Quantization in Hamiltonian particle mechanics

Sergey A. Rashkovskiy

The quantization process was always tightly connected to the Hamiltonian formulation of classical mechanics. For non-Hamiltonian systems, traditional quantization algorithms turn out to be unsuitable. Numerous attempts to quantize non-Hamiltonian systems have shown that this problem is nontrivial and requires the development of new approaches. In this paper, we present the quantization methods that do not depend upon the Hamiltonian formulation of classical mechanics. Two approaches to the quantization of mechanical systems are considered: axiomatic and hydrodynamic. It is shown that the formal application of these approaches to the classical Hamilton-Jacobi theory allows obtaining the wave equation for the corresponding quantum system in natural way. Examples are considered that show the effectiveness of the proposed approaches, both for Hamiltonian and non-Hamiltonian systems. The spinor form of the relativistic Hamilton-Jacobi theory for classical particles is considered. It is shown that it naturally leads to the Dirac equation for the corresponding quantum particle and to its non-Hamiltonian generalization, the bispinor relativistic Kostin equation.

A. Horzela

Alternative Hamiltonians of classical mechanics, i.e. , canonically inequivalent Hamiltonians leading to the same Newton equations of motion, generate different phase space descriptions of the same mechanical systems. Relation of such descriptions to different algebraic descriptions of quantum systems consistent with the same time evolution - so called Wigner quantization - is presented and illustrated on example of two particles system moving on a line.

E. I. Bogdanov

Gennadi Aleksandrovich Sardanashvily

Nathan M. Lewis

This thesis constructs quantum mechanical theories of finite simple graphs for both labeled and unlabeled graphs.These theories may provide a description for discrete spacetime in a quantum gravity theory. Finite simple graphs and their properties are introduced. The first and second quantization of the standard quantum mechanics of particle systems are reviewed. A quantum mechanical theory of graphs similar to first quantization of standard quantum mechanics is developed via a correspondence of graph edges to particles in particle systems. An algorithm for constructing quantum states of graphs which is independent of graph vertex labeling is developed. It builds antisymmetric graphstates which describe fermiongraphs, the analog of systems of fermions for graphs. A quantum mechanical theory of graphs similar to second quantization of standard quantum mechanics is developed via the same correspondence of graph edges and particles. Edge creation and annihilation operators are defined which insert or delete edges between any pair of grap hvertices. The commutation and anticommutation relations of these operators are defined to produce states which are symmetricand antisymmetric upon any vertex label permutation. Operators for certain graph properties and constructions are defined in this theory. Dynamics are then incorporated into the second quantization approach of the quantum mechanical graph theory by developing two model Hamiltonians, in analogy with the Hamiltonians of the Ising and Heisenberg model sof ferromagnetism. From these Hamiltonians, statistical mechanics can be used to describe the dynamics ofgraphs.