Forty years after the advent of quantum mechanics the problem of hidden variables, that is, the possibility of imbedding quantum theory into a classical theory, remains a controversial and obscure subject. Whereas to most physicists the possibility of a classical reinterpretation of quantum mechanics remains remote and perhaps irrelevant to current problems, a minority have kept the issue alive throughout this period. (See Freistadt [5] for a review of the problem and a comprehensive bibliography up to 1957.) As far as results are concerned there are on the one hand purported proofs of the non-existence of hidden variables, most notably von Neumann’s proof, and on the other, various attempts to introduce hidden variables such as de Broglie [4] and Bohm [1] and [2]. One of the difficulties in evaluating these contradictory results is that no exact mathematical criterion is given to enable one to judge the degree of success of these proposals.

The Problem of Hidden Variables in Quantum Mechanics

These notes are based on a course of lectures given by Professor Nelson at Princeton during the spring term of 1966. The subject of Brownian motion has long been of interest in mathematical probability. In these lectures, Professor Nelson traces the history of earlier work in Brownian motion, both the mathematical theory, and the natural phenomenon with its physical interpretations. He continues through recent dynamical theories of Brownian motion, and concludes with a discussion of the relevance of these theories to quantum field theory and quantum statistical mechanics.

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Dynamical Theories of Brownian Motion

Classical mechanics was first envisaged by Newton, formed into a powerful tool by Euler, and brought to perfection by Lagrange and Laplace. It has served as the paradigm of science ever since. Even the great revolutions of 19th century phys icsnamely, the FaradayMaxwell electro-magnetic theory and the kinetic t h e o r y w e r e viewed as further support for the complete adequacy of the mechanistic world view. The physicist at the end of the 19th century had a coherent conceptual scheme which, in principle at least, answered all his questions about the world. The only work left to be done was the computing of the next decimal. This consensus began to unravel at the beginning of the 20th century. The work of Planck, Einstein, and Bohr simply could not be made to fit. The series of ad hoc moves by Bohr, Eherenfest, et al., now called the old quantum theory, was viewed by all as, at best, a stopgap. In the period 1925-27 a new synthesis was formed by Heisenberg, Schr6dinger, Dirac and others. This new synthesis was so successful that even today, fifty years later, physicists still teach quantum mechanics as it was formulated by these men. Nevertheless, two foundational tasks remained: that of providing a rigorous mathematical formulation of the theory, and that of providing a systematic comparison with classical mechanics so that the full ramifications of the quantum revolution could be clearly revealed. These tasks are, of course, related, and a possible fringe benefit of the second task might be the pointing of the way 'beyond quantum theory'. These tasks were taken up by von Neumann as a consequence of a seminar on the foundations of quantum mechanics conducted by Hilbert in the fall of 1926. In papers published in 1927 and in his book, The Mathemat ical Foundations of Quantum Mechanics, von Neumann provided the first completely rigorous

The mathematical foundations of quantum mechanics

Although skeptical of the prohibitive power of no-hidden-variables theorems, John Bell was himself responsible for the two most important ones. I describe some recent versions of the lesser known of the two (familiar to experts as the "Kochen-Specker theorem") which have transparently simple proofs. One of the new versions can be converted without additional analysis into a powerful form of the very much better known "Bell's Theorem," thereby clarifying the conceptual link between these two results of Bell.

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Hidden variables and the two theorems of John Bell

From the Publisher:
A New York Times bestseller when it appeared in 1989, Roger Penrose's The Emperor's New Mind was universally hailed as a marvelous survey of modern physics as well as a brilliant reflection on the human mind, offering a new perspective on the scientific landscape and a visionary glimpse of the possible future of science. Now, in Shadows of the Mind, Penrose offers another exhilarating look at modern science as he mounts an even more powerful attack on artificial intelligence. But perhaps more important, in this volume he points the way to a new science, one that may eventually explain the physical basis of the human mind. Penrose contends that some aspects of the human mind lie beyond computation. This is not a religious argument (that the mind is something other than physical) nor is it based on the brain's vast complexity (the weather is immensely complex, says Penrose, but it is still a computable thing, at least in theory). Instead, he provides powerful arguments to support his conclusion that there is something in the conscious activity of the brain that transcends computation - and will find no explanation in terms of present-day science. To illuminate what he believes this "something" might be, and to suggest where a new physics must proceed so that we may understand it, Penrose cuts a wide swathe through modern science, providing penetrating looks at everything from Turing computability and Godel's incompleteness, via Schrodinger's Cat and the Elitzur-Vaidman bomb-testing problem, to detailed microbiology. Of particular interest is Penrose's extensive examination of quantum mechanics, which introduces some new ideas that differ markedly from those advanced in The Emperor's New Mind, especially concerning the mysterious interface where classical and quantum physics meet. But perhaps the most interesting wrinkle in Shadows of the Mind is Penrose's excursion into microbiology, where he examines cytoskeletons and microtubules, minute substructures lying dee

Shadows of the Mind: A Search for the Missing Science of Consciousness

The logical structures studied in this paper are generalizations of the propositional calculus. The classical propositional calculus is essentially Boolean algebra or, alternatively, the theory of functions on an arbitrary set S with values in a two-element set. The generalization consists in allowing partial functions on the set S, i.e., functions defined on certain subsets of S, and defining an equivalence relation among these functions such that any two constant functions with the same constant value belong to the same equivalence class. The generalization is equally natural for functions with values in the field of real numbers and we shall consider this case first.

E. P. Specker

Papers

The Problem of Hidden Variables in Quantum Mechanics

Logical Structures Arising in Quantum Theory

LOGICAL STRUCTURES ARISING IN QUANTUM THEORY**This work was supported in part by a U.S. National Science Foundation grant.

The Problem of Hidden f7ariables Quantum Mechanics . In