scholarly journals Generalized Fock space and contextuality

2019 ◽  
Vol 377 (2157) ◽  
pp. 20190096 ◽  
Author(s):  
Sergey Rashkovskiy ◽  
Andrei Khrennikov
Keyword(s):  
Quantum Mechanics ◽  
Linear Space ◽  
Wide Class ◽  
Fock Space ◽  
Kinetic Equations ◽  
Interference Term ◽  
Context Dependence ◽  
Total Probability ◽  
Theme Issue ◽  
Formula Of Total Probability

This paper is devoted to linear space representations of contextual probabilities—in generalized Fock space. This gives the possibility to use the calculus of creation and annihilation operators to express probabilistic dynamics in the Fock space (in particular, the wide class of classical kinetic equations). In this way, we reproduce the Doi–Peliti formalism. The context-dependence of probabilities can be quantified with the aid of the generalized formula of total probability—by the magnitude of the interference term. This article is part of the theme issue ‘Contextuality and probability in quantum mechanics and beyond'.

scholarly journals A search for a stochastic archetype of quantum probability

2021 ◽  
Author(s):  
Tim C Jenkins
Keyword(s):  
Probability Theory ◽  
Quantum Mechanics ◽  
Probability Density ◽  
Random Variables ◽  
Quantum Probability ◽  
Interference Term ◽  
Mathematical Foundation ◽  
Quantum Process ◽  
Hidden Variable ◽  
Probability Densities

Abstract Superposed wavefunctions in quantum mechanics lead to a squared amplitude that introduces interference into a probability density, which has long been a puzzle because interference between probability densities exists nowhere else in probability theory. In recent years, Man’ko and coauthors have successfully reconciled quantum and classic probability using a symplectic tomographic model. Nevertheless, there remains an unexplained coincidence in quantum mechanics, namely, that mathematically, the interference term in the squared amplitude of superposed wavefunctions gives the squared amplitude the form of a variance of a sum of correlated random variables, and we examine whether there could be an archetypical variable behind quantum probability that provides a mathematical foundation that observes both quantum and classic probability directly. The properties that would need to be satisfied for this to be the case are identified, and a generic hidden variable that satisfies them is found that would be present everywhere, transforming into a process-specific variable wherever a quantum process is active. Uncovering this variable confirms the possibility that it could be the stochastic archetype of quantum probability.

scholarly journals A search for a stochastic archetype of quantum probability

2021 ◽  
Author(s):  
Tim C Jenkins
Keyword(s):  
Probability Theory ◽  
Quantum Mechanics ◽  
Probability Density ◽  
Random Variables ◽  
Quantum Probability ◽  
Interference Term ◽  
Mathematical Foundation ◽  
Classical Probability ◽  
Quantum Process ◽  
Probability Densities

Abstract Superposed wavefunctions in quantum mechanics lead to a squared amplitude that introduces interference into a probability density, which has long been a puzzle because interference between probability densities exists nowhere else in probability theory. In recent years Man’ko and co-authors have successfully reconciled quantum and classical probability using a symplectic tomographic model. Nevertheless, there remains an unexplained coincidence in quantum mechanics, namely that mathematically the interference term in the squared amplitude of superposed wavefunctions has the form of a variance of a sum of correlated random variables and we examine whether there could be an archetypical variable behind quantum probability that provides a mathematical foundation that observes both quantum and classical probability directly. The properties that would need to be satisfied for this to be the case are identified, and a generic variable that satisfies them is found that would be present everywhere, transforming into a process-specific variable wherever a quantum process is active. This hidden generic variable appears to be such an archetype.

scholarly journals Modeling of quantum-like cognitive phenomena by the Fourier-holography technique under the choice of alternatives

2021 ◽  
Vol 45 (4) ◽  
pp. 551-561
Author(s):  
A.V. Pavlov
Keyword(s):  
Physical Nature ◽  
Correlation Radius ◽  
Reference Image ◽  
Total Probability ◽  
Natural Experiments ◽  
Classical Formula ◽  
Decision Logic ◽  
Ring Architecture ◽  
Logic Inference ◽  
Formula Of Total Probability

The article is dedicated to the search for a biologically motivated mechanism of the cognitive phenomenon of violation of the classical formula of total probability for the disjunction of incompatible events, which is considered by a number of researchers as a quantum-like phenomenon. A classical mechanism implemented by the 6f Fourier holography scheme of the resonant architecture that does not require reference to quantum mechanics either in its physical nature or at the level of formalism is demonstrated. In the analysis, the decision-making is interpreted as a choice of alternatives by using the non-cooperative game "Prisoner's Dilemma". The approach to the task is based on the search for a mechanism for forming a conditional estimate under a condition that contradicts the rule of monotonous decision logic. It is demonstrated that this estimate, in contrast to the unconditional and conditional one with a non-contradictory condition, is formed by logic with exception. The ring architecture of the holographic setup corresponds to the biologically inspired neural network concept of the excitation ring and implements cognitive dissonance on logic with exception. Conditions and ranges of violation of the classical formula of total probability in relation to the correlation radius of the reference image recorded in a hologram storing the monotone logic inference rule are analytically determined. The analytical model is confirmed by a quantitative coincidence of the results of numerical modeling with the published results of natural experiments.

scholarly journals Relating causal and probabilistic approaches to contextuality

2019 ◽  
Vol 377 (2157) ◽  
pp. 20190133
Author(s):  
Matt Jones
Keyword(s):  
Quantum Mechanics ◽  
Causal Model ◽  
Probability Distributions ◽  
Hidden Variables ◽  
Measurement Model ◽  
Direct Influence ◽  
Theme Issue ◽  
Full System ◽  
General Systems ◽  
New Interpretation

A primary goal in recent research on contextuality has been to extend this concept to cases of inconsistent connectedness, where observables have different distributions in different contexts. This article proposes a solution within the framework of probabi- listic causal models, which extend hidden-variables theories, and then demonstrates an equivalence to the contextuality-by-default (CbD) framework. CbD distinguishes contextuality from direct influences of context on observables, defining the latter purely in terms of probability distributions. Here, we take a causal view of direct influences, defining direct influence within any causal model as the probability of all latent states of the system in which a change of context changes the outcome of a measurement. Model-based contextuality (M-contextuality) is then defined as the necessity of stronger direct influences to model a full system than when considered individually. For consistently connected systems, M-contextuality agrees with standard contextuality. For general systems, it is proved that M-contextuality is equivalent to the property that any model of a system must contain ‘hidden influences’, meaning direct influences that go in opposite directions for different latent states, or equivalently signalling between observers that carries no information. This criterion can be taken as formalizing the ‘no-conspiracy’ principle that has been proposed in connection with CbD. M-contextuality is then proved to be equivalent to CbD-contextuality, thus providing a new interpretation of CbD-contextuality as the non-existence of a model for a system without hidden direct influences. This article is part of the theme issue ‘Contextuality and probability in quantum mechanics and beyond’.

scholarly journals Simple unity among the fundamental equations of science

2020 ◽  
Vol 375 (1797) ◽  
pp. 20190351 ◽  
Author(s):  
Steven A. Frank
Keyword(s):  
Information Theory ◽  
Natural Selection ◽  
Frame Of Reference ◽  
Price Equation ◽  
Physical Work ◽  
Total Probability ◽  
Theme Issue ◽  
Total Change ◽  
Fundamental Equations ◽  
Change Concerns

The Price equation describes the change in populations. Change concerns some value, such as biological fitness, information or physical work. The Price equation reveals universal aspects for the nature of change, independently of the meaning ascribed to values. By understanding those universal aspects, we can see more clearly why fundamental mathematical results in different disciplines often share a common form. We can also interpret more clearly the meaning of key results within each discipline. For example, the mathematics of natural selection in biology has a form closely related to information theory and physical entropy. Does that mean that natural selection is about information or entropy? Or do natural selection, information and entropy arise as interpretations of a common underlying abstraction? The Price equation suggests the latter. The Price equation achieves its abstract generality by partitioning change into two terms. The first term naturally associates with the direct forces that cause change. The second term naturally associates with the changing frame of reference. In the Price equation’s canonical form, total change remains zero because the conservation of total probability requires that all probabilities invariantly sum to one. Much of the shared common form for the mathematics of different disciplines may arise from that seemingly trivial invariance of total probability, which leads to the partitioning of total change into equal and opposite components of the direct forces and the changing frame of reference. This article is part of the theme issue ‘Fifty years of the Price equation’.

scholarly journals Indistinguishability and the origins of contextuality in physics

2019 ◽  
Vol 377 (2157) ◽  
pp. 20190150 ◽  
Author(s):  
J. Acacio de Barros ◽  
Federico Holik ◽  
Décio Krause
Keyword(s):  
Quantum Mechanics ◽  
Set Theory ◽  
Physical Quantity ◽  
Theoretical Framework ◽  
Quantum Systems ◽  
Strong Sense ◽  
Theme Issue ◽  
Identity Of Indiscernibles ◽  
Theoretical Structure

In this work, we discuss a formal way of dealing with the properties of contextual systems. Our approach is to assume that properties describing the same physical quantity, but belonging to different measurement contexts, are indistinguishable in a strong sense. To construct the formal theoretical structure, we develop a description using quasi-set theory, which is a set-theoretical framework built to describe collections of elements that violate Leibnitz's principle of identity of indiscernibles. This framework allows us to consider a new ontology in order to study the properties of quantum systems. This article is part of the theme issue ‘Contextuality and probability in quantum mechanics and beyond’.

scholarly journals CONSISTENT DEFORMED BOSONIC ALGEBRA IN NONCOMMUTATIVE QUANTUM MECHANICS

2008 ◽  
Vol 23 (09) ◽  
pp. 1393-1403 ◽  
Author(s):  
JIAN-ZU ZHANG
Keyword(s):  
Quantum Mechanics ◽  
Weyl Algebra ◽  
Fock Space ◽  
General Relation ◽  
Noncommutative Space ◽  
Two Dimensional ◽  
Noncommutative Quantum Mechanics ◽  
Commutative Space ◽  
Creation Operators ◽  
Noncommutative Quantum

In two-dimensional noncommutative space for the case of both position–position and momentum–momentum noncommuting, the consistent deformed bosonic algebra at the nonperturbation level described by the deformed annihilation and creation operators is investigated. A general relation between noncommutative parameters is fixed from the consistency of the deformed Heisenberg–Weyl algebra with the deformed bosonic algebra. A Fock space is found, in which all calculations can be similarly developed as if in commutative space and all effects of spatial noncommutativity are simply represented by parameters.

scholarly journals Quantum Bayesianism as the basis of general theory of decision-making

2016 ◽  
Vol 374 (2068) ◽  
pp. 20150245 ◽  
Author(s):  
Andrei Khrennikov
Keyword(s):  
Decision Making ◽  
Probability Theory ◽  
Subjective Probability ◽  
Quantum Probability ◽  
Total Probability ◽  
Interpretation Of Quantum Mechanics ◽  
Classical Probability ◽  
Probability Interpretation ◽  
Formula Of Total Probability

We discuss the subjective probability interpretation of the quantum-like approach to decision making and more generally to cognition. Our aim is to adopt the subjective probability interpretation of quantum mechanics, quantum Bayesianism (QBism), to serve quantum-like modelling and applications of quantum probability outside of physics. We analyse the classical and quantum probabilistic schemes of probability update, learning and decision-making and emphasize the role of Jeffrey conditioning and its quantum generalizations. Classically, this type of conditioning and corresponding probability update is based on the formula of total probability—one the basic laws of classical probability theory.

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