Diederik Aerts

Quantum Structures and the Nature of Reality is a collection of papers written for an interdisciplinary audience about the quantum structure research within the International Quantum Structures Association. The advent of quantum mechanics has changed our scientific worldview in a fundamental way. Many popular and semi-popular books have been published about the paradoxical aspects of quantum mechanics. Usually, however, these reflections find their origin in the standard views on quantum mechanics, most of all the wave-particle duality picture. Contrary to relativity theory, where the meaning of its revolutionary ideas was linked from the start with deep structural changes in the geometrical nature of our world, the deep structural changes about the nature of our reality that are indicated by quantum mechanics cannot be traced within the standard formulation. The study of the structure of quantum theory, its logical content, its axiomatic foundation, has been motivated primarily by the search for their structural changes. Due to the high mathematical sophistication of this quantum structure research, no books have been published which try to explain the recent results for an interdisciplinary audience. This book tries to fill this gap by collecting contributions from some of the main researchers in the field. They reveal the steps that have been taken towards a deeper structural understanding of quantum theory.

We investigate the question of ‘why customary macroscopic entities appear to us humans as they do, i.e., as bounded entities occupying space and persisting through time’, starting from our knowledge of quantum theory, how it affects the behavior of such customary macroscopic entities, and how it influences our perception of them. For this purpose, we approach the question from three perspectives. Firstly, we look at the situation from the standard quantum angle, more specifically the de Broglie wavelength analysis of the behavior of macroscopic entities, indicate how a problem with spin and identity arises, and illustrate how both play a fundamental role in well-established experimental quantum-macroscopical phenomena, such as Bose-Einstein condensates. Secondly, we analyze how the question is influenced by our result in axiomatic quantum theory, which proves that standard quantum theory is structurally incapable of describing separated entities. Thirdly, we put forward our new ‘conceptual quantum interpretation’, including a highly detailed reformulation of the question to confront the new insights and views that arise with the foregoing analysis. At the end of the final section, a nuanced answer is given that can be summarized as follows. The specific and very classical perception of human seeing—light as a geometric theory—and human touching—only ruled by Pauli's exclusion principle—plays a role in our perception of macroscopic entities as ontologically stable entities in space. To ascertain quantum behavior in such macroscopic entities, we will need measuring apparatuses capable of its detection. Future experimental research will have to show if sharp quantum effects—as they occur in smaller entities—appear to be ontological aspects of customary macroscopic entities. It remains a possibility that standard quantum theory is an incomplete theory, and hence incapable of coping ultimately with separated entities, meaning that a more general theory will be needed.

We analyze the meaning of the violation of the marginal probability law for situations of correlation measurements where entanglement is identified. We show that for quantum theory applied to the cognitive realm such a violation does not lead to the type of problems commonly believed to occur in situations of quantum theory applied to the physical realm. We briefly situate our quantum approach for modeling concepts and their combinations with respect to the notions of “extension” and “intension” in theories of meaning, and in existing concept theories.

The quantum logical `or' is analyzed from a physical perspective. We show that it is the existence of EPR-like correlation states for the quantum mechanical entity under consideration that make it nonequivalent to the classical situation. Specifically, the presence of potentiality in these correlation states gives rise to the quantum deviation from the classical logical `or'. We show how this arises not only in the microworld, but also in macroscopic situations where EPR-like correlation states are present. We investigate how application of this analysis to concepts could alleviate some well known problems in cognitive science.

The sets of contexts and properties of a concept are embedded in the complex Hilbert space of quantum mechanics. States are unit vectors or density operators, and contexts and properties are orthogonal projections. The way calculations are done in Hilbert space makes it possible to model how context influences the state of a concept. Moreover, a solution to the combination of concepts is proposed. Using the tensor product, a procedure for describing combined concepts is elaborated, providing a natural solution to the pet fish problem. This procedure allows the modeling of an arbitrary number of combined concepts. By way of example, a model for a simple sentence containing a subject, a predicate and an object, is presented.

A stream of conscious experience is extremely contextual; it is impacted by sensory stimuli, drives and emotions, and the web of associations that link, directly or indirectly, the subject of experience to other elements of the individual's worldview. The contextuality of one's conscious experience both enhances and constrains the contextuality of one's behavior. Since we cannot know first-hand the conscious experience of another, it is by way of behavioral contextuality that we make judgements about whether or not, and to what extent, a system is conscious. Thus we believe that a deep understanding of contextuality is vital to the study of consciousness. Methods have been developed for handling contextuality in the microworld of quantum particles. Our goal has been to investigate the extent to which these methods can be used to analyze contextuality in conscious experience.

We outline the rationale and preliminary results of using the State Context Property (SCOP) formalism, originally developed as a generalization of quantum mechanics, to describe the contextual manner in which concepts are evoked, used, and combined to generate meaning. The quantum formalism was developed to cope with problems arising in the description of (1) the measurement process, and (2) the generation of new states with new properties when particles become entangled. Similar problems arising with concepts motivated the formal treatment introduced here. Concepts are viewed not as fixed representations, but entities existing in states of potentiality that require interaction with a context---a stimulus or another concept---to `collapse' to observable form as an exemplar, prototype, or other (possibly imaginary) instance. The stimulus situation plays the role of the measurement in physics, acting as context that induces a change of the cognitive state from superposition state to collapsed state. The collapsed state is more likely to consist of a conjunction of concepts for associative than analytic thought because more stimulus or concept properties take part in the collapse. We provide two contextual measures of conceptual distance---one using collapse probabilities and the other weighted properties---and show how they can be applied to conjunctions using the pet fish problem.