scholarly journals An orthomodular poset which does not admit a normed orthovaluation

1970 ◽  
Vol 3 (2) ◽  
pp. 163-170 ◽  
Author(s):  
Peter D. Meyer ◽  
P.D. Finch
Keyword(s):  
Quantum Mechanics ◽  
Orthomodular Poset ◽  
Counter Example

It is of relevance to studies in the logic of quantum mechanics whether or not every separable completely orthomodular poset admits a normed σ-ortho-valuation. A finite orthomodular poset is constructed which is a counter-example to this proposition.

scholarly journals The logic of orthomodular posets of finite height

2020 ◽  
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Quantum Mechanics ◽  
Positive Answer ◽  
Complete List ◽  
Orthomodular Poset ◽  
Central Question ◽  
Implication Operator ◽  
Algebraizable Logic ◽  
Binary Operator ◽  
Orthomodular Posets ◽  
Finite Height

Abstract Orthomodular posets form an algebraic formalization of the logic of quantum mechanics. A central question is how to introduce implication in such a logic. We give a positive answer whenever the orthomodular poset in question is of finite height. The crucial advantage of our solution is that the corresponding algebra, called implication orthomodular poset, i.e. a poset equipped with a binary operator of implication, corresponds to the original orthomodular poset and that its implication operator is everywhere defined. We present here a complete list of axioms for implication orthomodular posets. This enables us to derive an axiomatization in Gentzen style for the algebraizable logic of orthomodular posets of finite height.

Weakly orthomodular and dually weakly orthomodular posets

2018 ◽  
Vol 11 (02) ◽  
pp. 1850093 ◽  
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Quantum Mechanics ◽  
Orthomodular Poset ◽  
Algebraic Semantic ◽  
Orthomodular Posets ◽  
Congruence Properties

Orthomodular posets form an algebraic semantic for the logic of quantum mechanics. We show several methods how to construct orthomodular posets via a representation within the powerset of a given set. Further, we generalize this concept to the concept of weakly orthomodular and dually weakly orthomodular posets where the complementation need not be antitone or an involution. We show several interesting examples of such posets and prove which intervals of these posets are weakly orthomodular or dually weakly orthomodular again. To every (dually) weakly orthomodular poset can be assigned an algebra with total operations, a so-called (dually) weakly orthomodular [Formula: see text]-lattice. We study properties of these [Formula: see text]-lattices and show that the variety of these [Formula: see text]-lattices has nice congruence properties.

scholarly journals Maxwell’s Demon in Quantum Mechanics

Entropy ◽  
2020 ◽  
Vol 22 (3) ◽  
pp. 269
Author(s):  
Orly Shenker ◽  
Meir Hemmo
Keyword(s):  
Quantum Mechanics ◽  
Thought Experiment ◽  
Classical Mechanics ◽  
Second Law Of Thermodynamics ◽  
Quantum Mechanical ◽  
Second Law ◽  
Arrow Of Time ◽  
Maxwell’S Demon ◽  
Counter Example ◽  
Maxwell's Demon

Maxwell’s Demon is a thought experiment devised by J. C. Maxwell in 1867 in order to show that the Second Law of thermodynamics is not universal, since it has a counter-example. Since the Second Law is taken by many to provide an arrow of time, the threat to its universality threatens the account of temporal directionality as well. Various attempts to “exorcise” the Demon, by proving that it is impossible for one reason or another, have been made throughout the years, but none of them were successful. We have shown (in a number of publications) by a general state-space argument that Maxwell’s Demon is compatible with classical mechanics, and that the most recent solutions, based on Landauer’s thesis, are not general. In this paper we demonstrate that Maxwell’s Demon is also compatible with quantum mechanics. We do so by analyzing a particular (but highly idealized) experimental setup and proving that it violates the Second Law. Our discussion is in the framework of standard quantum mechanics; we give two separate arguments in the framework of quantum mechanics with and without the projection postulate. We address in our analysis the connection between measurement and erasure interactions and we show how these notions are applicable in the microscopic quantum mechanical structure. We discuss what might be the quantum mechanical counterpart of the classical notion of “macrostates”, thus explaining why our Quantum Demon setup works not only at the micro level but also at the macro level, properly understood. One implication of our analysis is that the Second Law cannot provide a universal lawlike basis for an account of the arrow of time; this account has to be sought elsewhere.

scholarly journals Residuation in finite posets

2021 ◽  
Vol 71 (4) ◽  
pp. 807-820
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Quantum Mechanics ◽  
Effect Algebra ◽  
Algebraic Logic ◽  
Quantum Effects ◽  
Orthomodular Poset ◽  
Minimal Elements ◽  
Finite Posets ◽  
General Conditions

Abstract When an algebraic logic based on a poset instead of a lattice is investigated then there is a natural problem how to introduce implication to be everywhere defined and satisfying (left) adjointness with conjunction. We have already studied this problem for the logic of quantum mechanics which is based on an orthomodular poset or the logic of quantum effects based on a so-called effect algebra which is only partial and need not be lattice-ordered. For this, we introduced the so-called operator residuation where the values of implication and conjunction need not be elements of the underlying poset, but only certain subsets of it. However, this approach can be generalized for posets satisfying more general conditions. If these posets are even finite, we can focus on maximal or minimal elements of the corresponding subsets and the formulas for the mentioned operators can be essentially simplified. This is shown in the present paper where all theorems are explained by corresponding examples.

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.

How to introduce the connective implication in orthomodular posets

2020 ◽  
pp. 2150066
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Quantum Mechanics ◽  
Orthomodular Poset ◽  
Natural Question ◽  
Partial Operation ◽  
Orthomodular Posets ◽  
Residuated Poset ◽  
One To One ◽  
Additional Assumptions

Since orthomodular posets serve as an algebraic axiomatization of the logic of quantum mechanics, it is a natural question how the connective of implication can be defined in this logic. It should be introduced in such a way that it is related with conjunction, i.e. with the partial operation meet, by means of some kind of adjointness. We present here such an implication for which a so-called unsharp residuated poset can be constructed. Then this implication is connected with the operation meet by the so-called unsharp adjointness. We prove that also conversely, under some additional assumptions, such an unsharp residuated poset can be converted into an orthomodular poset and that this assignment is nearly one-to-one.

scholarly journals Wigner's theorem for an infinite set

2018 ◽  
Vol 68 (5) ◽  
pp. 1173-1222
Author(s):  
John Harding
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Automorphism Group ◽  
Direct Product ◽  
Orthomodular Lattice ◽  
Space Form ◽  
Orthomodular Poset ◽  
Wigner’S Theorem ◽  
Infinite Set ◽  
Wigner's Theorem

Abstract It is well known that the closed subspaces of a Hilbert space form an orthomodular lattice. Elements of this orthomodular lattice are the propositions of a quantum mechanical system represented by the Hilbert space, and by Gleason’s theorem atoms of this orthomodular lattice correspond to pure states of the system. Wigner’s theorem says that the automorphism group of this orthomodular lattice corresponds to the group of unitary and anti-unitary operators of the Hilbert space. This result is of basic importance in the use of group representations in quantum mechanics. The closed subspaces A of a Hilbert space ${\mathcal H}$ correspond to direct product decompositions $\mathcal{H}\simeq A\times A^\perp$ of the Hilbert space, a result that lies at the heart of the superposition principle. In [10] it was shown that the direct product decompositions of any set, group, vector space, and topological space form an orthomodular poset. This is the basis for a line of study in foundational quantum mechanics replacing Hilbert spaces with other types of structures. It is the purpose of this note to prove a version of Wigner’s theorem: for an infinite set X, the automorphism group of the orthomodular poset Fact(X) of direct product decompositions of X is isomorphic to the permutation group of X. The structure Fact(X) plays the role for direct product decompositions of a set that the lattice of equivalence relations plays for surjective images of a set. So determining its automorphism group is of interest independent of its application to quantum mechanics. Other properties of Fact(X) are determined in proving our version of Wigner’s theorem, namely that Fact(X) is atomistic in a very strong way.

Ambiguity: aspects of the wave–particle duality

1997 ◽  
Vol 30 (3) ◽  
pp. 375-385 ◽  
Author(s):  
BARBARA K. STEPANSKY
Keyword(s):  
Quantum Mechanics ◽  
Historical Development ◽  
Theoretical Physics ◽  
Particle Theory ◽  
Quantum Theories ◽  
Science Texts ◽  
Counter Example ◽  
Equivalent Quantum ◽  
Wave Particle Duality ◽  
Theoretical Foundations

As has become evident from historical studies, science does not proceed in the coherent and predictable way that basic science texts would have us believe. I will argue that an excellent counter-example is an episode from the historical development of quantum mechanics in which the incompatibility of the particle and the wave representations of the electron and light were destined to be encompassed by two mathematically equivalent, but conceptually quite different theories.I shall argue that the appearance of two such different, yet equivalent, quantum theories was not surprising at all and I claim even predictable. As Einstein himself wrote in 1909: ‘It is my opinion that the next phase in the development of theoretical physics will bring us a theory of light that can be interpreted as a kind of fusion of the wave and the [particle] theory.’ Certainly, the interpretative content of Werner Heisenberg's and Erwin Schrödinger's theories could not have been more different. By mid-1926, the theoretical foundations had been laid for a scientific and emotional battleground between the particle and the wave. I suggest that an important element in the debate was not the incompatibility itself but actually coming to terms with ambiguity in science. For, in the end, ambiguous and vague interpretations of the same phenomena became part of science, where science was supposed to give a clear and unambiguous description of nature.

On the structure of quantum logic

1969 ◽  
Vol 34 (2) ◽  
pp. 275-282 ◽  
Author(s):  
P. D. Finch
Keyword(s):  
Quantum Mechanics ◽  
Quantum Logic ◽  
Orthomodular Poset ◽  
Nonrelativistic Quantum ◽  
Nonrelativistic Quantum Mechanics

In the axiomatic development of the logic of nonrelativistic quantum mechanics it is not difficult to set down certain plausible axioms which ensure that the quantum logic of propositions has the structure of an orthomodular poset. This can be done in a number of ways, for example, as in Gunson [2], Mackey [4], Piron [5], Varadarajan [7] and Zierler [8], and we summarise one of these ways in §2 below.

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