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Gleason's Theorem and Its Applications

2013-06-29 Mathematics
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Release : 2013-06-29
Genre : Mathematics
Kind : eBook
Book Rating : 22X/5 ( reviews)

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Book Synopsis Gleason's Theorem and Its Applications by : Anatolij Dvurecenskij

Download or read book Gleason's Theorem and Its Applications written by Anatolij Dvurecenskij. This book was released on 2013-06-29. Available in PDF, EPUB and Kindle. Book excerpt: For many years physics and mathematics have had a fruitful influence on one another. Classical mechanics and celestial mechanics have produced very deep problems whose solutions have enhanced mathematics. On the other hand, mathematics itself has found interesting theories which then (sometimes after many years) have been reflected in physics, confirming the thesis that nothing is more practical than a good theory. The same is true for the younger physical discipline -of quantum mechanics. In the 1930s two events, not at all random, became: The mathematical back grounds of both quantum mechanics and probability theory. In 1936, G. Birkhoff and J. von Neumann published their historical paper "The logic of quantum mechanics", in which a quantum logic was suggested. The mathematical foundations of quantum mechanics remains an outstanding problem of mathematics, physics, logic and philosophy even today. The theory of quantum logics is a major stream in this axiomatical knowledge river, where L(H), the system of all closed subspaces of a Hilbert space H, due to J. von Neumann, plays an important role. When A.M. Gleason published his solution to G. Mackey's problem showing that any state (= probability measure) corresponds to a density operator, he probably did not anticipate that his solution would become a cornerstone of ax iomati cal theory of quantum mechanics nor that it would provide many interesting applications to mathematics.

Quantum Measure Theory

2013-03-14 Mathematics
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Release : 2013-03-14
Genre : Mathematics
Kind : eBook
Book Rating : 199/5 ( reviews)

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Book Synopsis Quantum Measure Theory by : J. Hamhalter

Download or read book Quantum Measure Theory written by J. Hamhalter. This book was released on 2013-03-14. Available in PDF, EPUB and Kindle. Book excerpt: This book is the first systematic treatment of measures on projection lattices of von Neumann algebras. It presents significant recent results in this field. One part is inspired by the Generalized Gleason Theorem on extending measures on the projection lattices of von Neumann algebras to linear functionals. Applications of this principle to various problems in quantum physics are considered (hidden variable problem, Wigner type theorems, decoherence functional, etc.). Another part of the monograph deals with a fascinating interplay of algebraic properties of the projection lattice with the continuity of measures (the analysis of Jauch-Piron states, independence conditions in quantum field theory, etc.). These results have no direct analogy in the standard measure and probability theory. On the theoretical physics side, they are instrumental in recovering technical assumptions of the axiomatics of quantum theories only by considering algebraic properties of finitely additive measures (states) on quantum propositions.

Mathematical Methods Of Theoretical Physics

2020-02-24 Science
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Release : 2020-02-24
Genre : Science
Kind : eBook
Book Rating : 425/5 ( reviews)

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Book Synopsis Mathematical Methods Of Theoretical Physics by : Karl Svozil

Download or read book Mathematical Methods Of Theoretical Physics written by Karl Svozil. This book was released on 2020-02-24. Available in PDF, EPUB and Kindle. Book excerpt: 'This book could serve either as a good reference to remind students about what they have seen in their completed courses or as a starting point to show what needs more investigation. Svozil (Vienna Univ. of Technology) offers a very thorough text that leaves no mathematical area out, but it is best described as giving a synopsis of each application and how it relates to other areas … The text is organized well and provides a good reference list. Summing Up: Recommended. Upper-division undergraduates and graduate students.'CHOICEThis book contains very explicit proofs and demonstrations through examples for a comprehensive introduction to the mathematical methods of theoretical physics. It also combines and unifies many expositions of this subject, suitable for readers with interest in experimental and applied physics.

An Introduction to Hilbert Space and Quantum Logic

2012-12-06 Science
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Release : 2012-12-06
Genre : Science
Kind : eBook
Book Rating : 414/5 ( reviews)

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Book Synopsis An Introduction to Hilbert Space and Quantum Logic by : David W. Cohen

Download or read book An Introduction to Hilbert Space and Quantum Logic written by David W. Cohen. This book was released on 2012-12-06. Available in PDF, EPUB and Kindle. Book excerpt: Historically, nonclassical physics developed in three stages. First came a collection of ad hoc assumptions and then a cookbook of equations known as "quantum mechanics". The equations and their philosophical underpinnings were then collected into a model based on the mathematics of Hilbert space. From the Hilbert space model came the abstaction of "quantum logics". This book explores all three stages, but not in historical order. Instead, in an effort to illustrate how physics and abstract mathematics influence each other we hop back and forth between a purely mathematical development of Hilbert space, and a physically motivated definition of a logic, partially linking the two throughout, and then bringing them together at the deepest level in the last two chapters. This book should be accessible to undergraduate and beginning graduate students in both mathematics and physics. The only strict prerequisites are calculus and linear algebra, but the level of mathematical sophistication assumes at least one or two intermediate courses, for example in mathematical analysis or advanced calculus. No background in physics is assumed.

Wigner-Type Theorems for Hilbert Grassmannians

2020-01-16 Mathematics
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Release : 2020-01-16
Genre : Mathematics
Kind : eBook
Book Rating : 397/5 ( reviews)

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Book Synopsis Wigner-Type Theorems for Hilbert Grassmannians by : Mark Pankov

Download or read book Wigner-Type Theorems for Hilbert Grassmannians written by Mark Pankov. This book was released on 2020-01-16. Available in PDF, EPUB and Kindle. Book excerpt: Wigner's theorem is a fundamental part of the mathematical formulation of quantum mechanics. The theorem characterizes unitary and anti-unitary operators as symmetries of quantum mechanical systems, and is a key result when relating preserver problems to quantum mechanics. At the heart of this book is a geometric approach to Wigner-type theorems, unifying both classical and more recent results. Readers are initiated in a wide range of topics from geometric transformations of Grassmannians to lattices of closed subspaces, before moving on to a discussion of applications. An introduction to all the key aspects of the basic theory is included as are plenty of examples, making this book a useful resource for beginning graduate students and non-experts, as well as a helpful reference for specialist researchers.

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