New Foundations for Physical Geometry

The Theory of Linear Structures

Author: Tim Maudlin

Publisher: Oxford University Press

ISBN: 0198701306

Category: Mathematics

Page: 363

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Topology is the mathematical study of the most basic geometrical structure of a space. Mathematical physics uses topological spaces as the formal means for describing physical space and time. Tim Maudlin proposes a completely new mathematical structure for describing geometrical notions such as continuity, connectedness, boundaries of sets, and so on, in order to provide a better mathematical tool for understanding space-time. He begins with a brief historicalreview of the development of mathematics as it relates to geometry and an overview of standard topology, and goes on to develop his original Theory of Linear Structures.
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Spooky Action at a Distance

The Phenomenon That Reimagines Space and Time--and What It Means for Black Holes, the Big Bang, and Theories of Everything

Author: George Musser

Publisher: Scientific American / Farrar, Straus and Giroux

ISBN: 0374713553

Category: Science

Page: 304

View: 5988

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Long-listed for the 2016 PEN/E. O. Wilson Literary Science Writing Award “An important book that provides insight into key new developments in our understanding of the nature of space, time and the universe. It will repay careful study.” —John Gribbin, The Wall Street Journal “An endlessly surprising foray into the current mother of physics' many knotty mysteries, the solving of which may unveil the weirdness of quantum particles, black holes, and the essential unity of nature.” —Kirkus Reviews (starred review) What is space? It isn't a question that most of us normally ask. Space is the venue of physics; it's where things exist, where they move and take shape. Yet over the past few decades, physicists have discovered a phenomenon that operates outside the confines of space and time: nonlocality-the ability of two particles to act in harmony no matter how far apart they may be. It appears to be almost magical. Einstein grappled with this oddity and couldn't come to terms with it, describing it as "spooky action at a distance." More recently, the mystery has deepened as other forms of nonlocality have been uncovered. This strange occurrence, which has direct connections to black holes, particle collisions, and even the workings of gravity, holds the potential to undermine our most basic understandings of physical reality. If space isn't what we thought it was, then what is it? In Spooky Action at a Distance, George Musser sets out to answer that question, offering a provocative exploration of nonlocality and a celebration of the scientists who are trying to explain it. Musser guides us on an epic journey into the lives of experimental physicists observing particles acting in tandem, astronomers finding galaxies that look statistically identical, and cosmologists hoping to unravel the paradoxes surrounding the big bang. He traces the often contentious debates over nonlocality through major discoveries and disruptions of the twentieth century and shows how scientists faced with the same undisputed experimental evidence develop wildly different explanations for that evidence. Their conclusions challenge our understanding of not only space and time but also the origins of the universe-and they suggest a new grand unified theory of physics. Delightfully readable, Spooky Action at a Distance is a mind-bending voyage to the frontiers of modern physics that will change the way we think about reality.
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Applying Mathematics

Immersion, Inference, Interpretation

Author: Otávio Bueno,Steven French

Publisher: Oxford University Press

ISBN: 0192558978

Category: Science

Page: 288

View: 8311

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How is that when scientists need some piece of mathematics through which to frame their theory, it is there to hand? What has been called 'the unreasonable effectiveness of mathematics' sets a challenge for philosophers. Some have responded to that challenge by arguing that mathematics is essentially anthropocentric in character, whereas others have pointed to the range of structures that mathematics offers. Otávio Bueno and Steven French offer a middle way, which focuses on the moves that have to be made in both the mathematics and the relevant physics in order to bring the two into appropriate relation. This relation can be captured via the inferential conception of the applicability of mathematics, which is formulated in terms of immersion, inference, and interpretation. In particular, the roles of idealisations and of surplus structure in science and mathematics respectively are brought to the fore and captured via an approach to models and theories that emphasize the partiality of the available information: the partial structures approach. The discussion as a whole is grounded in a number of case studies drawn from the history of quantum physics, and extended to contest recent claims that the explanatory role of certain mathematical structures in scientific practice supports a realist attitude towards them. The overall conclusion is that the effectiveness of mathematics does not seem unreasonable at all once close attention is paid to how it is actually applied in practice.
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Empty Ideas

A Critique of Analytic Philosophy

Author: Peter Unger

Publisher: Oxford University Press

ISBN: 0199330832

Category: Philosophy

Page: 288

View: 2103

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Peter Unger's provocative new book poses a serious challenge to contemporary analytic philosophy, arguing that to its detriment it focuses the predominance of its energy on "empty ideas." In the mid-twentieth century, philosophers generally agreed that, by contrast with science, philosophy should offer no substantial thoughts about the general nature of concrete reality. Leading philosophers were concerned with little more than the semantics of ordinary words. For example: Our word "perceives" differs from our word "believes" in that the first word is used more strictly than the second. While someone may be correct in saying "I believe there's a table before me" whether or not there is a table before her, she will be correct in saying "I perceive there's a table before me" only if there is a table there. Though just a parochial idea, whether or not it is correct does make a difference to how things are with concrete reality. In Unger's terms, it is a concretely substantial idea. Alongside each such parochial substantial idea, there is an analytic or conceptual thought, as with the thought that someone may believe there is a table before her whether or not there is one, but she will perceive there is a table before her only if there is a table there. Empty of import as to how things are with concrete reality, those thoughts are what Unger calls concretely empty ideas. It is widely assumed that, since about 1970, things had changed thanks to the advent of such thoughts as the content externalism championed by Hilary Putnam and Donald Davidson, various essentialist thoughts offered by Saul Kripke, and so on. Against that assumption, Unger argues that, with hardly any exceptions aside from David Lewis's theory of a plurality of concrete worlds, all of these recent offerings are concretely empty ideas. Except when offering parochial ideas, Peter Unger maintains that mainstream philosophy still offers hardly anything beyond concretely empty ideas.
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Relativity and Geometry

Author: Roberto Torretti

Publisher: Courier Corporation

ISBN: 0486690466

Category: Science

Page: 395

View: 3990

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Early in this century, it was shown that the new non-Newtonian physics -- known as Einstein's Special Theory of Relativity -- rested on a new, non-Euclidean geometry, which incorporated time and space into a unified "chronogeometric" structure. This high-level study elucidates the motivation and significance of the changes in physical geometry brought about by Einstein, in both the first and the second phase of Relativity. After a discussion of Newtonian principles and 19th-century views on electrodynamics and the aether, the author offers illuminating expositions of Einstein's electrodynamics of moving bodies, Minkowski spacetime, Einstein's quest for a theory of gravity, gravitational geometry, the concept of simultaneity, time and causality and other topics. An important Appendix -- designed to define spacetime curvature -- considers differentiable manifolds, fiber bundles, linear connections and useful formulae. Relativity continues to be a major focus of interest for physicists, mathematicians and philosophers of science. This highly regarded work offers them a rich, "historico-critical" exposition -- emphasizing geometrical ideas -- of the elements of the Special and General Theory of Relativity.
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Non-Linear Theory of Elasticity and Optimal Design

Author: L.W. Ratner

Publisher: Elsevier

ISBN: 9780080537603

Category: Science

Page: 279

View: 3128

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In order to select an optimal structure among possible similar structures, one needs to compare the elastic behavior of the structures. A new criterion that describes elastic behavior is the rate of change of deformation. Using this criterion, the safe dimensions of a structure that are required by the stress distributed in a structure can be calculated. The new non-linear theory of elasticity allows one to determine the actual individual limit of elasticity/failure of a structure using a simple non-destructive method of measurement of deformation on the model of a structure while presently it can be done only with a destructive test for each structure. For building and explaining the theory, a new logical structure was introduced as the basis of the theory. One of the important physical implications of this logic is that it describes mathematically the universal domain of the possible stable physical relations.
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The mathematical foundations of gauge theories

Author: Kishore B. Marathe,Giovanni Martucci

Publisher: North Holland

ISBN: N.A

Category: Science

Page: 372

View: 3753

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Theoretical physicists tend to discuss their theories in the language of mathematics. However, the adequate mathematical formulation may not yet be available when the physical law is first discovered. Mathematical physicists trying to develop the relevant mathematics for these theories, may obtain new insights into old mathematical structures or may even disclose entirely new structures. Gauge Theory is such a gift from physics to mathematics. This volume presents a self-contained development of a differential geometric formulation of gauge theories, in particular, the theory of Yang-Mills fields. The book is aimed at advanced graduate students and research workers in theoretical physics and pure and applied mathematics who are acquainted with the elements of the theory of differential manifolds. It enables the reader to apply this theory to gauge theories and to understand the role of gauge theories in high energy physics, gravitation theory and electromagnetism.
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Geometric Algebra for Physicists

Author: Chris Doran,Anthony Lasenby

Publisher: Cambridge University Press

ISBN: 1139643142

Category: Science

Page: 578

View: 9242

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Geometric algebra is a powerful mathematical language with applications across a range of subjects in physics and engineering. This book is a complete guide to the current state of the subject with early chapters providing a self-contained introduction to geometric algebra. Topics covered include new techniques for handling rotations in arbitrary dimensions, and the links between rotations, bivectors and the structure of the Lie groups. Following chapters extend the concept of a complex analytic function theory to arbitrary dimensions, with applications in quantum theory and electromagnetism. Later chapters cover advanced topics such as non-Euclidean geometry, quantum entanglement, and gauge theories. Applications such as black holes and cosmic strings are also explored. It can be used as a graduate text for courses on the physical applications of geometric algebra and is also suitable for researchers working in the fields of relativity and quantum theory.
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Mathematical Foundations of Quantum Theory

Author: A.R. Marlow

Publisher: Elsevier

ISBN: 0323141188

Category: Science

Page: 382

View: 5815

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Mathematical Foundations of Quantum Theory is a collection of papers presented at the 1977 conference on the Mathematical Foundations of Quantum Theory, held in New Orleans. The contributors present their topics from a wide variety of backgrounds and specialization, but all shared a common interest in answering quantum issues. Organized into 20 chapters, this book's opening chapters establish a sound mathematical basis for quantum theory and a mode of observation in the double slit experiment. This book then describes the Lorentz particle system and other mathematical structures with which fundamental quantum theory must deal, and then some unsolved problems in the quantum logic approach to the foundations of quantum mechanics are considered. Considerable chapters cover topics on manuals and logics for quantum mechanics. This book also examines the problems in quantum logic, and then presents examples of their interpretation and relevance to nonclassical logic and statistics. The accommodation of conventional Fermi-Dirac and Bose-Einstein statistics in quantum mechanics or quantum field theory is illustrated. The final chapters of the book present a system of axioms for nonrelativistic quantum mechanics, with particular emphasis on the role of density operators as states. Specific connections of this theory with other formulations of quantum theory are also considered. These chapters also deal with the determination of the state of an elementary quantum mechanical system by the associated position and momentum distribution. This book is of value to physicists, mathematicians, and researchers who are interested in quantum theory.
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Mathematical Foundations of Quantum Mechanics

Author: John Von Neumann

Publisher: Princeton University Press

ISBN: 9780691028934

Category: Mathematics

Page: 445

View: 2164

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This text shows that insights in quantum physics can be obtained by exploring the mathematical structure of quantum mechanics. It presents the theory of Hermitean operators and Hilbert spaces, providing the framework for transformation theory, and using th
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