Embedding Quantum Universes in Classical Ones

Do the partial order and ortholattice operations of a quantum logic correspond to the logical implication and connectives of classical logic? Rephrased, How far might a classical understanding of quantum mechanics be, in principle, possible? A celebrated result of Kochen and Specker answers the above question in the negative. However, this answer is just one among various possible ones, not all negative. It is our aim to discuss the above question in terms of mappings of quantum worlds into classical ones, more specifically, in terms of embeddings of quantum logics into classical logics; depending upon the type of restrictions imposed on embeddings, the question may get negative or positive answers.     

“Incerto Tempore, Incertisque Loci”: Can We Compute the Exact Time at Which a Quantum Measurement Happens?

Without addressing the measurement problem (i. e., what causes the wave function to “collapse,” or to ”branch,” or a history to become realized, or a property to actualize), I discuss the problem of the timing of the quantum measurement: Assuming that in an appropriate sense a measurement happens, when precisely does it happen? This question can be posed within most interpretations of quantum mechanics. By introducing the operator M, which measures whether or not the quantum measurement has happened, I suggest that, contrary to what is often claimed, quantum mechanics does provide a precise answer to this question, although a somewhat surprising one.     

A non-symmetric transition probability in quantum mechanics

The purpose of this paper is to show the possibility of introducing a non-symmetric transition probability in quantum mechanics, starting from the axiomatic basis of the theory known as the “quantum logic approach”. The quantum-mechanical postulates are later reformulated in the language of the transition probability and the operations (filtering procedures) transforming pure states of a physical system into themselves, ans some advantages of this formulation are shown consisting mainly in resolving the old troubles connected with quantum logics (e.g., the questions of the complete lattice structure of the logic, atomicity, the validity of the covering law).As a consequence of the axioms assumed, the representation theorem for the logic of proposition is deduced.     

WHEN DOES A MEASUREMENT OR EVENT OCCUR?

Within quantum mechanics, a complete set of commutting observables can be found which describe the attributes of a system at a given time. However, the correct way to describe attributes of a system in time is still an open question. We discuss the difficulties in extending the standard approach of quantum mechanics to describe attributes of a system in time. We find that measuring when an event occurred and measuring that it occurred, are complimentary in Bohr's sense. To exemplify the differences between measurements at a given time and in time, we will compare Rovelli's recent proposal (quant-ph/9802020), to determine “at what time does a measurement occurred” with another model of a continuous measurement in time. Rovelli's scheme answers the question “has the measurement already occurred at a certain time?”, but does not answer to the more difficult question: “when did the measurement occur?” We also discuss the use of the probability current to measure the time at which a particle arrives to a certain location.     

Quantum states and potentialities of quantum systems

In a previous article it was shown that in general quantum states represent perspectives on the potentialities of quantum systems, rather than the potentialities themselves. In the present paper the following questions are investigated in the context of this result: (1) How do quantum states which undergo collapse transform under pure translations? (2) Under what conditions do quantum states represent the potentialities themselves? Two alternatives are presented in response to the first question: (1) Quantum states are scalars under translations. (2) The collapse of a quantum state propagates between frames of reference at the speed of light. The advantages and disadvantages of the two alternatives are discussed. The response to the second question is shown to depend on the chosen alternative. In addition, the second alternative is shown to lead to a consistent view of quantum states as “potential perspectives on potentialities.”     

Theory of filters

We consider the following statistical problem: suppose we have a light beam and a collection of semi-transparent windows which can be placed in the way of the beam. Assume that we are colour blind and we do not possess any colour sensitive detector. The question is, whether by only measurements of the decrease in the beam intensity in various sequences of windows we can recognize which among our windows are light beam filters absorbing photons according to certain definite rules? To answer this question a definition of physical systems is formulated independent of “quantum logic” and lattice theory, and a new idea of quantization is proposed. An operational definition of filters is given: in the framework of this definition certain nonorthodox classes of filters are admissible with a geometry incompatible to that assumed in orthodox quantum mechanics. This leads to an extension of the existing quantum mechanical structure generalizing the schemes proposed by Ludwig [10] and the present author [13]. In the resulting theory, the quantum world of orthodox quantum mechanics is not the only possible but is a special member of a vast family of “quantum worlds” mathematically admissible. An approximate classification of these worlds is given, and their possible relation to the quantization of non-linear fields is discussed. It turns out to be obvious that the convex set theory has a similar significance for quantum physics as the Riemannian geometry for space-time physics.     

Quantum Conservative Gates for Finite-Valued Logics

We introduce some conservative gates for finite-valued logics which are able to realize all the main connectives of the many-valued logics of ?ukasiewicz, the MV-algebras of Chang and Brower–Zadeh algebras. After a brief exposition of the motivations for this work, the gates are defined and their properties are explored. Finally, a possible quantum realization of them is proposed, using three techniques: a “brute force” method--an extension of the Conditional Quantum Control argument, and a new technique which we call the Constants Method. For all these techniques, the unitary operator which describes the gate is a sum of local operators.     

Empirical reality,empirical causality,and the measurement problem

Does physics describe anything that can meaningfully be called “independent reality,” or is it merely operational? Most physicists implicitly favor an intermediate standpoint, which takes quantum physics into account, but which nevertheless strongly holds fast to quite strictly realistic ideas about apparently “obvious facts” concerning the macro-objects. Part 1 of this article, which is a survey of recent measurement theories, shows that, when made explicit, the standpoint in question cannot be upheld. Part 2 brings forward a proposal for making minimal changes to this standpoint in such a way as to remove such objections. The “empirical reality” thus constructed is a notion that, to some extent,does ultimately refer to the human means of apprehension and of data processing. It nevertheless cannot be said that it reduces to a mere name just labelling a “set of recipes that never fail.” It is shown that our usual notion of macroscopic causality must be endowed with similar features.     

What Is “System”: The Information-Theoretic Arguments

The problem of “what is ‘system’?” is in the very foundations of modern quantum mechanics. Here, we point out the interest in this topic in the information-theoretic context. E.g., we point out the possibility to manipulate a pair of mutually non-interacting, non-entangled systems to employ entanglement of the newly defined “(sub)systems” consisting the one and the same composite system. Given the different divisions of a composite system into “subsystems”, the Hamiltonian of the system may generate in general non-equivalent quantum computations. Redefinition of “subsystems” of a composite system may be regarded as a method for avoiding decoherence in the quantum hardware. In principle, all the notions refer to a composite system as simple as the hydrogen atom.     

Linearity of expectation functionals

LetB be the set of bounded observables on a quantum logic. A mapJ: B →R is called an expectation functional ifJ is normalized, positive, continuous, and compatibly linear. Two questions are considered. IsJ linear, and isJ an expectation relative to some state? It is shown that the answers are affirmative for hidden variable logics and most Hilbert space logics. An example is given which shows thatJ can be nonlinear on an arbitrary quantum logic.     

Partial and unsharp quantum logics

The total and the sharp character of orthodox quantum logic has been put in question in different contexts. This paper presents the basic ideas for a unified approach to partial and unsharp forms of quantum logic. We prove a completeness theorem for some partial logics based on orthoalgebras and orthomodular posets. We introduce the notion of unsharp orthoalgebra and of generalized MV algebra. The class of all effects of any Hilbert space gives rise to particular examples of these structures. Finally, we investigate the relationship between unsharp orthoalgebras, generalized MV algebras, and orthomodular lattices.     

Paraconsistent quantum logics

Paraconsistent quantum logics are weak forms of quantum logic, where the noncontradiction and the excluded-middle laws are violated. These logics find interesting applications in the operational approach to quantum mechanics. In this paper, we present an axiomatization, a Kripke-style, and an algebraic semantical characterization for two forms of paraconsistent quantum logic. Further developments are contained in Giuntini and Greuling's paper in this issue.     

Do fuzzy quantum structures exist?

Recently, some fuzzy quantum structures were introduced. We focus on the fuzzy quantum logics arising from the isomorphism of some quantum logics and some systems of fuzzy subsets of the ordering sets of states. In general, a fuzzy quantum logic is equipped with the pointwise-defined fuzzy connectives generated by a common generator g. Stressing the pointwise nature of fuzzy structures and omitting the global properties of quantum elements, we find that only crisp values of elements of a fuzzy quantum logic are allowed. Consequently, fuzzy quantum structures do not exist! However, there exist quantum structures of fuzzy subsets.     

Description of physical systems

A general “logical” scheme, containing both classical and quantum mechanics, is developed on the basis of plausible axioms. We introduce the division of states and yes-no measurements into sharp and diffuse ones, and prove that sharp states possess their carriers. Owing to this result, the existence of lattice joins and meets is proved for a wide class of elements of the logic. This “semi-lattice” structure gives the familiar lattice picture for special cases of classical and quantum mechanics. The notion of quantum superposition is introduced in this general scheme. It is proved that if in a theory appear nontrivial quantum superpositions, then this theory is “undeterministic” and vise versa. Further analysis of the pure state space leads to the construction of the canonical embedding of the general logic into an orthomodular complete ortho-lattice. After defining the probability of transition between pure states, the pure state space appears to be a generalization of Mielnik's “probability space” of quantum mechanics.     

Compact quantum systems and the Pauli data problem

Compact quantum systems have underlying compact kinematical Lie algebras, in contrast to familiar noncompact quantum systems built on the Weyl-Heisenberg algebra. Pauli asked in the latter case: to what extent does knowledge of the probability distributions in coordinate and momentum space determine the state vector? The analogous question for compact quantum systems is raised, and some preliminary results are obtained.     

Quantum Logics that are Symmetric-difference-closed

International Journal of Theoretical Physics - In this note we contribute to the recently developing study of “almost Boolean” quantum logics (i.e. to the study of orthomodular...     

Differential Structure of Greechie Logics

A liaison between quantum logics and non-commutative differential geometry isoutlined: a class of quantum logics are proved to possess the structure of discretedifferential manifolds. We show that the set of proper elements of an arbitraryatomic Greechie logic is naturally endowed by Koszul's differential calculus.     

Fuzzy set ideas in quantum logics

Fuzzy set theory language and ideas are used to express basic quantum logic notions. The possibility of replacing probabilistic interpretation of quantum mechanics by interpretation based on infinite-valued logics and fuzzy set theory is outlined. Short review of various structures encountered in the fuzzy set approach to quantum logics is given.     

Coincidental spectral lines for the hydrogen atom

Asim Barut once,en passant, asked the question “For what transitions of the hydrogen atom do the spectral lines coincide”? It is a pleasure for us to give the answer in this paper.