Mathematical Apparatus Of Quantum Mechanics Research Articles

Model of cognitive activity of the human brain based on the mathematical apparatus of quantum mechanics

Describing the transmission and processing of information by an individual is a fundamental issue in modern cognitive science. Previously, unique scientific models for information transfer between individuals [1], as well as models for cognitive activity [2], were developed. However, numerous presented models lack scalability and formalization, hindering their ability to provide a comprehensive explanation for information transfer processes and their distortion due to interaction with the external communicative environment. One aspect of human cognitive activity is that individuals think not based on a code, as a computer does, but through the interaction of various mental images. Despite the fact that these images have a specific material foundation in the form of electrical and chemical activity in the human brain, describing them using conventional mathematical models presents several difficulties [3]. Therefore, this paper suggests novel techniques for describing information images/representations’ functioning, which simulate an individual’s cognitive activity. These methods rely on the mathematical models of self-oscillatory and conventional quantum physics (including potential wells and virtual particles conventionally used in the physical domain) for characterizing basic interactions [4]. The authors adopt a phenomenological perspective and do not regard cognitive systems as quantum. The aim of this study ject is to establish a model for cognitive function in the human brain using the mathematical principles of self-oscillating quantum mechanics from the perspective of information imaging and representation. Methods. The theory proposes [5] that information images/representations (IR) share characteristic properties with virtual Feymann particles and other elementary particles. The human mind is presented as a one-dimensional potential well with finite walls of varying sizes and an internal potential barrier that simulates the boundary between consciousness and subconsciousness. The authors have applied parametrization based on this theory. As a result, we propose the foundations of the mathematical apparatus based on classical quantum mechanics, followed by the mathematical apparatus of self-oscillating quantum mechanics. The latter, though little known, may allow to predict certain cognitive functions of the human brain by modifying and applying it to non-quantum settings. An equation is derived for the state function of the information image of an individual engaging in cognitive activity. Primary calculations were conducted on the state functions of information images using a computer model. The movement patterns of the information image within and outside were then deduced.

Результативність процесу: системний, енергетичний та еконофізичний підходи у контексті моделювання

In the article from the point of view of the system, energy and econophysic approaches, certain aspects of the theory and practice of measuring the effectivity (in particular, efficiency, performance) of the processes of functioning of the system are outlined. It is emphasized that modern realities point to the actualization of issues in quantum econophysics, in particular the adaptation and use of the mathematical apparatus of quantum mechanics for modeling processes in the economy, the use of quantum mechanical models and analogies, the application of quantum mechanical ideology, since modern socio-ecological-economic systems are complex, nonlinear, synergistic phenomena are observed in their dynamics, instability and poor predictability occur. The aim of the publication is to highlight some analogues of concepts and indicators similar to fundamental physical concepts in the paradigm of «cognition – measurement – evaluation – management» regarding the functioning of socio-ecological-economic systems and processes in them along with outlining approaches to determining the performance of these processes on the basis of modeling using the system, energy and econophysic approaches. It is proposed in the paradigm of «cognition – measurement – evaluation – management» regarding the performance of the processes of functioning of systems to take into account the economic interpretation of the Heisenberg uncertainty principle and the concept of the economic analogue of the Planck constant. It is substantiated that the theoretical and methodological analysis of some basic fundamental physical concepts (energy, momentum, etc.) and their formal and informal connections with real economic concepts and processes will contribute to the improvement of the procedure for determining and applying the energy efficiency coefficient of the processes of functioning of systems based on the authors’ indicators of the components of the performance of processes, taking into account the economic interpretation of the Heisenberg uncertainty principle and the concept of an economic analogue of the Planck constant.

Operatorial Formulation of a Model of Spatially Distributed Competing Populations

This paper deals with the application of the mathematical apparatus of quantum mechanics for the formulation of an operatorial model of a couple of populations spatially distributed over a one-dimensional region. The two populations interact with a competitive mechanism and are able to diffuse over the region. A nonlocal competition effect is also included. In more detail, we consider a one-dimensional region divided in N cells where the actors, represented by annihilation, creation, and a number fermionic operators, interact. The dynamics is governed by a self-adjoint and time-independent Hamiltonian operator describing the various interactions. The results of some numerical simulations are presented and discussed. The recently introduced variant of the standard Heisenberg approach, named (H,ρ)-induced dynamics, is also used in order to take into account some changes in time of the attitudes of the two populations, and obtain more realistic dynamical outcomes.

On the space of Schwartz operators in the symmetric Fock space and its dual

A long-standing problem that arises when constructing the mathematical apparatus of quantum mechanics is the need to work with unbounded operators. Since the space of nuclear operators is preconjugate for the algebra of all bounded operators, we can consider the states of a quantum system as nuclear operators, and consider bounded operators as observables. In this case, taking the trace for the product of a nuclear operator (a quantum state) and a bounded operator (a quantum observable) gives the average value of a quantum observable in a fixed state. The existence of such an average for unbounded operators is not guaranteed. If we want to define a space of observables that includes such naturally occurring unbounded operators as the position and momentum operators, for which average values are always determined, we should consider a space of states smaller than all nuclear operators. Recently, this approach has been mathematically correct implemented in the Hilbert space H = L^2(R^N). The so-called space of Schwarz operators, equipped with a system of semi-norms and being a Frechet space, was chosen as the space of states. Schwarz operators appeared to be integral operators whose kernels are functions belonging to the usual Schwarz space. The dual space to the space of Schwarz operators should be considered as the space of quantum observables and it really includes such standard observables as polynomials from the products of the position and momentum operators. In the present work we transfer this approach to the symmetric Fock space H = F(H) over an infinite dimensional separable Hilbert space H. We introduce the space of Schwartz operators in F(H) over an infinite dimensional separable space H and investigate which of the standard operators of quantum white noise belong to the space dual to the space of Schwartz operators.

Modeling Cognitive Activity of the Human Brain by the Mathematical Apparatus of Quantum Mechanics

This paper discusses the possible approaches to modeling the cognitive activity of the human brain using the mathematical apparatus of quantum mechanics (primarily – potential wells and virtual particles) in terms of the theory of information representations. The article briefly describes the proposed theory of information representations, draws analogies, and identifies common features of information representations of the human mind and Feynman’s virtual particles. The human mind is represented as a one-dimensional potential well with finite walls of different sizes and internal potential barrier simulating the boundary between consciousness and subconsciousness. This creates a foundation for a mathematical apparatus that can make it possible to forecast particular cognitive functions of the human brain. The results of these studies can be used to create predictive models of various cognitive disorders (diseases) and to be used in diagnostics.

Deterministic systems with natural quantization

Subject of Research. The research of deterministic systems is a topical problem of natural science. The paper presents an approach for behavior study of the deterministic systems. The work is aimed at creation of the evolution equation for deterministic systems, which shows that part of the macroscopic processes complies with quantum logic. Method. A new algebraic approach is proposed based on non-commutative algebra. It is shown that for any deterministic system in case of setting its change for a short time period, it is possible to create a system of differential equations describing the evolution of a given system in time. Implemented apparatus of non-commutative multiplication is an alternative to the operator calculus for quantum mechanics. Main Results. An associative non-commutative ring is built describing the evolution of arbitrary deterministic system. The proposed algebra is an isomorphic Heisenberg algebra. It is shown that all elements of algebraic rings are functions of numerical variables unlike the mathematical apparatus of quantum mechanics and, therefore, it is possible to give them different physical meaning. Practical Relevance. An example of differential equation creation is considered describing the motion of a classical particle in the presence of random forces. The obtained equation describes the probability density of a classical particle location at an arbitrary point of time in phase space.

Generalized Hamiltonian for a two-mode fermionic model and asymptotic equilibria

In some recent papers, the so called (H,ρ)-induced dynamics of a system S whose time evolution is deduced adopting an operatorial approach, has been introduced. According to the formal mathematical apparatus of quantum mechanics, H denotes the Hamiltonian for S, while ρ is a certain rule applied periodically on S. In this approach the rule acts at specific times kτ, with k integer and τ fixed, by modifying some of the parameters entering H according to the state variation of the system. As a result, a dynamics admitting an asymptotic equilibrium state can be obtained. Here, we consider the limit for τ→0, so that we introduce a generalized model leading to asymptotic equilibria. Moreover, in the case of a two-mode fermionic model, we are able to derive a relation linking the parameters involved in the Hamiltonian to the asymptotic equilibrium states.

Semantic shifts of terms: Analysis of relationships and a quantum-mechanical model

A model of a semantic shift is described, which is based on the probabilistic concept of meanings. Employing the mathematical apparatus of quantum mechanics and defining the equivalents of such physical parameters as the mass, energy, and tunneling distance helped to introduce a measure of the probability of the shift of a term into another semantic field. The experimental testing of the model was conducted for the keyword lexics of the IAEA's INIS abstract database.

Quantum-like modeling of cognition

This paper begins with a historical review of the mutual influence of physics and psychology, from Freud's invention of psychic energy inspired by von Boltzmann' thermodynamics to the enrichment quantum physics gained from the side of psychology by the notion of complementarity (the invention of Niels Bohr who was inspired by William James), besides we consider the resonance of the correspondence between Wolfgang Pauli and Carl Jung in both physics and psychology. Then we turn to the problem of development of mathematical models for laws of thought starting with Boolean logic and progressing towards foundations of classical probability theory. Interestingly, the laws of classical logic and probability are routinely violated not only by quantum statistical phenomena but by cognitive phenomena as well. This is yet another common feature between quantum physics and psychology. In particular, cognitive data can exhibit a kind of the probabilistic interference effect. This similarity with quantum physics convinced a multi-disciplinary group of scientists (physicists, psychologists, economists, sociologists) to apply the mathematical apparatus of quantum mechanics to modeling of cognition. We illustrate this activity by considering a few concrete phenomena: the order and disjunction effects, recognition of ambiguous figures, categorization-decision making. In Appendix 1 we briefly present essentials of theory of contextual probability and a method of representations of contextual probabilities by complex probability amplitudes (solution of the ``inverse Born's problem'') based on a quantum-like representation algorithm (QLRA).

A Survey of the ESR Model for an Objective Reinterpretation of Quantum Mechanics

Most scholars concerned with the foundations of quantum mechanics (QM) think that contextuality and nonlocality (hence nonobjectivity of physical properties) are unavoidable features of QM which follow from the mathematical apparatus of QM. Moreover these features are usually considered as basic in quantum information processing. Nevertheless they raise still unsolved problems, as the objectification problem in the quantum theory of measurement. The extended semantic realism (ESR) model offers a possible way out from these difficulties by embedding the mathematical formalism of QM into a broader mathematical formalism and reinterpreting quantum probabilities as conditional on detection rather than absolute. The embedding allows to recover the formal apparatus of QM within the ESR model, and the reinterpretation of QM allows to construct a noncontextual hidden variables theory which justifies the assumptions introduced in the ESR model and proves its objectivity. According to the ESR model both linear and nonlinear time evolution occur, depending on the physical environment, as in QM. In addition, the ESR model, though objective, implies modified Bell's inequalities that do not conflict with QM, supplies different mathematical representations of proper and improper mixtures, provides a general framework in which the local interpretations of the GHZ experiment obtained by other authors are recovered and explained, and supports an interpretation of quantum logic which avoids the introduction of the problematic notion of quantum truth.

Numerical Simulation of Asano-Khrennikov-Ohya Quantum-like Decision Making Model

In classical game theory, the rational behavior for the player is to make the decision which is approaching Nash equilibrium. The Prisoner's Dilemma, which is a canonical game, is often used to present the rationality. In real experiment in cognitive psychology which were performed by Tversky and Shafir, the statistical data show the existence of the irrational behaviors in reality. The phenomenon is called disjunction effect. To explain why it probably happens, we review the Asano-Khrennikov-Ohya model which is the mathematical modeling of the process of decision making in the game of Prisoner's Dilemma. It applies only the mathematical apparatus of quantum mechanics to the decision making process rather than the quantum physical model. In this paper, we present several numerical simulations for the Asano-Khrennikov-Ohya model together with the graphs of the von Neumann entropy for the solutions. By analyzing the simulation results, we explicitly and numerically present the existence of the irrational behavior for the player which is generated by the Asano-Khrennikov-Ohya model.

Embedding Quantum Mechanics into a Broader Noncontextual Theory

Scholars concerned with the foundations of quantum mechanics (QM) usually think that contextuality (hence nonobjectivity of physical properties, which implies numerous problems and paradoxes) is an unavoidable feature of QM which directly follows from the mathematical apparatus of QM. Based on some previous papers on this issue, we criticize this view and supply a new informal presentation of the extended semantic realism (ESR) model which embodies the formalism of QM into a broader mathematical formalism and reinterprets quantum probabilities as conditional on detection rather than absolute. Because of this reinterpretation a hidden variables theory can be constructed which justifies the assumptions introduced in the ESR model and proves its objectivity. When applied to special cases the ESR model settles long-standing conflicts (it reconciles Bell’s inequalities with QM), provides a general framework in which previous results obtained by other authors (as local interpretations of the GHZ experiment) are recovered and explained, and supports an interpretation of quantum logic which avoids the introduction of the problematic notion of quantum truth.

The mathematical apparatus of quantum mechanics as applied to investigation into biological systems

Diffusion-Schrodinger quantum mechanics unusual as to its physics has been constructed by introducing a diffusion analog of the Planck constant and generalizing the diffusion equation. The mechanics can be widely used in quantum biology, biological electronics, synthetic biology, quantum geometrodynamics, and in other fields of science and technology.

Quantum mechanics as applied mathematical statistics

Basic mathematical apparatus of quantum mechanics like the wave function, probability density, probability density current, coordinate and momentum operators, corresponding commutation relation, Schrödinger equation, kinetic energy, uncertainty relations and continuity equation is discussed from the point of view of mathematical statistics. It is shown that the basic structure of quantum mechanics can be understood as generalization of classical mechanics in which the statistical character of results of measurement of the coordinate and momentum is taken into account and the most important general properties of statistical theories are correctly respected.

Time asymmetry in quantum mechanics: a pure mathematical point of view

In the paper it is pointed out that ‘time asymmetry in quantum mechanics’ (TAQM) is an intrinsic element of the mathematical apparatus of quantum mechanics for semibounded Hamiltonians H with absolutely continuous spectrum coinciding with the positive half-line and of constant multiplicity. It is shown that the TAQM-semigroups are unsuitable for a spectral characterization of the resonances in terms of H.

Quantum mechanical approach to fuzzy logic modelling

This paper proposes a new method to construct a degree of membership in fuzzy modelling. The method is based on similarity between fuzzy and quantum-mechanical logic and suggests using the mathematical apparatus of quantum mechanics (Hermit's operators and wave-functions) for fuzzy models. This approach first requires the construction of a special “quasi-potential” function based upon the knowledge of an analyzed system. How this method works is illustrated with the help of three simple examples concerning hormone normal concentrations, an optimal number of hospitals, and the distance to healthcare.

The conceptual analysis (CA) method in theories of microchannels: Application to quantum theory. Part III. Idealizations. Hilbert space representation

We illustrate the application of the conceptual analysis (CA) method outlined in Part I by the example of quantum mechanics. In the present part the Hilbert space structure of conventional quantum mechanics is deduced as a consequence of postulates specifying further idealized concepts. A critical discussion of the idealizations of quantum mechanics is proposed. Quantum mechanics is characterized as a “statistically complete” theory and a simple and elegant formal recipe for the construction of the fundamental mathematical apparatus of quantum mechanics is formulated. Our analysis may also lead to a criticism of quantum mechanics as a “strongly idealized” theory. A critical analysis of the fundamental structure of quantum mechanics seems an indispensable and natural starting point for the construction of new theories. A major technical problem in a more general application of the CA method is the lack of mathematical representation theorems for more general algebraic structures.

Quantum Mechanics and Dialectical Materialism

O f all the philosophic issues posed by modern scientific theory, those involving quantum mechanics have been the most pressing and obstinate. Several problems in the philosophy of science of the past generation-such as the interpretation of special relativity-held the attention of scholars for a decade or more but have now lost much of their allure; other issues-such as the interpretation of cybernetics and information theory-have gained prominence only recently. But forty years after the publication of the essential mathematical apparatus of quantum mechanics the controversy swirls unabated, even intensified.' It is a debate in which the scholars of many nations have participated, including those of the USSR. The structure of quantum mechanics may be divided into a mathematical formalism and a physical interpretation of that formalism. The mathematical formalism which is the core of quantum mechanics is a differential wave equation, the solution of which is usually termed the psi (I) function; the wave equation was first developed by Erwin Schrodinger, who pursued Louis de Broglie's extension of the concept of wave-corpuscle duality from light to elementary particles of matter. The advantage of this formalism is that it yields, on a probabilistic basis, numerical values permitting a more complete mathematical description of microphysical states, including prediction of future states, than has any other formalism so far. The disadvantage of the mathematical apparatus of quantum mechanics is that the only widely accepted (somne would say the only possible) physical interpretation for it contradicts several of man's most basic intuitions concerning matter. Spe-