Mathematical Formalism Of Quantum Theory Research Articles

Quantum Value Valuation Continuum

Price, cost, and income (PCI) methods are traditionally used to approximate the value state of an economic commodity such as a property. Based on the estimates of these methods, we explore how quantum theory represents the fundamental process of value valuation in practice. We propose that the mathematical formalism of quantum theory is a promising view and measure of economic value. To ground our exploration, we first map traditional PCI estimates onto three-dimensional spherical coordinates, which were then transformed into two-dimensional quantum states using the Bloch sphere. This step enabled the computation of eigenvalues and eigenvectors of the Hamiltonian matrix, from which the value state measures were derived. The results exhibit practical applications as well as fundamental insights into potential connections between economic and quantum value states.

On "Decisions and Revisions Which a Minute Will Reverse": Consciousness, The Unconscious and Mathematical Modeling of Thinking.

This article considers a partly philosophical question: What are the ontological and epistemological reasons for using quantum-like models or theories (models and theories based on the mathematical formalism of quantum theory) vs. classical-like ones (based on the mathematics of classical physics), in considering human thinking and decision making? This question is only partly philosophical because it also concerns the scientific understanding of the phenomena considered by the theories that use mathematical models of either type, just as in physics itself, where this question also arises as a physical question. This is because this question is in effect: What are the physical reasons for using, even if not requiring, these types of theories in considering quantum phenomena, which these theories predict fully in accord with the experiment? This is clearly also a physical, rather than only philosophical, question and so is, accordingly, the question of whether one needs classical-like or quantum-like theories or both (just as in physics we use both classical and quantum theories) in considering human thinking in psychology and related fields, such as decision science. It comes as no surprise that many of these reasons are parallel to those that are responsible for the use of QM and QFT in the case of quantum phenomena. Still, the corresponding situations should be understood and justified in terms of the phenomena considered, phenomena defined by human thinking, because there are important differences between these phenomena and quantum phenomena, which this article aims to address. In order to do so, this article will first consider quantum phenomena and quantum theory, before turning to human thinking and decision making, in addressing which it will also discuss two recent quantum-like approaches to human thinking, that by M. G. D’Ariano and F. Faggin and that by A. Khrennikov. Both approaches are ontological in the sense of offering representations, different in character in each approach, of human thinking by the formalism of quantum theory. Whether such a representation, as opposed to only predicting the outcomes of relevant experiments, is possible either in quantum theory or in quantum-like theories of human thinking is one of the questions addressed in this article. The philosophical position adopted in it is that it may not be possible to make this assumption, which, however, is not the same as saying that it is impossible. I designate this view as the reality-without-realism, RWR, view and in considering strictly mental processes as the ideality-without-idealism, IWI, view, in the second case in part following, but also moving beyond, I. Kant’s philosophy.

A Unified Theory of Human Judgements and Decision-Making under Uncertainty.

Growing empirical evidence reveals that traditional set-theoretic structures cannot in general be applied to cognitive phenomena. This has raised several problems, as illustrated, for example, by probability judgement errors and decision-making (DM) errors. We propose here a unified theoretical perspective which applies the mathematical formalism of quantum theory in Hilbert space to cognitive domains. In this perspective, judgements and decisions are described as intrinsically non-deterministic processes which involve a contextual interaction between a conceptual entity and the cognitive context surrounding it. When a given phenomenon is considered, the quantum-theoretic framework identifies entities, states, contexts, properties and outcome statistics, and applies the mathematical formalism of quantum theory to model the considered phenomenon. We explain how the quantum-theoretic framework works in a variety of judgement and decision situations where systematic and significant deviations from classicality occur.

A Quantum-Like multimodal network framework for modeling interaction dynamics in multiparty conversational sentiment analysis

Sentiment analysis in conversations is an emerging yet challenging artificial intelligence (AI) task. It aims to discover the affective states and emotional changes of speakers involved in a conversation on the basis of their opinions, which are carried by different modalities of information (e.g., a video associated with a transcript). There exists a wealth of intra- and inter-utterance interaction information that affects the emotions of speakers in a complex and dynamic way. How to accurately and comprehensively model complicated interactions is the key problem of the field. To fill this gap, in this paper, we propose a novel and comprehensive framework for multimodal sentiment analysis in conversations, called a quantum-like multimodal network (QMN), which leverages the mathematical formalism of quantum theory (QT) and a long short-term memory (LSTM) network. Specifically, the QMN framework consists of a multimodal decision fusion approach inspired by quantum interference theory to capture the interactions within each utterance (i.e., the correlations between different modalities) and a strong-weak influence model inspired by quantum measurement theory to model the interactions between adjacent utterances (i.e., how one speaker influences another). Extensive experiments are conducted on two widely used conversational sentiment datasets: the MELD and IEMOCAP datasets. The experimental results show that our approach significantly outperforms a wide range of baselines and state-of-the-art models.

Entanglement is Necessary for Emergent Classicality in All Physical Theories.

One of the most striking features of quantum theory is the existence of entangled states, responsible for Einstein's so called "spooky action at a distance." These states emerge from the mathematical formalism of quantum theory, but to date we do not have a clear idea of the physical principles that give rise to entanglement. Why does nature have entangled states? Would any theory superseding classical theory have entangled states, or is quantum theory special? One important feature of quantum theory is that it has a classical limit, recovering classical theory through the process of decoherence. We show that any theory with a classical limit must contain entangled states, thus establishing entanglement as an inevitable feature of any theory superseding classical theory.

The Real and the Mathematical in Quantum Modeling: From Principles to Models and from Models to Principles

The history of mathematical modeling outside physics has been dominated by the use of classical mathematical models, C-models, primarily those of a probabilistic or statistical nature. More recently, however, quantum mathematical models, Q-models, based in the mathematical formalism of quantum theory have become more prominent in psychology, economics, and decision science. The use of Q-models in these fields remains controversial, in part because it is not entirely clear whether Q-models are necessary for dealing with the phenomena in question or whether C-models would still suffice. My aim, however, is not to assess the necessity of Q-models in these fields, but instead to reflect on what the possible applicability of Q-models may tell us about the corresponding phenomena there, vis-a-vis quantum phenomena in physics. In order to do so, I shall first discuss the key reasons for the use of Q-models in physics. In particular, I shall examine the fundamental principles that led to the development of quantum mechanics. Then I shall consider a possible role of similar principles in using Q-models outside physics. Psychology, economics, and decision science borrow already available Q-models from quantum theory, rather than derive them from their own internal principles, while quantum mechanics was derived from such principles, because there was no readily available mathematical model to handle quantum phenomena, although the mathematics ultimately used in quantum did in fact exist then. I shall argue, however, that the principle perspective on mathematical modeling outside physics might help us to understand better the role of Q-models in these fields and possibly to envision new models, conceptually analogous to but mathematically different from those of quantum theory, helpful or even necessary there or in physics itself. I shall suggest one possible type of such models, singularized probabilistic, SP, models, some of which are time-dependent, TDSP-models. The necessity of using such models may change the nature of mathematical modeling in science and, thus, the nature of science, as it happened in the case of Q-models, which not only led to a revolutionary transformation of physics but also opened new possibilities for scientific thinking and mathematical modeling beyond physics.

Constructibility of the Universal Wave Function

This paper focuses on a constructive treatment of the mathematical formalism of quantum theory and a possible role of constructivist philosophy in resolving the foundational problems of quantum mechanics, particularly, the controversy over the meaning of the wave function of the universe. As it is demonstrated in the paper, unless the number of the universe's degrees of freedom is fundamentally upper bounded (owing to some unknown physical laws) or hypercomputation is physically realizable, the universal wave function is a non-constructive entity in the sense of constructive recursive mathematics. This means that even if such a function might exist, basic mathematical operations on it would be undefinable and subsequently the only content one would be able to deduce from this function would be pure symbolical.

On the Foundations of the Brussels Operational-Realistic Approach to Cognition

The scientific community is becoming more and more interested in the research that applies the mathematical formalism of quantum theory to model human decision-making. In this paper, we provide the theoretical foundations of the quantum approach to cognition that we developed in Brussels. These foundations rest on the results of two decade studies on the axiomatic and operational-realistic approaches to the foundations of quantum physics. The deep analogies between the foundations of physics and cognition lead us to investigate the validity of quantum theory as a general and unitary framework for cognitive processes, and the empirical success of the Hilbert space models derived by such investigation provides a strong theoretical confirmation of this validity. However, two situations in the cognitive realm, 'question order effects' and 'response replicability', indicate that even the Hilbert space framework could be insufficient to reproduce the collected data. This does not mean that the mentioned operational-realistic approach would be incorrect, but simply that a larger class of measurements would be in force in human cognition, so that an extended quantum formalism may be needed to deal with all of them. As we will explain, the recently derived 'extended Bloch representation' of quantum theory (and the associated 'general tension-reduction' model) precisely provides such extended formalism, while remaining within the same unitary interpretative framework.

Editorial: Quantum Structures in Cognitive and Social Science.

A fundamental problem in cognitive and social science concerns the identification of the principles guiding human cognitive acts such as decision-making, categorization, and behavior under uncertainty. Identifying these mechanisms would have manifold implications for fields ranging from psychology to economics, finance, politics, computer science, and artificial intelligence. The predominant theoretical paradigm rests on a classical conception of logic and probability theory. According to this paradigm people make decisions by following the rules of Boole's logic, while the probabilistic aspects of these decisions can be formalized by Kolmogorov's probability theory. This classical approach was believed to provide a quite complete and accurate account of human decision-making at both a normative level (describing what people should do) and a descriptive level (describing what people actually do). However, starting from the seventies, experimental studies of conceptual categorization, human judgment and perception, and behavioral economics have revealed that this classical conception is fundamentally problematical, in the sense that the cognitive models based on these mathematical structures are not capable of capturing how humans make decisions in situations involving uncertainty. In the last decade, an alternative scientific paradigm has arisen that employs a different and more general modeling scheme; it uses the mathematical formalism of quantum theory to model situations and processes in cognitive and social science. This new approach has not only met with considerable success but is becoming increasingly accepted in the scientific community, having attracted interest from important scientists, top journals, funding institutions, and media. Prisoners' dilemmas, conjunction and disjunction fallacies, disjunction effects, violations of the Sure-Thing principle, Allais, Ellsberg and Machina paradoxes, are only some of the examples where the application of the quantum mechanical formalism has shown significant effectiveness over traditional modeling schemes of a classical type. The Frontiers Research Topic “Quantum Structures in Cognitive and Social Science” present an overview of current research that applies the formalism of quantum theory to cognitive and socio-economic domains. The term “quantum” may be misleading. The aim here is not to investigate the microphysical processes occurring in the human brain and, as a consequence, driving human judgments. Rather, we inquire into the validity of quantum theory as a general, coherent, and unitary paradigm for human cognition. In this respect, this research benefits from studies into the axiomatic and operational foundations of quantum physics. The scope of this bold approach to human cognition is discovering general rules that associate the empirical phenomenology in these domains with states, measurements, and probabilities of outcomes in such a way that these entities are represented exactly as quantum theory in Hilbert space represents states, measurements, and probabilities of outcomes in the phenomenology of microphysics. The ensuing modeling is theory-based, not experiment-based; that is, the models are not built around a specific effect or experiment, although they are sometimes used in conjunction with empirical data to build a stronger case. The models are constructed following the general epistemological and technical constraints of quantum theory; hence the successes of this quantum theory-based modeling suggest that it might provide a general theory for human cognition. This Research Topic develops around three main directions of research, as follows. The deep reasons underlying the success of the quantum paradigm in cognitive and socio-economic domains are investigated. Further empirical situations are identified in these domains where the quantum formalism presents advantages with respect to traditional modeling schemes, and new genuine quantum structures appear. The application of this quantum paradigm is extended to novel and barely explored domains. The first set of results concern knowledge representation and conceptual categorization. Aerts et al. analyze the results of a cognitive test on conjunctions and negations of natural concepts, showing that a quantum-theoretic probabilistic model in Hilbert space faithfully represents the collected data, at variance with a set-theoretic Kolmogorovian model. This result is explained by assuming the existence of two types of reasoning in human cognition, a dominant emergent reasoning, and a secondary logical reasoning. Some mathematical aspects of this quantum-theoretic model on conceptual conjunctions and negations are developed in Veloz and Desjardins through the introduction of unitary operators in Hilbert space. Aerts et al. show instead that the quantum-theoretic approach Aerts et al. can be interpreted as a suitable generalization of Rosch's prototype theory, where prototypes are context-dependent and may interfere when concepts combine. The second set of results concern the modeling of human decision-making. Moreira and Wichert explore an alternative quantum-theoretic approach, the quantum-like Bayesian network, to describe the paradoxes related to the violation of the Sure-Thing Principle in experiments on human judgments. Their model is in a good agreement with different sets of empirical data. The opinion paper in Pothos et al. reviews some current progress on the quantum similarity model in Hilbert space recently proposed by Pothos et al. which correctly represents human similarity judgments. Decision-making errors and preference reversal are also investigated in Yukalov and Sornette within their quantum decision theory. Wang and Busemeyer analyze the notion of complementarity in human cognition, and claim that the way in which it is used in quantum physics can also be helpful in cognitive science. Human perception is the object of the study in Khrennikov, where the author develops a quantum model of the sensation-perception dynamics, illustrating it by means of the model of bistable perception of a specific ambiguous figure, the Schroder stair. Finally, Tressoldi et al. identify a significant violation of temporal Bell inequalities in a set of cognitive tests. The violation indicates, according to the authors, the presence of temporal entanglement between binary human behavioral unconscious choices at a given time and binary random outcomes at a different time. In all these approaches, the presence of quantum structures in cognition is determined by the fact that the cognitive systems under investigation share a common feature, namely contextuality. A different position with respect to the presence of contextuality in cognition is assumed in Zhang and Dzhafarov, where the authors apply a theory of (non)contextuality to analyze series-parallel (SP) mental architectures. The third set of results concern advanced applications of the quantum-mathematical formalism to wider ranges of social science. Bisconti et al. propose an inverse Potts model, typically used in statistical quantum field theory, to reconstruct the node states in a real-world social network. Haven explores the properties of two types of potential functions, inspired by classical and quantum physics that can be potentially employed to model financial information, including preferences toward risk and uncertainty. Finally, Dalla Chiara et al. investigate different, but mutually related, aspects of parallelism within the framework of quantum computation, cognition and music, and study potential applications of quantum computational semantics in both natural and musical language. Leaving aside the specific differences between the approaches above, most of them agree in claiming that quantum structures are systematically present in cognitive and social science phenomena, and that quantum-inspired models are more efficient than traditional set-theoretic models of probability. Is “quantum” the end of the story? Is Hilbert space really the place where all these phenomena can be modeled? Is there any empirical deviation from quantum predictions? We do not have yet an answer to these questions. This is why we believe that the road that will lead further to possibly a generally accepted quantum theory of human decision-making will still be full of fascinating surprises.

Generalizing Prototype Theory: A Formal Quantum Framework.

Theories of natural language and concepts have been unable to model the flexibility, creativity, context-dependence, and emergence, exhibited by words, concepts and their combinations. The mathematical formalism of quantum theory has instead been successful in capturing these phenomena such as graded membership, situational meaning, composition of categories, and also more complex decision making situations, which cannot be modeled in traditional probabilistic approaches. We show how a formal quantum approach to concepts and their combinations can provide a powerful extension of prototype theory. We explain how prototypes can interfere in conceptual combinations as a consequence of their contextual interactions, and provide an illustration of this using an intuitive wave-like diagram. This quantum-conceptual approach gives new life to original prototype theory, without however making it a privileged concept theory, as we explain at the end of our paper.

First Time Demonstration of the Quantum Interference Effect during Integration of Cognition and Emotion in Children

Quantum cognition is a scientific approach to cognitive phenomena which makes use of the mathematical formalism of quantum theory. Quantum interference effect constitutes one of this theory’s main tenets and has been repeatedly demonstrated experimentally, in the last decade, in adult subjects. In the present paper, we aim to demonstrate, for the first time, the existence of thequantum interference effect on children during an experiment involving an integration of cognition and emotion. Our positive results consolidate the presuppositions of quantum cognition, enlarging its field of application to children’s mental apparatus and evidence the important question to consider the quantum model in the current investigated question of the interaction of cognition and emotion in children at neurological and psychological levels.

Quantum structure in competing lizard communities

Almost two decades of research on applications of the mathematical formalism of quantum theory as a modeling tool in domains different from the micro-world has given rise to many successful applications in situations related to human behavior and thought, more specifically in cognitive processes of decision-making and the ways concepts are combined into sentences. In this article, we extend this approach to animal behavior, showing that an analysis of an interactive situation involving a mating competition between certain lizard morphs allows to identify a quantum theoretic structure. More in particular, we show that when this lizard competition is analyzed structurally in the light of a compound entity consisting of subentities, the contextuality provided by the presence of an underlying rock-paper-scissors cyclic dynamics leads to a violation of Bell's inequality, which means it is of a non-classical type. We work out an explicit quantum-mechanical representation in Hilbert space for the lizard situation and show that it faithfully models a set of experimental data collected on three throat-colored morphs of a specific lizard species. Furthermore, we investigate the Hilbert space modeling, and show that the states describing the lizard competitions contain entanglement for each one of the considered confrontations of lizards with different competing strategies, which renders it no longer possible to interpret these states of the competing lizards as compositions of states of the individual lizards.

A violation of the uncertainty principle implies a violation of the second law of thermodynamics

Uncertainty relations state that there exist certain incompatible measurements, to which the outcomes cannot be simultaneously predicted. While the exact incompatibility of quantum measurements dictated by such uncertainty relations can be inferred from the mathematical formalism of quantum theory, the question remains whether there is any more fundamental reason for the uncertainty relations to have this exact form. What, if any, would be the operational consequences if we were able to go beyond any of these uncertainty relations? Here we give a strong argument that justifies uncertainty relations in quantum theory by showing that violating them implies that it is also possible to violate the second law of thermodynamics. More precisely, we show that violating the uncertainty relations in quantum mechanics leads to a thermodynamic cycle with positive net work gain, which is very unlikely to exist in nature.

Origin of complex quantum amplitudes and Feynman’s rules

Complex numbers are an intrinsic part of the mathematical formalism of quantum theory and are perhaps its most characteristic feature. In this article, we show that the complex nature of the quantum formalism can be derived directly from the assumption that a pair of real numbers is associated with each sequence of measurement outcomes, with the probability of this sequence being a real-valued function of this number pair. By making use of elementary symmetry conditions, and without assuming that these real number pairs have any other algebraic structure, we show that these pairs must be manipulated according to the rules of complex arithmetic. We demonstrate that these complex numbers combine according to Feynman's sum and product rules, with the modulus-squared yielding the probability of a sequence of outcomes.

Notes on von Neumann measurement scheme

A quantum measurement scheme devised by von Neumann plays a central role in all considerations relevant to the measurement problem in quantum mechanics. Despite the fact that the scheme ultimately did not reach the anticipated goal, a view is still maintained, that at least the very approach to the solution of the problem is correct and does not rise any doubts about its consistency. However, a little critical check of a validity of the key assumptions, lying in the foundations of the scheme, casts some shadow on its physically unimpeachable status. Speaking explicitly, the use of such physically bad established notions as a quantization of a purely macroscopic object and an “ideal” measurement, although embodied in the proper mathematical formalism of quantum theory, as well as the over-simplified interpretation of the theory, characterize this measurement scheme as a purely theoretical construction, which is far away from the situation occurring in the case of real measurements.

The nature of particle–wave complementarity

To resolve the challenges (paradoxes) attending the genesis of quantum theory, Niels Bohr in 1927 proposed a generalization of the conceptual framework of classical physics, termed complementarity, which posited pairs of classical concepts that could be manifested only in mutually exclusive experimental arrangements with both needed to embrace the full range of physical experience. In this context particle–wave complementarity has encountered two major obstacles; (1) No formulation of it has commanded a general consensus; and (2) empirical evidence abounds that contradicts the claim of mutual exclusivity of particle and wave properties of quantum objects in a single experiment. These problems are rooted in the unrestricted scope of this claim. It is shown that a restricted version of particle–wave complementarity, here called “which value–interference” complementarity, is consistent with observations, is naturally contained in the mathematical formalism of quantum theory and, in fact, holds in any representation of quantum theory defined by the eigenvectors of a complete set of commuting observables, such as the configuration space position (kinematic) variables or the complementary linear momentum (dynamic) variables. Thus, the which value–interference form of particle–wave complementarity exists within kinematic–dynamic complementarity, a “layered” complementarity, so to speak. It is hoped that a consistent and valid exposition of particle–wave complementarity will be helpful to those who find the framework of complementarity useful and appropriate for some purpose.