Nonstandard extension of quantum logic and Dirac's bra-ket formalism of quantum mechanics

1994 ◽  
Vol 33 (2) ◽  
pp. 307-338 ◽  
Author(s):  
Arye Friedman
Keyword(s):  
Quantum Mechanics ◽  
Quantum Logic

A THEOREM OF SOLÉR, THE THEORY OF SYMMETRY AND QUANTUM MECHANICS

2012 ◽  
Vol 09 (02) ◽  
pp. 1260005 ◽  
Author(s):  
GIANNI CASSINELLI ◽  
PEKKA LAHTI
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Logic ◽  
Hilbert Spaces ◽  
Classical Problem ◽  
Partial Solution ◽  
Space Realization ◽  
Symmetry Transformations ◽  
Axiomatic Quantum Mechanics

A classical problem in axiomatic quantum mechanics is deducing a Hilbert space realization for a quantum logic that admits a vector space coordinatization of the Piron–McLaren type. Our aim is to show how a theorem of M. Solér [Characterization of Hilbert spaces by orthomodular spaces, Comm. Algebra23 (1995) 219–243.] can be used to get a (partial) solution of this problem. We first derive a generalization of the Wigner theorem on symmetry transformations that holds already in the Piron–McLaren frame. Then we investigate which conditions on the quantum logic allow the use of Solér's theorem in order to obtain a Hilbert space solution for the coordinatization problem.

scholarly journals Classical Logic in the Quantum Context

2020 ◽  
Vol 2 (4) ◽  
pp. 600-616
Author(s):  
Andrea Oldofredi
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Quantum Logic ◽  
Classical Logic ◽  
Theory Of Measurement ◽  
Propositional Calculus ◽  
Bohmian Mechanics ◽  
Physical Content ◽  
Present Essay ◽  
Classical Propositional Calculus

It is generally accepted that quantum mechanics entails a revision of the classical propositional calculus as a consequence of its physical content. However, the universal claim according to which a new quantum logic is indispensable in order to model the propositions of every quantum theory is challenged. In the present essay, we critically discuss this claim by showing that classical logic can be rehabilitated in a quantum context by taking into account Bohmian mechanics. It will be argued, indeed, that such a theoretical framework provides the necessary conceptual tools to reintroduce a classical logic of experimental propositions by virtue of its clear metaphysical picture and its theory of measurement. More precisely, it will be shown that the rehabilitation of a classical propositional calculus is a consequence of the primitive ontology of the theory, a fact that is not yet sufficiently recognized in the literature concerning Bohmian mechanics. This work aims to fill this gap.

Quantum logic and operational quantum mechanics

1984 ◽  
Vol 19 (3) ◽  
pp. 383-406 ◽  
Author(s):  
Maria Cristina Abbati ◽  
Alessandro Manià
Keyword(s):  
Quantum Mechanics ◽  
Quantum Logic

Simon Kochen and E. P. Specker. Logical structures arising in quantum theory. A reprint of XL 507(21). The logieo-algebraic approach to quantum mechanics, Volume I, Historicale evolution, edited by C. A. Hooker, The University of Western Ontario series in philosophy of science, vol. 5, D. Reidel Publishing Company, Dordrecht and Boston1975, pp. 263–276. - Simon Kochen and E. P. Specker. The calculus of partial propositional functions. A reprint of XL 508(20). The logieo-algebraic approach to quantum mechanics, Volume I, Historical evolution, edited by C. A. Hooker, The University of Western Ontario series in philosophy of science, vol. 5, D. Reidel Publishing Company, Dordrecht and Boston1975, pp. 277–292. - P.D. Finch. On the structure of quantum logic. The logieo-algebraic approach to quantum mechanics, Volume I, Historical evolution, edited by C. A. Hooker, The University of Western Ontario series in philosophy of science, vol. 5, D. Reidel Publishing Company, Dordrecht and Boston1975, pp. 415–425. (Reprinted from The journal of symbolic logic, vol. 34 (1969), pp. 275–282.) - Stanley P. Gudder. Partial algebraic structures associated with orthomodular posets. Pacific journal of mathematics, vol. 41 (1972), pp. 717–730. - Janusz Czelakowski. Logics based on partial Boolean σ-algebras (1). Studia logica, vol. 33 (1974), pp. 371–396. - Gary M. Hardegree and Patricia J. Frazer. Charting the labyrinth of quantum logics: a progress report. Current issues in quantum logic, edited by Enrico G. Beltrametti and Bas C. van Fraassen, Ettore Majorana international science series, physical sciences, vol. 8, Plenum Press, New York and London1981, pp. 53–76. - Janusz Czelakowski. Partial referential matrices for quantum logics. Current issues in quantum logic, edited by Enrico G. Beltrametti and Bas C. van Fraassen, Ettore Majorana international science series, physical sciences, vol. 8, Plenum Press, New York and London1981, pp. 131–146.

1985 ◽  
Vol 50 (2) ◽  
pp. 558-566 ◽  
Author(s):  
R. I. G. Hughes
Keyword(s):  
New York ◽  
Quantum Mechanics ◽  
Philosophy Of Science ◽  
Quantum Logic ◽  
Algebraic Approach ◽  
Publishing Company ◽  
Reidel Publishing Company ◽  
Quantum Logics ◽  
Ettore Majorana ◽  
The University

scholarly journals Transfer principle in quantum set theory

2007 ◽  
Vol 72 (2) ◽  
pp. 625-648 ◽  
Author(s):  
Masanao Ozawa
Keyword(s):  
Quantum Mechanics ◽  
Quantum Logic ◽  
Set Theory ◽  
Von Neumann Algebra ◽  
Transfer Principle ◽  
Real Numbers ◽  
Von Neumann ◽  
Equality Relation ◽  
Adjoint Operators ◽  
Transfer Theorems

AbstractIn 1981, Takeuti introduced quantum set theory as the quantum counterpart of Boolean valued models of set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed subspaces in a Hilbert space and showed that appropriate quantum counterparts of ZFC axioms hold in the model. Here, Takeuti's formulation is extended to construct a model of set theory based on the logic represented by the lattice of projections in an arbitrary von Neumann algebra. A transfer principle is established that enables us to transfer theorems of ZFC to their quantum counterparts holding in the model. The set of real numbers in the model is shown to be in one-to-one correspondence with the set of self-adjoint operators affiliated with the von Neumann algebra generated by the logic. Despite the difficulty pointed out by Takeuti that equality axioms do not generally hold in quantum set theory, it is shown that equality axioms hold for any real numbers in the model. It is also shown that any observational proposition in quantum mechanics can be represented by a corresponding statement for real numbers in the model with the truth value consistent with the standard formulation of quantum mechanics, and that the equality relation between two real numbers in the model is equivalent with the notion of perfect correlation between corresponding observables (self-adjoint operators) in quantum mechanics. The paper is concluded with some remarks on the relevance to quantum set theory of the choice of the implication connective in quantum logic.

Popper and the Quantum Theory

1995 ◽  
Vol 39 ◽  
pp. 163-176
Author(s):  
Michael Redhead
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Quantum Logic ◽  
Scientific Discovery ◽  
Von Neumann ◽  
Interpretation Of Probability ◽  
Propensity Interpretation

Popper wrote extensively on the quantum theory. In Logic der Forschung (LSD) he devoted a whole chapter to the topic, while the whole of Volume 3 of the Postscript to the Logic of Scientific Discovery is devoted to the quantum theory. This volume entitled Quantum Theory and the Schism in Physics (QTSP) incorporated a famous earlier essay, ‘Quantum Mechanics without “the Observer”’ (QM). In addition Popper's development of the propensity interpretation of probability was much influenced by his views on the role of probability in quantum theory, and he also wrote an insightful critique of the 1936 paper of Birkhoff and von Neumann on nondistributive quantum logic (BNIQM).

The center of an orthologic

1972 ◽  
Vol 37 (4) ◽  
pp. 641-645 ◽  
Author(s):  
Barbara Jeffcott
Keyword(s):  
Quantum Mechanics ◽  
Boolean Algebra ◽  
Quantum Logic ◽  
Decisive Role ◽  
Orthomodular Poset ◽  
Fundamental Relation ◽  
Von Neumann ◽  
Modular Ortholattice ◽  
Probability And Statistics ◽  
Kolmogorov Theory

Since 1933, when Kolmogorov laid the foundations for probability and statistics as we know them today [1], it has been recognized that propositions asserting that such and such an event occurred as a consequence of the execution of a particular random experiment tend to band together and form a Boolean algebra. In 1936, Birkhoff and von Neumann [2] suggested that the so-called logic of quantum mechanics should not be a Boolean algebra, but rather should form what is now called a modular ortholattice [3]. Presumably, the departure from Boolean algebras encountered in quantum mechanics can be attributed to the fact that in quantum mechanics, one must consider more than one physical experiment, e.g., an experiment measuring position, an experiment measuring charge, an experiment measuring momentum, etc., and, because of the uncertainty principle, these experiments need not admit a common refinement in terms of which the Kolmogorov theory is directly applicable.Mackey's Axioms I–VI for quantum mechanics [4] imply that the logic of quantum mechanics should be a σ-orthocomplete orthomodular poset [5]. Most contemporary practitioners of quantum logic seem to agree that a quantum logic is (at least) an orthomodular poset [6], [7], [8], [9], [10] or some variation thereof [11]. P. D. Finch [12] has shown that every completely orthomodular poset is the logic arising from sets of Boolean logics, where these sets have a structure similar to the structures generally given to quantum logic. In all of these versions of quantum logic, a fundamental relation, the relation of compatibility or commutativity, plays a decisive role.

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