Finite groups and quantum physics
- PDF / 895,116 Bytes
- 18 Pages / 612 x 792 pts (letter) Page_size
- 70 Downloads / 257 Views
EMENTARY PARTICLES AND FIELDS Theory
Finite Groups and Quantum Physics V. V. Kornyak* Laboratory of Information Tecnnologies, Joint Institute for Nuclear Physics, Dubna, Moscow oblast, 141980 Russia Received February 2, 2012
Abstract—Concepts of quantum theory are considered from the constructive “finite” point of view. The introduction of a continuum or other actual infinities in physics destroys constructiveness without any need for them in describing empirical observations. It is shown that quantum behavior is a natural consequence of symmetries of dynamical systems. The underlying reason is that it is impossible in principle to trace the identity of indistinguishable objects in their evolution—only information about invariant statements and values concerning such objects is available. General mathematical arguments indicate that any quantum dynamics is reducible to a sequence of permutations. Quantum phenomena, such as interference, arise in invariant subspaces of permutation representations of the symmetry group of a dynamical system. Observable quantities can be expressed in terms of permutation invariants. It is shown that nonconstructive number systems, such as complex numbers, are not needed for describing quantum phenomena. It is sufficient to employ cyclotomic numbers—a minimal extension of natural numbers that is appropriate for quantum mechanics. The use of finite groups in physics, which underlies the present approach, has an additional motivation. Numerous experiments and observations in the particle physics suggest the importance of finite groups of relatively small orders in some fundamental processes. The origin of these groups is unclear within the currently accepted theories—in particular, within the Standard Model. DOI: 10.1134/S1063778813010079
1. INTRODUCTION Universality is a distinguishing feature of quantum mechanics. It is applicable to systems of totally different physical nature and scales: from subatomic particles to large molecules.1) This universality is characteristic of theories based on some a priori mathematical principles. In the case of quantum mechanics, symmetry is an underlying principle. Only systems containing indistinguishable particles exhibit quantum behavior: any deviation from the identity of particles destroys quantum interference. The indistinguishability of elements of a system means that they lie on the same orbit of the symmetry group of the system. For systems featuring symmetries, only relations and statements that are independent of the “rearrangement” of “homogeneous” elements—that is, invariant relations and statements—are objective. For example, electric potentials ϕ and ψ or points of space that are denoted by vectors a and b cannot be endowed with any physical meaning. However, their combinations denoted by ψ − ϕ or b − a (ϕ−1 ψ and a−1 b according to more general symbolic notation of group theory) are meaningful. * 1)
E-mail: [email protected] In particular, experiments in which quantum interference between fullerene molecules C60 was observed were described in
Data Loading...
