几何动量、几何势能与约束体系中一种新的代数对称性
|
摘要
约束无处不在。在微观层次，石墨烯、C_(60)分子、纳米管等低维系统都是典型的约束系统。不但其中载流子的运动被约束在二维弯曲的曲面上，而且这些体系的一些整体运动例如转动也是约束运动。在宇观层次，整个宇宙就是一个体积有限体系，任何物理过程和信息交换都发生在这个受限体系内。
近年来的理论和实验研究表明，对约束在二维曲面上的运动，约束能导致新的量子效应：几何动量和几何势能。这种几何效应的最先导出需要用到薛定谔方程和所谓的限制势技术。这种技术的核心是薛定谔方程必须首先写在三维平直空间中，然后通过限制势把运动约束在二维弯曲曲面附近。这个技术有一个明显的理论缺陷：为什么不能把薛定谔方程直接写到曲面上？换言之，尽管知道这样做的后果得不到几何动量和几何势能，但理论本身并不排斥这种作法。不妨把这个问题称之为问题一。
由于量子化是建立薛定谔方程的基础，而正则量子化又是所有量子化的基础。在通常的正则量子化理论中，把正则坐标和正则动量之间的对易关系假定为基本量子条件。把一些重要的力学量(例如哈密顿量)和一些重要的关系(例如运动方程)的量子化作为推论而导出。但这些导出性或从属性的理论之间的自洽性要求另一个规则：存在一个平直的底空间并利用笛卡尔坐标系进行量子化。这一点早在1920年代就被狄拉克注意到。这个运算规则不像是物理学基本理论的一部分，而且在运用到弯曲曲面时会导致无法获得体系哈密顿量的唯一形式。不妨把这个问题称之为问题二。
文献中对问题一、二有许多研究，有群量子化，几何量子化，增强量子化等等，但都只能解决部分的问题而非全部。我们提出的解决方案如下。将在狄拉克正则量子化理论的基础上进行了如下增强：将坐标，动量和粒子哈密顿量之间的对易关系定义为体系的第二类基本对易关系，而且在量子化的过程中不改变坐标，动量和哈密顿量之间的基本代数关系，从而使得坐标，动量和哈密顿量同时量子化。我们把这个扩展后的量子化理论称之为增强型正则量子化方案(SCQS)，这种代数量子化方案，在量子化过程中，保持了坐标、动量和哈密顿量之间的代数关系不变，反映了一种特殊的对称性，即代数对称性。这一理论方案并不能解决全部困难，但能较好地解决问题一，部分解决问题二，不过最重要的是可以得到几何动量和几何势能的唯一形式。这一理论也需要平直的外部空间和笛卡尔坐标系。所以几何动量和几何势能是一嵌入效应。
本研究主要利用增强型正则量子化方案处理平直空间中的约束在如下曲面上的量子自由运动：二维球面、任意维球面、二维圆环面、轴对称极小曲面等。表明新的代数对称性在量子化之后仍然保持，从而我们将假设SCQS普遍成立。全文总共分为五章。
第1章，首先综述了约束体系量子化及其研究进展，然后简单回顾了薛定谔理论处理二维曲面上粒子的量子运动，最后介绍了狄拉克约束体系正则量子化理论。
第2章，首先介绍了二维曲面的SCQS，然后将利用SCQS处理的二维球面和三维球面上粒子的量子运动作为两个示例简单介绍了一下，发现三维空间中的二维球面的几何动量在Monge参数形式下具有几何不变性；得到了在准内禀几何中三维球面的SCQS不自洽，它在量子化过程中动量与哈密顿量之间的对易关系破坏了它们之间的基本代数关系，所以只能从外部几何的角度构建SCQS，并且得到了嵌入在四维欧几里得空间的三维球面上粒子的几何势能与几何动量，此时的坐标、动量和哈密顿量之间的代数关系保守不变，并且得到的几何势能与薛定谔理论的结论完全一致。
第3章，研究了约束在任意维球面上粒子的量子运动。发现在准内禀几何中n-1维球面上运动粒子的正则动量与哈密顿量之间的对易关系破坏了它们之间的代数关系，从而导致SCQS不自洽，所以用准内禀几何描述n-1维球面上粒子的量子运动是不适合的；而当将n-1维球面嵌入在n维欧几里得空间中考虑粒子的量子运动时，利用代数对称性进行量子化，得到了粒子的几何动量和几何势能的具体形式，并且哈密顿量与薛定谔理论得到的结果完全一致。
第4章，研究了约束在二维圆环面上粒子的量子运动。由于二维圆环面正则动量与哈密顿量的对易关系破坏了它们之间的代数关系，表明了狄拉克正则量子化不能在内禀几何的框架下构建；而从三维欧几里得空间中考虑二维圆环面上粒子的量子运动，利用代数对称性进行量子化时，得到了粒子的几何势能和几何动量，并且它们也与薛定谔理论的结论完全相符，所以二维圆环面上粒子的量子运动可以在三维平直空间中构建。
第5章，研究了约束在二维轴对称极小曲面——悬链面上粒子的量子运动。通过自洽的方式得到了在准内禀几何中SCQS不适用；而在三维欧几里得空间中，利用代数对称性进行量子化时，得到的几何势能和几何动量都与薛定谔理论的结果相同。
本文的最后对全文进行了简单的总结。
Constrained motion is ubiquitous. In the microcosmic, low-dimensional systems suchas Graphene, C_(60)and Carbon nanotubes are of typical constrained system. Not only thecarrier's motion is constrained on a two-dimensional curved surface, but also the overallmotion of these systems such as rotation. The universe is a system of finite volume, anyphysical process or information exchange occurs in the restricted system.
In recent years, both the theoretical and experimental investigations have shown thatthe constraints may lead to new quantum effects,such as geometric momentum andgeometric potential for a particle moving on a two-dimensional curved surface. The processof achieving these geometric effects utilizes the Schrodinger equation and the so-calledConfning Potential Approach Method. The core of this technique is to set up theSchrodinger equation in three-dimensional Cartesian space, and then limit the motion in thelayer of the two-dimensional curved surface by Confning Potential Approach Method. Thisapproach leads an obvious theoretical question: why the Schr dinger equation can not beentirely formulated on the two-dimensional curved surface without considering anyembedding? In other words, we can not obtain the Geometric momentum and Geometricpotential in this way, but the theory itself does not directly disagree with it. This is thequestion one.
The quantization is the basis to construct the Schrodinger equation, and the canonicalquantization is the basis of all quantizations. In common canonical quantization theories, therelation between the canonical coordinates and canonical momenta is assumed to be thefundamental quantum condition. And the quantization of some important mechanicalquantities (for example the Hamiltonian) and some important relations (such as theequations of motion) are the inference of the theory. And the leading or sub theoriesself-consistently should consistent theirself by satisfying another rule: the quantization is inflat space and must utilize the Cartesian coordinate system. That has been noticed by Diracin the1920s. The calculation rule in the Dirac's theory does not seem a part of the basictheory of physics. And when it is applied to curved surface, unambiguous quantized form ofthe Hamiltonian cannot be obtained. This is the question two.
For questions one and two, there are many explorations in references, for instance thegroup of quantization, the geometric quantization, the enhanced Quantization. However theycan only solve parts of the problems rather than all of them. We give a proposal as follows: a generalization of the Dirac’s canonical quantization theory. The fundamental commutationrelations that are constituted by all commutators between positions, momenta andHamiltonian, called the second category of the fundamental commutation relations, and wepostulate a simultaneous quantization for positions, momenta, and Hamiltonian whilepreserving the formal algebraic structure between them. We call it Strengthened CanonicalQuantization Scheme (SCQS). This is an algebraic quantization solution. This scheme cannot solve all problems, but can solve the question one better, and partly for question two. Itcan get the geometric momentum. This is the most important thing. This theory also requiresto be constructed in the flat space with cartesian coordinate system. So the geometricmomentum and geometric potential are embedding effects.
This research mainly discusses the quantum motions on the following surfacesembedded in three-dimensional flat space with SCQS: the two-dimensional sphere, then-dimensional sphere, the two-dimensional torus, the axisymmetric minimal surface. Inthese examples we will prove the algebraic symmetry reserves after the quantization, as theevidence to show the SCQS is universal. The dissertation is organized as the followingdivided to five parts.
In the first part, we review the present progress on the study of quantization for theconstraints systems, and then the quantum motion of a particle on two-dimensional curvedsurface with the Schr dinger theory. In the end, we introduce Dirac's theory of the canonicalquantization.
At beginning of the second part, we introduce the SCQS on the2D surface. We willbriefly introduce the quantum motion on2D and3D sphere with SCQS as examples, andshow the geometric momentum in the Monge parametrization on the2D sphere embedded in3D flat space is a geometric invariant. And for the3D sphere, we see that the SCQS in thequasi intrinsic geometry is not self-consistent because the fundamental algebraic relationbreaks in quantization. We therefore need to construct SCQS with extrinsic geometry. Andwe get the geometric momentum and geometric potential of a particle moving on the3Dsphere that is embedded in4D Euclidian space. In this case, the algebraic relation amongpositions, momenta and Hamiltonian will reserve invariant and the geometric potential arethen in agreement with those given by the Schr dinger theory.
In the third part, we research the motion of a particle constrained to the n-1dimensionalsphere embedded in the n dimensional Euclidean space. The results illustrates that there is aobvious breakdown of the formal algebraic structure within purely intrinsic geometry, itdemonstrates that the purely intrinsic geometry does not satisfy the general theory of thecanonical quantization and cannot be self-consistently formulated; taking the sphere as a submanifold in n-dimensional space, we obtain unique forms of the geometric momentumand geometric potential that is compatible with Schr dinger’s results. When n3, thegeometric momentum is repulsive. This means that when a curved space embedded in flatspace, it can lead to the curved space itself with energy. The results provide a inspiration forthe geometric energy due to the embedding four dimensional space-time in thehigher-dimensional space-time.
The fourth part investigates the quantum mechanics on a torus. The theory formulatedpurely on the torus, i.e., and based on the so-called the purely intrinsic geometry, conflictswith itself, because of a manifest breakdown of the formal algebraic structure between
[p_θ,H]﹛p_θ,H﹜and D. An extrinsic examination of the torus as a submanifold in threedimensional flat space turns out to be self-consistent and the derived momenta andHamiltonian are satisfactory all around.
The fifth part investigates the quantum mechanics on a catenoid. Results show that thegeneral theory of the canonical quantization can be established in a self-consistent way forquantum motions on catenoid, but it is not compatible with Schr dinger theory. In contrast,in three-dimensional Euclidean space, the geometric momentum and potential are then inagreement with those given by the Schr dinger theory.
In the last part of this dissertation, we give a summary to the above-mentioned works.
近年来的理论和实验研究表明，对约束在二维曲面上的运动，约束能导致新的量子效应：几何动量和几何势能。这种几何效应的最先导出需要用到薛定谔方程和所谓的限制势技术。这种技术的核心是薛定谔方程必须首先写在三维平直空间中，然后通过限制势把运动约束在二维弯曲曲面附近。这个技术有一个明显的理论缺陷：为什么不能把薛定谔方程直接写到曲面上？换言之，尽管知道这样做的后果得不到几何动量和几何势能，但理论本身并不排斥这种作法。不妨把这个问题称之为问题一。
由于量子化是建立薛定谔方程的基础，而正则量子化又是所有量子化的基础。在通常的正则量子化理论中，把正则坐标和正则动量之间的对易关系假定为基本量子条件。把一些重要的力学量(例如哈密顿量)和一些重要的关系(例如运动方程)的量子化作为推论而导出。但这些导出性或从属性的理论之间的自洽性要求另一个规则：存在一个平直的底空间并利用笛卡尔坐标系进行量子化。这一点早在1920年代就被狄拉克注意到。这个运算规则不像是物理学基本理论的一部分，而且在运用到弯曲曲面时会导致无法获得体系哈密顿量的唯一形式。不妨把这个问题称之为问题二。
文献中对问题一、二有许多研究，有群量子化，几何量子化，增强量子化等等，但都只能解决部分的问题而非全部。我们提出的解决方案如下。将在狄拉克正则量子化理论的基础上进行了如下增强：将坐标，动量和粒子哈密顿量之间的对易关系定义为体系的第二类基本对易关系，而且在量子化的过程中不改变坐标，动量和哈密顿量之间的基本代数关系，从而使得坐标，动量和哈密顿量同时量子化。我们把这个扩展后的量子化理论称之为增强型正则量子化方案(SCQS)，这种代数量子化方案，在量子化过程中，保持了坐标、动量和哈密顿量之间的代数关系不变，反映了一种特殊的对称性，即代数对称性。这一理论方案并不能解决全部困难，但能较好地解决问题一，部分解决问题二，不过最重要的是可以得到几何动量和几何势能的唯一形式。这一理论也需要平直的外部空间和笛卡尔坐标系。所以几何动量和几何势能是一嵌入效应。
本研究主要利用增强型正则量子化方案处理平直空间中的约束在如下曲面上的量子自由运动：二维球面、任意维球面、二维圆环面、轴对称极小曲面等。表明新的代数对称性在量子化之后仍然保持，从而我们将假设SCQS普遍成立。全文总共分为五章。
第1章，首先综述了约束体系量子化及其研究进展，然后简单回顾了薛定谔理论处理二维曲面上粒子的量子运动，最后介绍了狄拉克约束体系正则量子化理论。
第2章，首先介绍了二维曲面的SCQS，然后将利用SCQS处理的二维球面和三维球面上粒子的量子运动作为两个示例简单介绍了一下，发现三维空间中的二维球面的几何动量在Monge参数形式下具有几何不变性；得到了在准内禀几何中三维球面的SCQS不自洽，它在量子化过程中动量与哈密顿量之间的对易关系破坏了它们之间的基本代数关系，所以只能从外部几何的角度构建SCQS，并且得到了嵌入在四维欧几里得空间的三维球面上粒子的几何势能与几何动量，此时的坐标、动量和哈密顿量之间的代数关系保守不变，并且得到的几何势能与薛定谔理论的结论完全一致。
第3章，研究了约束在任意维球面上粒子的量子运动。发现在准内禀几何中n-1维球面上运动粒子的正则动量与哈密顿量之间的对易关系破坏了它们之间的代数关系，从而导致SCQS不自洽，所以用准内禀几何描述n-1维球面上粒子的量子运动是不适合的；而当将n-1维球面嵌入在n维欧几里得空间中考虑粒子的量子运动时，利用代数对称性进行量子化，得到了粒子的几何动量和几何势能的具体形式，并且哈密顿量与薛定谔理论得到的结果完全一致。
第4章，研究了约束在二维圆环面上粒子的量子运动。由于二维圆环面正则动量与哈密顿量的对易关系破坏了它们之间的代数关系，表明了狄拉克正则量子化不能在内禀几何的框架下构建；而从三维欧几里得空间中考虑二维圆环面上粒子的量子运动，利用代数对称性进行量子化时，得到了粒子的几何势能和几何动量，并且它们也与薛定谔理论的结论完全相符，所以二维圆环面上粒子的量子运动可以在三维平直空间中构建。
第5章，研究了约束在二维轴对称极小曲面——悬链面上粒子的量子运动。通过自洽的方式得到了在准内禀几何中SCQS不适用；而在三维欧几里得空间中，利用代数对称性进行量子化时，得到的几何势能和几何动量都与薛定谔理论的结果相同。
本文的最后对全文进行了简单的总结。
Constrained motion is ubiquitous. In the microcosmic, low-dimensional systems suchas Graphene, C_(60)and Carbon nanotubes are of typical constrained system. Not only thecarrier's motion is constrained on a two-dimensional curved surface, but also the overallmotion of these systems such as rotation. The universe is a system of finite volume, anyphysical process or information exchange occurs in the restricted system.
In recent years, both the theoretical and experimental investigations have shown thatthe constraints may lead to new quantum effects,such as geometric momentum andgeometric potential for a particle moving on a two-dimensional curved surface. The processof achieving these geometric effects utilizes the Schrodinger equation and the so-calledConfning Potential Approach Method. The core of this technique is to set up theSchrodinger equation in three-dimensional Cartesian space, and then limit the motion in thelayer of the two-dimensional curved surface by Confning Potential Approach Method. Thisapproach leads an obvious theoretical question: why the Schr dinger equation can not beentirely formulated on the two-dimensional curved surface without considering anyembedding? In other words, we can not obtain the Geometric momentum and Geometricpotential in this way, but the theory itself does not directly disagree with it. This is thequestion one.
The quantization is the basis to construct the Schrodinger equation, and the canonicalquantization is the basis of all quantizations. In common canonical quantization theories, therelation between the canonical coordinates and canonical momenta is assumed to be thefundamental quantum condition. And the quantization of some important mechanicalquantities (for example the Hamiltonian) and some important relations (such as theequations of motion) are the inference of the theory. And the leading or sub theoriesself-consistently should consistent theirself by satisfying another rule: the quantization is inflat space and must utilize the Cartesian coordinate system. That has been noticed by Diracin the1920s. The calculation rule in the Dirac's theory does not seem a part of the basictheory of physics. And when it is applied to curved surface, unambiguous quantized form ofthe Hamiltonian cannot be obtained. This is the question two.
For questions one and two, there are many explorations in references, for instance thegroup of quantization, the geometric quantization, the enhanced Quantization. However theycan only solve parts of the problems rather than all of them. We give a proposal as follows: a generalization of the Dirac’s canonical quantization theory. The fundamental commutationrelations that are constituted by all commutators between positions, momenta andHamiltonian, called the second category of the fundamental commutation relations, and wepostulate a simultaneous quantization for positions, momenta, and Hamiltonian whilepreserving the formal algebraic structure between them. We call it Strengthened CanonicalQuantization Scheme (SCQS). This is an algebraic quantization solution. This scheme cannot solve all problems, but can solve the question one better, and partly for question two. Itcan get the geometric momentum. This is the most important thing. This theory also requiresto be constructed in the flat space with cartesian coordinate system. So the geometricmomentum and geometric potential are embedding effects.
This research mainly discusses the quantum motions on the following surfacesembedded in three-dimensional flat space with SCQS: the two-dimensional sphere, then-dimensional sphere, the two-dimensional torus, the axisymmetric minimal surface. Inthese examples we will prove the algebraic symmetry reserves after the quantization, as theevidence to show the SCQS is universal. The dissertation is organized as the followingdivided to five parts.
In the first part, we review the present progress on the study of quantization for theconstraints systems, and then the quantum motion of a particle on two-dimensional curvedsurface with the Schr dinger theory. In the end, we introduce Dirac's theory of the canonicalquantization.
At beginning of the second part, we introduce the SCQS on the2D surface. We willbriefly introduce the quantum motion on2D and3D sphere with SCQS as examples, andshow the geometric momentum in the Monge parametrization on the2D sphere embedded in3D flat space is a geometric invariant. And for the3D sphere, we see that the SCQS in thequasi intrinsic geometry is not self-consistent because the fundamental algebraic relationbreaks in quantization. We therefore need to construct SCQS with extrinsic geometry. Andwe get the geometric momentum and geometric potential of a particle moving on the3Dsphere that is embedded in4D Euclidian space. In this case, the algebraic relation amongpositions, momenta and Hamiltonian will reserve invariant and the geometric potential arethen in agreement with those given by the Schr dinger theory.
In the third part, we research the motion of a particle constrained to the n-1dimensionalsphere embedded in the n dimensional Euclidean space. The results illustrates that there is aobvious breakdown of the formal algebraic structure within purely intrinsic geometry, itdemonstrates that the purely intrinsic geometry does not satisfy the general theory of thecanonical quantization and cannot be self-consistently formulated; taking the sphere as a submanifold in n-dimensional space, we obtain unique forms of the geometric momentumand geometric potential that is compatible with Schr dinger’s results. When n3, thegeometric momentum is repulsive. This means that when a curved space embedded in flatspace, it can lead to the curved space itself with energy. The results provide a inspiration forthe geometric energy due to the embedding four dimensional space-time in thehigher-dimensional space-time.
The fourth part investigates the quantum mechanics on a torus. The theory formulatedpurely on the torus, i.e., and based on the so-called the purely intrinsic geometry, conflictswith itself, because of a manifest breakdown of the formal algebraic structure between
[p_θ,H]﹛p_θ,H﹜and D. An extrinsic examination of the torus as a submanifold in threedimensional flat space turns out to be self-consistent and the derived momenta andHamiltonian are satisfactory all around.
The fifth part investigates the quantum mechanics on a catenoid. Results show that thegeneral theory of the canonical quantization can be established in a self-consistent way forquantum motions on catenoid, but it is not compatible with Schr dinger theory. In contrast,in three-dimensional Euclidean space, the geometric momentum and potential are then inagreement with those given by the Schr dinger theory.
In the last part of this dissertation, we give a summary to the above-mentioned works.
引文
[1]李子平，李爱民著。约束系统的量子对称性质[M].北京工业大学出版社，2011
[2] R.P. Feynman. The quantum theory of gravitation. Acta Physica Polonica,1963,24:697
[3] Wang Anmin, Ruan Tunan. Extension of the Dirac-Bergmann theory of constrainedsystems. Physics Review A,1996,54(1):57–80.
[4] J.L.Anderson, and P.G.Bergmann. Constraints in covariant field theories. PhysicalReview,1951,83(5):1018-1025
[5] P.G.Bergmann, and J.H.M.Brunings. Non-Linear Field Theories II. Canonical Equati-ons and Quantization. Review of Moderm Physics,1949,21(3):480
[6] P.A.M. Dirac. Generalized Hamiltonion dynamics. Canadian Journal of Mathematics,1950,2:129-148
[7] P.A.M. Dirac. The Hamiltonian form of field dynamics Canadian Journal of Mathema-tics,1951,3:1-23
[8] P.A.M. Dirac. Forms of Relativistic Dynamics. Review of Moderm Physics,1949,21(3):392-399
[9] P.A.M. Dirac. The Theory of Gravitation in Hamiltonian Form. Proceedings of the RoyalSociety of London A,1958,246(246):333-343
[10] P.A.M Dirac. Fixation of Coordinates in the Hamiltonian Theory of Gravitation.Physical Review,1959,114(3):924-930
[11] R.P. Feynman. Space-time approach to non-relativistic quantum mechanics. Reviews ofModem Physics,1948,20:367-387
[12] P.A.M. Dirac. Lecture on quantum mechanics. New York: Yeshiva University,1964.
[13] R.P.Feynman, A.R.Hibbs. Quantum mechanics and path integrals. New York: McGraw-Hill,1965
[14] L.D.Faddeev, V.N.Popov. Feynman diagrams for the Yang-Mills field. Physics Letters,1967,25:29-30
[15] L.D.Faddeev. Feynman integral for singular Lagrangian. Theoretical and Mathema-tical Physics,1970,1:1-13
[16] P.Senjanovic. Path integral quantization of field theories with second-class constraints.Annals of Physics,1976,100:227-261
[17] M. Henneaus. Hamiltonian form of the path integral for theories with a gauge freed-om.Physics Reports,1985,126(1):1-66
[18] M. Henneaus, C. Teitelboim. Quantization of gauge system. Princeton: PrincetonUniversity Press,1992
[19] J. Gomis, J. Paris, S. Samuel. Antibracket antifields and gauge-theory quantization.Physics Reports,1995,259:1-145
[20] B. Podolsky. Quantum-Mechanically Correct Form of Hamiltonian Function forConservative Systems. Physics Review,1928,32:812
[21] L. I. Schiff. Quantum Mechanics. New York: McGraw-Hill,1949,135
[22] P. W. Higgs. Dynamical symmetries in a spherical geometry. International Journalof Modern Physics A,1979,12:309
[23] H. Kleinert. Path Integrals in Quantum Mechanics. Singapore: Statistics and PolymerPhysics,5th ed.2009,1-670
[24] V. Aldaya, M. Calixto, J. Guerrero, and F. F. Lopez-ruiz. Group-quantization ofnonlinear sigma models: particle on S2revisited. Reports on Mathematical Physics,2009,64(1):49-58
[25] B. S. DeWitt. Dynamical Theory in Curved Spaces. I. A Review of the Classical andQuantum Action Principles. Reviews of Modern Physics,1957,29:377.
[26] K. S. Cheng. Quantization of a General Dynamical System by Feynman's PathIntegration Formulation. Journal of Mathematical Physics,1972,13(11):1723-1726
[27] H. Dekker. On the path integral for diffusion in curved spaces. Physica A: StatisticalMechanics and its Applications,1980,103(3):586
[28] H. Kleinert and S. V. Shabanov. Proper Dirac quantization of a free particle on aD-dimensional sphere. Physics Letters A,1997,232:327-332
[29] R. H. Dicke and J. P. Wittke. Introduction to Quantum Mechanics. Massachusetts:Addison-Wesley,1963,143
[30] J. M. Domingos, and M. H. Caldeira. The quantization of the Hamiltonian in curvedspace. Foundations of Physics,1984,14(7):147
[31] Makoto Kato. Momentum and Hamiltonian Operators in Generalized Coordinates. In-ternational Journal of Theoretical Physics,1994,33(4):857
[32] H. Grundling. Quantum constraints. Reports on Mathematical Physics,2006,57:97-120
[33] H. J. Rothe and K. D. Rothe. Classical and Quantum Dynamics of Constrained Ham-iltonian Systems. Singapore: World Scientifc,2010
[34] L. S. Schulman. Techniques and Applications of Path Integration. New York: Wiley,1981
[35] M. Henneaux and C. Teitelboim. Quantization of Gauge Systems. Princeton: Prince-ton University Press,1992
[36] V.I. Arnold, S.P. Novikov, G. Wassermann et al. Dynamical Systems IV: SymplecticGeometry and Its Applications. English: Springer-Verlag,1996
[37] Gamkrelidze, Revaz V. Encyclopaedia of Mathematical Sciences, Vol.4. NewYork:Springer-Verlag,1990
[38] N. M. J. Woodhouse. Geometric Quantization. Clarendon: Oxford,1992
[39] P.A.M. Dirac. The Principles of Quantum Mechanics.3rd Edition. Oxford: OxfordUniv. Press,1947,114.
[40]王正行.谨与简洁之美:王竹溪一生的物理追求，北京大学出版社，2008
[41] L. Kaplan, N. T. Maitra, and E. J. Heller. Quantizing constrained systems. PhysicsReview A,1997,56:2592
[42] J. R. Klauder and S. V. Shabanov. Coordinate-free quantization of second-class con-straints. Nuclear Physics,1998,511:713-736
[43] D. Baleanu and Y. Güler. Hamilton-Jacobi treatment of the non-relativistic particle onthe curved space. Journal of Physics A,2001,34:73-80
[44] H. J. Schnitzer. Chiral perturbation theory for the pion-skyrmion system. Nuclear Phy-sics B,1985,261:546-572
[45] M. Ikegami, Y. Nagaoka, S. Takagi, and T. Tanzawa. Quantum Mechanics of a Particleon a Curved Surface. Progress of Theoretical Physics,1992,88:229
[46] P. Maraner. Monopole gauge fields and quantum potentials induced by the geometry insimple dynamical systems. Annals of Physics,1996,246:325-346
[47] L. Kaplan, N. T. Maitra, and E. J. Heller. Quantizing constrained systems. Physics Re-view A,1997,56(4):2592-2599
[48] A. G. Nuramatov, l. V. Prokhorov. On quantization of systems with second class con-straints. International Journal of Geometric Methods in Modern Physics,2006,3:1459-1467
[49] G. Gyorgyi and S. Kovesi-Domokos, Nuovo Cimento B58,191(1968)
[50] N. K. Falck and A. C. Hirshfeld. Dirac-bracket quantisation of a constrained non-linearsystem: the rigid rotator. European Journal of Physics,1983,4(1):5
[51] S. Ishikawa, T.Miyazaki, K. Yamamoto, andM. Yamanobe. How to construct a coor-dinate representation of a hamiltonian operator on a torus. International Journalof Modern Physics A,1996,11:3363
[52] S. T. Hong, W. T. Kim, and Y. J. Park. Improved dirac quantization of a free particle.Mod. Physics Letters A,2000,15:1915-1922
[53] T. Homma, T. Inamoto, and T. Miyazaki. Schrodinger equation for the nonrelativisticparticle constrained on a hypersurface in a curved space. Physics Review D,1990,42:2049
[54] T. Homma, T. Inamoto, and T. Miyazaki. Alternative formulation in the constrainedhamiltonian system of a rigid rotator. Zeitschrift für Physik C,1990,48:105-108
[55] N. Ogawa, K. Fujii, A. Kobushukin. Quantum Mechanics in Riemannian Manifold.Progress of Theoretical Physics,1990,83(5):894-905
[56] N. Ogawa, K. Fujii, N. Chepilko and A. Kobushukin. Quantum Mechanics in Rie-mannian Manifold. II. Progress of Theoretical Physics,1991,85(6):1189-1201
[57] N. Ogawa. The Difference of Effective Hamiltonian in Two Methods in QuantumMechanics on Submanifold. Progress of Theoretical Physics,1992,87(2):513-517
[58] A. V. Golovnev. Dirac quantization of free motion on curved surfaces. InternationalJournal of Geometric Methods in Modern Physics,2006,3:655
[59] A. V. Golovnev. Canonical quantization of motion on submanifolds. Reports on Mathe-matical Physics,2009,64:59
[60] P. C. Schuster and R. L. Jaffe. Quantum mechanics on manifolds embedded ineuclidean space. Annals of Physics,2003,307:132-143
[61] H. Jensen and H. Koppe. Quantum mechanics with constraints. Ann. Phys.(NY).1971,63:586
[62] R. C. T. da Costa. Quantum mechanics of a constrained particle. Physics Review A.1981,23:1982
[63] A. V. Chaplik and R. H. Blick. On geometric potentials in quantum-electro mech-anical circuits. New Journal of Physics,2004,6:33
[64] G. Ferrari and G. Cuoghi. Schr dinger Equation for a Particle on a Curved Surface inan Electric and Magnetic Field. Physics Review Letters,2008,100:230403
[65] Q. H. Liu, C. L. Tong and M. M. Lai. Constraint-induced mean curvature dependenceof cartesian momentum operators. Journal of Physics A: Mathematical and Theoretical,2007,40:4161
[66] Q. H. Liu, L. H. Tang and D. M. Xun. Geometric momentum: The proper momentumfor a free particle on a two-dimensional sphere. Physics Review A,2011,84:042101
[67] S. Matsutani. The physical meaning of the embedded effect in the quantum sub-manifold system. Journal of Physics A: Mathematical and General,1993,26(19):5133
[68] D. M. Xun, and Q. H. Liu. Geometric momentum in the monge parametrization oftwo-dimensional sphere. International Journal of Geometric Methods in ModernPhysics,2013,10:1220031
[69] Q. H. Liu, Y. Shen, D. M. Xun and X. Wang. On relation between geometricmomentum and annihilation operators on a two-dimensional sphere. InternationalJournal of Geometric Methods in Modern Physics,2013,10:1320007
[70] V. Atanasov, R. Dandoloff, and A. Saxena. Geometry induced charge separation on ahelicoidal ribbon. Physics Review B,2009,79:033404
[71] R. Dandoloff, A. Saxena, and B. Jensen. Geometry-induced potential on a two-di-mensional section of a wormhole: Catenoid. Physics Review A,2010,81(1):014102
[72] I. Bialynicki-Birula, M. A. Cirone, J. P. Dahl, M. Fedorov, and W. P. Schleich. In-and Outbound Spreading of a Free-Particle s-Wave. Physics Review Letters,2002,89(6):060404-060407
[73] M. A. Cirone, K. Rzazewski, W. P. Schleich, F. Straub, and J. A. Wheeler. Quantumanticentrifugal force. Physics Review A,2001,65(2):022101-022106
[74] A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann and S. Longhi.Geometric Potential and Transport in Photonic Topological Crystals. Physics ReviewLetters,2010,104(15):150403-150406
[75] J. Onoe, T. Ito, H. Shima, H. Yoshioka and S. Kimura. Observation of Riemanniangeometric effects on electronic states. Europhysics Letters,2012,98(2):27001
[76] J. K. david. Canonical Quantization with Quadratic Constraints. General Relati-vity and Gravitation,2,(1971),247-279
[77] I.A. Batalin and E.S. Fradkin. Operatorial quantization of dynamical systems Subject tosecond class constraints. Nuclear Physics,1987,279:514-528
[78] J.Barcelos-Neto, Ashok Das, W. Scherer. Canonical quantization of constrained sys-tems. Acta physica polonica,1987,18:269-288
[79] R. Banerjee. Hamiltonian embedding of a second-class system with a Chern-Simonsterm. Physical Review D,1993,48:5467-5470
[80] Jorge Ananias Neto. Remarks on the collective coordinates quantization of the SU(2)Skyrme model. Journal of Physics G: Nuclear and Particle Physics,1995,21:695-699
[81] Petre Dita. Quantization of the motion of a particle on an n-dimensional sphere. Phy-sical Review A,1997,56:2574-2578.
[82] Jorge Ananias Neto, Wilson Oliveira. Does the Weyl ordering prescription lead to thecorrect energy levels for the quantum particle on the D-dimensional sphere? Inter-national Journal of Modern Physics A,1999,14:3699-3713
[83] Bill Poirier. Wigner-Weyl correspondence and semiclassical quantization in sphericalcoordinates. Journal of mathematical physics,1999,40:6302-6318
[84] Soon-Tae Hong, Yong-Wan Kim, Young-Jai Park. Consistent Dirac quantization ofSU(2) Skyrmion equivalent to the BFT scheme. Physical Review D.1999,59:114026
[85] John R. Klauder. Quantization of Constrained Systems. Lecture Notes in Physics,2001,572:143-182
[86] K.D. Rothe, and F.G. Scholtz. On the Hamilton–Jacobi equation for second-classconstrained systems. Annals of Physics,2003,308:639–651.
[87] C. Benavides and A.F. Reyes-Lega. Canonical Group Quantization, Rotation Gene-rators and Quantum Indistinguishability. Geometric and Topological Methods forQuantum Field Theory,2010.344-367
[88] Naoko Okamoto and Masayoshi Nakamura. Quantum Corrections in ConstrainedSystems. Progress of Theoretical Physics,1996,96:235-246.
[89] do Carmo M P. Differential Geometry of Curves and Surfaces. Englewood Cliffs, NJ:Prentice-Hall,1976
[90] Ou-Yang Z C, Liu J X and Xie Y Z. Geometrical Methods in the Elastic Theory ofMembranes in Liquid Crystal Phases. Singapore: World Scientific,1999
[91] Q. H. Liu, Geometric Momentum for a Particle on a Curved Surface, http://arXiv:1109.0153
[92] G. Cantele, D. Ninno, and G. Iadonisi. Topological surface states in deformed quan-tum wires. Physics Review B,2000,61:13730
[93] H. Aoki, M. Koshino, D. Takeda, H. Morise, and K. Kuroki. Electronic structure ofperiodic curved surfaces: Topological band structure. Physics Review B,2001,65:035102
[94] M.Encinosa and L. Mott. Curvature-induced toroidal bound states. Physics Review A,2003,68:014102
[95] N. Fujita and O. Terasaki. Band structure of the P, D, and G surfaces. Physics ReviewB,2005,72:085459
[96] M. Koshino and H. Aoki. Electronic structure of an electron on the gyroid surface: Ahelical labyrinth. Physics Review B,2005,71:073405
[97] J. Gravesen and M. Willatzen. Eigenstates of M bius nanostructures including curva-ture effects. Physics Review A,2005,72:032108
[98] A. Marchi, S. Reggiani, M. Rudan, and A. Bertoni. Coherent electron transport in bentcylindrical surfaces. Physics Review B,2005,72:035403
[99] B. Jensen and R. Dandoloff. Quantum mechanics of a constrained electrically chargedparticle in the presence of electric currents. Physics Review A,2009,80:052109
[100]B. Jensen and R. Dandoloff. Publisher's Note: Physiologically driven avian vocalsynthesizer.2010,81:049905(E)
[101]B. Jensen and R. Dandoloff. Remarks on quantum mechanics on surfaces. PhysicsLetters A,2011,375:448
[102]H. D. Doebner, P. st'ovicek and J. Tolar. Quantization of kinematics on confgurationmanifolds. Reviews in Mathematical Physics,2001,13:1–47
[103]M. Englis. Berezin quantization and reproducing kernels on complex domains. Tran-sactions of the American Mathematical Society,1996,348:411–479.
[104]G. A. Goldin, R. Menikoff and D. H. Sharp. Particle statistics from induced represen-tations of a local current group. Journal of Mathematical Physics,1980,21:650–664
[105]W. Pauli, General principles of quantum mechanics, Translated by P. Achuthan and K.Venkatesan,(Springer-Verlag, Berlin,1980) p.41
[106]C. Neves and C. Wotzasek. Stückelberg field-shifting quantization of a free particle ona D-dimensional sphere. Journal of Physics A: Mathematical and General,2000,33:6447–6456
[107]H. Grundling, C. A. Hurst. Constrained dynamics for quantum mechanics. I. Restri-cting a particle to a surface. Journal of Mathematical Physics,1998,39:3091–3119
[108]R. Froese, I. Herbst. Realizing Holonomic Constraints in Classical and QuantumMechanics. Communications in Mathematical Physics,2001,220:489–535
[109]W.S.Massey. A Basic Coursein Algebraic Topology. New York Berlin Heidelberg:Springer-Verlag,1991
[110]K. Kowalski and J. Rembielinski. Coherent states for the quantum mechanics on atorus. Physics Review A,2007,75:052102
[111]R. Dandoloff. Geometry Induced Potential on a2D-section of a Wormhole: Catenoid.Physics Letters A,2009,373:2667
[112]M. Morris and K. Thorne. Wormholes in spacetime and their use for interstellartravel:A tool for teaching general relativity. American Journal of Physics,1988,56(5):395
[113]B. Jensen. Electronic states on the helicoidal surface. Physics Review A,2009,80(2):022101-022107
[2] R.P. Feynman. The quantum theory of gravitation. Acta Physica Polonica,1963,24:697
[3] Wang Anmin, Ruan Tunan. Extension of the Dirac-Bergmann theory of constrainedsystems. Physics Review A,1996,54(1):57–80.
[4] J.L.Anderson, and P.G.Bergmann. Constraints in covariant field theories. PhysicalReview,1951,83(5):1018-1025
[5] P.G.Bergmann, and J.H.M.Brunings. Non-Linear Field Theories II. Canonical Equati-ons and Quantization. Review of Moderm Physics,1949,21(3):480
[6] P.A.M. Dirac. Generalized Hamiltonion dynamics. Canadian Journal of Mathematics,1950,2:129-148
[7] P.A.M. Dirac. The Hamiltonian form of field dynamics Canadian Journal of Mathema-tics,1951,3:1-23
[8] P.A.M. Dirac. Forms of Relativistic Dynamics. Review of Moderm Physics,1949,21(3):392-399
[9] P.A.M. Dirac. The Theory of Gravitation in Hamiltonian Form. Proceedings of the RoyalSociety of London A,1958,246(246):333-343
[10] P.A.M Dirac. Fixation of Coordinates in the Hamiltonian Theory of Gravitation.Physical Review,1959,114(3):924-930
[11] R.P. Feynman. Space-time approach to non-relativistic quantum mechanics. Reviews ofModem Physics,1948,20:367-387
[12] P.A.M. Dirac. Lecture on quantum mechanics. New York: Yeshiva University,1964.
[13] R.P.Feynman, A.R.Hibbs. Quantum mechanics and path integrals. New York: McGraw-Hill,1965
[14] L.D.Faddeev, V.N.Popov. Feynman diagrams for the Yang-Mills field. Physics Letters,1967,25:29-30
[15] L.D.Faddeev. Feynman integral for singular Lagrangian. Theoretical and Mathema-tical Physics,1970,1:1-13
[16] P.Senjanovic. Path integral quantization of field theories with second-class constraints.Annals of Physics,1976,100:227-261
[17] M. Henneaus. Hamiltonian form of the path integral for theories with a gauge freed-om.Physics Reports,1985,126(1):1-66
[18] M. Henneaus, C. Teitelboim. Quantization of gauge system. Princeton: PrincetonUniversity Press,1992
[19] J. Gomis, J. Paris, S. Samuel. Antibracket antifields and gauge-theory quantization.Physics Reports,1995,259:1-145
[20] B. Podolsky. Quantum-Mechanically Correct Form of Hamiltonian Function forConservative Systems. Physics Review,1928,32:812
[21] L. I. Schiff. Quantum Mechanics. New York: McGraw-Hill,1949,135
[22] P. W. Higgs. Dynamical symmetries in a spherical geometry. International Journalof Modern Physics A,1979,12:309
[23] H. Kleinert. Path Integrals in Quantum Mechanics. Singapore: Statistics and PolymerPhysics,5th ed.2009,1-670
[24] V. Aldaya, M. Calixto, J. Guerrero, and F. F. Lopez-ruiz. Group-quantization ofnonlinear sigma models: particle on S2revisited. Reports on Mathematical Physics,2009,64(1):49-58
[25] B. S. DeWitt. Dynamical Theory in Curved Spaces. I. A Review of the Classical andQuantum Action Principles. Reviews of Modern Physics,1957,29:377.
[26] K. S. Cheng. Quantization of a General Dynamical System by Feynman's PathIntegration Formulation. Journal of Mathematical Physics,1972,13(11):1723-1726
[27] H. Dekker. On the path integral for diffusion in curved spaces. Physica A: StatisticalMechanics and its Applications,1980,103(3):586
[28] H. Kleinert and S. V. Shabanov. Proper Dirac quantization of a free particle on aD-dimensional sphere. Physics Letters A,1997,232:327-332
[29] R. H. Dicke and J. P. Wittke. Introduction to Quantum Mechanics. Massachusetts:Addison-Wesley,1963,143
[30] J. M. Domingos, and M. H. Caldeira. The quantization of the Hamiltonian in curvedspace. Foundations of Physics,1984,14(7):147
[31] Makoto Kato. Momentum and Hamiltonian Operators in Generalized Coordinates. In-ternational Journal of Theoretical Physics,1994,33(4):857
[32] H. Grundling. Quantum constraints. Reports on Mathematical Physics,2006,57:97-120
[33] H. J. Rothe and K. D. Rothe. Classical and Quantum Dynamics of Constrained Ham-iltonian Systems. Singapore: World Scientifc,2010
[34] L. S. Schulman. Techniques and Applications of Path Integration. New York: Wiley,1981
[35] M. Henneaux and C. Teitelboim. Quantization of Gauge Systems. Princeton: Prince-ton University Press,1992
[36] V.I. Arnold, S.P. Novikov, G. Wassermann et al. Dynamical Systems IV: SymplecticGeometry and Its Applications. English: Springer-Verlag,1996
[37] Gamkrelidze, Revaz V. Encyclopaedia of Mathematical Sciences, Vol.4. NewYork:Springer-Verlag,1990
[38] N. M. J. Woodhouse. Geometric Quantization. Clarendon: Oxford,1992
[39] P.A.M. Dirac. The Principles of Quantum Mechanics.3rd Edition. Oxford: OxfordUniv. Press,1947,114.
[40]王正行.谨与简洁之美:王竹溪一生的物理追求，北京大学出版社，2008
[41] L. Kaplan, N. T. Maitra, and E. J. Heller. Quantizing constrained systems. PhysicsReview A,1997,56:2592
[42] J. R. Klauder and S. V. Shabanov. Coordinate-free quantization of second-class con-straints. Nuclear Physics,1998,511:713-736
[43] D. Baleanu and Y. Güler. Hamilton-Jacobi treatment of the non-relativistic particle onthe curved space. Journal of Physics A,2001,34:73-80
[44] H. J. Schnitzer. Chiral perturbation theory for the pion-skyrmion system. Nuclear Phy-sics B,1985,261:546-572
[45] M. Ikegami, Y. Nagaoka, S. Takagi, and T. Tanzawa. Quantum Mechanics of a Particleon a Curved Surface. Progress of Theoretical Physics,1992,88:229
[46] P. Maraner. Monopole gauge fields and quantum potentials induced by the geometry insimple dynamical systems. Annals of Physics,1996,246:325-346
[47] L. Kaplan, N. T. Maitra, and E. J. Heller. Quantizing constrained systems. Physics Re-view A,1997,56(4):2592-2599
[48] A. G. Nuramatov, l. V. Prokhorov. On quantization of systems with second class con-straints. International Journal of Geometric Methods in Modern Physics,2006,3:1459-1467
[49] G. Gyorgyi and S. Kovesi-Domokos, Nuovo Cimento B58,191(1968)
[50] N. K. Falck and A. C. Hirshfeld. Dirac-bracket quantisation of a constrained non-linearsystem: the rigid rotator. European Journal of Physics,1983,4(1):5
[51] S. Ishikawa, T.Miyazaki, K. Yamamoto, andM. Yamanobe. How to construct a coor-dinate representation of a hamiltonian operator on a torus. International Journalof Modern Physics A,1996,11:3363
[52] S. T. Hong, W. T. Kim, and Y. J. Park. Improved dirac quantization of a free particle.Mod. Physics Letters A,2000,15:1915-1922
[53] T. Homma, T. Inamoto, and T. Miyazaki. Schrodinger equation for the nonrelativisticparticle constrained on a hypersurface in a curved space. Physics Review D,1990,42:2049
[54] T. Homma, T. Inamoto, and T. Miyazaki. Alternative formulation in the constrainedhamiltonian system of a rigid rotator. Zeitschrift für Physik C,1990,48:105-108
[55] N. Ogawa, K. Fujii, A. Kobushukin. Quantum Mechanics in Riemannian Manifold.Progress of Theoretical Physics,1990,83(5):894-905
[56] N. Ogawa, K. Fujii, N. Chepilko and A. Kobushukin. Quantum Mechanics in Rie-mannian Manifold. II. Progress of Theoretical Physics,1991,85(6):1189-1201
[57] N. Ogawa. The Difference of Effective Hamiltonian in Two Methods in QuantumMechanics on Submanifold. Progress of Theoretical Physics,1992,87(2):513-517
[58] A. V. Golovnev. Dirac quantization of free motion on curved surfaces. InternationalJournal of Geometric Methods in Modern Physics,2006,3:655
[59] A. V. Golovnev. Canonical quantization of motion on submanifolds. Reports on Mathe-matical Physics,2009,64:59
[60] P. C. Schuster and R. L. Jaffe. Quantum mechanics on manifolds embedded ineuclidean space. Annals of Physics,2003,307:132-143
[61] H. Jensen and H. Koppe. Quantum mechanics with constraints. Ann. Phys.(NY).1971,63:586
[62] R. C. T. da Costa. Quantum mechanics of a constrained particle. Physics Review A.1981,23:1982
[63] A. V. Chaplik and R. H. Blick. On geometric potentials in quantum-electro mech-anical circuits. New Journal of Physics,2004,6:33
[64] G. Ferrari and G. Cuoghi. Schr dinger Equation for a Particle on a Curved Surface inan Electric and Magnetic Field. Physics Review Letters,2008,100:230403
[65] Q. H. Liu, C. L. Tong and M. M. Lai. Constraint-induced mean curvature dependenceof cartesian momentum operators. Journal of Physics A: Mathematical and Theoretical,2007,40:4161
[66] Q. H. Liu, L. H. Tang and D. M. Xun. Geometric momentum: The proper momentumfor a free particle on a two-dimensional sphere. Physics Review A,2011,84:042101
[67] S. Matsutani. The physical meaning of the embedded effect in the quantum sub-manifold system. Journal of Physics A: Mathematical and General,1993,26(19):5133
[68] D. M. Xun, and Q. H. Liu. Geometric momentum in the monge parametrization oftwo-dimensional sphere. International Journal of Geometric Methods in ModernPhysics,2013,10:1220031
[69] Q. H. Liu, Y. Shen, D. M. Xun and X. Wang. On relation between geometricmomentum and annihilation operators on a two-dimensional sphere. InternationalJournal of Geometric Methods in Modern Physics,2013,10:1320007
[70] V. Atanasov, R. Dandoloff, and A. Saxena. Geometry induced charge separation on ahelicoidal ribbon. Physics Review B,2009,79:033404
[71] R. Dandoloff, A. Saxena, and B. Jensen. Geometry-induced potential on a two-di-mensional section of a wormhole: Catenoid. Physics Review A,2010,81(1):014102
[72] I. Bialynicki-Birula, M. A. Cirone, J. P. Dahl, M. Fedorov, and W. P. Schleich. In-and Outbound Spreading of a Free-Particle s-Wave. Physics Review Letters,2002,89(6):060404-060407
[73] M. A. Cirone, K. Rzazewski, W. P. Schleich, F. Straub, and J. A. Wheeler. Quantumanticentrifugal force. Physics Review A,2001,65(2):022101-022106
[74] A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann and S. Longhi.Geometric Potential and Transport in Photonic Topological Crystals. Physics ReviewLetters,2010,104(15):150403-150406
[75] J. Onoe, T. Ito, H. Shima, H. Yoshioka and S. Kimura. Observation of Riemanniangeometric effects on electronic states. Europhysics Letters,2012,98(2):27001
[76] J. K. david. Canonical Quantization with Quadratic Constraints. General Relati-vity and Gravitation,2,(1971),247-279
[77] I.A. Batalin and E.S. Fradkin. Operatorial quantization of dynamical systems Subject tosecond class constraints. Nuclear Physics,1987,279:514-528
[78] J.Barcelos-Neto, Ashok Das, W. Scherer. Canonical quantization of constrained sys-tems. Acta physica polonica,1987,18:269-288
[79] R. Banerjee. Hamiltonian embedding of a second-class system with a Chern-Simonsterm. Physical Review D,1993,48:5467-5470
[80] Jorge Ananias Neto. Remarks on the collective coordinates quantization of the SU(2)Skyrme model. Journal of Physics G: Nuclear and Particle Physics,1995,21:695-699
[81] Petre Dita. Quantization of the motion of a particle on an n-dimensional sphere. Phy-sical Review A,1997,56:2574-2578.
[82] Jorge Ananias Neto, Wilson Oliveira. Does the Weyl ordering prescription lead to thecorrect energy levels for the quantum particle on the D-dimensional sphere? Inter-national Journal of Modern Physics A,1999,14:3699-3713
[83] Bill Poirier. Wigner-Weyl correspondence and semiclassical quantization in sphericalcoordinates. Journal of mathematical physics,1999,40:6302-6318
[84] Soon-Tae Hong, Yong-Wan Kim, Young-Jai Park. Consistent Dirac quantization ofSU(2) Skyrmion equivalent to the BFT scheme. Physical Review D.1999,59:114026
[85] John R. Klauder. Quantization of Constrained Systems. Lecture Notes in Physics,2001,572:143-182
[86] K.D. Rothe, and F.G. Scholtz. On the Hamilton–Jacobi equation for second-classconstrained systems. Annals of Physics,2003,308:639–651.
[87] C. Benavides and A.F. Reyes-Lega. Canonical Group Quantization, Rotation Gene-rators and Quantum Indistinguishability. Geometric and Topological Methods forQuantum Field Theory,2010.344-367
[88] Naoko Okamoto and Masayoshi Nakamura. Quantum Corrections in ConstrainedSystems. Progress of Theoretical Physics,1996,96:235-246.
[89] do Carmo M P. Differential Geometry of Curves and Surfaces. Englewood Cliffs, NJ:Prentice-Hall,1976
[90] Ou-Yang Z C, Liu J X and Xie Y Z. Geometrical Methods in the Elastic Theory ofMembranes in Liquid Crystal Phases. Singapore: World Scientific,1999
[91] Q. H. Liu, Geometric Momentum for a Particle on a Curved Surface, http://arXiv:1109.0153
[92] G. Cantele, D. Ninno, and G. Iadonisi. Topological surface states in deformed quan-tum wires. Physics Review B,2000,61:13730
[93] H. Aoki, M. Koshino, D. Takeda, H. Morise, and K. Kuroki. Electronic structure ofperiodic curved surfaces: Topological band structure. Physics Review B,2001,65:035102
[94] M.Encinosa and L. Mott. Curvature-induced toroidal bound states. Physics Review A,2003,68:014102
[95] N. Fujita and O. Terasaki. Band structure of the P, D, and G surfaces. Physics ReviewB,2005,72:085459
[96] M. Koshino and H. Aoki. Electronic structure of an electron on the gyroid surface: Ahelical labyrinth. Physics Review B,2005,71:073405
[97] J. Gravesen and M. Willatzen. Eigenstates of M bius nanostructures including curva-ture effects. Physics Review A,2005,72:032108
[98] A. Marchi, S. Reggiani, M. Rudan, and A. Bertoni. Coherent electron transport in bentcylindrical surfaces. Physics Review B,2005,72:035403
[99] B. Jensen and R. Dandoloff. Quantum mechanics of a constrained electrically chargedparticle in the presence of electric currents. Physics Review A,2009,80:052109
[100]B. Jensen and R. Dandoloff. Publisher's Note: Physiologically driven avian vocalsynthesizer.2010,81:049905(E)
[101]B. Jensen and R. Dandoloff. Remarks on quantum mechanics on surfaces. PhysicsLetters A,2011,375:448
[102]H. D. Doebner, P. st'ovicek and J. Tolar. Quantization of kinematics on confgurationmanifolds. Reviews in Mathematical Physics,2001,13:1–47
[103]M. Englis. Berezin quantization and reproducing kernels on complex domains. Tran-sactions of the American Mathematical Society,1996,348:411–479.
[104]G. A. Goldin, R. Menikoff and D. H. Sharp. Particle statistics from induced represen-tations of a local current group. Journal of Mathematical Physics,1980,21:650–664
[105]W. Pauli, General principles of quantum mechanics, Translated by P. Achuthan and K.Venkatesan,(Springer-Verlag, Berlin,1980) p.41
[106]C. Neves and C. Wotzasek. Stückelberg field-shifting quantization of a free particle ona D-dimensional sphere. Journal of Physics A: Mathematical and General,2000,33:6447–6456
[107]H. Grundling, C. A. Hurst. Constrained dynamics for quantum mechanics. I. Restri-cting a particle to a surface. Journal of Mathematical Physics,1998,39:3091–3119
[108]R. Froese, I. Herbst. Realizing Holonomic Constraints in Classical and QuantumMechanics. Communications in Mathematical Physics,2001,220:489–535
[109]W.S.Massey. A Basic Coursein Algebraic Topology. New York Berlin Heidelberg:Springer-Verlag,1991
[110]K. Kowalski and J. Rembielinski. Coherent states for the quantum mechanics on atorus. Physics Review A,2007,75:052102
[111]R. Dandoloff. Geometry Induced Potential on a2D-section of a Wormhole: Catenoid.Physics Letters A,2009,373:2667
[112]M. Morris and K. Thorne. Wormholes in spacetime and their use for interstellartravel:A tool for teaching general relativity. American Journal of Physics,1988,56(5):395
[113]B. Jensen. Electronic states on the helicoidal surface. Physics Review A,2009,80(2):022101-022107