Quantum Logic In Algebraic Approach
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| Author | : Miklós Rédei |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 244 |
| Release | : 2013-03-09 |
| Genre | : Science |
| ISBN | : 9401590265 |
This work has grown out of the lecture notes that were prepared for a series of seminars on some selected topics in quantum logic. The seminars were delivered during the first semester of the 1993/1994 academic year in the Unit for Foundations of Science of the Department of History and Foundations of Mathematics and Science, Faculty of Physics, Utrecht University, The Netherlands, while I was staying in that Unit on a European Community Research Grant, and in the Center for Philosophy of Science, University of Pittsburgh, U. S. A. , where I was staying during the 1994/1995 academic year as a Visiting Fellow on a Fulbright Research Grant, and where I also was supported by the Istvan Szechenyi Scholarship Foundation. The financial support provided by these foundations, by the Center for Philosophy of Science and by the European Community is greatly acknowledged, and I wish to thank D. Dieks, the professor of the Foundations Group in Utrecht and G. Massey, the director of the Center for Philosophy of Science in Pittsburgh for making my stay at the respective institutions possible. I also wish to thank both the members of the Foundations Group in Utrecht, especially D. Dieks, C. Lutz, F. Muller, J. Uffink and P. Vermaas and the participants in the seminars at the Center for Philosophy of Science in Pittsburgh, especially N. Belnap, J. Earman, A. Janis, J. Norton, and J.
| Author | : C.A. Hooker |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 611 |
| Release | : 2012-12-06 |
| Genre | : Science |
| ISBN | : 9401017956 |
The twentieth century has witnessed a striking transformation in the un derstanding of the theories of mathematical physics. There has emerged clearly the idea that physical theories are significantly characterized by their abstract mathematical structure. This is in opposition to the tradi tional opinion that one should look to the specific applications of a theory in order to understand it. One might with reason now espouse the view that to understand the deeper character of a theory one must know its abstract structure and understand the significance of that struc ture, while to understand how a theory might be modified in light of its experimental inadequacies one must be intimately acquainted with how it is applied. Quantum theory itself has gone through a development this century which illustrates strikingly the shifting perspective. From a collection of intuitive physical maneuvers under Bohr, through a formative stage in which the mathematical framework was bifurcated (between Schrödinger and Heisenberg) to an elegant culmination in von Neumann's Hilbert space formulation the elementary theory moved, flanked even at the later stage by the ill-understood formalisms for the relativistic version and for the field-theoretic altemative; after that we have a gradual, but constant, elaboration of all these quantal theories as abstract mathematical struc tures (their point of departure being von Neumann's formalism) until at the present time theoretical work is heavily preoccupied with the manip ulation of purely abstract structures.
| Author | : Miklos Redei |
| Publisher | : Springer |
| Total Pages | : 243 |
| Release | : 2013-01-22 |
| Genre | : Science |
| ISBN | : 9789401590273 |
This work has grown out of the lecture notes that were prepared for a series of seminars on some selected topics in quantum logic. The seminars were delivered during the first semester of the 1993/1994 academic year in the Unit for Foundations of Science of the Department of History and Foundations of Mathematics and Science, Faculty of Physics, Utrecht University, The Netherlands, while I was staying in that Unit on a European Community Research Grant, and in the Center for Philosophy of Science, University of Pittsburgh, U. S. A. , where I was staying during the 1994/1995 academic year as a Visiting Fellow on a Fulbright Research Grant, and where I also was supported by the Istvan Szechenyi Scholarship Foundation. The financial support provided by these foundations, by the Center for Philosophy of Science and by the European Community is greatly acknowledged, and I wish to thank D. Dieks, the professor of the Foundations Group in Utrecht and G. Massey, the director of the Center for Philosophy of Science in Pittsburgh for making my stay at the respective institutions possible. I also wish to thank both the members of the Foundations Group in Utrecht, especially D. Dieks, C. Lutz, F. Muller, J. Uffink and P. Vermaas and the participants in the seminars at the Center for Philosophy of Science in Pittsburgh, especially N. Belnap, J. Earman, A. Janis, J. Norton, and J.
| Author | : C.A. Hooker |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 498 |
| Release | : 1979-05-31 |
| Genre | : Philosophy |
| ISBN | : 9789027707079 |
The twentieth century has witnessed a striking transformation in the understanding of the theories of mathematical physics. There has emerged clearly the idea that physical theories are significantly characterized by their abstract mathematical structure. This is in opposition to the tradi tional opinion that one should look to the specific applications of a theory in orrter to understand it. One might with reason now espouse the view that to understand the deeper character of a theory one must know its abstract structure and understand the significance of that structure, while to understand how a theory might be modified in light of its experimental inadequacies one must be intimately acquainted with how it is applied. Quantum theory itself has gone through a development this century which illustrates strikingly the shifting perspective. From a collection of intuitive physical manoeuvers under Bohr, through a formative stage in which the mathematical framework was bifurcated (between Schrodinger and Heisenberg) to an elegant culmination in von Neumann's Hilbert space formulation, the elementary theory moved, flanked even at this later stage by the ill-understood formalisms for the relativistic version and for the field-theoretic alternative; after that we have a gradual, but constant, elaboration of all these quantal theories as abstract mathematical structures (their point of departure being von Neumann's formalism) until at the present time theoretical work is heavily preoccupied with the manipulation of purely abstract structures.
| Author | : Timothy Scott Thomas |
| Publisher | : |
| Total Pages | : 83 |
| Release | : 1982 |
| Genre | : Quantum theory |
| ISBN | : |
This paper develops some of the work of Foulis, Randall, Aerts, and Piron in the fields of empirical science and quantum logic from an algebraic point of view. More specifically, it begins with three axioms of what is called a 'subtraction algebra, ' and generates various theorems associated with properties which are useful in empirical science. After a foundation is established, it moves on to define the term manual. This term is defined as a dominated, atomic, semi-Boolean algebra which satisfies an additional condition called condition M. Several properties of the manual are discussed, and different types of manuals are given: classical semi-classical and non-classical. The paper defines operational complements, operational perspectively, atoms, events, and tests, before moving on to define a logic, and how it is derived from a manual. Properties of the logic are discussed, including a subtraction operation, a partial order, and an ortho complement. Next, a computer program is presented. Its purpose is to take a finite semi-Boolean algebra and decide if the algebra is a manual. This is followed by a brief non-classical probabilistic discussion, which includes topics such as weights, pure states, and dispersion-free states. Aerts' and Piron's work with properties, states, and questions is briefly discussed before moving on to several examples, some of them arising from navigation problems. The examples include the hook, the square, the Wright Triangle, and the free algebra. Empirical techniques are demonstrated on these examples. The examples comprise the bulk of this paper.
| Author | : C.A. Hooker |
| Publisher | : Springer |
| Total Pages | : 0 |
| Release | : 1975-09-30 |
| Genre | : Science |
| ISBN | : 9789027706133 |
The twentieth century has witnessed a striking transformation in the un derstanding of the theories of mathematical physics. There has emerged clearly the idea that physical theories are significantly characterized by their abstract mathematical structure. This is in opposition to the tradi tional opinion that one should look to the specific applications of a theory in order to understand it. One might with reason now espouse the view that to understand the deeper character of a theory one must know its abstract structure and understand the significance of that struc ture, while to understand how a theory might be modified in light of its experimental inadequacies one must be intimately acquainted with how it is applied. Quantum theory itself has gone through a development this century which illustrates strikingly the shifting perspective. From a collection of intuitive physical maneuvers under Bohr, through a formative stage in which the mathematical framework was bifurcated (between Schrödinger and Heisenberg) to an elegant culmination in von Neumann's Hilbert space formulation the elementary theory moved, flanked even at the later stage by the ill-understood formalisms for the relativistic version and for the field-theoretic altemative; after that we have a gradual, but constant, elaboration of all these quantal theories as abstract mathematical struc tures (their point of departure being von Neumann's formalism) until at the present time theoretical work is heavily preoccupied with the manip ulation of purely abstract structures.
| Author | : Clifford Alan Hooker |
| Publisher | : |
| Total Pages | : 607 |
| Release | : 1975 |
| Genre | : Logic, Symbolic and mathematical |
| ISBN | : 9789401017961 |
| Author | : Peter Mittelstaedt |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 168 |
| Release | : 2012-12-06 |
| Genre | : Science |
| ISBN | : 9400998716 |
In 1936, G. Birkhoff and J. v. Neumann published an article with the title The logic of quantum mechanics'. In this paper, the authors demonstrated that in quantum mechanics the most simple observables which correspond to yes-no propositions about a quantum physical system constitute an algebraic structure, the most important proper ties of which are given by an orthocomplemented and quasimodular lattice Lq. Furthermore, this lattice of quantum mechanical proposi tions has, from a formal point of view, many similarities with a Boolean lattice L8 which is known to be the lattice of classical propositional logic. Therefore, one could conjecture that due to the algebraic structure of quantum mechanical observables a logical calculus Q of quantum mechanical propositions is established, which is slightly different from the calculus L of classical propositional logic but which is applicable to all quantum mechanical propositions (C. F. v. Weizsacker, 1955). This calculus has sometimes been called 'quan tum logic'. However, the statement that propositions about quantum physical systems are governed by the laws of quantum logic, which differ from ordinary classical logic and which are based on the empirically well-established quantum theory, is exposed to two serious objec tions: (a) Logic is a theory which deals with those relationships between various propositions that are valid independent of the content of the respective propositions. Thus, the validity of logical relationships is not restricted to a special type of proposition, e. g. to propositions about classical physical systems.
| Author | : |
| Publisher | : |
| Total Pages | : |
| Release | : 1975 |
| Genre | : |
| ISBN | : |
| Author | : Jennifer Chubb |
| Publisher | : Cambridge University Press |
| Total Pages | : 355 |
| Release | : 2016-02-26 |
| Genre | : Mathematics |
| ISBN | : 1316654060 |
Arising from a special session held at the 2010 North American Annual Meeting of the Association for Symbolic Logic, this volume is an international cross-disciplinary collaboration with contributions from leading experts exploring connections across their respective fields. Themes range from philosophical examination of the foundations of physics and quantum logic, to exploitations of the methods and structures of operator theory, category theory, and knot theory in an effort to gain insight into the fundamental questions in quantum theory and logic. The book will appeal to researchers and students working in related fields, including logicians, mathematicians, computer scientists, and physicists. A brief introduction provides essential background on quantum mechanics and category theory, which, together with a thematic selection of articles, may also serve as the basic material for a graduate course or seminar.
