Geometric Momentum, Geometric Potential And A Novel Algebraic Symmetry In Constrained Systems

Constrained motion is ubiquitous. In the microcosmic, low-dimensional systems suchas Graphene, C₆₀and 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.