On the isomorphism of a quantum logic with the logic of the projections in a Hilbert space. II

1974 ◽  
Vol 11 (2) ◽  
pp. 135-144 ◽  
Author(s):  
R. Cirelli ◽  
P. Cotta-Ramusino ◽  
E. Novati
Keyword(s):  
Hilbert Space ◽  
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.

Efficient quantum logic circuits: or, How I Learned to Stop Worrying and Love Hilbert Space

2007 ◽  
Author(s):  
A. G. White ◽  
M. Pereira de Almeida ◽  
M. Barbieri ◽  
D. N. Biggerstaff ◽  
R. B. Dalton ◽  
...  
Keyword(s):  
Hilbert Space ◽  
Quantum Logic ◽  
Logic Circuits

An Introduction to Hilbert Space and Quantum Logic

1990 ◽  
Vol 74 (470) ◽  
pp. 418
Author(s):  
R. L. Hudson ◽  
D. W. Cohen
Keyword(s):  
Hilbert Space ◽  
Quantum Logic

Abacus logic: The lattice of quantum propositions as the poset of a theory

1994 ◽  
Vol 59 (2) ◽  
pp. 501-515
Author(s):  
Othman Qasim Malhas
Keyword(s):  
Hilbert Space ◽  
Quantum Logic ◽  
Hidden Variables ◽  
Propositional Calculus ◽  
Real Numbers ◽  
Quantum Observable ◽  
Quantum Observables ◽  
Classical Propositional Calculus ◽  
Classical Observable ◽  
Graphic Interpretation

AbstractWith a certain graphic interpretation in mind, we say that a function whose value at every point in its domain is a nonempty set of real numbers is an Abacus. It is shown that to every collection C of abaci there corresponds a logic, called an abacus logic, i.e.. a certain set of propositions partially ordered by generalized implication. It is also shown that to every collection C of abaci there corresponds a theory Jc in a classical propositional calculus such that the abacus logic determined by C is isomorphic to the poset of Jc. Two examples are given. In both examples abacus logic is a lattice in which there happens to be an operation of orthocomplementation. In the first example abacus logic turns out to be the Lindenbaum algebra of Jc. In the second example abacus logic is a lattice isomorphic to the ortholattice of subspaces of a Hilbert space. Thus quantum logic can be regarded as an abacus logic. Without suggesting “hidden variables” it is finally shown that the Lindenbaum algebra of the theory in the second example is a subalgebra of the abacus logic B of the kind studied in example 1. It turns out that the “classical observables” associated with B and the “quantum observables” associated with quantum logic are not unrelated. The value of a classical observable contains, in coded form, information about the “uncertainty” of a quantum observable. This information is retrieved by decoding the value of the corresponding classical observable.

scholarly journals An Application of Quantum Logic to Experimental Behavioral Science

2021 ◽  
Vol 3 (4) ◽  
pp. 643-655
Author(s):  
Louis Narens
Keyword(s):  
Hilbert Space ◽  
Quantum Logic ◽  
Additive Function ◽  
Quantum Physics ◽  
Behavioral Science ◽  
Slight Modification ◽  
Quantum Probability ◽  
Rational Approach ◽  
Dutch Book Argument ◽  
Von Neumann

In 1933, Kolmogorov synthesized the basic concepts of probability that were in general use at the time into concepts and deductions from a simple set of axioms that said probability was a σ-additive function from a boolean algebra of events into [0, 1]. In 1932, von Neumann realized that the use of probability in quantum mechanics required a different concept that he formulated as a σ-additive function from the closed subspaces of a Hilbert space onto [0,1]. In 1935, Birkhoff & von Neumann replaced Hilbert space with an algebraic generalization. Today, a slight modification of the Birkhoff-von Neumann generalization is called “quantum logic”. A central problem in the philosophy of probability is the justification of the definition of probability used in a given application. This is usually done by arguing for the rationality of that approach to the situation under consideration. A version of the Dutch book argument given by de Finetti in 1972 is often used to justify the Kolmogorov theory, especially in scientific applications. As von Neumann in 1955 noted, and his criticisms still hold, there is no acceptable foundation for quantum logic. While it is not argued here that a rational approach has been carried out for quantum physics, it is argued that (1) for many important situations found in behavioral science that quantum probability theory is a reasonable choice, and (2) that it has an arguably rational foundation to certain areas of behavioral science, for example, the behavioral paradigm of Between Subjects experiments.

Strong Projections in Hilbert Space and Quantum Logic

2019 ◽  
Vol 40 (10) ◽  
pp. 1521-1531
Author(s):  
M. S. Matvejchuk ◽  
E. V. Vladova
Keyword(s):  
Hilbert Space ◽  
Quantum Logic

Jauch–Piron states on quantum logics

2019 ◽  
Vol 19 (01) ◽  
pp. 2050017
Author(s):  
Michal Hroch ◽  
Pavel Pták
Keyword(s):  
Hilbert Space ◽  
Quantum Logic ◽  
Quantum Logics ◽  
Logic Formula ◽  
Boolean Subalgebra ◽  
Lattice Quantum ◽  
State Formula ◽  
Space Logic

We show in this note that if [Formula: see text] is a Boolean subalgebra of the lattice quantum logic [Formula: see text], then each state on [Formula: see text] can be extended over [Formula: see text] as a Jauch–Piron state provided [Formula: see text] is Jauch–Piron unital with respect to [Formula: see text] (i.e. for each nonzero [Formula: see text], there is a Jauch–Piron state [Formula: see text] on [Formula: see text] such that [Formula: see text]). We then discuss this result for the case of [Formula: see text] being the Hilbert space logic [Formula: see text] and [Formula: see text] being a set-representable logic.

scholarly journals Quantum logic, Hilbert space, revision theory

2002 ◽  
Vol 136 (1) ◽  
pp. 61-100 ◽  
Author(s):  
Kurt Engesser ◽  
Dov M. Gabbay
Keyword(s):  
Hilbert Space ◽  
Quantum Logic ◽  
Revision Theory
Sign in / Sign up

Export Citation Format

Share Document