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We discuss the problem of interpretation of the wave function. The latest version of the Va¨xjo¨ interpretation is presented. It differs essentially from the original Va¨xjo¨ interpretation (2001). The main distinguishing feature of the present Va¨xjo¨ interpretation is the combination of realism on the subquantum level with nonobjectivity of quantum observables (i.e., impossibility to assign their values before measurements). Hence, realism is destroyed by detectors transforming continuous subquantum reality into discrete events, clicks of detectors. The Va¨xjo¨ interpretation-2012 is fundamentally contextual in the sense that the value of an observable depends on measurement context. This is contextuality in Bohr's sense. It is more general than Bell's contextuality based on joint measurements of compatible observables. [ABSTRACT FROM AUTHOR]

We discuss a generalization of POVM which is used in quantum-like modeling of mental processing. [ABSTRACT FROM AUTHOR]

This paper gives a detailed derivation of the classical electromagnetic modes of a layered circularly symmetrical dielectric waveguide. The corresponding Hamilton function is derived by using suitable canonical observables and the standard analogy to the classical harmonic oscillator. The derivation is generic in the sense that it can be used as an "algorithm" to compute the electromagnetic field of the waveguide. The associated Hamilton operator can then be obtained by using the standard quantization procedure where the canonical observables are replaced by the corresponding operators i.e., the creation and the annihilation operators of the photon (or equivalently, the position and the momentum operators of the harmonic oscillator) and by taking the appropriate commutation relations into account. [ABSTRACT FROM AUTHOR]

We apply theory of open quantum systems to modeling of epigenetic evolution. This is an attempt to unify Darwinian and Lamarckian viewpoints on evolution on the basis of a quantum-like model. The state of uncertainty of cell's epigenome is resolved to a stable and inherited epigenetic configuration. This process of evolution and stabilization is described by the quantum master equation (the Gorini-Kossakowski-Sudarshan-Lindblad equation). The initial state of epigenome starting interaction with a new environment is represented as a pure quantum state. It evolves to a steady state solution of the quantum master equation given by a diagonal density matrix. The latter represents the state resulting from a series of epimutations induced by the environment. We use the information interpretation of the wave function which was elaborated by C. Fuchs and A. Zeilinger. [ABSTRACT FROM AUTHOR]

Motivated by the novel applications of the mathematical formalism of quantum theory and its generalizations in cognitive science, psychology, social and political sciences, and economics, we extend the notion of the tensor product and entanglement. We also study the relation between conventional entanglement of complex qubits and our generalized entanglement. Our construction can also be used to describe entanglement in the framework of non-Archimedean physics. It is also possible to construct tensor products of non-Archimedean (e.g., p-adic) and complex Hilbert spaces. [ABSTRACT FROM AUTHOR]

We discuss a possibility of resolution nonobjectivity-nonlocality dilemma in the light of experimental tests of the Bell inequality for two entangled photons and Bell-like inequality for a single neutron. Our conclusion is that on the basis of these experiments we can conclude that quantum mechanics is nonobjective, i.e., values of physical observables cannot be assigned to a system before measurement. The Bell's assumption of nonlocality has to be rejected as having no direct experimental confirmation. We discuss inter-relation nonobjectivity/contextuality. We analyze the impact of the Kochen-Specker theorem to the problem of contextuality of quantum observables. Our conclusion is that as well as von Neumann no-go theorem the the Kochen-Specker theorem is based on assumptions which do not match the real physical situation. Finally, we present theory of measurements for a classical purely wave model (prequantum classical statistical field theory) reproducing quantum probabilities. Here continuous fields are transformed into discrete clicks of detectors. The model is classical. However, it is nonobjective. Here, nonobjectivity is the result of contextuality - dependence on the context of measurement (in complete accordance with Bohr's views). [ABSTRACT FROM AUTHOR]

We consider the grading structure on the tensor product corresponding to the tensor rank; the relation to the notion of entanglement is discussed. We also study a complex problem of finding the minimal (with respect to the aforementioned grading structure) representation of elements of the tensor product. The general construction is presented over an arbitrary number field. Hence, it can be applied not only to the conventional notion of entanglement over the field of complex numbers, but even for models of non-Archimedean (in particular, p-adic) quantum physics. The problem of tensor reduction is also studied for an arbitrary number field, but only in the two dimensional case. [ABSTRACT FROM AUTHOR]

Recently, we proposed a new method to compute probabilities which do not satisfy basic law in classical probability theory. In this note, we analyze glucose effect in Escherichia coli's growth with the method, and we show an invariant quantity which Escherichia coli has. [ABSTRACT FROM AUTHOR]

We propose a double blinding-attack on entangled-based quantum key distribution protocols. The principle of the attack is the same as in existing blinding attack except that instead of blinding the detectors on one side only, Eve is blinding the detectors of both Alice and Bob. In the BBM92 protocol, the attack allows Eve to get a full knowledge of the key and remain undetected even if Alice and Bob are using 100% efficient detectors. The attack can be easily extended to Ekert protocol, with an efficiency as high as 85.3%. [ABSTRACT FROM AUTHOR]

A preface to the book "Foundations of Probability and Physics-6" by Andrie Khrennikov is presented.

In cognitive psychology, some experiments of games were reported [1, 2, 3, 4], and these demonstrated that real players did not use the "rational strategy" provided by classical game theory. To discuss probabilities of such "irrational choice", recently, we proposed a decision-making model which is based on the formalism of quantum mechanics [5, 6, 7, 8]. In this paper, we briefly explain the above model and calculate the probability of irrational choice in several prisoner's dilemma (PD) games. [ABSTRACT FROM AUTHOR]

We present fundamentals of a prequantum model with hidden variables of the classical field type. In some sense this is the comeback of classical wave mechanics. Our approach also can be considered as incorporation of quantum mechanics into classical signal theory. All quantum averages (including correlations of entangled systems) can be represented as classical signal averages and correlations. [ABSTRACT FROM AUTHOR]

Recently a new model with hidden variables of the wave type was elaborated, so called prequantum classical statistical field theory (PCSFT). Roughly speaking PCSFT is a classical signal theory applied to a special class of signals-'quantum systems'. PCSFT reproduces successfully all probabilistic predictions of QM, including correlations for entangled systems. This model peacefully coexists with all known no-go theorems, including Bell's theorem. In our approach QM is an approximate model. All probabilistic predictions of QM are only (quite good) approximations of 'real physical averages'. The latter are averages with respect to fluctuations of prequantum fields. In particular, Born's rule is only an approximate rule. More precise experiments should demonstrate its violation. We present a simple experiment which has to produce statistical data violating Born's rule. Since the PCSFT-presentation of this experiment may be difficult for experimenters, we reformulate consequences of PCSFT in terms of the conventional wave function. In general, deviation from Born's rule is rather small. We found an experiment amplifying this deviation. We start with a toy example. Then we present a more realistic example based on Gaussian states with very small dispersion. [ABSTRACT FROM AUTHOR]

It took two millennia after Euclid and until in the early 1880s, when we went beyond the ancient axiom of parallels, and inaugurated geometries of curved spaces. In less than one more century, General Relativity followed. At present, physical thinking is still beheld by the yet deeper and equally ancient Archimedean assumption. In view of that, it is argued with some rather easily accessible mathematical support that Theoretical Physics may at last venture into the non-Archimedean realms. In this introductory paper we stress two fundamental consequences of the non-Archimedean approach to Theoretical Physics: one of them for quantum theory and another for relativity theory. From the non-Archimedean viewpoint, the assumption of the existence of minimal quanta of light (of the fixed frequency) is an artifact of the present Archimedean mathematical basis of quantum mechanics. In the same way the assumption of the existence of the maximal velocity, the velocity of light, is a feature of the real space-time structure which is fundamentally Archimedean. Both these assumptions are not justified in corresponding non-Archimedean models. [ABSTRACT FROM AUTHOR]

We present a quantum-like model of decision making in games of the Prisoner's Dilemma type. By this model the brain processes information by using representation of mental states in complex Hilbert space. Driven by the master equation the mental state of a player, say Alice, approaches an equilibrium point in the space of density matrices. By using this equilibrium point Alice determines her mixed (i.e., probabilistic) strategy with respect to Bob. Thus our model is a model of thinking through decoherence of initially pure mental state. Decoherence is induced by interaction with memory and external environment. In this paper we study (numerically) dynamics of quantum entropy of Alice's state in the process of decision making. Our analysis demonstrates that this dynamics depends nontrivially on the initial state of Alice's mind on her own actions and her prediction state (for possible actions of Bob.) [ABSTRACT FROM AUTHOR]

We show that an explicit mixture of separable states can exhibit a correlation that differs from the prediction for an entangled state only by its visibility (1/2 instead of 1). We thus argue that experiments with low visibility (about 1/2 or less) between remote systems can be given a local interpretation within the framework of Quantum Mechanics, and should thus only be regarded as a proof of entanglement with caution. [ABSTRACT FROM AUTHOR]

Recently, applications of quantum mechanics to coginitive psychology have been discussed, see [1]-[11]. It was known that statistical data obtained in some experiments of cognitive psychology cannot be described by classical probability model (Kolmogorov's model) [12]-[15]. Quantum probability is one of the most advanced mathematical models for non-classical probability. In the paper of [11], we proposed a quantum-like model describing decision-making process in a two-player game, where we used the generalized quantum formalism based on lifting of density operators [16]. In this paper, we discuss the quantum-like representation of Bayesian inference, which has been used to calculate probabilities for decision making under uncertainty. The uncertainty is described in the form of quantum superposition, and Bayesian updating is explained as a reduction of state by quantum measurement. [ABSTRACT FROM AUTHOR]

Prequantum classical statistical field theory (PCSFT) is a new attempt to consider quantum mechanics (QM) as an emergent phenomenon, cf. with De Broglie’s “double solution” approach, Bohmian mechanics, stochastic electrodynamics (SED), Nelson’s stochastic QM and its generalization by Davidson, ‘t Hooft’s models and their development by Elze. PCSFT is a comeback to a purely wave viewpoint on QM, cf. with early Schrödinger. There is no quantum particles at all, only waves. In particular, photons are simply wave-pulses of the classical electromagnetic field, cf. SED. Moreover, even massive particles are special “prequantum fields”: the electron field, the neutron field and so on. PCSFT claims that (soon or later) people will be able to measure components of these fields: components of the “photonic field” (the classical electromagnetic field of low intensity), electronic field, neutronic field and so on. However, at the moment (in this paper) we restrict our efforts to reproduce “simply” predictions of QM in the classical field framework. We will show that correlations of entangled systems can be obtained from fluctuations of the prequantum field. We consider the most general case: in QM the state is given by the density operator. [ABSTRACT FROM AUTHOR]

In number theory, a partition of a positive integer n is a way of writing n as a sum of positive integers. The number of partitions of n is given by the partition function p(n). Inspired by quantum information processing, we extend the concept of partitions in number theory as follows: for an integer n, we treat each partition as a basis state of a quantum system representing that number n, so that the Hilbert-space that corresponds to that integer n is of dimension p(n); the “classical integer” n can thus be generalized into a (pure) quantum state | |ψ(n) > which is a superposition of the partitions of n, in the same way that a quantum bit (qubit) is a generalization of a classical bit. More generally, ρ(n) is a density matrix in that same Hilbert-space (a probability distribution over pure states). Inspired by the notion of quantum numbers in quantum theory (such as in Bohr’s model of the atom), we then try to go beyond the partitions, by defining (via recursion) the notion of “sub-partitions” in number theory. Combining the two notions mentioned above, sub-partitions and quantum integers, we finally provide an alternative definition of the quantum integers [the pure-state |ψ′(n)> and the mixed-state ρ′(n), ] this time using the sub-partitions as the basis states instead of the partitions, for describing the quantum number that corresponds to the integer n. [ABSTRACT FROM AUTHOR]

This notes clarity authors positions on a possibility to go beyond quantum mechanics. [ABSTRACT FROM AUTHOR]

This paper is an attempt to reproduce predictions of quantum mechanics (QM) on probability of detection by starting with classical random field (e.g. random electromagnetic field). Thus it is an attempt to go beyond QM. The crucial point of our approach is that density operators of QM can be represented as covariance operators of classical random fields. Born’s rule can be obtained from measurement theory for classical random field under the assumption that the probability of detection of field is proportional to the power of this field. [ABSTRACT FROM AUTHOR]

Classical mechanics implicitly use Archimedes’ axiom, according to that everything may be measured by a rigid scale. Uncertainty in quantum mechanics shows the limits of applying Archimedes’ axiom. The negation of Archimedes’ axiom is derivative from the negation of the set-theoretic axiom of foundation. The latter postulates that the set-membership relation is well-founded: for every set there exists no infinitely descending chain. Denying the foundation axiom in number systems implies setting a non-Archimedean ordering structure. The main claim of our paper is that physical reality may be regarded as non-well-founded in the framework of probabilities distributed on non-Archimedean ordering structures, in particular, distributed on p-adic numbers. [ABSTRACT FROM AUTHOR]

In this paper we would like to stress that, besides two commonly discussed conditions inducing violation of Bell’s inequality—nonlocality and death of realism—there is the third condition having the same consequence. This is the condition of probabilistic incompatibility (PI) of random variables—impossibility to realize them on a single probability space—“non-Kolmogorovness.” This additional source of violation of Bell’s inequality should be taken into account. We remark that PI can be a consequence of nonlocality or impossibility to use the realistic model. However, PI is essentially more general condition. Random variables can be of the PI-type even in absence of nonolocal effects. [ABSTRACT FROM AUTHOR]

We show that paradoxical consequences of violations of Bell’s inequality are induced by the use of an unsuitable probabilistic description for the EPR-Bohm-Bell experiment. The conventional description (due to Bell) is based on a combination of statistical data collected for different settings of polarization beam splitters (PBSs). In fact, such data consists of some conditional probabilities which only partially define a probability space. Ignoring this conditioning leads to apparent contradictions in the classical probabilistic model (due to Kolmogorov). We show how to make a completely consistent probabilistic model by taking into account the probabilities of selecting the settings of the PBSs. Our model matches both the experimental data and is consistent with classical probability theory. [ABSTRACT FROM AUTHOR]

We remind the viewpoint that violation of Bell's inequality might be interpreted not only as an evidence of the alternative—either nonlocality or “death of reality” (under the assumption the quantum mechanics is incomplete). Violation of Bell's type inequalities is a well known sufficient condition of incompatibility of random variables—impossibility to realize them on a single probability space. Thus, in fact, we should take into account an additional interpretation of violation of Bell's inequality—a few pairs of random variables (two dimensional vector variables) involved in the EPR-Bohm experiment are incompatible. They could not be realized on a single Kolmogorov probability space. Thus one can choose between: a) completeness of quantum mechanics; b) nonlocality; c) “death of reality”; d) non-Kolmogorovness. In any event, violation of Bell's inequality has a variety of possible interpretations. Hence, it could not be used to derive the unique conclusion on the relation between quantum and classical models. [ABSTRACT FROM AUTHOR]

Textbooks on quantum mechanics often give the impression that the Schrödinger equation can be solved exactly only for a few simple potential models. However, exact solutions are available in terms of hypergeometric functions and their confluent variants for the so-called Natanzon potentials. These potentials include the Pöschl-Teller, Manning-Rosen and Rosen-Morse potentials that are also special cases of the Eckart potential. The Natanzon potentials are reviewed and connections are made to problems in classical physics like propagation of electromagnetic waves in inhomogeneous media and of acoustic waves in a variable speed profile. The availability of exact solutions is of particular interest for the explicit construction of time evolution operators and in the solution of inverse scattering problems. [ABSTRACT FROM AUTHOR]

We present a detailed analysis of assumptions that J. Bell used to show that local realism contradicts QM. We find that Bell’s viewpoint on realism is nonphysical, because it implicitly assume that observed physical variables coincides with ontic variables (i.e., these variables before measurement). The real physical process of measurement is a process of dynamical interaction between a system and a measurement device. Therefore one should check the adequacy of QM not to “Bell’s realism,” but to adaptive realism (chameleon realism). Dropping Bell’s assumption we are able to construct a natural representation of the EPR-Bohm correlations in the local (adaptive) realistic approach. © 2007 American Institute of Physics [ABSTRACT FROM AUTHOR]

In this paper dedicated to the memory of Walter Philipp, we formalize the rules of classical→ quantum correspondence and perform a rigorous mathematical analysis of the assumptions in Bell’s NO-GO arguments. © 2007 American Institute of Physics [ABSTRACT FROM AUTHOR]

We present a trivial probabilistic illustration for representation of quantum mechanics as an algorithm for approximative calculation of averages. © 2007 American Institute of Physics [ABSTRACT FROM AUTHOR]

In this paper we discuss the rationale why sub(super)-additive probabilities in a psychological setting could be explained via the use of quantum probability interference. We propose to measure the complementarity of two variables: i) time of processing (by experiment participants) of (non-moving) images and ii) the ability (by experiment participants) of recognizing deformations of (non-moving) pictures. We argue in the paper why we can not find this complementarity using the Heisenberg Uncertainty Principle. The paper provides for the details on the experimental set up to test the complementarity. © 2007 American Institute of Physics [ABSTRACT FROM AUTHOR]

We perform geometrization of genetics by representing genetic information by points of the 4-adic information space. By well known theorem of number theory this space can also be represented as the 2-adic space. The process of DNA-reproduction is described by the action of a 4-adic (or equivalently 2-adic) dynamical system. As we know, the genes contain information for production of proteins. The genetic code is a degenerate map of codons to proteins. We model this map as functioning of a polynomial dynamical system. The purely mathematical problem under consideration is to find a dynamical system reproducing the degenerate structure of the genetic code. We present one of possible solutions of this problem.. © 2007 American Institute of Physics [ABSTRACT FROM AUTHOR]

We show that in the opposition to a rather common opinion quantum mechanics is not complete. It is possible to introduce so called hidden variable — in our model classical fields — and combine the statistical predictions of quantum mechanics with deterministic dynamics of those hidden variables. Quantum mechanics can be considered as an approximative description of physical processes based on neglecting by quantities of the magnitude o(α), where α is the dispersion of fluctuations of the Gaussian background field. In this paper we present the detailed presentation of theory of infinite-dimensional phase space and derive main equations of quantum mechanics (e.g., Schrödinger’s equation, Heisenberg’s equation and von Neumann equation) from the Hamilton equation on the infinite-dimensional symplectic space. We emphasize (to escape misunderstanding) that our paper is not about quantization of systems with infinite number of degrees of freedom, but about representation of quantum systems as classical systems with infinite number of degrees of freedom. We also investigate the purely mathematical problem of preserving of the dispersion of Gaussian fluctuations by Hamiltonian flows. © 2006 American Institute of Physics [ABSTRACT FROM AUTHOR]

We present the results of computer simulations which give the evidence for the existence of an interesting symmetry in a multi-agent model demonstrating, in special cases, both Bose-Einstein and Fermi-Dirac statistics. This symmetry is expressed in the coincidence of the degree of ultrametricity and the fraction of isosceles of the sets of agents memories coded by two different information loss coding schemes. © 2006 American Institute of Physics [ABSTRACT FROM AUTHOR]

This is a review about p-adic valued probabilities, applications to quantum physics and cognitive sciences. We also do the next step in development of non-Kolmogorovian models and consider an analogue of probability theory for probabilities taking values in topological groups. The main attention is paid to statistical interpretation of p-adic probabilities as well as more general probabilities with values in topological groups. © 2006 American Institute of Physics [ABSTRACT FROM AUTHOR]

We show that quantum mechanics can be represented as an asymptotic projection of statistical mechanics of classical fields. Our approach does not contradict to a rather common opinion that quantum mechanics could not be reduced to statistical mechanics of classical particles. Notions of a system and causality can be reestablished on the prequantum level, but the price is sufficiently high — the infinite dimension of the phase space. In our approach quantum observables, symmetric operators in the Hilbert space, are obtained as the second order derivatives of functionals of classical fields: f(ψ) → Aquant = f″(0). Statistical states are given by Gaussian ensembles of classical fields with zero mean value (so these are vacuum fluctuations) and dispersion α which plays the role of a small parameter of the model (so these are small vacuum fluctuations). Our approach can be called Prequantum Classical Statistical Field Theory — PCSFT. © 2006 American Institute of Physics [ABSTRACT FROM AUTHOR]

A new analysis of the Weihs et al 1998 experimental raw data is given. Using a new normalization procedure based on the statistics of the single counts, we investigate seemingly unimportant anomalies in the coincidence count rates. We show that interpreting these anomalies within a quantum mechanical framework is problematic as they imply that the fair sampling assumption is untenable. © 2006 American Institute of Physics [ABSTRACT FROM AUTHOR]

We analyze dynamical consequences of a conjecture that there exists a fundamental (indivisible) quant of time. In particular we study the problem of discrete energy levels of hydrogen atom. We are able to reconstruct potential which in discrete time formalism leads to energy levels of unperturbed hydrogen atom. We also consider linear energy levels of quantum harmonic oscillator and show how they are produced in the discrete time formalism. More generally, we show that in discrete time formalism finite motion in central potential leads to discrete energy spectrum, the property which is common for quantum mechanical theory. Thus deterministic (but discrete time!) dynamics is compatible with discrete energy levels. © 2006 American Institute of Physics [ABSTRACT FROM AUTHOR]

The purpose of this paper is to advocate for more thorough investigations of EPR experiments with photons, and in particular of the main remaining loophole (i.e., the low detection efficiency loophole) using some recent proposals to circumvent this problem. © 2005 American Institute of Physics [ABSTRACT FROM AUTHOR]

The notion of context (complex of physical conditions) is basic in this paper. We show that the main structures of quantum theory (interference of probabilities, Born’s rule, complex probabilistic amplitudes, Hilbert state space, representation of observables by operators) are present in a latent form in the classical Kolmogorov probability model. However, this model should be considered as a calculus of contextual probabilities. In our approach it is forbidden to consider abstract context independent probabilities: “first context and then probability.” We start with the conventional formula of total probability for contextual (conditional) probabilities and then we rewrite it by eliminating combinations of incompatible contexts from consideration. In this way we obtain interference of probabilities without to appeal to the Hilbert space formalism or wave mechanics. However, we did not just reconstruct the probabilistic formalism of conventional quantum mechanics. Our contextual probabilistic model is essentially more general and, besides the projection to the complex Hilbert space, it has other projections. The most important new prediction is the possibility (at least theoretical) of appearance of hyperbolic interference. A projection of the classical contextual probabilistic model to the hyperbolic Hilbert space (a module over the commutative two dimensional Clifford algebra) has some similarities with the projection to the complex Hilbert space. However, in the hyperbolic quantum mechanics the principle of superposition is violated. Our realistic (but contextual!) approach to quantum mechanics does not contradict to various “no-go theorems”, e.g., von Neumann, Bell, Kochen-Specker. We emphasize that our projection of the classical probabilistic model to the complex Hilbert space is based on two incompatible observables (“reference observables”), e.g., the position and the momentum, or the position and the energy. Only these two observables can be considered as objective properties of quantum systems. © 2005 American Institute of Physics [ABSTRACT FROM AUTHOR]

In this short note we present (without to go into mathematical details) the main notions of Prequantum Classical Statistical Field Theory (PCSFT): theory of classical fields which fluctuations produce averages approximatively coinciding with averages given by the quantum formalism. Thus the conventional quantum mechanics can be considered as a probabilistic approximation of a more detailed classical field model. Experimental predictions of PCSFT are discussed [ABSTRACT FROM AUTHOR]