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Snapshots on quantum probability

Luigi Accardi

Published Jan 1, 2008• 1 citation
OpenAlex

Abstract

As suggested by the title, the goal of the present talk is to describe some casual photographs of different parts of quantum probability without pretense of completeness. I will choose three topics which, in my opinion, efficiently illustrate the fruitful interplay between mathematics and physics which has characterized the development of quantum probability in the past thirty years: (i) the description of a recent experiment which has brought to a conclusion the long standing debate about possible non local effects as necessary consequences of the basic principles of quantum mechanics; (ii) the notion of interacting Fock space, which emerged from quantum electrodynamics without dipole approximation and turned out to be a fruitful tool in such disparate fields as orthogonal polynomials, asymptotics of graphs, quantum structure of classical probability measures, exclusion statistics, . . . ; (iii) the square (and higher powers) of white noise and its relation to renormalization theory and infinitely divisible processes.

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As suggested by the title, the goal of the present talk is to describe some casual photographs of different parts of quantum probability without pretense of completeness. I will choose three topics which, in my opinion, efficiently illustrate the fruitful interplay between mathematics and physics which has characterized the development of quantum probability in the past thirty years: (i) the description of a recent experiment which has brought to a conclusion the long standing debate about possible non local effects as necessary consequences of the basic principles of quantum mechanics; (ii) the notion of interacting Fock space, which emerged from quantum electrodynamics without dipole approximation and turned out to be a fruitful tool in such disparate fields as orthogonal polynomials, asymptotics of graphs, quantum structure of classical probability measures, exclusion statistics, . . . ; (iii) the square (and higher powers) of white noise and its relation to renormalization theory and infinitely divisible processes.
The present volume contains contributions of the 32nd International Conference on Quantum Probability and Related Topics held in Levico Terme (Italy) in June 2011 as the annual meeting of the Association for Quantum Probability and Innite Dimensional Analysis (AQPIDA). The\nconference was sponsored by CIRM Bruno Kessler Foundation, Centro V. Volterra and MIUR PRIN 2009 project "Quantum Probability and Applications\nto Information Theory".\nThe goal of the conference was to communicate new results in quantum probability and infinite dimensional analysis and related elds. A wide range of topics was discussed: white noise and higher powers of white\nnoise, quantum Markov semigroups and applications to quantum statistical mechanics, quantum stochastic calculus, stochastic differential equations, quantum Levy processes, product systems and interacting Fock spaces, algebraic probability, realizations of classical processes, quantum probability in bio-systems, complexity, entanglement and measurement showing the highly interdisciplinary activity of the field and deep connections with many areas of mathematics.\nSelected contributions highlighting the latest developments, collected after the conference, refereed by one or more experts, are now presented in this volume that will be of great value for researchers, applied mathematicians and graduate students.
This author has been engaged over the past several years in the effort to develop certain basic intuitions into a fully mathematical theory. These evolving intuitions first began with the idea that the economist's notion of utility might more truly be represented mathematically by a normed measure on the commodity space rather than by a point function, the value of the measure on a particular set of goods having the sense of a relative propensity of the individual to select a good in that set. With this idea there came in turn, quite naturally, the insight that utility and probability are one and the same thing. And accordingly it emerged that the theory we were seeking would not be restricted to the economic behavior of human beings, nor even to the behavior in general of human beings: it would pertain, in utmost generality, to all behavior of all systems. Thus, it must affirm that all behavior is the evolution of stochastic processes, and that all systems are stochastic processes. Consistent with this the structural elements of stochastic processes are seen as the fundamental, objective, real entities, the ultimate building-blocks of reality. These are eventualities and acts and-according to the traditional mathematical representation of stochastic processes, also-probabilities. This structural aspect of the theory under development puts into sharp evidence its wide divergence from the classical scientific conception of reality. Our intensive search failed to discover a dynamical law for the proposed new theory within the mathematical context of the traditional, σ-field-real-valued-measure representation of stochastic processes. We therefore have gone on to a higher level of possible mathematical representation. This particular representation would have collections of eventualities represented by lattices of certain configurations in a unitary space. Closed linear manifolds are among these configurations, and indeed the formalism of quantum mechanics falls into the context of this representation. But the general lattice element in the kinds of lattices being investigated is more complicated than a linear manifold, being heuristically comparable with an acute-angle cone of vectors. These configurations are formulated by means of a ƒlanking relation between linear manifolds. One such lattice has been developed. Others are currently being sought through variations on the notion of flanking. The fact that probabilities present themselves in the unitary space representation-as is familiar from quantum mechanics-as parameters of the relationship between linear manifolds suggests that probability is not a fundamental structural concept. It is consequently indicated that eventualities and acts alone are the basic elements of real structure. Accordingly, the dynamical law must be formulable in terms of these elements only. There is a natural suggestion of what the law might look like. Even at the present early stage of development of the new theory being sought it is possible to cite many pieces of empirical evidence vouching for its exclusive validity. Such evidence is particularly noteworthy in the domain of human behavior.
The theory of probability and the quantum theory, the one mathematical and the other physical, are related in that each admits a number of very different interpretations. It has been proposed that the conceptual problems of the quantum theory could be, if not resolved, at least mitigated by a proper interpretation of probability. We rather show, through a historical and analytical overview of probability and quantum theory, that if some interpretations of the one and the other go along particularly well, none follows in a unique way.
The context of an operational description is given by the distinction between what we consider as relevant and what as irrelevant for a particular experiment or observation. A rigorous description of a context in terms of a mathematically formulated context-independent fundamental theory is possible by the restriction of the domain of the basic theory and the introduction of a new coarser topology. Such a new topology is never given by first principles, but depends in a crucial way on the abstractions made by the cognitive apparatus or the pattern recognition devices used by the experimentalist. A consistent mathematical formulation of a higher-level theory requires the closure of the restriction of the basic theory in the new contextual topology. The validity domain of the so constructed higher-level theory intersects nontrivially with the validity domain of the basic theory: neither domain is contained in the other. Therefore, higher-level theories cannot be totally ordered and theory reduction is not transitive. The emergence of qualitatively new properties is a necessary consequence of such a formulation of theory reduction (which does not correspond to the traditional one). Emergent properties are not manifest on the level of the basic theory, but they can be derived rigorously by imposing new, contextually selected topologies upon context-independent first principles. Most intertheoretical relations are mathematically describable as singular asymptotic expansions which do not converge in the topology of the primary theory, or by choosing one of the infinitely many possible, physically inequivalent representations of the primary theory (Gelfand Naimark Segal-construction of algebraic quantum mechanics). As examples we discuss the emergence of shadows, inductors, capacitors and resistors from Maxwell's electrodynamics, the emergence of order parameters in statistical mechanics, the emergence of mass as a classical observable in Galilei-relativistic theories, the emergence of the shape of molecules in quantum mechanics, the emergence of temperature and other classical observables in algebraic quantum mechanics.