Mathematical Foundations of Quantum Mechanics
01 Jan 1996-
TL;DR: The Mathematical Foundations of Quantum Mechanics as discussed by the authors is a seminal work in theoretical physics that introduced the theory of Hermitean operators and Hilbert spaces and provided a mathematical framework for quantum mechanics.
Abstract: Mathematical Foundations of Quantum Mechanics was a revolutionary book that caused a sea change in theoretical physics. Here, John von Neumann, one of the leading mathematicians of the twentieth century, shows that great insights in quantum physics can be obtained by exploring the mathematical structure of quantum mechanics. He begins by presenting the theory of Hermitean operators and Hilbert spaces. These provide the framework for transformation theory, which von Neumann regards as the definitive form of quantum mechanics. Using this theory, he attacks with mathematical rigor some of the general problems of quantum theory, such as quantum statistical mechanics as well as measurement processes. Regarded as a tour de force at the time of publication, this book is still indispensable for those interested in the fundamental issues of quantum mechanics.
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TL;DR: The problem of hidden variables in quantum theory has been a controversial and obscure subject for decades as mentioned in this paper, and there are many proofs of the non-existence of such variables, most notably von Neumann's proof, and various attempts to introduce hidden variables such as de Broglie [4] and Bohm [1] and [2].
Abstract: 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.
2,007 citations
Book•
21 Feb 1967
TL;DR: In a course of lectures given by Professor Nelson at Princeton during the spring term of 1966, the authors traces the history of earlier work in Brownian motion, both the mathematical theory, and the natural phenomenon with its physical interpretations.
Abstract: 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.
1,517 citations
TL;DR: In a Hilbert space H, discrete families of vectors {hj} with the property that f = ∑j〈hj ǫ à à hj à f à for every f in H are considered.
Abstract: In a Hilbert space H, discrete families of vectors {hj} with the property that f=∑j〈hj‖ f〉hj for every f in H are considered. This expansion formula is obviously true if the family is an orthonormal basis of H, but also can hold in situations where the hj are not mutually orthogonal and are ‘‘overcomplete.’’ The two classes of examples studied here are (i) appropriate sets of Weyl–Heisenberg coherent states, based on certain (non‐Gaussian) fiducial vectors, and (ii) analogous families of affine coherent states. It is believed, that such ‘‘quasiorthogonal expansions’’ will be a useful tool in many areas of theoretical physics and applied mathematics.
1,508 citations
TL;DR: Bell's theorem represents a significant advance in understanding the conceptual foundations of quantum mechanics as mentioned in this paper, showing that essentially all local theories of natural phenomena that are formulated within the framework of realism may be tested using a single experimental arrangement.
Abstract: Bell's theorem represents a significant advance in understanding the conceptual foundations of quantum mechanics. The theorem shows that essentially all local theories of natural phenomena that are formulated within the framework of realism may be tested using a single experimental arrangement. Moreover, the predictions by those theories must significantly differ from those by quantum mechanics. Experimental results evidently refute the theorem's predictions for these theories and favour those of quantum mechanics. The conclusions are philosophically startling: either one must totally abandon the realistic philosophy of most working scientists, or dramatically revise out concept of space-time.
1,285 citations
31 Jan 2015
TL;DR: This volume contains Dr. Everett's short paper from 1957, "'Relative State' Formulation of Quantum Mechanics," and a far longer exposition of his interpretation, entitled "The Theory of the Universal Wave Function," never before published.
Abstract: A novel interpretation of quantum mechanics, first proposed in brief form by Hugh Everett in 1957, forms the nucleus around which this book has developed In his interpretation, Dr Everett denies the existence of a separate classical realm and asserts the propriety of considering a state vector for the whole universe Because this state vector never collapses, reality as a whole is rigorously deterministic This reality, which is described jointly by the dynamical variables and the state vector, is not the reality customarily perceived; rather, it is a reality composed of many worlds By virtue of the temporal development of the dynamical variables, the state vector decomposes naturally into orthogonal vectors, reflecting a continual splitting of the universe into a multitude of mutually unobservable but equally real worlds, in each of which every good measurement has yielded a definite result, and in most of which the familiar statistical quantum laws hold The volume contains Dr Everett's short paper from 1957, "'Relative State' Formulation of Quantum Mechanics," and a far longer exposition of his interpretation, entitled "The Theory of the Universal Wave Function," never before published In addition, other papers by Wheeler, DeWitt, Graham, and Cooper and Van Vechten provide further discussion of the same theme Together, they constitute virtually the entire world output of scholarly commentary on the Everett interpretation Originally published in 1973 The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905
1,198 citations