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arXiv:1803.00408 · Journal of Physics Conference Series

Quantum models based on finite groups

Vladimir V Kornyak

Published Feb 1, 2018• 15 citations
arXivHTMLPDF

Abstract

We consider a constructive modification of quantum-mechanical formalism. Replacement of a general unitary group by unitary representations of finite groups makes it possible to reproduce quantum formalism without loss of its empirical content. Since any linear representation of a finite group can be implemented as a subrepresentation of a permutation representation, quantum-mechanical problems can be formulated in terms of permutation groups. Reproducing quantum behavior in the framework of permutation representations of finite groups makes it possible to clarify the meaning of a number of physical concepts.

DOI
10.1088/1742-6596/965/1/012023

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We consider a constructive modification of quantum-mechanical formalism. Replacement of a general unitary group by unitary representations of finite groups makes it possible to reproduce quantum formalism without loss of its empirical content. Since any linear representation of a finite group can be implemented as a subrepresentation of a permutation representation, quantum-mechanical problems can be formulated in terms of permutation groups. Reproducing quantum behavior in the framework of permutation representations of finite groups makes it possible to clarify the meaning of a number of physical concepts.
Quantum-mechanical concepts can be formulated in constructive finite terms without loss of their empirical content if we replace a general unitary group by a unitary representation of a finite group. Any linear representation of a finite group can be realized as a subrepresentation of a permutation representation. Thus, quantum-mechanical problems can be expressed in terms of permutation groups. This approach allows us to clarify the meaning of a number of physical concepts. Combining methods of computational group theory with Monte Carlo simulation we study a model based on representations of permutation groups.
We study quantum behavior from a constructive "finite" point of view, since the introduction of continuum or other actual infinities into physics poses serious conceptual and technical difficulties without any need for these concepts in physics as an empirical science. Taking this approach, we can show that the quantum-mechanical problems can be formulated in the invariant subspaces of permutation representations of finite groups, while the quantum interferences occur as phenomena that are observable in these subspaces. The scalar products in the invariant subspaces (which are needed for formulating the Born rule – the main postulate of quantum mechanics that links mathematical description with experiment) are linear combinations of independent bilinear invariant forms of the permutation representation. A complete set of such forms for any permutation group can be easily calculated by a simple algorithm. Slightly more sophisticated algorithms are required for expressing quantum observables in terms of these forms.
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.
Mathematical core of quantum mechanics is the theory of unitary representations of symmetries of physical systems. We argue that quantum behavior is a natural result of extraction of "observable" information about systems containing "unobservable" elements in their descriptions. Since our aim is physics where the choice between finite and infinite descriptions can not have any empirical consequences, we consider the problem in the finite background. Besides, there are many indications from observations - from the lepton mixing data, for example - that finite groups underly phenomena in particle physics at the deep level. The "finite" approach allows to reduce any quantum dynamics to the simple permutation dynamics, and thus to express quantum observables in terms of permutation invariants of symmetry groups and their integer characteristics such as sizes of conjugate classes, sizes of group orbits, class coefficients, dimensions of representations. Our study has been accompanied by computations with finite groups, their representations and invariants. We have used both our C implementation of algorithms for working with groups and computer algebra system GAP.
Concepts of quantum theory are considered from the constructive 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.