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Journal of Stochastic Analysis

Emergence of Quantum Theories from Classical Probability: Historical Origins, Developments, and Open Problems

Luigi Accardi, Yun Gang Lu

Published Dec 6, 2020• 0 citations
Publisher

Abstract

After briefly mentioning some achievements of quantum probability during the past 20 years, we concentrate our attention on the emergence of natural extensions of usual quantum theory from the combination of classical probability with the theory of orthogonal polynomials and we discuss some implications, both for mathematics and physics of this fact.

DOI
10.31390/josa.1.4.09

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After briefly mentioning some achievements of quantum probability during the past 20 years, we concentrate our attention on the emergence of natural extensions of usual quantum theory from the combination of classical probability with the theory of orthogonal polynomials and we discuss some implications, both for mathematics and physics of this fact.
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.
We survey some results obtained in the last 25 years to illustrate the following probabilistic quantization statement: quantum probability is not a generalization of classical probability but a deeper level of it. Generalizing the classical theory of orthogonal polynomials, one can show that any random field XXX admitting all the moments can be represented as the sum of three operators, which are natural extensions of the creation, annihilation, and preservation operators in the usual boson Fock quantum theory. These operators generate a noncommutative ∗*∗-algebra on which the quantum extension of the expected value relative to the probability distribution of the field XXX induces a quantum state. So, any classical algebraic probability space generates a quantum space with its own commutation relations. The Heisenberg commutation relation characterizes the classical fields, while the new type of commutation relations (of type II) appears in non-Gaussian cases. The same machinery, but applied to Bernoulli fields, leads to Fermi-Dirac anticommutation relations (see the introduction).
We survey some results obtained in the last 25 years to illustrate the following probabilistic quantization statement: quantum probability is not a generalization of classical probability but a deeper level of it. Generalizing the classical theory of orthogonal polynomials, one can show that any random field XXX admitting all the moments can be represented as the sum of three operators, which are natural extensions of the creation, annihilation, and preservation operators in the usual boson Fock quantum theory. These operators generate a noncommutative ∗*∗-algebra on which the quantum extension of the expected value relative to the probability distribution of the field XXX induces a quantum state. So, any classical algebraic probability space generates a quantum space with its own commutation relations. The Heisenberg commutation relation characterizes the classical fields, while the new type of commutation relations (of type II) appears in non-Gaussian cases. The same machinery, but applied to Bernoulli fields, leads to Fermi--Dirac anticommutation relations (see the introduction).
The report presents an exhaustive review of the recent attempt to overcome the difficulties that standard quantum mechanics meets in accounting for the measurement (or macro-objectification) problem, an attempt based on the consideration of nonlinear and stochastic modifications of the Schroedinger equation. The proposed new dynamics is characterized by the feature of not contradicting any known fact about microsystems and of accounting, on the basis of a unique, universal dynamical principle, for wavepacket reduction and for the classical behavior of macroscopic systems. We recall the motivations for the new approach and we briefly review the other proposals to circumvent the above mentioned difficulties which appeared in the literature. In this way we make clear the conceptual and historical context characterizing the new approach. After having reviewed the mathematical techniques (stochastic differential calculus) which are essential for the rigorous and precise formulation of the new dynamics, we discuss in great detail its implications and we stress its relevant conceptual achievements. The new proposal requires also to work out an appropriate interpretation; a procedure which leads us to a reconsideration of many important issues about the conceptual status of theories based on a genuinely Hilbert space description of natural processes. We also discuss the possibility and the problems one meets in trying to develop an analogous formalism for the relativistic case. Finally we discuss the experimental implications of the new dynamics for various physical processes which should allow, in principle, to test it against quantum mechanics.
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.