arXiv subjects
Publications and source records attributed to Vladimir V. Kornyak.
Any Hilbert space with composite dimension can be factorized into a tensor product of smaller Hilbert spaces. This allows to decompose a quantum system into subsystems. We propose a simple tractable model for a constructive study of decompositions of quantum systems.
We describe an algorithm for computing the complete set of primitive orthogonal idempotents in the centralizer ring of the permutation representation of a wreath product. This set of idempotents determines the decomposition of the representation into irreducible components. In the formalism of quantum mechanics, these idempotents are projection operators into irreducible invariant subspaces of the Hilbert space of a multipartite quantum system. The C implementation of the algorithm constructs irreducible decompositions of high-dimensional representations of wreath products. Examples of computations are given.
We describe an algorithm for splitting permutation representations of finite group over fields of characteristic zero into irreducible components. The algorithm is based on the fact that the components of the invariant inner product in invariant subspaces are operators of projection into these subspaces. An important element of the algorithm is the calculation of Gröbner bases of polynomial ideals. A preliminary implementation of the algorithm splits representations up to dimensions of tens of thousands. Some examples of computations are given in appendix.
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.
Using the fact that any linear representation of a group can be embedded into permutations, we propose a constructive description of quantum behavior that provides, in particular, a natural explanation of the appearance of complex numbers and unitarity in the formalism of quantum mechanics. In our approach, the quantum behavior can be explained by the fundamental impossibility to trace the identity of indistinguishable objects in their evolution. Any observation only provides information about the invariant relations between such objects. The trajectory of a quantum system is a sequence of unitary evolutions interspersed with observations -- non-unitary projections. We suggest a scheme to construct combinatorial models of quantum evolution. The principle of selection of the most likely trajectories in such models via the large numbers approximation leads in the continuum limit to the principle of least action with the appropriate Lagrangians and deterministic evolution equations.
This essay advocates the view that any problem that has a meaningful empirical content, can be formulated in constructive, more definitely, finite terms. We consider combinatorial models of dynamical systems and approaches to statistical description of such models. We demonstrate that many concepts of continuous physics --- such as continuous symmetries, the principle of least action, Lagrangians, deterministic evolution equations --- can be obtained from combinatorial structures as a result of the large number approximation. We propose a constructive description of quantum behavior that provides, in particular, a natural explanation of appearance of complex numbers in the formalism of quantum mechanics. Some approaches to construction of discrete models of quantum evolution that involve gauge connections are discussed.
We study deterministic and quantum dynamics from a constructive "finite" point of view, since the introduction of a continuum, or other actual infinities in physics poses serious conceptual and technical difficulties, without any need for these concepts to physics as an empirical science. For a consistent description of the symmetries of dynamical systems at different times and the symmetries of the various parts of such systems, we introduce discrete analogs of the gauge connections. Gauge structures are particularly important to describe the quantum behavior. We show that quantum behavior is the result of a fundamental inability to trace the identity of indistinguishable objects in the process of evolution. Information is available only on invariant statements and values, relating to such objects. Using mathematical arguments of a general nature we can show that any quantum dynamics can be reduced to a sequence of permutations. Quantum interferences occur in the invariant subspaces of permutation representations of symmetry groups of dynamical systems. The observable values can be expressed in terms of permutation invariants. We also show that for the description of quantum phenomena, instead of a nonconstructive number system --- the field of complex numbers, it is enough to use cyclotomic fields --- the minimal extentions of natural numbers suitable for quantum mechanics. Finite groups of symmetries play a central role in this article. The interest in such groups has an additional motivation in physics. Numerous experiments and observations in particle physics point to an important role of finite groups of relatively low orders in a number of fundamental processes.
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.
We consider the problem of quantum behavior in the finite background. Introduction of continuum or other infinities into physics leads only to technical complications without any need for them in description of empirical observations. The finite approach makes the problem constructive and more tractable. We argue that quantum behavior is a natural consequence of symmetries of dynamical systems. It is a result of fundamental impossibility to trace identity of indistinguishable objects in their evolution - only information about invariant combinations of such objects is available. We demonstrate that any quantum dynamics can be embedded into a simple permutation dynamics. Quantum phenomena, such as interferences, arise in invariant subspaces of permutation representations of the symmetry group of a system. Observable quantities can be expressed in terms of the permutation invariants.
To study discrete dynamical systems of different types --- deterministic, statistical and quantum --- we develop various approaches. We introduce the concept of a system of discrete relations on an abstract simplicial complex and develop algorithms for analysis of compatibility and construction of canonical decompositions of such systems. To illustrate these techniques we describe their application to some cellular automata. Much attention is paid to study symmetries of the systems. In the case of deterministic systems, we reveale some important relations between symmetries and dynamics. We demonstrate that moving soliton-like structures arise inevitably in deterministic dynamical system whose symmetry group splits the set of states into a finite number of group orbits. We develop algorithms and programs exploiting discrete symmetries to study microcanonical ensembles and search phase transitions in mesoscopic lattice models. We propose an approach to quantization of discrete systems based on introduction of gauge connection with values in unitary representations of finite groups --- the elements of the connection are interpreted as amplitudes of quantum transitions. We discuss properties of a quantum description of finite systems. In particular, we demonstrate that a finite quantum system can be embedded into a larger classical system. Computer algebra and computational group theory methods were useful tools in our study.
Universality of quantum mechanics -- its applicability to physical systems of quite different nature and scales -- indicates that quantum behavior can be a manifestation of general mathematical properties of systems containing indistinguishable, i.e. lying on the same orbit of some symmetry group, elements. In this paper we demonstrate, that quantum behavior arises naturally in systems with finite number of elements connected by non-trivial symmetry groups. The "finite" approach allows to see the peculiarities of quantum description more distinctly without need for concepts like "wave function collapse", "Everett's multiverses" etc. In particular, under the finiteness assumption any quantum dynanics is reduced to the simple permutation dynamics. The advantage of the finite quantum models is that they can be studied constructively by means of computer algebra and computational group theory methods.
Gauge invariance in discrete dynamical systems and its connection with quantization are considered. For a complete description of gauge symmetries of a system we construct explicitly a class of groups unifying in a natural way the space and internal symmetries. We describe the main features of the gauge principle relevant to the discrete and finite background. Assuming that continuous phenomena are approximations of more fundamental discrete processes, we discuss -- with the help of a simple illustration -- relations between such processes and their continuous approximations. We propose an approach to introduce quantum structures in discrete systems, based on finite gauge groups. In this approach quantization can be interpreted as introduction of gauge connection of a special kind. We illustrate our approach to quantization by a simple model and suggest generalization of this model. One of the main tools for our study is a program written in C.
This paper has been withdrawn by the author, since proposed extensions are equivalent at any k to either wreath or direct product.
We consider discrete dynamical systems and lattice models in statistical mechanics from the point of view of their symmetry groups. We describe a C program for symmetry analysis of discrete systems. Among other features, the program constructs and investigates phase portraits of discrete dynamical systems modulo groups of their symmetries, searches dynamical systems possessing specific properties, e.g., reversibility, computes microcanonical partition functions and searches phase transitions in mesoscopic systems. Some computational results and observations are presented. In particular, we explain formation of moving soliton-like structures similar to ``spaceships'' in cellular automata.
The cellular automata with local permutation invariance are considered. We show that in the two-state case the set of such automata coincides with the generalized Game of Life family. We count the number of equivalence classes of the rules under consideration with respect to permutations of states. This reduced number of rules can be efficiently generated in many practical cases by our C program. Since a cellular automaton is a combination of a local rule and a lattice, we consider also maximally symmetric two-dimensional lattices. In addition, we present the results of compatibility analysis of several rules from the Life family.
An approach to compatibility analysis of systems of discrete relations is proposed. Unlike the Groebner basis technique, the proposed scheme is not based on the polynomial ring structure. It uses more primitive set-theoretic and topological concepts and constructions. We illustrate the approach by application to some two-state cellular automata. In the two-state case the Groebner basis method is also applicable, and we compare both approaches.
We demonstrate advantages of non-standard grading for computing cohomology of restricted Hamiltonian and Poisson algebras. These algebras contain the inner grading element in the properly defined symmetric grading compatible with the symplectic structure. Using modulo p analog of the theorem on the structure of cohomology of Lie algebra with inner grading element, we show that all nontrivial cohomology classes are located in the grades which are the multiples of the characteristic p. Besides, this grading implies another symmetries in the structure of cohomology. These symmetries are based on the Poincare duality and symmetry with respect to transpositions of conjugate variables of the symplectic space. Some results obtained by computer program utilizing these peculiarities in the cohomology structure are presented.
We describe an essential improvement of our recent algorithm for computing cohomology of Lie (super)algebra based on partition of the whole cochain complex into minimal subcomplexes. We replace the arithmetic of rational numbers or integers by a much cheaper arithmetic of a modular field and use the inequality between the dimensions of cohomology H over any modular field F_p = Z/pZ and over Q: dim H(F_p) >= dim H(Q). With this inequality we can, by computing over arbitrary F_p, quickly find the (usually, rare) subcomplexes for which dim H(F_p) > 0 and then carry out the full computation over Q within these subcomplexes. We also present the results of application of the corresponding C program to the Lie superalgebra of special vector fields preserving an "odd-symplectic" structure on the (2|2)-dimensional supermanifold. For this algebra, we found some new basis elements of the cohomology in the trivial module.