Quantum logic and operational quantum mechanics

We characterize the class of the μ-complete F-spaces with unit corresponding to the observables of a quantum logic. We show that, conversely, every μ-complete F-space satisfying Axiom I and Axiom II corresponds to a quantum logic. The latter class of F-spaces generalizes that of “spectral F-spaces” introduced by Alfsen and Shultz and by Edwards.     

Brouwer-Zadeh logic and the operational approach to quantum mechanics

This paper is concerned with a logical system, called Brouwer-Zadeh logic, arising from the BZ poset of all effects of a Hilbert space. In particular, we prove a representation theorem for Brouwer-Zadeh lattices, and we show that Brouwer-Zadeh logic is not characterized by the MacNeille completions of all BZ posets of effects.     

Quantum probability and operational statistics

We develop the concept of quantum probability based on ideas of R. Feynman. The general guidelines of quantum probability are translated into rigorous mathematical definitions. We then compare the resulting framework with that of operational statistics. We discuss various relationship between measurements and define quantum stochastic processes. It is shown that quantum probability includes both conventional probability theory and traditional quantum mechanics. Discrete quantum systems, transition amplitudes, and discrete Feynman amplitudes are treated. We close with some examples that illustrate previously defined concepts.     

Orthospaces and quantum logic

In this paper we construct the ortholattices arising in quantum logic starting from the phenomenologically plausible idea of a collection of ensembles subject to passing or failing various tests. A collection of ensembles forms a certain kind of preordered set with extra structure called anorthospace; we show that complete ortholattices arise as canonical completions of orthospaces in much the same way as arbitrary complete lattices arise as canonical completions of partially ordered sets. We also show that the canonical completion of an orthospace of ensembles is naturally identifiable as the complete lattice of properties of the ensembles, thereby revealing exactlywhy ortholattices arise in the analysis of tests or experimental propositions. Finally, we axiomatize the hitherto implicit concept of test and show how they may be correlated with properties of ensembles.     

Renormalizations in quantum statistics

A simple proof is given that quantum statistics is multiplicatively renormalizable (if the corresponding quantum field theory is), that the renormalization constants are independent of the temperature and the parameters in the theory with the dimensions of mass, and that the renormalization constants (of the counterterms) are equal to the renormalization constants of the corresponding quantum field theory.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 26–30, January, 1981.I should like to thank B. L. Voronov, O. K. Kalashnikov, A. D. Linde, E. S. Fradkin, and A. E. Shabad for helpful discussions.     

Unified quantum logic

Unified quantum logic based on unified operations of implication is formulated as an axiomatic calculus. Soundness and completeness are demonstrated using standard algebraic techniques. An embedding of quantum logic into a new modal system is carried out and discussed.On leave of absence from Department of Mathematics, University of Zagreb, Pot. Pret. 165, YU-41001 Zagreb, Yugoslavia.     

Automaton partition logic versus quantum logic

The propositional system of a general class of discrete deterministic systems is formally characterized. We find that any finite prime orthomodular lattice allowing two-valued states can be represented by an automaton logic.     

On quantum logic

The status and justification of quantum logic are reviewed. On the basis of several independent arguments it is concluded that it cannot be a logic in the philosophical sense of a general theory concerning the structure of valid inferences. Taken as a calculus for combining quantum mechanical propositions, it leaves a number of significant aspects of quantum physics unaccounted for. It is shown, moreover, that quantum logic, far from being more general than Boolean logic, forms a subset of a slight and natural extension of Boolean logic, a subset which corresponds to incomplete statements. The philosophical background of this unsatisfactory state of affairs is briefly explored.     

Coherent quantum logic

The von Neumann quantum logic lacks two basic symmetries of classical logic, that between sets and classes, and that between lower and higher order predicates. Similarly, the structural parallel between the set algebra and linear algebra of Grassmann and Peano was left incomplete by them in two respects. In this work a linear algebra is constructed that completes this correspondence and is interpreted as a new quantum logic that restores these invariances, and as a quantum set theory. It applies to experiments with coherent quantum phase relations between the quantum and the apparatus. The quantum set theory is applied to model a Lorentz-invariant quantum time-space complex.     

Relativistic quantum logic

On the basis of the well-known quantum logic and quantum probability a formal language of relativistic quantum physics is developed. This language incorporates quantum logical as well as relativistic restrictions. It is shown that relativity imposes serious restrictions on the validity regions of propositions in space-time. By an additional postulate this relativistic quantum logic can be made consistent. The results of this paper are derived exclusively within the formal quantum language; they are, however, in accordance with well-known facts of relativistic quantum physics in Hilbert space.     

Propositions and orthocomplementation in quantum logic

We briefly analyze two partial order relations that are usually introduced in quantum logic by making use of the concepts of yes-no experiment and of preparation as fundamental. We show that two distinct posetsE andL can be defined, the latter being identifiable with the lattice of quantum logic. We consider the posetE and find that it contains a subsetE ₀ which can easily be orthocomplemented. These results are used, together with suitable assumptions, in order to show that an Orthocomplementation inL can be deduced by the Orthocomplementation defined inE ₀, and also to give a rule to find the orthocomplement of any element ofL.Research sponsored by C.N.R. (Italy).     

Properties and operational propositions in quantum mechanics

In orthodox quantum mechanics, it has virtually become the custom to identify properties of a physical system with operationally testable propositions about the system. The causes and consequences of this practice are explored mathematically in this paper. Among other things, it is found that such an identification imposes severe constraints on the admissible states of the physical system.     

Logical independence in quantum logic

The projection latticesP(₁),P(₂) of two von Neumann subalgebras ₁, ₂ of the von Neumann algebra are defined to be logically independent if A B0 for any 0AP(₁), 0BP(₂). After motivating this notion in independence, it is shown thatP(₁),P(₂) are logically independent if ₁ is a subfactor in a finite factor andP(₁),P(₂ commute. Also, logical independence is related to the statistical independence conditions called C^*-independence W^*-independence, and strict locality. Logical independence ofP(₁,P(₂ turns out to be equivalent to the C^*-independence of (₁,₂) for mutually commuting ₁,₂ and it is shown that if (₁,₂) is a pair of (not necessarily commuting) von Neumann subalgebras, thenP(₁,P(₂ are logically independent in the following cases: is a finite-dimensional full-matrix algebra and ₁,₂ are C^*-independent; (₁,₂) is a W^*-independent pair; ₁,₂ have the property of strict locality.     

Produced entropy in quantum statistics

Hg-films were condensed with admixtures of HgBr₂ or HgCl₂ onto a substrate at 4.2 °K. The superconducting transition temperature is measured as a function of the impurity concentration. The transition temperature increases from 3.96 °K of the pure quenched film to 4.06 °K at a content of 0.3 mol.% HgBr₂. Higher impurity concentrations lower the transition temperature and superconductivity vanishes at about 16 mol.% HgBr₂ or 23 mol.% HgCl₂ respectively. Above these concentrations the electrical resistivity shows a negative temperature coefficient. By means of Debye-Scherrer diagrams there is found a diminishing of the crystallite size with increasing impurity concentration.     

Convex structures and operational quantum mechanics

A general mathematical framework called a convex structure is introduced. This framework generalizes the usual concept of a convex set in a real linear space. A metric is constructed on a convex structure and it is shown that mappings which preserve the structure are contractions. Convex structures which are isomomorphic to convex sets are characterized and for such convex structures it is shown that the metric is induced by a norm and that structure preserving mappings can be extended to bounded linear operators.Convex structures are shown to give an axiomatization of the states of a physical system and the metric is physically motivated. We demonstrate how convex structures give a generalizing and unifying formalism for convex set and operational methods in axiomatic quantum mechanics.     

Consistent histories and operational quantum theory

In this work a generalization of the consistent histories approach to quantum mechanics is presented. We first critically review the consistent histories approach to nonrelativistic quantum mechanics in a mathematically rigorous way and give some general comments about it. We investigate to what extent the consistent histories scheme is compatible with the results of the operational formulation of quantum mechanics. According to the operational approach, nonrelativistic quantum mechanics is most generally formulated in terms of effects, states, and operations. We formulate a generalized consistent histories theory using the concepts and the terminology which have proven useful in the operational formulation of quantum mechanics. The logical rule of the logical interpretation of quantum mechanics is generalized to the present context. The algebraic structure of the generalized theory is studied in detail.     

Reflections on quantum logic

This is a short, self-contained summary of problems connected with the interpretation of state vectors in quantum mechanics. We discuss the reconstruction of the “ψ function” from statistical data, some related mathematical questions, the classical “paradoxes,” the probability interpretation of the state vectors, and, finally, quantum logic in relation to hidden variable theories and Hilbert space formalism, to build up a consistent framework for the indeterministic quantum picture of nature.     

Fuzzy quantum logic I

The paper gives a review of the application of fuzzy set ideas in quantum logics. After a brief introduction to the fuzzy set theory, the historical development of the main attempts to utilize fuzzy set ideas in quantum logics are presented. Results of investigations of all major researchers (except the Italian group discussed elsewhere), who work or worked in the field, are discussed.