphere in high fields. Pis’ma Zh. éksp. Teor. Fiz. 70, No. 6, 405–409 (25 September 1999) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Curvature-induced noncommutativity of two different components of momentum for a particle on a hypersurface

As a nonrelativistic particle constrained to remain on an(N?1)-dimensional((N≥2))hypersurface embedded in an N-dimensional Euclidean space,two different components pi and pj(i,j=1,2,3,...N)of the Cartesian momentum of the particle are not mutually commutative,and explicitly commutation relations[p^{^}_i,p^{^}_j](≠0) depend on products of positions and momenta in uncontrollable ways.The generalized Dupin indicatrix of the hypersurface,a local analysis technique,is utilized to explore the dependence of the noncommutativity on the curvatures around a local point of the hypersurface.The first finding is that the noncommutativity can be grouped into two categories;one is the product of a sectional curvature and the angular momentum,and another is the product of a principal curvature and the momentum.The second finding is that,for a small circle lying a tangential plane covering the local point,the noncommutativity leads to a rotation operator and the amount of the rotation is an angle anholonomy;and along each of the normal sectional curves centering the given point the noncommutativity leads to a translation plus an additional rotation and the amount of the rotation is one half of the tangential angle change of the arc.