Showing papers in "Foundations of probability and physics in 2012"

Einstein-Podolsky-Rosen-Bohm laboratory experiments: Data analysis and simulation

Abstract: Data produced by laboratory Einstein-Podolsky-Rosen-Bohm (EPRB) experiments is tested against the hypothesis that the statistics of this data is given by quantum theory of this thought experiment. Statistical evidence is presented that the experimental data, while violating Bell inequalities, does not support this hypothesis. It is shown that an event-based simulation model, providing a cause-and-effect description of real EPRB experiments at a level of detail which is not covered by quantum theory, reproduces the results of quantum theory of this thought experiment, indicating that there is no fundamental obstacle for a real EPRB experiment to produce data that can be described by quantum theory.

Informational axioms for quantum theory

Abstract: It was recently proved that quantum theory can be derived from six axioms about information processing. Here we review these axioms, discussing various facets of their information-theoretical nature, and illustrating the general picture of quantum physics that emerges from them.

Bell argument: Locality or realism? Time to make the choice

Abstract: 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).

A contextual extension of Spekkens' toy model

Abstract: Quantum systems show contextuality. More precisely, it is impossible to reproduce the quantum-mechanical predictions using a non-contextual realist model, i.e., a model where the outcome of one measurement is independent of the choice of compatible measurements performed in the measurement context. There has been several attempts to quantify the amount of contextuality for specific quantum systems, for example, in the number of rays needed in a KS proof, or the number of terms in certain inequalities, or in the violation, noise sensitivity, and other measures. This paper is about another approach: to use a simple contextual model that reproduces the quantum-mechanical contextual behaviour, but not necessarily all quantum predictions. The amount of contextuality can then be quantified in terms of additional resources needed as compared with a similar model without contextuality. In this case the contextual model needs to keep track of the context used, so the appropriate measure would be memory. Another wa...

Quantum tomography of current and voltage states in nanoelectric circuits

Abstract: Nanoelectric circuit with small variable capacitance C(t) and inductance L(t) considered as quantum parametric oscillator with varying frequency is studied. Current and voltage in quantum circuits play the role of conjugate “position” and “momentum” of the parametric oscillator with the wave function (density matrix) depending on current (voltage). The physical realization of electric circuits with varying capacitance and inductance by Josephson junctions used in superconducting devices is discussed in the context of possibilities to use an analog of nonstationary Casimir effect to create current (voltage) in the circuits by varying the Josephson junction parameters. The idea to observe the analog of nonstationary Casimir effect proposed in [1]-[5] is accompanied by the possibility to introduce the tomographic probability description of the current and voltage state. The equation for optical and symplectic quantum tomograms of evolving current (voltage) states in nanoelectric circuits is obtained in explicit form.

Probability in quantum theory from observer's mathematics point of view

Abstract: This work considers the Hamilton equations of general relativity and quantum theory in a setting of arithmetic, algebra, and topology provided by Observer's Mathematics, see [1], [2], [3]. The probability appears automatically, without apriori assumptions. Certain results and communications pertaining to solutions of these problems are also provided.

Distance dependence of entangled photons in waveguides

Abstract: The distance dependence of the probability of observing two photons in a waveguide is investigated and the Glauber correlation functions of the entangled photons are considered. First the case of a hollow waveguide with modal dispersion is treated in detail: the spatial and temporal dependence of the correlation functions is evaluated and the distance dependence of the probability of observing two photons upper bounds and asymptotic expressions valid for large distances are derived. Second the generalization to a real fibre with both material and modal dispersion, allowing dispersion shift, is discussed.

Quantum-like dynamics of decision-making in prisoner's dilemma game

Abstract: 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

The instrumentalist aspects of quantum mechanics stem from probability theory

Abstract: The aim of the article is to argue that the interpretations of quantum mechanics and of probability are much closer than usually thought. Indeed, a detailed analysis of the concept of probability (within the standard frequency interpretation of R. von Mises) reveals that this notion always refers to an observing system. Therefore the instrumentalist aspects of quantum mechanics, and in particular the enigmatic role of the observer in the Copenhagen interpretation, derive from a precise understanding of probability.

Physics as quantum information processing: Quantum fields as quantum automata

Abstract: Can we reduce Quantum Field Theory (QFT) to a quantum computation? Can physics be simulated by a quantum computer? Do we believe that a quantum field is ultimately made of a numerable set of quantum systems that are unitarily interacting? A positive answer to these questions corresponds to substituting QFT with a theory of quantum cellular automata (QCA), and the present work is examining this hypothesis. These investigations are part of a large research program on a quantum-digitalization of physics, with Quantum Theory as a special theory of information, and Physics as emergent from the same quantum-information processing. A QCA-based QFT has tremendous potential advantages compared to QFT, being quantum ab-initio and free from the problems plaguing QFT due to the continuum hypothesis. Here I will show how dynamics emerges from the quantum processing, how the QCA can reproduce the Dirac-field phenomenology at large scales, and the kind of departures from QFT that that should be expected at a Planckscale discreteness. I will introduce the notions of linear field quantum automaton and local-matrix quantum automaton, in terms of which I will provide the solution to the Feynman's problem about the possibility of simulating a Fermi field with a quantum computer.

Lower bounds of concurrence for multipartite states

Abstract: We study the entanglement of multipartite quantum states. Some lower bounds of the multipartite concurrence are reviewed. We further present more effective lower bounds for detecting and qualifying entanglement, by establishing functional relations between the concurrence and the generalized partial transpositions of the multipartite systems.

Mean field description of the Dicke model

Abstract: A mean field description of the Dicke model is presented, employing the Holstein-Primakoff realization of the angular momentum algebra. It is shown that, in the thermodynamic limit, when the number of atoms interacting with the photons goes to infinity the energy surface takes a simple form, allowing for a direct description of many observables.

Statistics of observables in the probability representation of quantum and classical system states

Abstract: A review of the probability representation of quantum and classical states is presented. Inequalities corresponding to nonnegativity of the states' density operators and the presence of quantum correlations are considered. Comparison with some classical correlations is discussed. Entropic uncertainty relations along with other uncertainty relations are studied.

Potentiality and causation

Abstract: It is argued here that a causal account of the behavior of quantum systems can coherently be given within the context of the Basic interpretation of quantum mechanics of Dirac and von Neumann when extended to include positive-operator-valued measures and to include effects as observables. This account is most naturally provided via the notion of probabilistic causation, in particular, as an extension of von Neumann's conception of mathematical causation in mechanics which was intended to generalize aspects of the Laplacian notion to include quantum mechanics.

An example of the stochastic dynamics of a causal set

Abstract: An example of a discrete pregeometry on a microscopic scale is introduced. The model is a directed dyadic acyclic graph. This is the particular case of a causal set. The particles in this model must be self-organized repetitive structures. The dynamics of this model is a stochastic sequential growth dynamics. New vertexes are added one by one. The probability of this addition depends on the structure of existed graph. The particular case of the dynamics is considered. The numerical simulation provides some symptoms of self-organization.

System with classical and quantum subsystems in tomographic probability representation

Abstract: Description of system containing classical and quantum subsystems by means of tomographic probability distributions is considered. Evolution equation of the system states is studied.

Quantum-like model of glucose effect on Escherichia coli growth

Abstract: 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

On foundational thinking in fundamental physics, from Riemann to Einstein to Heisenberg

Abstract: This paper considers the nature of foundational thinking in fundamental physics, most especially in quantum mechanics. By “fundamental physics” I mean those areas of experimental and theoretical physics that deal with the ultimate constitution of nature, for example, as defined by the so-called elementary particles in the case of quantum physics. By “foundational thinking” I mean thinking that concerns fundamental physics itself. First, I argue, following Riemann, that our foundational thinking is based on hypotheses that we form and test. Second, I argue that foundational thinking in physics is defined by concepts, and that in modern physics foundational concepts always contains physical, mathematical, and philosophical components. Third, finally, I argue that the relationships between these components and, hence, our foundational thinking, are different in quantum mechanics than they are in classical physics and relativity. In these theories mathematics describes, by way of idealized models, physical re...

The two limits of the Schrödinger equation in the semi-classical approximation: discerned and non-discerned particles in classical mechanics

Abstract: We study, in the semi-classical approximation, the convergence of the quantum density and the quantum action, solutions to the Madelung equations, when the Planck constant h tends to 0. We find two different solutions which depend on the initial density . In the first case where the initial quantum density is a classical density ρ0(X), the quantum density and the quantum action converge to a classical action and a classical density which satisfy the statistical Hamilton-Jacobi equations. These are the equations of a set of classical particles whose initial positions are known only by the density ρ0(X). In the second case, where initial density converges to a Dirac density, the density converges to the Dirac function which corresponds to a unique classical trajectory. Therefore we introduce into classical mechanics non-discerned particles (case 1), which explain the Gibbs paradox, and discerned particles (case 2). Finally, we deduce a quantum mechanics interpretation which depends on the initial conditions (preparation), the Broglie-Bohm interpretation in the first case and the Schrodinger interpretation in the second case.

Decoherence time and spin measurement in the Stern-Gerlach experiment

Abstract: We present an analytical solution to the decoherence time for the spin measurement and the diagonalization of the density matrix of spin variables in the Stern-Gerlach experiment. This solution requires the calculation of the Pauli spinor with a spatial extension, which is not found in quantum mechanics textbooks. With this full spinor and the measured position of the particle we demonstrate the three postulates of quantum measurement: quantization, spectral decomposition and wave function reduction. The transition from a quantum superposition to a statistical mixture is well explained in this way, but not the single result that always emerges from a particular experiment. The spinor spatial extension also allows the introduction of the de Broglie-Bohm trajectories which give a very simple explanation of the particles' impact.

Quantum structure in economics: The Ellsberg paradox

Abstract: The expected utility hypothesis and Savage's Sure-Thing Principle are violated in real life decisions, as shown by the Allais and Ellsberg paradoxes. The popular explanation in terms of ambiguity aversion is not completely accepted. As a consequence, uncertainty is still problematical in economics. To overcome these difficulties a distinction between risk and ambiguity has been introduced which depends on the existence of a Kolmogorovian probabilistic structure modeling these uncertainties. On the other hand, evidence of everyday life suggests that context plays a fundamental role in human decisions under uncertainty. Moreover, it is well known from physics that any probabilistic structure modeling contextual interactions between entities structurally needs a non-Kolmogorovian framework admitting a quantum-like representation. For this reason, we have recently introduced a notion of contextual risk to mathematically capture situations in which ambiguity occurs. We prove in this paper that the contextual r...

Derivation of Dirac's equation from conformal differential geometry

Abstract: A rigorous ab initio derivation of the (square of) Dirac's equation for a single particle with spin is presented. The theory is carried out in the framework of an approach where it is assumed that quantum phenomena originate from the interplay between the motion of a relativistic spherical top and the non trivial background geometry of its configuration space. We require full conformal invariance in each step of the theory, which is achieved by replacing the mass of the top with Weyl's curvature. The curvature acts on the particle as a scalar potential and the particle, in turn, acts back on curvature modifying Weyl's pre-potential. The mechanism is similar to the one at the basis of the general relativity, with the difference that curvature is originated here by the affine connections of space rather than by the metric tensor, which can be prescribed at will. The theory is intrinsically nonlinear, but it is linearized, exactly and in closed form, by an ansatz solution that can be straightforwardly interp...

Gravity versus quantum theory: Is electron really pointlike?

Abstract: Quantum theory claims that electron is pointlike and structureless. Contrary, the consistent with Gravity electron is considered as a gravitating soliton based on the Kerr-Newman (KN) solution. The KN electron model indicates its extended structure of the Compton size rc = ħ/m and sheds a new light on some old puzzles of quantum theory. In particular, the KN Gravity predicts the topologically nontrivial background of the electron, formed by a closed string of the heterotic type. We argue that this string can be detected by the novel experimental tool – nonforward Compton Scattering. If it will be confirmed, it would be of primary importance for foundations of Quantum theory and unification of Quantum theory with Gravity.

A quantum-like approach to the stock market

Abstract: Modern approaches to stock pricing in quantitative finance are typically founded on the Black-Scholes model and the underlying random walk hypothesis. Empirical data indicate that this hypothesis works well in stable situations but, in abrupt transitions such as during an economical crisis, the random walk model fails and alternative descriptions are needed. For this reason, several proposals have been recently forwarded which are based on the formalism of quantum mechanics. In this paper we apply the SCoP formalism, elaborated to provide an operational foundation of quantum mechanics, to the stock market. We argue that a stock market is an intrinsically contextual system where agents' decisions globally influence the market system and stocks prices, determining a nonclassical behavior. More specifically, we maintain that a given stock does not generally have a definite value, e.g., a price, but its value is actualized as a consequence of the contextual interactions in the trading process. This contextual influence is responsible of the non-Kolmogorovian quantumlike behavior of the market at a statistical level. Then, we propose a sphere model within our hidden measurement formalism that describes a buying/selling process of a stock and shows that it is intuitively reasonable to assume that the stock has not a definite price until it is traded. This result is relevant in our opinion since it provides a theoretical support to the use of quantum models in finance.

A few remarks on microscopes and telescopes for theoretical physics: How rich locally and large globally is the geometric straight line?

Abstract: One is reminded in this paper of the often overlooked fact that the geometric straight line, or GSL, of Euclidean geometry is not necessarily identical with its usual Cartesian coordinatisation given by the real numbers in R. Indeed, the GSL is an abstract idea, while the Cartesian, or for that matter, any other specific coordinatisation of it is but a mathematical model chosen upon certain reasons. And as is known, there are a a variety of mathematical models of GSL, among them given by nonstandard analysis, reduced power algebras, the topological long line, or the surreal numbers. As shown in this paper, the GSL can allow coordinatisations which are arbitrarily more rich locally and also more large globally, being given by corresponding linearly ordered sets of no matter how large cardinal. Thus one can obtain in relatively simple ways structures which are more rich locally and large globally than in nonstandard analysis, or in various reduced power algebras. Furthermore, vector space structures can be defined in such coordinatisations. This fact can have a major importance in physics, since such locally more rich and globally more large coordinatisations of the GSL do allow new physical insights, just as the introduction of various microscopes and telescopes have done. Among others, it can reassess special relativity with respect to its independence of the mathematical models used for the GSL. Also, it can allow the more appropriate modelling of certain physical phenomena. One of the long vexing issue of so called “infinities in physics” can obtain a clarifying reconsideration. It indeed all comes down to looking at the GSL with suitably constructed microscopes and telescopes, and apply the resulted new modelling possibilities in theoretical physics. One may as well consider that in string theory, for instance, where several dimensions are supposed to be compact to the extent of not being observable on classical scales, their mathematical modelling may benefit from the presence of infinitesimals in the mathematical models of the GSL presented here. However, beyond all such particular considerations, and not unlikely also above them, is the following one: theories of physics should be not only background independent, but quite likely, should also be independent of the specific mathematical models used when representing geometry, numbers, and in particular, the GSL.

Graded tensor products and entanglement

Abstract: 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.

"To be. To be. What does it mean to be?": On quantum-like literary models

Abstract: This paper discusses a particular type of quantum-like literary models, which are conceptual, rather than mathematical, in character. These models share with quantum mechanics the difficulties of applying the concepts of reality and causality at the ultimately ontological levels they consider, analogous to the level of quantum objects and processes in quantum mechanics. They respond to this difficulty by suspending and even precluding the application of both concepts, as do certain interpretations of quantum mechanics. I call such models and such interpretations "nonclassical ", in juxtaposition to "classical " models, which retain realism and causality at the ultimate level of description, even when considering random events. While I offer a sketch of Western thinking concerning the subject, I focus on certain philosophical and literary quantum-like thinking of the late-eighteenth and early-nineteenth centuries, associated with Romantic literature, which shows particular affinities with quantum-theoretic...

Possibility, probability and entanglement: Non-contextuality in modal quantum mechanics

Abstract: Modal quantum theory (MQT) is a simplified cousin of ordinary Hilbert space quantum theory. We show that two important theorems of actual quantum theory, the Kochen-Specker theorem excluding non-contextual hidden variables and the Conway-Kochen "free will theorem" about entangled systems, have direct analogues in MQT. The structure of possible measurement results for an entangled system in MQT cannot be represented by probability assignments satisfying the no-signaling principle, such as those given by ordinary quantum theory.

How do classical and quantum probabilities differ

Abstract: I show that the classical coin tossing experiment involves two distinct definitions of probability, one ontological (the relative frequencies of initial deterministic states) and another empirical (the relative frequencies of observations). In quantum theory, I argue, only the latter definition can be invoked, since a single superposition state can give rise to multiple observation experiences. This difference explains why the present statistical quantum mechanics is an ontological dead end, despite its enormous pragmatic success. To get at the ontological content of quantum theory, I propose that the observations themselves can be interpreted on a different footing, without reference to determinate detector states.