On The Enlarged Canonical Quantization Scheme

Although many concepts in classical mechanics and quantum mechanics are contradictory, there are connections in between. Quantization is one of the bridges connecting these two systems. Quantization has many different schemes, among them the representative ones include the group quantization, the geometrical quantization, the path integral quantization, etc. However, these quantization schemes cannot determine the form of geometric momentum, potential energy or the Hamiltonian geometry uniquely. Thus a new quantization method is needed to solve the above problems.Recently, we put forward a new method of quantizati on—the Enlarged Canonical Quantization Scheme(ECQS). It is based on the basis of Dirac's theory of canonical quantization which is a kind of algebraic properties of quantization scheme. In ECQS, the algebraic relations between positions, momenta and Hamil tonian remain the same during the quantization system process. In fact, it reflects a kind of algebraic symmetry. The core part of ECQS is to quantize the positions, momenta and Hamiltonian simultaneously. The fundamental commutation relations consist of commutators between positions and positions, positions and momenta, and momenta and momenta, and it is classified to be the first category of the fundamental commutation relations, which is defined by Dirac. Moreover, the ECQS regards commutators between positions and Hamiltonian and momenta and Hamiltonian as the second category of the fundamental commutation relations. These two categories, instead of the first category, are used to quantize positions, momenta and Hamiltonian simultaneously. A very interesting yet surprising consequence of the ECQS is the choice of the space during the quantization.The most interesting result is that ECQS can pick out the space quantized completely.This thesis mainly consists of three parts.In the first part, we study the relation of the ECQS and the Dirac's canonical quantization theory to explain the success and the defects of the universal hypothesis of quantization. To understand its defects, different physics scientists, for example, Dirac and Pauli, have difference.In the second part, we study quantization problem of the motion constrained on the two-dimensional sphere and develop a discriminant that can be used to show how the quantization within the intrinsic geometry is improper.In the third part, we research the universal discriminant through specific parameterization. More importantly, we study that how to make the quantization drastically. The results show that it must make the two-dimensional sphere embedded in three-dimensional flat space, and use the Cartes ian coordinates of three-dimensional flat space.This study shows that the ECQS can pick up the appropriate coordinate system for quantization. For the non-constraint system, ECQS thinks that the appropriate coordinate system is the Cartesian coordinate system. However, for the motion of quantum mechanics constrained on the two-dimensional sphere, ECQS thinks that it must be embedded in the three-dimensional Cartesian space, and use the three-dimensional Cartesian coordinate system.