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Geometric quantization rules in QCPB theory

Gen Wang

Published May 10, 2020• 0 citations
OpenAlex

Abstract

Using the QCPB theory, we can accomplish the compatible combination of the quantum mechanics and general relativity supported by the G-dynamics. We further study the generalized quantum harmonic oscillator, such as geometric creation and annihilation operators, especially, the geometric quantization rules based on the QCPB theory.

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Using the quantum covariant Poisson bracket (QCPB) theory, we can accomplish much more compatible explanations of the quantum mechanics supported by the G-dynamics. We further study the generalized quantum harmonic oscillator equipped with the G-dynamics of type I, such as geometric creation and annihilation operators, and the geometric number operator as an extension of the number operator is well given for the deep discussions, the geometric Hamiltonian operator is expressed as another form. Especially, the geometric quantization rules based on the QCPB theory is then calculated.
In this paper, we further study the G-dynamics newly emerged in the covariant dynamics defined by the QCPB theory. Then we want to seek the precise formulations of the G-dynamics to calculate the practical quantum problems. We explicitly verify how the role of G-dynamics plays in quantum mechanics, and then we propose three new operators based on the G-dynamics: D-operator, T-operator and G-operator, we find that the non-Hermitian operators are inevitable to appear in the framework of the QCPB. Then the eigenvalue equation of them are discussed accordingly by using the Schrodinger equation and geometrinetic energy operator (GEO) which leads to the Ri-operator, respectively, we obtain some useful results between two quantum situations. Meanwhile, we present generalized covariant wave equation based on the G-operator. The GEO for quantum harmonic oscillator as an application is further calculated.
In this paper, we further study the G-dynamics newly emerged in the covariant dynamics defined by the quantum covariant Poisson bracket (QCPB) theory. We propose three new operators based on the G-dynamics, we find that the non-Hermitian operators are inevitable to appear in the framework of the QCPB. Then we seek the precise formulations of the G-dynamics to calculate the practical quantum problems, the G-dynamics of three types in terms of different Hamiltonian are discovered. The geometrinetic energy operator for quantum harmonic oscillator as an application is further calculated.
In the framework of geometric quantization we extend the Bohr-Sommerfeld rules to a full quantization theory which resembles Heisenberg's matrix theory. This extension is possible because Bohr-Sommerfeld rules not only provide an orthogonal basis in the space of quantum states, but also give a lattice structure to this basis. This permits the definition of appropriate shifting operators. As examples, we discuss the 1-dimensional harmonic oscillator and the coadjoint orbits of the rotation group.
A method is proposed to obtain the dynamics of a system which only makes use of the group law. It incorporates many features of the traditional geometric quantization program as well as the possibility of obtaining the classical dynamics: The classical or quantum character of the theory is related to the choice of the group, avoiding thus the need of quantizing preexisting classical systems and providing a group connection between the quantum and classical systems, i.e., the classical limit. The method is applied to the free-particle dynamics and the harmonic oscillator.
The conditions that must be fulfilled by a certain physical system to apply geometric quantization prescription on it are investigated. These terms are sought as mathematical requirements, which can be traced in an analysis of integrable systems, from the perspective of both potential function and Hamiltonian vector field. The answer is found in momentum map critical points. Basically, a certain disposal of points that allow a momentum map C∗C^*C∗ isomorphism, of observables, with harmonic oscillator momentum map enforce geometric quantization rules. Following the general theory, two newly presented examples, which exhibits these properties, are quantified through geometric quantization prescription. The Lennard-Jones' type potential is one of the examples, it is known as describing molecules in interaction. We end with a third example that shows the local isomorphism of potentials do not induce C∗C^*C∗ isomorphism of observables.