形式化量子力学不必使用量子逻辑

一般认为,标准量子力学需要使用一套它自己的逻辑系统,即量子逻辑。量子逻辑采用与一般逻辑系统不同的语义规则,因此和古典逻辑无法兼容。此篇文章将呈现一套量子力学的严格形式基础,它是对古典二值逻辑之保守扩充;保守扩充意指比原先之逻辑系统强,但较强的原因为它有较多之词汇。此套逻辑为三值逻辑。古典逻辑中为真的句子仍然为真。古典逻辑中为假的句子将被区分为强性假与中性。第三个真值一中性一考虑了非本征态情况中之观察句。本文详列了物理的公理并显示它们具有一个模型。此提案的可行性说明了量子逻辑是不必要的,并且存在一个共同的逻辑架构可提供给数学、非量子物理及量子力学使用。     

Quantum Cooperation

In a theoretical simulation the cooperation of two insects is investigated who share a large number of maximally entangled EPR-pairs to correlate their probabilistic actions. Specifically, two distant butterflies must find each other. Each butterfly moves in a chaotic form of short flights, guided only by the weak scent emanating from the other butterfly. The flight directions result from classical random choices. Each such decision of an individual is followed by a read-out of an internal quantum measurement on a spin, the result of which decides whether the individual shall do a short flight or wait. These assumptions reflect the scarce environmental information and the small brains’ limited computational capacity. The quantum model is contrasted to two other cases: In the classical case the coherence between the spin pairs gets lost and the two butterflies act independently. In the super classical case the two butterflies read off their decisions of whether to fly or to wait from the same internal list so that they always take the same decision as if they were super correlated. The numerical simulation reveals that the quantum entangled butterflies find each other with a much shorter total flight path than in both classical models.     

Quantum Vagueness

It has been suggested that quantum particles are genuinelyvague objects (Lowe 1994a). The present work explores thissuggestion in terms of the various metaphysical packages that areavailable for describing such particles. The formal frameworksunderpinning such packages are outlined and issues of identityand reference are considered from this overall perspective. Indoing so we hope to illuminate the diverse ways in whichvagueness can arise in the quantum context.     

Quantum perichoresis: Quantum field theory and the Trinity

The main thesis of this article is that the Trinitarian theological doctrine of perichoresis can be metaphorically interpreted as a form of Divine phase entanglement with the world. Such entanglement would entail non-local, relational holism and superposition through which the immanent unity of the Trinity is economically present in creation. Christ kenoticly empties himself of the immanent perichoresis of the Trinity in order to enter the economic perichoresis of the creation. The Spirit is then the continuing perichoretic love of God sanctifying the creation toward life and fulfillment from within. It is the Trinity in ongoing perichoretic entanglement with the creation, affirming Divine ubiquity and panentheism.     

Quantum sortal predicates

Sortal predicates have been associated with a counting process, which acts as a criterion of identity for the individuals they correctly apply to. We discuss in what sense certain types of predicates suggested by quantum physics deserve the title of ‘sortal’ as well, although they do not characterize either a process of counting or a criterion of identity for the entities that fall under them. We call such predicates ‘quantum-sortal predicates’ and, instead of a process of counting, to them is associated a ‘criterion of cardinality’. After their general characterization, it is discussed how these predicates can be formally described. To Patrick Suppes on his 80th birthday.     

Quantum metaphysical indeterminacy

Philosophical Studies - On a wide variety of presently live interpretations, quantum mechanics violates the classical supposition of ‘value definiteness’, according to which the...     

Quantum MV algebras

We introduce the notion of quantum MV algebra (QMV algebra) as a generalization of MV algebras and we show that the class of all effects of any Hilbert space gives rise to an example of such a structure. We investigate some properties of QMV algebras and we prove that QMV algebras represent non-idempotent extensions of orthomodular lattices.I should like to thank Prof. M.L. Dalla Chiara and Dr. P. Minari for many interesting comments and remarks. Daniele Mundici     

Axiomatic Quantum Theory

The basis of a rigorous formal axiomatization of quantum mechanics is constructed, built upon Dirac's bra–ket notation. The system is three-sorted, with separate variables for scalars, vectors and operators. First-order quantification over all three types of variable is permitted. Economy in the axioms is effected by, e.g., assigning a single logical function * to transform (i) a scalar into its complex conjugate, (ii) a ket vector into a bra and a bra into a ket, (iii) an operator into its adjoint. The system is accompanied by a formal semantics. Further papers will deal with vector subspaces and projection operators, operators with continuous spectra, tensor products, observables, and quantum mechanical probabilities.     

Extended Quantum Logic

The concept of quantum logic is extended so that it covers a more general set of propositions that involve non-trivial probabilities. This structure is shown to be embedded into a multi-modal framework, which has desirable logical properties such as an axiomatization, the finite model property and decidability.     

Stabilizing Quantum Disjunction

Since the appearance of Prior’s tonk, inferentialists tried to formulate conditions that a collection of inference rules for a logical constant has to satisfy in order to succeed in conferring an acceptable meaning to it. Dummett proposed a pair of conditions, dubbed ‘harmony’ and ‘stability’ that have been cashed out in terms of the existence of certain transformations on natural deduction derivations called reductions and expansions. A long standing open problem for this proposal is posed by quantum disjunction: although its rules are intuitively unstable, they pass the test of existence of expansions. Although most authors view instabilities of this kind as too subtle to be detected by the requirement of existence of expansions, we first discuss a case showing that this requirement can indeed detect instabilities of this kind, and then show how the expansions for disjunction-like connectives have to be reformulated to rule out quantum disjunction. We show how the alternative pattern for expansions can be formulated for connectives and quantifiers whose rules satisfy a scheme originally developed by Prawitz and Schroeder-Heister. Finally we compare our proposal with a recent one due to Jacinto and Read.     

Search for Quantum Reality

We summarize a recent search for quantum reality. The full anhomomorphic logic of coevents for an event set is introduced. The quantum integral over an event with respect to a coevent is defined. Reality filters such as preclusivity and regularity of coevents are considered. A quantum measure that can be represented as a quantum integral with respect to a coevent is said to 1-generate that coevent. This gives a stronger filter that may produce a unique coevent called the “actual reality” for a physical system. What we believe to be a more general filter is defined in terms of a double quantum integral and is called 2-generation. It is noted that ordinary measures do not 1 or 2-generate coevents except in a few simple cases. Various general results are stated.     

Quantum measurements and supertasks

This article addresses the question whether supertasks are possible within the context of non-relativistic quantum mechanics. The supertask under consideration consists of performing an infinite number of quantum mechanical measurements in a finite amount of time. Recent arguments in the physics literature claim to show that continuous measurements, understood as N discrete measurements in the limit where N goes to infinity, are impossible. I show that there are certain kinds of measurements in quantum mechanics for which these arguments break down. This suggests that there is a new context in which quantum mechanics, in principle, permits the performance of a supertask.     

Weakly Intuitionistic Quantum Logic

In this article von Neumann’s proposal that in quantum mechanics projections can be seen as propositions is followed. However, the quantum logic derived by Birkhoff and von Neumann is rejected due to the failure of the law of distributivity. The options for constructing a distributive logic while adhering to von Neumann’s proposal are investigated. This is done by rejecting the converse of the proposal, namely, that propositions can always be seen as projections. The result is a weakly Heyting algebra for describing the language of quantum mechanics.     

Systems of Quantum Logic

Logical implications are closely related to modal operators. Lattice-valued logic LL and quantum logic QL were formulated in Titani S (1999) Lattice Valued Set Theory. Arch Math Logic 38:395–421, Titani S (2009) A Completeness Theorem of Quantum Set Theory. In: Engesser K, Gabbay DM, Lehmann D (eds) Handbook of Quantum Logic and Quantum Structures: Quantum Logic. Elsevier Science Ltd., pp. 661–702, by introducing the basic implication → which represents the lattice order. In this paper, we fomulate a predicate orthologic provided with the basic implication, which corresponds to complete ortholattices, and then formulate a quantum logic which is equivalent to QL, by using a modal operator instead of the basic implication.     

Remarks On Quantum Ontology

Despite the discrepancies between quantum objects and `classical' ones, mainly with regard to the fact that the latter may be thought of as `individuals', contrary to the former, we still regard the quanta as `things' in our ordinary discourse as well as in the logico-mathematical basis of quantum theories. This paper considers some possibilities for accomodating the logico-mathematical framework of the theories which deal with such a strange ontology where the inhabitants are things devoid of identity and both having and not having certain properties.``All right', said the Cat; and this time it vanished quite slowly, beginning with the end of the tail, and ending with the grin, which remained some time after the rest of it had gone.``Well! I've often seen a cat without a grin', thought Alice, ``but a grin without a cat! It's the most curious thing I ever saw in all my life!'Lewis Carroll     

Against Radical Quantum Ontologies

Some theories of quantum mechanical phenomena endorse wave function realism, according to which the physical space we inhabit is very different from the physical space we appear to inhabit. In this paper I explore an argument against wave function realism that appeals to a type of simplicity that, although often overlooked, plays a crucial role in scientific theory choice. The type of simplicity in question is simplicity of fit between the way a theory says the world is and the way the world appears to be. This argument can be understood as one way of spelling out the so‐called “incredulous stare objection” that is sometimes leveled against surprising metaphysical theories.     

MV-Algebras and Quantum Computation

We introduce a generalization of MV algebras motivated by the investigations into the structure of quantum logical gates. After laying down the foundations of the structure theory for such quasi-MV algebras, we show that every quasi-MV algebra is embeddable into the direct product of an MV algebra and a “flat” quasi-MV algebra, and prove a completeness result w.r.t. a standard quasi-MV algebra over the complex numbers. Presented by Heinrich Wansing