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Boletín de la Asociación Matemática Venezolana

Classical analogues of quantum paradoxes

Henryk Gzyl

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

In this note we discuss a few simple classical (as opposed to quantum) prediction problems. The thrust of this work is to examine the predictive role of probability theory and we shall see that some situations are not re-ally paradoxical. In particular we want to separate the role of probability as a predictive tool from one of the basic sources of strangeness in quan-tum mechanics, namely the principle of superposition of states. For that we shall re-examine some of the quantum paradoxes, recast as a problem about making a prediction about the result of a random experiment in classical physics when some information is known. 1

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In this note we discuss a few simple classical (as opposed to quantum) prediction problems. The thrust of this work is to examine the predictive role of probability theory and we shall see that some situations are not re-ally paradoxical. In particular we want to separate the role of probability as a predictive tool from one of the basic sources of strangeness in quan-tum mechanics, namely the principle of superposition of states. For that we shall re-examine some of the quantum paradoxes, recast as a problem about making a prediction about the result of a random experiment in classical physics when some information is known. 1
In classical probability theory, the best predictor of a future observation of a random variable X,X,X, is its expected value EP[X]E_P[X]EP​[X] when no other information is available When information consisting in the observation of another random variable YYY is available, then the best predictor of XXX is another random variable EP[X∣Y].E_P[X|Y].EP​[X∣Y]. It is the purpose of this note to explore the analogue of this in the case of quantum mechanics. We shall see that exactly as in classical prediction theory, when the result of an observation is taken into account by means of a non-commutative conditional expectation, some of the usual paradoxes cease to be such.
Looking at conditional probability from the right perspective allows one to avoid classic paradoxical aspects of quantum mechanics. We discuss, as an elementary analogue of quan-tum states, a classical example (an urn of unknown composition) with similar (putative) paradoxical situ-ations, despite the fact that there is no quantum effect to which they could be ascribed. A crucial feature is that we look at them from an epis-temic point of view and not from the viewpoint of a (seemingly) ontologi-cal reality.
In classical probability theory, the best predictor of a future observation of a random variable X,X,X, is its expected value EP[X]E_P[X]EP​[X] when no other information is available, whereas if information consisting in the observation of another random variable is available, then the best predictor is is another random variable EP[X∣Y].E_P[X|Y].EP​[X∣Y]. It is the purpose of this note to explore the analogue of this in the case of quantum mechanics. we shall see that when the result of an observation is taken into account by means of a non-commutative conditional expectation, exactly as in classical prediction theory, some of the usual paradoxes cease to be so. The moral of this note is that the mystery of the probabilistic interpretation of quantum mechanics lies in the superposition principle
This book deals with applications of quantum mechanical techniques to areas outside of quantum mechanics, so-called quantum-like modeling. Research in this area has grown over the last 15 years. But even already more than 50 years ago, the interaction between Physics Nobelist Pauli and the psychologist Carl Jung in the 1950's on seeking to find analogous uses of the complementarity principle from quantum mechanics in psychology needs noting. This book does NOT want to advance that society is quantum mechanical! The macroscopic world is manifestly not quantum mechanical. But this rules not out that one can use concepts and the mathematical apparatus from quantum physics in a macroscopic environment. A mainstay ingredient of quantum mechanics, is 'quantum probability' and this tool has been proven to be useful in the mathematical modelling of decision making. In the most basic experiment of quantum physics, the double slit experiment, it is known (from the works of A. Khrennikov) that the law of total probability is violated. It is now well documented that several decision making paradoxes in psychology and economics (such as the Ellsberg paradox) do exhibit this violation of the law of total probability. When data is collected with experiments which test 'non-rational' decision making behaviour, one can observe that such data often exhibits a complex non-commutative structure, which may be even more complex than if one considers the structure allied to the basic two slit experiment. The community exploring quantum-like models has tried to address how quantum probability can help in better explaining those paradoxes. Research has now been published in very high standing journals on resolving some of the paradoxes with the mathematics of quantum physics. The aim of this book is to collect the contributions of world's leading experts in quantum like modeling in decision making, psychology, cognition, economics, and finance.
Ever since the beginning of quantum mechanics, the measurement problem has been a subject of sometimes discontinued but nevertheless recurrent concern. As students, most physicists have raised questions about it, but the spectacular and renewed successes of quantum theory were a strong incentive for them to work out applications without further worrying about foundations. In each generation, however, some dissatisfaction reemerges, which leads to slow progress, and we may expect that the measurement problem will some day be cleared up rather than forgotten. The situation recalls statistical mechanics, the foundations of which still give rise to scientific discussions, in spite of more than one century’s achievements. Actually, the paradoxes of irreversibility and of Maxwell’s demon share some common features with quantum measurements, to wit, they involve probabilities in a crucial fashion and their explanation requires elucidating the role of the observer.