Quantum mechanics was still in its infancy in 1932 when the young John von Neumann, who would go on to become one of the greatest mathematicians of the twentieth century, published Mathematical Foundations of Quantum Mechanics —a revolutionary book that for the first time provided a rigorous mathematical framework for the new science. Robert Beyer's 1955 English translation, which von Neumann reviewed and approved, is cited more frequently today than ever before. But its many treasures and insights were too often obscured by the limitations of the way the text and equations were set on the page. This new edition of this classic work has been completely reset in TeX, making the text and equations far easier to read. The book has also seen the correction of a handful of typographic errors, revision of some sentences for clarity and readability, provision of an index for the first time, and prefatory remarks drawn from the writings of Léon Van Hove and Freeman Dyson have been added. The result brings new life to an essential work in theoretical physics and mathematics.
Quantum mechanics was still in its infancy in 1932 when the young John von Neumann, who would go on to become one of the greatest mathematicians of the twentieth century, published Mathematical Foundations of Quantum Mechanics —a revolutionary book that for the first time provided a rigorous mathematical framework for the new science. Robert Beyer's 1955 English translation, which von Neumann reviewed and approved, is cited more frequently today than ever before. But its many treasures and insights were too often obscured by the limitations of the way the text and equations were set on the page. This new edition of this classic work has been completely reset in TeX, making the text and equations far easier to read. The book has also seen the correction of a handful of typographic errors, revision of some sentences for clarity and readability, provision of an index for the first time, and prefatory remarks drawn from the writings of Léon Van Hove and Freeman Dyson have been added. The result brings new life to an essential work in theoretical physics and mathematics.
Quantum mechanics was still in its infancy in 1932 when the young John von Neumann, who would go on to become one of the greatest mathematicians of the twentieth century, published Mathematical Foundations of Quantum Mechanics--a revolutionary book that for the first time provided a rigorous mathematical framework for the new science. Robert Beyer's 1955 English translation, which von Neumann reviewed and approved, is cited more frequently today than ever before. But its many treasures and insights were too often obscured by the limitations of the way the text and equations were set on the page. In this new edition of this classic work, mathematical physicist Nicholas Wheeler has completely reset the book in TeX, making the text and equations far easier to read. He has also corrected a handful of typographic errors, revised some sentences for clarity and readability, provided an index for the first time, and added prefatory remarks drawn from the writings of Léon Van Hove and Freeman Dyson. The result brings new life to an essential work in theoretical physics and mathematics.
Interest in quantum mechanics is growing in connection with new experiments with quantum particles and a deeper knowledge of nano- and subatomic principles. The practical use of quantum mechanic methods goes in parallel with the constant rethinking of its foundations, its ideological and epistemological role. A rigorous exposition of the mathematical foundations of quantum mechanics in abstract-Hilbert spaces was given in the book of J. Von Neumann in 1933. It is the first and only experiment that has been brought to the end, the presentation of the apparatus of quantum mechanics with the sequence and rigor which usually presented in a purely mathematical theory. Later in 1949 the group of French mathematicians under the pseudonym N. Bourbaki introduced the concept of "mathematical structure" into mathematics. To date, there is no narration of quantum mechanics, where mathematical rigor is combined with the concept of "mathematical structure". In the framework of the approach formulated by the group of N. Burbaki, quantum mechanics is a composite structure consisting of three simple mathematical structures: the Hilbert space of complex-valued vectors, the space of linear self-adjoint operators and the structure of classical mechanics are showed in this article.
Quantum mechanics and the theory of operator algebras have been intertwined since their origin.In the 1930s [20] von Neumann laid the foundations for the theory of (what are now known as) von Neumann algebras with the explicit goal of establishing Heisenberg's matrix mechanics on a rigorous footing (quoting from the preface, in the translation by Beyer: "The object of this book is to present the new quantum mechanics in a unified representation which, so far as it is possible and useful, is mathematically rigorous").Following the initial explorations of Murray and von Neumann, the new theory took on a life of its own, eventually leading to multiple applications unrelated to quantum mechanics, such as to free probability or noncommutative geometry.In his 1976 paper completing the classification of injective von Neumann algebras [6] Connes made a casual
The proposed book consists of two parts and is devoted to the systematic analysis of quantum mechanics and the modern theory of electromagnetic phenomena. Some inaccuracies of the application of mathematics in theoretical physics are discussed. This first part examines in detail the basic concepts of quantum mechanics, its apparatus, methods and applications and demonstrates their groundlessness, and often the internal inconsistency of the most common version of the Copenhagen interpretation of quantum mechanics. The book contains a critical analysis of the experiments underlying this theory. All this together proves that the generally accepted version of quantum mechanics is a temporary construction, and the creation of a new consistent theory of the microcosm is required. The book also discusses some alternative ideas applicable to the microcosm. This book may be useful for students, postgraduates, teachers, scientific and technical workers and anyone interested in the basics of physics. To the English version of the book. This is an English translation of the Russian version of the book. Quotations were also translated from Russian-language publications (therefore, for English-language works they are not a literal original, but they convey the essence of the quotations). However, references to pages or chapters have been corrected for the indicated English-language sources.
. We put forward a new take on the logic of quantum mechanics, following Schrödinger's point of view that it is composition which makes quantum theory what it is, rather than its particular propositional structure due to the existence of superpositions, as proposed by Birkhoff and von Neumann. This gives rise to an intrinsically quantitative kind of logic, which truly deserves the name ‘logic’ in that it also models meaning in natural language, the latter being the origin of logic, that it supports automation, the most prominent practical use of logic, and that it supports probabilistic inference. The physics and the logic of quantum-ish logic. In 1932 John von Neumann formalized Quantum Mechanics in his book “Mathematische Grundlagen der Quantenmechanik”. This was effectively the official birth of the quantum mechanical formalism which until now, some 75 years later, has remained the same. Quantum theory underpins so many things in our daily lives including chemical industry, energy production and information technology, which arguably makes it the most technologically successful theory of physics ever. However, in 1935, merely three years after the birth of his brainchild, von Neumann wrote in a letter to American mathematician Garrett Birkhoff: “I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space no more.” (sic)—for more details see [73]. Soon thereafter they published a paper entitled “The Logic of Quantum Mechanics” [13]. Their ‘quantum logic’ was cast in order-theoretic terms, very much in the spirit of the then reigning algebraic view of logic, with the distributive law being replaced with a weaker (ortho)modular law. This resulted in a research community of quantum logicians [68, 71, 47, 30]. However, despite von Neumann's reputation, and the large body of research that has been produced in the area, one does not find a trace of this activity neither in the mainstream physics, mathematics, nor logic literature. Hence, 75 years later one may want to conclude that this activity was a failure. What went wrong? The mathematics of it. Let us consider the raison d'être for the Hilbert space formalism.