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Jurnal Fisika dan Aplikasinya

History-like Path of Identical Particle Systems in Topological SpaceM

Akhmad Aminudin Bama

Published Jun 10, 2010• 0 citations
Publisher

Abstract

In this paper, a new concept has been introduced related to the construction of inequivalent quantizations for a system, which is called “history-like path” for particles. This is a collection of non-causal homotopical paths. Each homotopy class of paths will be labelled with a “word” constructed from the generators of fundamental group 1(QN(i)) of a configuration space of the system in a space-like i which is a deformation retract of Mi region that is a sub-space-time of M in which there is no singular slice (a slice Mc in M that contains a singular point). The labels determine the generators and their relations in constructing the fundamental group 1(QN) of the system.

DOI
10.12962/j24604682.v6i2.924

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In this paper, a new concept has been introduced related to the construction of inequivalent quantizations for a system, which is called “history-like path” for particles. This is a collection of non-causal homotopical paths. Each homotopy class of paths will be labelled with a “word” constructed from the generators of fundamental group 1(QN(i)) of a configuration space of the system in a space-like i which is a deformation retract of Mi region that is a sub-space-time of M in which there is no singular slice (a slice Mc in M that contains a singular point). The labels determine the generators and their relations in constructing the fundamental group 1(QN) of the system.
Details of the many-particle system can also be read from my book titled “Quantum Field Theory: Feynman Path Integrals and Diagrammatic Techniques in Condensed Matter” [ 23 ]. The fundamental difference between classical and quantum mechanics relates the concept of indistinguishability of identical particles where each particle can be equipped with an identifying marker without influencing its behavior in classical mechanics. Besides in classical mechanics, the particle follows its own continuous path in phase space and so it is possible to determine the particular process taking place. So, principally, each particle in a group of identical particles can be identified which is not so in quantum mechanics. It is not possible to mark a particle without influencing its physical state, and worse, if a number of identical particles are brought to the same region in space, their wave functions will rapidly spread out and overlap with one another, thereby soon rendering it impossible to say which particle is where. One of the fundamental assumptions for many-particle systems is therefore that identical particles (i.e., particles characterized by the same quantum numbers such as mass, charge and spin) are, in principle, indistinguishable.
Quantization is defined as the act of assigning an appropriate C*-algebra [Formula: see text] to a given configuration space Q, along with a prescription mapping self-adjoint elements of [Formula: see text] into physically interpretable observables. This procedure is adopted to solve the problem of quantizing a particle moving on a homogeneous locally compact configuration space Q=G/H. Here [Formula: see text] is chosen to be the transformation group C*-algebra corresponding to the canonical action of G on Q. The structure of these algebras and their representations are examined in some detail. Inequivalent quantizations are identified with inequivalent irreducible representations of the C*-algebra corresponding to the system, hence with its superselection sectors. Introducing the concept of a pre-Hamiltonian, we construct a large class of G-invariant time-evolutions on these algebras, and find the Hamiltonians implementing these time-evolutions in each irreducible representation of [Formula: see text]. “Topological” terms in the Hamiltonian (or the corresponding action) turn out to be representation-dependent, and are automatically induced by the quantization procedure. Known “topological” charge quantization or periodicity conditions are then identically satisfied as a consequence of the representation theory of [Formula: see text].
For a quantum field living on a nonstatic space–time no instantaneous Hamiltonian is definable, for this generically necessitates a choice of inequivalent representation of the canonical commutation relations at each instant of time. This fact suggests a description in terms of time-dependent Hilbert spaces, a concept that fits naturally in a (consistent) histories framework. Our primary tool for the construction of the quantum theory in a continuous-time histories format is the recently developed formalism based on the notion of the history group. This we employ to study a model system involving a 1+1 scalar field in a cavity with moving boundaries. The instantaneous (smeared) Hamiltonian and a decoherence functional are then rigorously defined so that finite values for the time-averaged particle creation rate are obtainable through the study of energy histories. We also construct the Schwinger–Keldysh closed-time-path generating functional as a “Fourier transform” of the decoherence functional and evaluate the corresponding n-point functions.
In [1], a new approach was suggested for finding quantum structures that might, in particular, be used in potential approaches to quantum gravity that involve non-manifold models for space and/or space-time (for example, causal sets).This involved developing a procedure for quantising a system whose configuration space-or historytheory analogue-is the set of objects in a (small) category Q.In the present paper, we show how this theory can be applied to the special case when Q is a category of sets.