Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences最新文献

After briefly recalling some of the main theses of quantum probability concerning the foundational aspects of quantum mechanics and of probability theory, we describe the various categories of axioms in classical probability theory and focus our attention on the classical conditioning axioms. Four classical conditioning axioms are proposed, their probabilistic and model-independent meaning is discussed and it is proven that: (i) the only mathematical realization of these axioms is the classical Bayes rule (or equivalently the classical theorem of composite probabilities) and (ii) von Neumann's projection postulate, which can be considered one of the first examples of quantum conditioning axioms, violates two of these axioms. Finally, we construct an example showing that, even in classical probability, an uncritical application of Bayes' rule can lead to non-optimal predictions. This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.

In recent years there has been interest in the relationship between intransitivity and the presence of true (Type II) contextuality (contextuality with context independent marginals). The latter has been considered to be a sine qua non of quantum mechanics, although it has been observed in experiments on human decision making. Transitivity has long been viewed as essential to rational decision making although intransitivity appears to be ubiquitous in the natural world. An analogue of a preference graph is introduced for non-deterministic dynamical systems (of which decision making is an example) and used to identify several conditions which appear necessary for true contextuality to be present. Comparing the preference graph for the system against a reference preference graph formed from the marginals of the observables, one sees losses of options and intransitivity as necessary conditions.This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 1)'.

The conceptuality interpretation of quantum mechanics proposes that quantum entities have a conceptual nature, interacting with the material world through processes that are the physical counterpart of the meaning-based processes, which typically occur in human cognition. This interpretation emerged from the early developments in quantum cognition, a field that uses quantum mathematics to model human cognitive activity. It benefited from the specific approach taken by the Brussels research group, modelling concepts themselves as quantum entities and minds as measuring apparatus. The article sketches the essential steps of the intellectual journey, going from the first applications of quantum notions and formalisms to human cognition to the proposal of a potentially groundbreaking interpretation of quantum mechanics, offering profound explanations for major quantum phenomena. This was done by drawing numerous parallels with the human conceptual domain and suggesting the existence of a level of cognitive activity that would underlie our physical reality. This means that an increased cross-fertilization between the conceptuality interpretation and quantum cognition is to be expected in the future, both of which are synergistic in furthering our understanding of the nature of reality. This is the first part of a two-part article. In the second part, which can be read independently of the first, the successes of the interpretation will be described in a more systematic way, providing a brief overview of what has been achieved so far.This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 1)'.

An overview of the conceptuality interpretation of quantum mechanics is presented, along with an explanation of how it sheds light on key quantum and relativistic phenomena. In particular, we show how the interpretation clarifies Heisenberg's uncertainty principle, wave function-based and entanglement-based non-locality, interference effects resulting from the superposition principle, delayed choice experiments, quantum measurements, the mechanism of quantization, the reason why entities can establish entanglement bonds and the statistical behaviour of indistinguishable entities. We further argue that the interpretation can also elucidate relativistic effects, focusing on time dilation. Finally, we suggest that it can provide a novel and challenging perspective on evolution. This article is the second in a two-part series devoted to exploring this promising approach to reality. The first part, which serves as a companion to this discussion, outlines the intellectual trajectory leading from the first applications of quantum notions to human cognition to the bold rethinking suggested by the conceptuality interpretation.This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 1)'.

We are rapidly moving to a future in which our information environment is saturated by artificial intelligence (AI), and humans and AI agents will routinely engage in shared decision making even in conditions of high uncertainty and risk (such as natural disasters or nuclear accidents). Trust is fundamental to the effectiveness of these interactions. A key challenge in modelling the dynamics of trust in human-AI interactions is to provide a means to integrate the diversity of human trust fluctuations found empirically. In this article, we explore the ability of quantum random walk (QRW) models to model this dynamism of trust found in empirical human-AI interactions. Specifically, we manipulate certain features of the QRW to explore its ability to provide the necessary agility and sensitivity to fluctuations in trust judgments. The goal is to incorporate the nature of the interaction itself into the evolution of the model, with the features stemming from empirically derived parameters. We found that using empirical parameters to inform the use of different Hamiltonians throughout the interaction markedly influences the modelled trust dynamics and can provide a promising means to model the evolution of trust in human-AI interactions.This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 1)'.

The natural numbers play a key role in axiomatic set theory and hence in the foundations of mathematics. The natural numbers also have a major role in quantum mechanics. The idea is explored that von Neumann's construction (vNC) of the natural numbers, within the framework of Zermelo-Fraenkel (ZF) axiomatic set theory, can serve as a blueprint for deriving some key basic equations of quantum mechanics. This approach obviates any need for quantizing classical mechanics and provides further support for the view that quantum mechanics is perfectly applicable to fields beyond those in physics for which it was originally intended.This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 1)'.

This article considers the relationships between quantum theory (QT) and quantum-like theories (QLTs), theories using mathematical models based on the formalism of QT, from a reverse perspective, that of QLTs. The article argues that QT is no longer a theory of the behaviour, in particular motion, of physical objects, as was the case in classical physics and relativity. Instead, QT is a form of decision theory, involving a special 'topology' of decisions, using the term topology in part metaphorically, but only in part, because it applies in its proper mathematical sense to the formalism of QT. Part of this topology is the concept of free will, reconsidered through the concept of decision. This character of QT is grounded in a particular type of interpretations of QT, 'reality without realism' (RWR) interpretations. To address the affinities and differences between QT and QLTs, the article introduces two new principles: 'the unambiguity principle', equally applicable in QT and QLTs, or in mathematics and science in general, and 'the free will principle', only applicable in QLTs and not in QT. The article also reflects on the limits of quantum-like sciences (QLSs) and mathematical-experimental science in general in dealing with human thinking and decision making.This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 1)'.

An important challenge for quantum theories of cognition and decision concerns the incorporation of memory for recently made judgements and their effects on later judgements. First, we review a general approach to measurement based on system plus environment representations of states and measurement instruments. These more general measurement models provide ways to incorporate the effects of recent judgements on later judgements. Then we compare three different measurement models that are based on these more general measurement operations to account for a puzzling collection of question order effect findings.This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 1)'.

The innovative application of quantum mechanical concepts of probability to cognitive science has provided new, meaningful ways to more accurately model human decision making. Leveraging the dynamic geometric principles associated with the mathematics of quantum theory, we use the idea of probability interference to explain projection bias and its violations of the classical law of total probability and expected utility theory. In particular, Khrennikov's contextual probability model is successfully applied to the study by Read & van Leeuwen on hunger and projection bias. We conclude that probability interference provides an effective, accurate and meaningful way to model projection bias, thus broadening the reach of quantum cognition in economics.This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 1)'.