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.

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.

scholarly journals p-ADIC AND ADELIC MINISUPERSPACE QUANTUM COSMOLOGY

2002 ◽  
Vol 17 (10) ◽  
pp. 1413-1433 ◽  
Author(s):  
GORAN S. DJORDJEVIĆ ◽  
BRANKO DRAGOVICH ◽  
LJUBIŠA D. NEŠIĆ ◽  
IGOR V. VOLOVICH
Keyword(s):  
Quantum Mechanics ◽  
Cosmological Constant ◽  
Quantum Cosmology ◽  
Quantum Effects ◽  
De Sitter Space ◽  
Cosmological Models ◽  
De Sitter ◽  
Adelic Quantum Mechanics ◽  
Matter Fields

We consider the formulation and some elaboration of p-adic and adelic quantum cosmology. The adelic generalization of the Hartle–Hawking proposal does not work in models with matter fields. p-adic and adelic minisuperspace quantum cosmology is well defined as an ordinary application of p-adic and adelic quantum mechanics. It is illustrated by a few cosmological models in one, two and three minisuperspace dimensions. As a result of p-adic quantum effects and the adelic approach, these models exhibit some discreteness of the minisuperspace and cosmological constant. In particular, discreteness of the de Sitter space and its cosmological constant is emphasized.

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.

Properties of implication in effect algebras

2021 ◽  
Vol 71 (3) ◽  
pp. 523-534
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
State Space ◽  
Effect Algebra ◽  
Modus Ponens ◽  
Effect Algebras ◽  
Algebraic Semantics ◽  
Deductive Systems ◽  
Algebraic Description ◽  
Residuated Poset

Abstract Effect algebras form a formal algebraic description of the structure of the so-called effects in a Hilbert space which serve as an event-state space for effects in quantum mechanics. This is why effect algebras are considered as logics of quantum mechanics, more precisely as an algebraic semantics of these logics. Because every productive logic is equipped with implication, we introduce here such a concept and demonstrate its properties. In particular, we show that this implication is connected with conjunction via a certain “unsharp” residuation which is formulated on the basis of a strict unsharp residuated poset. Though this structure is rather complicated, it can be converted back into an effect algebra and hence it is sound. Further, we study the Modus Ponens rule for this implication by means of so-called deductive systems and finally we study the contraposition law.

scholarly journals Equations of quantum theory in the space of randomly joint quantum events

2019 ◽  
Vol 222 ◽  
pp. 03005
Author(s):  
Alexander Biryukov
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Quantum Effects ◽  
Symmetric Difference ◽  
System Transition ◽  
Additional Axiom ◽  
Random Events ◽  
Kolmogorov Axioms

The dynamics of the system in the space of random joint events is considered. The symmetric difference of events is introduced in space based on the Kolmogorov axioms. To describe quantum effects in the dynamics of the system, an additional axiom is introduced for random joint events: “the symmetric sum of random events.” In the generated space of random joint events, an equation is constructed for the probability of a system transition between two events. It is shown that for pairwise joint events it is equivalent to the equation of quantum mechanics.

Dynamic effect algebras

2012 ◽  
Vol 62 (3) ◽  
Author(s):  
Ivan Chajda ◽  
Miroslav Kolařík
Keyword(s):  
Quantum Mechanics ◽  
Effect Algebra ◽  
Dynamic Effect ◽  
Effect Algebras ◽  
Lattice Effect Algebra ◽  
Tense Operators ◽  
Lattice Effect

AbstractWe introduce the so-called tense operators in lattice effect algebras. Tense operators express the quantifiers “it is always going to be the case that” and “it has always been the case that” and hence enable us to express the dimension of time in the logic of quantum mechanics. We present an axiomatization of these tense operators and prove that every lattice effect algebra whose underlying lattice is complete can be equipped with tense operators. Such an effect algebra is called dynamic since it reflects changes of quantum events from past to future.

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.

A review of quantum transport in field-effect transistors

2021 ◽  
Author(s):  
David K Ferry ◽  
Josef Weinbub ◽  
Mihail Nedjalkov ◽  
Siegfried Selberherr
Keyword(s):  
Quantum Mechanics ◽  
Quantum Transport ◽  
Quantum Confinement ◽  
Field Effect ◽  
Field Effect Transistor ◽  
Field Effect Transistors ◽  
Quantum Effects ◽  
The Past ◽  
Effect Transistor

Abstract Confinement in small structures has required quantum mechanics, which has been known for a great many years. This leads to quantum transport. The field-effect transistor has had no need to be described by quantum transport over most of the century for which it has existed. But, this has changed in the past few decades, as modern versions tend to be absolutely controlled by quantum confinement and the resulting modifications to the normal classical descriptions. In addition, correlation and confinement lead to a need for describing the transport by quantum methods as well. In this review, we describe the quantum effects and the method of treating by various approaches to quantum transport.

scholarly journals Quantum violation of the pigeonhole principle and the nature of quantum correlations

2016 ◽  
Vol 113 (3) ◽  
pp. 532-535 ◽  
Author(s):  
Yakir Aharonov ◽  
Fabrizio Colombo ◽  
Sandu Popescu ◽  
Irene Sabadini ◽  
Daniele C. Struppa ◽  
...  
Keyword(s):  
Quantum Mechanics ◽  
Quantum Effects ◽  
Quantum Correlations ◽  
Fundamental Principle ◽  
Pigeonhole Principle ◽  
Quantum Particles ◽  
Interesting Structure

The pigeonhole principle: “If you put three pigeons in two pigeonholes, at least two of the pigeons end up in the same hole,” is an obvious yet fundamental principle of nature as it captures the very essence of counting. Here however we show that in quantum mechanics this is not true! We find instances when three quantum particles are put in two boxes, yet no two particles are in the same box. Furthermore, we show that the above “quantum pigeonhole principle” is only one of a host of related quantum effects, and points to a very interesting structure of quantum mechanics that was hitherto unnoticed. Our results shed new light on the very notions of separability and correlations in quantum mechanics and on the nature of interactions. It also presents a new role for entanglement, complementary to the usual one. Finally, interferometric experiments that illustrate our effects are proposed.

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