Logic Of Quantum Mechanics Research Articles

Algebraic Structures Formalizing the Logic of Quantum Mechanics Incorporating Time Dimension

AbstractAs Classical Propositional Logic finds its algebraic counterpart in Boolean algebras, the logic of Quantum Mechanics, as outlined within G. Birkhoff and J. von Neumann’s approach to Quantum Theory (Birkhoff and von Neumann in Ann Math 37:823–843, 1936) [see also (Husimi in I Proc Phys-Math Soc Japan 19:766–789, 1937)] finds its algebraic alter ego in orthomodular lattices. However, this logic does not incorporate time dimension although it is apparent that the propositions occurring in the logic of Quantum Mechanics are depending on time. The aim of the present paper is to show that tense operators can be introduced in every logic based on a complete lattice, in particular in the logic of quantum mechanics based on a complete orthomodular lattice. If the time set is given together with a preference relation, we introduce tense operators in a purely algebraic way. We derive several important properties of such operators, in particular we show that they form dynamic pairs and, altogether, a dynamic algebra. We investigate connections of these operators with logical connectives conjunction and implication derived from Sasaki projections in an orthomodular lattice. Then we solve the converse problem, namely to find for given time set and given tense operators a time preference relation in order that the resulting time frame induces the given operators. We show that the given operators can be obtained as restrictions of operators induced by a suitable extended time frame.

Tense Logic Based on Finite Orthomodular Posets

It is widely accepted that the logic of quantum mechanics is based on orthomodular posets. However, such a logic is not dynamic in the sense that it does not incorporate time dimension. To fill this gap, we introduce certain tense operators on such a logic in an inexact way, but still satisfying requirements asked on tense operators in the classical logic based on Boolean algebras or in various non-classical logics. Our construction of tense operators works perfectly when the orthomodular poset in question is finite. We investigate the behaviour of these tense operators, e.g. we show that some of them form a dynamic pair. Moreover, we prove that if the tense operators preserve one of the inexact connectives conjunction or implication as defined by the authors recently in another paper, then they also preserve the other one. Finally, we show how to construct the binary relation of time preference on a given time set provided the tense operators are given, up to equivalence induced by natural quasi-orders.

The logic induced by effect algebras

Effect algebras form an algebraic formalization of the logic of quantum mechanics. For lattice effect algebras {mathbf {E}}, we investigate a natural implication and prove that the implication reduct of {mathbf {E}} is term equivalent to {mathbf {E}}. Then, we present a simple axiom system in Gentzen style in order to axiomatize the logic induced by lattice effect algebras. For effect algebras which need not be lattice-ordered, we introduce a certain kind of implication which is everywhere defined but whose result need not be a single element. Then, we study effect implication algebras and prove the correspondence between these algebras and effect algebras satisfying the ascending chain condition. We present an axiom system in Gentzen style also for not necessarily lattice-ordered effect algebras and prove that it is an algebraic semantics for the logic induced by finite effect algebras.

Semimodules over commutative semirings and modules over unitary commutative rings

It is well-known that the lattice of all submodules of a module is modular. However, this is not the case for the lattice of subsemimodules of a semimodule. We show examples and describe these lattices for a given semimodule. We study closed and splitting subsemimodules and submodules of a given semimodule or module , respectively. We derive a sufficient condition under which the lattice of closed subsemimodules is a homomorphic image of the lattice of all subsemimodules. We describe the ordered set of splitting submodules of a module and show a natural bijective correspondence between this poset and the poset of all projections of this module. We show that this poset is orthomodular. This result extends the case known for the poset of closed subspaces of a Hilbert space which is used in the logic of quantum mechanics.

An unpublished debate brought to light: Karl Popper's enterprise against the logic of quantum mechanics

Karl Popper published, in 1968, a paper that allegedly found a flaw in a very influential article of Birkhoff and von Neumann, which pioneered the field of “quantum logic”. Nevertheless, nobody rebutted Popper's criticism in print for several years. This has been called in the historiographical literature an “unsolved historical issue”. Although Popper's proposal turned out to be merely based on misinterpretations and was eventually abandoned by the author himself, this paper aims at providing a resolution to such historical open issues. I show that (i) Popper's paper was just the tip of an iceberg of a much vaster campaign conducted by Popper against quantum logic (which encompassed several more unpublished papers that I retrieved); and (ii) that Popper's paper stimulated a heated debate that remained however confined within private correspondence.

Basic semirings

AbstractBasic algebras were introduced by Chajda, Halaš and Kühr as a common generalization of MV-algebras and orthomodular lattices, i.e. algebras used for formalization of non-classical logics, in particular the logic of quantum mechanics. These algebras were represented by means of lattices with section involutions. On the other hand, classical logic was formalized by means of Boolean algebras which can be converted into Boolean rings. A natural question arises if a similar representation exists also for basic algebras. Several attempts were already realized by the authors, see the references. Now we show that if a basic algebra is commutative then there exists a representation via certain semirings with involution similarly as it was done for MV-algebras by Belluce, Di Nola and Ferraioli. These so-called basic semirings, their ideals and congruences are studied in the paper.

Left residuated operators induced by posets with a unary operation

The concept of operator left residuation has been introduced by the authors in their previous paper (Chajda and Länger in Asian Eur J Math 11:1850097, 2018). Modifications of so-called quantum structures, in particular orthomodular posets, like pseudo-orthomodular, pseudo-Boolean and Boolean posets are investigated here in order to show that they are operator left residuated or even operator residuated. In fact, they satisfy more general sufficient conditions for operator residuation assumed for bounded posets equipped with a unary operation. It is shown that these conditions may be also necessary if a generalized version using subsets instead of single elements is considered. The above-listed posets can serve as an algebraic semantics for the logic of quantum mechanics in a broad sense. Moreover, our approach shows connections to substructural logics via the considered residuation.

Ideals and their complements in commutative semirings

We study conditions under which the lattice {{mathrm{mathbf {Id}}}}mathbf R of ideals of a given a commutative semiring {mathbf {R}} is complemented. At first we check when the annihilator I^* of a given ideal I of {mathbf {R}} is a complement of I. Further, we study complements of annihilator ideals. Next we investigate so-called Łukasiewicz semirings. These form a counterpart to MV-algebras which are used in quantum structures as they form an algebraic semantic of many-valued logics as well as of the logic of quantum mechanics. We describe ideals and congruence kernels of these semirings with involution. Finally, using finite unitary Boolean rings, a construction of commutative semirings with complemented lattice of ideals is presented.

Weakly orthomodular and dually weakly orthomodular posets

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.

Wrong Potential Energy Term in Schrödinger’s Equation for Hydrogen Atom

Even after nine decades of successful run of the Quantum Mechanics (QM), different viewpoints on foundational problems of Quantum Physics are still being actively debated. That is because mathematical logic of QM often defies the physical intuition which constitutes the main spirit of Physics. De Broglie’s hypothesis of matter waves implied that the dynamic characteristics of a micro particle in motion, can be ascribed to the wave characteristics of the wavelet accompanying the particle. The Schrodinger equation models the matter-wave interactions through wavefunction ψ and effectively serves as the foundation of QM. Even though mathematical structure of the Schrodinger equation is sound and elegant, here we show a conceptual mistake in the development of this equation wherein the physical situation has not been correctly modeled in the equation. The Coulomb potential energy of the proton electron pair in Hydrogen atom is essentially the negative interaction energy between their superposed electrostatic fields which is inversely proportional to their instantaneous separation distance. Assuming the proton to be relatively fixed at the origin of an appropriate coordinate system, the potential energy of the orbiting electron will be a function of instantaneous position coordinates of the electron. This has not been properly modeled in the Schrodinger equation. The resulting errors in the solution have been quantitatively demonstrated in this paper. We have stressed the necessity of incorporating a specific correction in the potential energy term of the Schrodinger equation, after which it may facilitate the adoption of Bohmian QM.

Basic algebras, logics, trends and applications

The classical logic was axiomatized algebraically by means of Boolean algebras in 19th century by George Boole. Similar attempts went on 20th century for algebraic axiomatization of non-classical logics, e.g. intuitionistic logics (Brouwer and Heyting algebras), many-valued logics (Łukasiewicz, Chang’s MV-algebras, Post algebras), the logic of quantum mechanics (orthomodular lattices and posets) and fuzzy logics (residuated lattices). In this paper, we are focused in a common generalization of MV-algebras and orthomodular lattices. The resulting algebras, called basic algebras, have surprisingly strong and interesting properties and they can be investigated in their own. The aim of the paper is to get an overview of results reached during the last decade.

Galois connections and tense operators on q-effect algebras

For effect algebras, the so-called tense operators were already introduced by Chajda and Paseka. They presented also a canonical construction of them using the notion of a time frame.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 both in the logic of quantum mechanics and in the many-valued logic.A crucial problem concerning tense operators is their representation. Having an effect algebra with tense operators, we can ask if there exists a time frame such that each of these operators can be obtained by the canonical construction. To approximate physical real systems as best as possible, we introduce the notion of a q-effect algebra and we solve this problem for q-tense operators on q-representable q-Jauch–Piron q-effect algebras.

On order types of linear basic algebras

Basic algebras form a common generalization of MV-algebras and of orthomodular lattices, the algebraic tool for axiomatization of many-valued Łukasiewicz logic and the logic of quantum mechanics. Hence, they are included among the socalled quantum structures. An important role is played by linearly ordered basic algebras because every subdirectly irreducible MV-algebra and every subdirectly irreducible commutative basic algebra is linearly ordered. Since subdirectly irreducible linearly ordered basic algebras exist of any infinite cardinality, the natural question is to describe all possible order types of these algebras. This problem is solved in the paper.

Separability and Non-Individuality: Is It Possible to Conciliate (At Least A Form Of) Einstein’s Realism with Quantum Mechanics?

In this paper we argue that physical theories, including the most recent ones, even if only implicitly, talk of `objects' (or `things') of some sort (really, of several sorts), and question the logico-mathematical apparatus we still use to formulate them, taking into account what such theories presuppose about these entities. I shall point out that despite the discourse (or at least some discourses) goes in the direction of assuming that these quantum objects would be `new entities' of some kind, distinct from the traditional physical objects of classical physics, the logico-mathematical framework we use is still the old one, grounded on classical logic and set theory, which are committed to atavistic concepts based on individuals and distinguishable things, in complete disagreement with our present day conception of quanta. So, the use of such apparatus would impede us to be in complete agreement with the ontological commitment the theories of \textit{quanta} seem to propose. Thus, I move in the direction of joining those who try to question the `logic of quantum mechanics' from a different point of view, looking for a formal rationale for a new ontology. As a consequence of this move, we can revisit Einstein's ideas on physical reality and see that, from the perspective of considering a new kind of object, here termed `non-individuals', it is possible to sustain that they still obey some of Einstein's conditions for `physical realities', so that it will be possible to talk of a `principle of separability' in a sense which is not in complete disagreement with quantum mechanics. So, Einstein's departure from quantum mechanics might be softened at least concerning a form of his realism (locality still remains a challenge of course), for we guess that the incompatibility between quantum mechanics (field theories included) and some form of `separability' makes sense only if the objects of discourse are thought as `classical' objects, typical of classical ontology.

An algebraic axiomatization of orthogonal posets

The so-called orthogonal posets form an important tool for some investigations in the logic of quantum mechanics because they can be recognized as so-called quantum structures. The motivation for studying quantum structures is included e.g. in the monograph by Dvureă?enskij and Pulmannova or in the papers by Beltrametti and Maczynski. It is shown that every space of numerical events [see Chajda and Langer (Intern J Theor Phys 50:2403, 2011b), Dorninger and Langer (Intern J Theor Phys 52:1141---1147, 2013) and references therein] forms an orthogonal poset. Hence, orthogonal posets should be axiomatized by standard algebraic machinery. However, considering supremum as a binary operation, they form only partial algebras. The aim of the paper is to involve a different way which enables us to describe orthogonal posets as total algebras and get an algebraic axiomatization as an equational theory.

Dynamic effect algebras and their representations

For lattice effect algebras, the so-called tense operators were already introduced by Chajda and Kolařik. 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 in every effect algebra can be introduced tense operators which, for non-complete lattice effect algebras, can be only partial mappings. An effect algebra equipped with tense operators reflects changes of quantum events from past to future. A crucial problem concerning tense operators is their representation. Having an effect algebra with tense operators, we can ask if there exists a frame such that each of these operators can be obtained by our construction. We solve this problem for (strict) dynamic effect algebras having a full set of homorphisms into a complete lattice effect algebra.

Dynamic effect algebras

Abstract We 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.

A non-associative generalization of effect algebras

Effect algebras play an important role in the logic of quantum mechanics. The aim of this paper is to drop the associativity of addition. However, some important properties of effect algebras are preserved, e.g. every so-called skew effect algebra is still a bounded poset with an antitone involution. Moreover, skew effect algebras are fully characterized as certain bounded posets with sectionally switching involutions.

De-linearizing Linearity: Projective Quantum Axiomatics From Strong Compact Closure

Elaborating on our joint work with Abramsky in [S. Abramsky, B. Coecke, B. (2004) A categorical semantics of quantum protocols. Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS'04), IEEE Computer Science Press. Extended version including proofs at arXiv:quant-ph/0402130, S. Abramsky, B. Coecke, (2005) Abstract physical traces. Theory and Applications of Categories 14, 111–124] we further unravel the linear structure of Hilbert spaces into several constituents. Some prove to be very crucial for particular features of quantum theory while others obstruct the passage to a formalism which is not saturated with physically insignificant global phases.First we show that the bulk of the required linear structure is purely multiplicative, and arises from the strongly compact closed tensor which, besides providing a variety of notions such as scalars, trace, unitarity, self-adjointness and bipartite projectors [S. Abramsky, B. Coecke, B. (2004) A categorical semantics of quantum protocols. Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS'04), IEEE Computer Science Press. Extended version including proofs at arXiv:quant-ph/0402130, S. Abramsky, B. Coecke, (2005) Abstract physical traces. Theory and Applications of Categories 14, 111–124], also provides Hilbert-Schmidt norm, Hilbert-Schmidt inner-product, and in particular, the preparation-state agreement axiom which enables the passage from a formalism of the vector space kind to a rather projective one, as it was intended in the (in)famous Birkhoff & von Neumann paper [G. Birkhoff, J. von Neumann, (1936) The logic of quantum mechanics. Annals of Mathematics 37, 823–843].Next we consider additive types which distribute over the tensor, from which measurements can be build, and the correctness proofs of the protocols discussed in [S. Abramsky, B. Coecke, (2004) A categorical semantics of quantum protocols. Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS'04), IEEE Computer Science Press. Extended version including proofs at arXiv:quant-ph/0402130] carry over to the resulting weaker setting. A full probabilistic calculus is obtained when the trace is moreover linear and satisfies the diagonal axiom, which brings us to a second main result, characterization of the necessary and sufficient additive structure of a both qualitatively and quantitatively effective categorical quantum formalism without redundant global phases. Along the way we show that if in a category a (additive) monoidal tensor distributes over a strongly compact closed tensor, then this category is always enriched in commutative monoids.

Superposition, Entropy and Schmidt Decomposition of States

Superposition and entropy are compared using the language of the logic of quantum mechanics. It is pointed out that a finite value of the relative quantum entropy of states implies a superposition relation between the states themselves. The superposition relation is then studied by comparing the pure state of the compound system with the product of the reduced states and an intermediate “Schmidt” state. All the corresponding relative quantum entropies are evaluated in terms of the Schmidt coefficients of the global pure state. Some of the results are extended in case the compound system is in a state represented by a general density operator.