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Chinese Physics Letters

Integer and Half-Integer Quantization Conditions in Quantum Mechanics

Yi-Shi Duan, Duojie Jia

Published Mar 21, 2001• 0 citations
Publisher

Abstract

The integer and half-integer quantization conditions are found in quantum mechanics based on the topological structure of symmetry group of the singlet and spinor wavefunction. The internal symmetry of the physical system is shown to be sufficient to determine the topological structure in quantum mechanics without taking into account the dynamical details about the interaction.

DOI
10.1088/0256-307x/18/4/302

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The integer and half-integer quantization conditions are found in quantum mechanics based on the topological structure of symmetry group of the singlet and spinor wavefunction. The internal symmetry of the physical system is shown to be sufficient to determine the topological structure in quantum mechanics without taking into account the dynamical details about the interaction.
This paper proposes a basic theory on physical reality, and a new foundation for quantum mechanics and classical mechanics. It does not only solve the problem of the arbitrariness on the operator ordering for the quantization procedure, but also clarifies how the classical-limit occurs. It further compares the new theory with the known quantization methods, and proposes a self-consistent interpretation for quantum mechanics. It also provides the internal structure inducing half-integer spin of a particle, the sense of the regularization in the quantum field theory, the quantization of a phenomenological system, the causality in quantum mechanics and the origin of the thermodynamic irreversibility under the new insight.
Discrete measurement outcomes are a defining feature of quantum mechanics, yet the origin of quantization itself remains conceptually unresolved. This work presents an interpretive and structural framework in which quantization arises from global constraint enforcement—due to symmetry, topology, and conservation laws—on continuous state manifolds. Standard quantized observables are reinterpreted as admissible resolutions under these constraints. A formal analogy with elliptic divisibility sequences illustrates rigid integer enforcement arising from global structure alone. The framework does not modify quantum dynamics, introduce collapse mechanisms, or posit hidden variables. The manuscript is currently under consideration at Foundations of Physics.
It is shown that a recently derived relativistic Schr\"odinger equation for free particles of arbitrary spin can be consistently quantized in the case of half-integer spins (but not for integer spins) by invoking the microcausality condition and using the role of certain expectation values of the ccc-number theory as generators of transformations of the Poincar\'e group in the qqq-number theory. The correct type of statistics (Fermi-Dirac) for half-integer-spin particles is obtained as a consequence of the theory. The way to handle the integer-spin case is indicated, but details are left for future presentation.
An attempt is made to study and understand the behavior of quantization of geometric phase of a quantum Ising chain with long range interaction. We show the existence of integer and fractional topological characterization for this model Hamiltonian with different quantization condition and also the different quantized value of geometric phase. The quantum critical lines behave differently from the perspective of topological characterization. The results of duality and its relation to the topological quantization is presented here. The symmetry study for this model Hamiltonian is also presented. Our results indicate that the Zak phase is not the proper physical parameter to describe the topological characterization of system with long range interaction. We also present quite a few exact solutions with physical explanation. Finally we present the relation between duality, symmetry and topological characterization. Our work provides a new perspective on topological quantization.
Abstract Study of symmetry, topology and geometric phase can reveal many new and interesting results on the topological states of matter. Here we present a completely new and interesting result of symmetry, topology and quantization of geometric phase along with the physical explanation for two different symmetry classes. We present a detailed study of the auxiliary space for two different symmetry classes of Hamiltonians. We show explicitly that the origin of the auxiliary space inside the curve is only a necessary condition but it is not a sufficient condition for the topological state. One of the most interesting results is that same symmetry-class Hamiltonians show different behavior in topology and quantized geometric phase.
This paper is concerned with the relationship between the group G of all symmetries of an unconstrained dynamical system and the space of its quantum states. The chief aim is to establish the validity of the claim that "quantizing a system" means deciding what G is and determining all projective unitary representations of G. The mathematical tool suited to this purpose is the theory of central extensions of an arbitrary group G by the circle group T (and the closely related cohomology group H2 (G, T)). The fundamental structural property is that there is a universal state space [Formula: see text] embedding all states, a unitary representation space of the universal central extension [Formula: see text] of G, on which the Pontryagin dual H2 (G, T)⋆ of H2 (G, T) operates as a group of su-perselection rules, so that the superselection sectors are [Formula: see text] and observables are self-adjoint operators Vh → Vh. This leads to a classification of time-evolution operators respecting the symmetries by H2 (G, T) and a general understanding of "anomalies" as arising from a mismatch of sectors to which symmetry transformations and time-evolution refer. For the special case of a Lie group G of symmetries, [Formula: see text] of G coincides with [Formula: see text] the universal covering group, if G is semisimple. More generally, depending on whether or not G is semisimple and/or simply connected, inequivalent quantizations are of topological, algebraic or "mixed" origin; in the algebraic case, the relationship between anomalous conservation laws and invariant dynamics is explicitly given. Illustrative examples worked out include the Euclidean group E (d) and the (connected) Poincaré group P (d, 1) for arbitrary spatial dimension d and the discrete groups (Z/N)2 and Z2 appropriate for planar square crystals — in particular, the absence of "fractional" angular momentum in the plane is established and the difficulties specific to (2,1)-dimensional quantum field theories precisely formulated. The paper concludes by noting that a number of ambiguities generally supposed to affect traditional procedures of quantization resolve themselves satisfactorily in this approach. Mathematically, the paper is self-contained as far as the concepts and results are concerned; occasionally, proofs of well-known facts are reworked when they serve to illustrate the value of the symmetry point of view. The theory of universal central extensions of groups, including important special cases, is treated in detail in an appendix by M. S. Raghunathan, in view of its significance for quantization and the absence of a similar account in the literature.