Koopman-von Neumann in the 30’s gave an operatorial formulation of Classical Mechanics. It was shown later on that this formulation could also be written in a path-integral form. We will label this functional approach as CPI (for classical path-integral) to distinguish it from the quantum mechanical one, which we will indicate with QPI. In the CPI two Grassmannian partners of time make their natural appearance and in this manner time becomes something like a three dimensional supermanifold. Next we (...) introduce a metric in this supermanifold and show that a particular choice of the supermetric reproduces the CPI while a different one gives the QPI. (shrink)

We use the system of p-adic numbers for the description of information processes. Basic objects of our models are so-called transformers of information, basic processes are information processes and statistics are information statistics (thus we present a model of information reality). The classical and quantum mechanical formalisms on information p-adic spaces are developed. It seems that classical and quantum mechanical models on p-adic information spaces can be applied for the investigation of flows of information in cognitive and social (...) systems, since a p-adic metric gives a quite natural description of the ability to form associations. (shrink)

This article focuses on two of the main problems raising interpretational issues in quantum mechanics, namely the notorious measurement problem and the equally important but not quite as widely discussed problem of the classical regime. The two problems are distinct, but they are both intimately related to some of the issues arising from entanglement and density operators. The article aims to be fairly non-technical in language, but modern in outlook and covering the chosen topics in more depth than (...) most introductory treatments. (shrink)

A quantum mechanical model of a counter monitoring the decay of an unstable microsystem is constructed. Detailed investigation of the time evolution of this model shows that the counter behaves essentially classically; i.e., its discharges may be considered as objective, observer-independent events. The possible relevance of this result for the physical interpretation of quantum mechanics is discussed.

Time is often said to play in quantum mechanics an essentially different role from position: whereas position is represented by a Hermitian operator, time is represented by a c-number. This discrepancy has been found puzzling and has given rise to a vast literature and many efforts at a solution. In this paper it is argued that the discrepancy is only apparent and that there is nothing in the formalism of quantum mechanics that forces us to treat (...) position and time differently. The apparent problem is caused by the dominant role point particles play in physics and can be traced back to classical mechanics. (shrink)

The philosophy of quantum mechanics has often been conceived by physicists as a collection of dogmas concerning what can be measured, observed and known. To this branch of dialectics the present paper does not attempt to contribute, chiefly because it is written from the conviction that no part of science, nor any philosophy, can safely predict what may be feasible or knowable. Rather, this brief essay endeavors to expose the epistemology of quantum physics in a way which (...) allows it to be recognized as an extension of the ordinary processes of cognition, as the continuation of a line which starts in sensory perception, leads straight through the methods of classical physics into quantum mechanics and shows no signs of terminating even there. (shrink)

The problem of indeterminism in quantum mechanics usually being considered as a generalization determinism of classical mechanics and physics for the case of discrete (quantum) changes is interpreted as an only mathematical problem referring to the relation of a set of independent choices to a well-ordered series therefore regulated by the equivalence of the axiom of choice and the well-ordering “theorem”. The former corresponds to quantum indeterminism, and the latter, to classical determinism. No other premises (...) (besides the above only mathematical equivalence) are necessary to explain how the probabilistic causation of quantum mechanics refers to the unambiguous determinism of classical physics. The same equivalence underlies the mathematical formalism of quantum mechanics. It merged the well-ordered components of the vectors of Heisenberg’s matrix mechanics and the non-ordered members of the wave functions of Schrödinger’s undulatory mechanics. The mathematical condition of that merging is just the equivalence of the axiom of choice and the well-ordering theorem implying in turn Max Born’s probabilistic interpretation of quantum mechanics. Particularly, energy conservation is justified differently than classical physics. It is due to the equivalence at issue rather than to the principle of least action. One may involve two forms of energy conservation corresponding whether to the smooth changes of classical physics or to the discrete changes of quantum mechanics. Further both kinds of changes can be equated to each other under the unified energy conservation as well as the conditions for the violation of energy conservation to be investigated therefore directing to a certain generalization of energy conservation. (shrink)

At the onset of quantum mechanics, it was argued that the new theory would entail a rejection of classical logic. The main arguments to support this claim come from the non-commutativity of quantum observables, which allegedly would generate a non-distributive lattice of propositions, and from quantum superpositions, which would entail new rules for quantum disjunctions. While the quantum logic program is not as popular as it once was, a crucial question remains unsettled: what is (...) the relationship between the logical structures of classical and quantum mechanics? In this essay we answer this question by showing that the original arguments promoting quantum logic contain serious flaws, and that quantum theory does satisfy the classical distributivity law once the full meaning of quantum propositions is properly taken into account. Moreover, we show that quantum mechanics can generate a distributive lattice of propositions, which, unlike the one of quantum logic, includes statements about expectation values which are of undoubtable physical interest. Lastly, we show that the lattice of statistical propositions in classical mechanics follows the same structure, yielding an analogue non-commutative sublattice of classical propositions. This fact entails that the purported difference between classical and quantum logic stems from a misconstructed parallel between the two theories. (shrink)

The paper interprets the concept “operator in the separable complex Hilbert space” (particalry, “Hermitian operator” as “quantity” is defined in the “classical” quantum mechanics) by that of “quantum information”. As far as wave function is the characteristic function of the probability (density) distribution for all possible values of a certain quantity to be measured, the definition of quantity in quantum mechanics means any unitary change of the probability (density) distribution. It can be represented as a (...) particular case of “unitary” qubits. The converse interpretation of any qubits as referring to a certain physical quantity implies its generalization to non-Hermitian operators, thus neither unitary, nor conserving energy. Their physical sense, speaking loosely, consists in exchanging temporal moments therefore being implemented out of the space-time “screen”. “Dark matter” and “dark energy” can be explained by the same generalization of “quantity” to non-Hermitian operators only secondarily projected on the pseudo-Riemannian space-time “screen” of general relativity according to Einstein's “Mach’s principle” and his field equation. (shrink)

We formulate from first principles a theory of stochastic processes in configuration space. The fundamental equations of the theory are an equation of motion which generalizes Newton's second law and an equation which expresses the condition of conservation of matter. Two types of stochastic motion are possible, both described by the same general equations, but leading in one case to classical Brownian motion behavior and in the other to quantum mechanical behavior. The Schrödinger equation, which is derived here with (...) no further assumption, is thus shown to describe a specific stochastic process. It is explicitly shown that only in the quantum mechanical process does the superposition of probability amplitudes give rise to interference phenomena; moreover, the presence of dissipative forces in the Brownian motion equations invalidates the superposition principle. At no point are any special assumptions made concerning the physical nature of the underlying stochastic medium, although some suggestions are discussed in the last section. (shrink)

Within the traditional Hilbert space formalism of quantum mechanics, it is not possible to describe a particle as possessing, simultaneously, a sharp position value and a sharp momentum value. Is it possible, though, to describe a particle as possessing just a sharp position value (or just a sharp momentum value)? Some, such as Teller, have thought that the answer to this question is No - that the status of individual continuous quantities is very different in quantum (...) class='Hi'>mechanics than in classical mechanics. On the contrary, I shall show that the same subtle issues arise with respect to continuous quantities in classical and quantum mechanics; and that it is, after all, possible to describe a particle as possessing a sharp position value without altering the standard formalism of quantum mechanics. (shrink)

I have written extensively of the topic of this tutorial. But in order to reach a broad audience I have in many of my more recent works refrained from using equations. That approach makes those works accessible in principle both to readers who are repelled by equations, and also to quantum physicists who are sufficiently familiar with the details of the quantum theory of measurement to be able to fill in for themselves the omitted equations. However, that approach (...) means also that an important class of potential readers is not being optimally reached. This class consists of persons having an engineering or mathematical background, and possessing perhaps even some familiarity with classical physics, but who have little knowledge of quantum mechanics. (shrink)

The present study attempts to provide a consistent and coherent account of what the world could be like, given the conceptual framework and results of contemporary quantum theory. It is suggested that standard quantum mechanics can, and indeed should, be understood as a realist theory within its domain of application. It is pointed out, however, that a viable realist interpretation of quantum theory requires the abandonment or radical revision of the classical conception of physical reality and (...) its traditional philosophical presuppositions. It is argued, in this direction, that the conceptualization of the nature of reality, as arising out of our most basic physical theory, calls for a kind of contextual realism. Within the domain of quantum mechanics, knowledge of 'reality in itself', 'the real such as it truly is' independent of the way it is contextualized, is impossible in principle. In this connection, the meaning of objectivity in quantum mechanics is analyzed, whilst the important question concerning the nature of quantum objects is explored. (shrink)

At the 1927 Como conference Bohr spoke the famous words “It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature.” However, if the Copenhagen interpretation really adheres to this motto, why then is there this nagging feeling of conflict when comparing it with realist interpretations? Surely what one can say about nature should in a certain sense be interpretation independent. In this paper I take Bohr’s (...) motto seriously and develop a quantum logic that avoids assuming any form of realism as much as possible. To illustrate the non-triviality of this motto, a similar result is first derived for classical mechanics. It turns out that the logic for classical mechanics is a special case of the quantum logic thus derived. Some hints are provided as to how these logics are to be used in practical situations and finally, I discuss how some realist interpretations relate to these logics. (shrink)

We attempt to clarify the main conceptual issues in approaches to ‘objectification’ or ‘measurement’ in quantum mechanics which are based on superselection rules. Such approaches venture to derive the emergence of classical ‘reality’ relative to a class of observers; those believing that the classical world exists intrinsically and absolutely are advised against reading this paper. The prototype approach (K. Hepp, Helv. Phys. Acta45 (1972), 237–248) where superselection sectors are assumed in the state space of the apparatus is shown (...) to be untenable. Instead, one should couple system and apparatus to an environment, and postulate superselection rules for the latter. These are motivated by the locality of any observer or other (actual or virtual) monitoring system. In this way ‘environmental’ solutions to the measurement problem (H.D. Zeh, Found. Phys.1 (1970), 69–76; W. H. Zurek, Phys. Rev.D26 (1982), 1862–1880 and Progr. Theor. Phys.89 (1993), 281–312) become consistent and acceptable, too. Points of contact with the modal interpretation are briefly discussed. We propose a minimal value attribution to observables in theories with superselection rules, in which only central observables have properties. In particular, the eigenvector-eigenvalue link is dropped. This is mainly motivated by Ockham's razor. (shrink)

Some of the problems associated with the construction of a manifestly covariant relativistic quantum theory are discussed. A resolution of this problem is given in terms of the off mass shell classical and quantum mechanics of Stueckelberg, Horwitz and Piron. This theory contains many questions of interpretation, reaching deeply into the notions of time, localizability and causality. A proper generalization of the Maxwell theory of electromagnetic interaction, required for the well-posed formulation of dynamical problems of systems with (...) electromagnetic interaction is discussed, and some of the significance of recently found (classical) relativistic chaotic behavior is pointed out. Many results of a technical nature have been achieved in this framework; in this paper, some of these are reviewed, but I shall concentrate on a discussion of the basic ideas and foundations of the theory. (shrink)

Some authors, inspired by the theoretical requirements for the formulation of a quantum theory of gravity, proposed a relational reconstruction of the quantum parameter-time—the time of the unitary evolution, which would make quantum mechanics compatible with relativity. The aim of the present work is to follow the lead of those relational programs by proposing a relational reconstruction of the event-time—which orders the detection of the definite values of the system’s observables. Such a reconstruction will be based (...) on the modal-Hamiltonian interpretation of quantum mechanics, which provides a clear criterion to select which observables acquire a definite value and to specify in what situation they do so. (shrink)

General metaphysical arguments have been proposed in favour of the thesis that all dispositions have categorical bases (Armstrong; Prior, Pargetter, Jackson). These arguments have been countered by equally general arguments in support of ungrounded dispositions (Molnar, Mumford). I believe that this controversy cannot be settled purely on the level of abstract metaphysical considerations. Instead, I propose to look for ungrounded dispositions in specific physical theories, such as quantum mechanics. I explain why non-classical properties such as spin are best (...) interpreted as irreducible dispositional properties, and I give reasons why even seemingly classical properties, for instance position or momentum, should receive a similar treatment when interpreted in the quantum realm. Contrary to the conventional wisdom, I argue that quantum dispositions should not be limited to probabilistic dispositions (propensities) by showing reasons why even possession of well-defined values of parameters should qualify as a dispositional property. I finally discuss the issue of the actuality of quantum dispositions, arguing that it may be justified to treat them as potentialities whose being has a lesser degree of reality than that of classical categorical properties, due to the incompatibility relations between non-commuting observables. (shrink)

Classical mechanics is empirically successful because the probabilistic mean values of quantum mechanical observables follow the classical equations of motion to a good approximation (Messiah 1970, 215). We examine this claim for the one-dimensional motion of a particle in a box, and extend the idea by deriving a special case of the ideal gas law in terms of the mean value of a generalized force used to define "pressure." The examples illustrate the importance of probabilistic averaging as a (...) method of abstracting away from the messy details of microphenomena, not only in physics, but in other sciences as well. (shrink)

We revisit the nonrelativistic problem of a bound, charged particle subject to the random zero-point radiation field, with the purpose of revealing the mechanism that takes it from the initially classical description to the final quantum-mechanical one. The combined effect of the zpf and the radiation reaction force results, after a characteristic time lapse, in the loss of the initial conditions and the concomitant irreversible transition of the dynamics to a stationary regime controlled by the field. In this regime, (...) the canonical variables x, p become expressed in terms of the dipolar response functions to a set of field modes. A proper ordering of the response coefficients leads to the matrix representation of quantum mechanics, as was proposed in the early days of the theory, and to the basic commutator x^,p^=iħ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[ {\hat{x}},{\hat{p}}\right] =i\hbar $$\end{document}. Further, the connection with the corresponding Fokker–Planck equation valid in the Markov approximation, allows one to obtain the radiative corrections of qed. These results reaffirm the essentially electrodynamic and stochastic nature of the quantum phenomenon, as proposed by stochastic electrodynamics. (shrink)

Maxwell’s Demon is a thought experiment devised by J. C. Maxwell in 1867 in order to show that the Second Law of thermodynamics is not universal, since it has a counter-example. Since the Second Law is taken by many to provide an arrow of time, the threat to its universality threatens the account of temporal directionality as well. Various attempts to “exorcise” the Demon, by proving that it is impossible for one reason or another, have been made throughout the years, (...) but none of them were successful. We have shown (in a number of publications) by a general state-space argument that Maxwell’s Demon is compatible with classical mechanics, and that the most recent solutions, based on Landauer’s thesis, are not general. In this paper we demonstrate that Maxwell’s Demon is also compatible with quantum mechanics. We do so by analyzing a particular (but highly idealized) experimental setup and proving that it violates the Second Law. Our discussion is in the framework of standard quantum mechanics; we give two separate arguments in the framework of quantum mechanics with and without the projection postulate. We address in our analysis the connection between measurement and erasure interactions and we show how these notions are applicable in the microscopic quantum mechanical structure. We discuss what might be the quantum mechanical counterpart of the classical notion of “macrostates”, thus explaining why our Quantum Demon setup works not only at the micro level but also at the macro level, properly understood. One implication of our analysis is that the Second Law cannot provide a universal lawlike basis for an account of the arrow of time; this account has to be sought elsewhere. (shrink)

Using the mathematical notion of an entity to represent states in quantum and classical mechanics, we show that, in a strict sense, proper superpositions are possible in classical mechanics.

Nonobjectivity of physical properties enters physics with the standard interpretation of quantum mechanics (QM), and a number of paradoxes of this theory follow from it. It seems, however, based on sound physical arguments (double slit experiment, Heisenberg's principle, Bell–Kochen–Specker theorem, etc.), so that most physicists think that avoiding it is impossible. We discuss these arguments here and show that they can be criticized from a physical viewpoint. Our criticism proves that nonobjectivity must be considered an epistemological choice rather (...) than an unavoidable feature of QM, so that an objective interpretation of QM is not a priori impossible, which justifies our attempt at providing it in some previous papers. This interpretation is based on a classical language in which the language of the standard interpretation (Quantum Logic) is embedded as a subset of statements that are directly testable according to QM. (shrink)

This work first explicates the philosophy of classical and quantum fields and particles. I am interested in determining how science can have a metaphysical dimension, and then with the claim that the quantum revolution has an important metaphysical component. I argue that the metaphysical implications of a theory are properties of its models, as classical mechanics determines properties of atomic diversity and temporal continuity with its representations of distinct, continuous trajectories. ;It is often suggested that classical statistical (...) physics requires that many particle states be represented so that permuting properties leads to distinct states; this implies that individuals can be reidentified across possible worlds in a non-qualitative way. I show there is no evidence for this conclusion, an important result, for it is claimed that quantum particles are not individuals. This claim is based on the misconception about classical statistics, but also on a conflation of notions of identity; I show that, while transworld identity is incompatible with quantum mechanics, other classical notions may be consistently ascribed. I also give a field-particle distinction that applies usefully in both quantum and classical domains. In the former the distinction helps defeat claims of underdetermined by data, in the latter it helps provide a minimal field metaphysics. ;Next I tackle renormalisation: I show how divergences occur in approximate, perturbative calculations, and demonstrate how finite, empirically verified, answers are obtained. These techniques seem to show that the predictions are not logical consequences of the exact theory. I use the techniques of the renormalisation group to establish that perturbative renormalised quantum field theory does indeed approximate the consequences of field theory. ;Finally, I discuss the idea that renormalisation proves that there can be no quantum theory of everything, only a patchwork of effective theories. The preceding chapter shows that renormalisation demonstrates only that the picture is consistent, and this is insufficient to show that physics must be phenomenological. (shrink)

This book concerns the metasemantics of quantum mechanics (QM). Roughly, it pursues an investigation at an intersection of the philosophy of physics and the philosophy of semantics, and it offers a critical analysis of rival explanations of the semantic facts of standard QM. Two problems for such explanations are discussed: categoricity and permanence of rules. New results include 1) a reconstruction of Einstein's incompleteness argument, which concludes that a local, separable, and categorical QM cannot exist, 2) a reinterpretation (...) of Bohr's principle of correspondence, grounded in the principle of permanence, 3) a meaning-variance argument for quantum logic, which follows a line of critical reflections initiated by Weyl, and 4) an argument for semantic indeterminacy leveled against inferentialism about QM, inspired by Carnap's work in the philosophy of classical logic. (shrink)

Why does time reversal involve two operations, a temporal reflection and the operation of complex conjugation? Why is it that time reversal preserves position and reverses momentum and spin? This puzzle of time reversal in quantum mechanics has been with us since Wigner’s first presentation. In this paper, I propose a new solution to this puzzle. First, it is shown that the standard account of time reversal can be derived based on the assumption that the probability current is (...) reversed by the time reversal transformation. Next, this assumption is justified and the meaning of time reversal is clarified by analyzing the relationship between the rates of change and the instantaneous quantities which determine them. Finally, I explain how the new analysis help solve the puzzle of time reversal in quantum mechanics. (shrink)

We review the notion of complementarity of observables in quantum mechanics, as formulated and studied by Paul Busch and his colleagues over the years. In addition, we provide further clarification on the operational meaning of the concept, and present several characterisations of complementarity—some of which new—in a unified manner, as a consequence of a basic factorisation lemma for quantum effects. We work out several applications, including the canonical cases of position–momentum, position–energy, number–phase, as well as periodic observables (...) relevant to spatial interferometry. We close the paper with some considerations of complementarity in a noisy setting, focusing especially on the case of convolutions of position and momentum, which was a recurring topic in Paul’s work on operational formulation of quantum measurements and central to his philosophy of unsharp reality. (shrink)

Non-commuting quantities and hidden parameters – Wave-corpuscular dualism and hidden parameters – Local or nonlocal hidden parameters – Phase space in quantum mechanics – Weyl, Wigner, and Moyal – Von Neumann’s theorem about the absence of hidden parameters in quantum mechanics and Hermann – Bell’s objection – Quantum-mechanical and mathematical incommeasurability – Kochen – Specker’s idea about their equivalence – The notion of partial algebra – Embeddability of a qubit into a bit – Quantum (...) computer is not Turing machine – Is continuality universal? – Diffeomorphism and velocity – Einstein’s general principle of relativity – „Mach’s principle“ – The Skolemian relativity of the discrete and the continuous – The counterexample in § 6 of their paper – About the classical tautology which is untrue being replaced by the statements about commeasurable quantum-mechanical quantities – Logical hidden parameters – The undecidability of the hypothesis about hidden parameters – Wigner’s work and и Weyl’s previous one – Lie groups, representations, and psi-function – From a qualitative to a quantitative expression of relativity − psi-function, or the discrete by the random – Bartlett’s approach − psi-function as the characteristic function of random quantity – Discrete and/ or continual description – Quantity and its “digitalized projection“ – The idea of „velocity−probability“ – The notion of probability and the light speed postulate – Generalized probability and its physical interpretation – A quantum description of macro-world – The period of the as-sociated de Broglie wave and the length of now – Causality equivalently replaced by chance – The philosophy of quantum information and religion – Einstein’s thesis about “the consubstantiality of inertia ant weight“ – Again about the interpretation of complex velocity – The speed of time – Newton’s law of inertia and Lagrange’s formulation of mechanics – Force and effect – The theory of tachyons and general relativity – Riesz’s representation theorem – The notion of covariant world line – Encoding a world line by psi-function – Spacetime and qubit − psi-function by qubits – About the physical interpretation of both the complex axes of a qubit – The interpretation of the self-adjoint operators components – The world line of an arbitrary quantity – The invariance of the physical laws towards quantum object and apparatus – Hilbert space and that of Minkowski – The relationship between the coefficients of -function and the qubits – World line = psi-function + self-adjoint operator – Reality and description – Does a „curved“ Hilbert space exist? – The axiom of choice, or when is possible a flattening of Hilbert space? – But why not to flatten also pseudo-Riemannian space? – The commutator of conjugate quantities – Relative mass – The strokes of self-movement and its philosophical interpretation – The self-perfection of the universe – The generalization of quantity in quantum physics – An analogy of the Feynman formalism – Feynman and many-world interpretation – The psi-function of various objects – Countable and uncountable basis – Generalized continuum and arithmetization – Field and entanglement – Function as coding – The idea of „curved“ Descartes product – The environment of a function – Another view to the notion of velocity-probability – Reality and description – Hilbert space as a model both of object and description – The notion of holistic logic – Physical quantity as the information about it – Cross-temporal correlations – The forecasting of future – Description in separable and inseparable Hilbert space – „Forces“ or „miracles“ – Velocity or time – The notion of non-finite set – Dasein or Dazeit – The trajectory of the whole – Ontological and onto-theological difference – An analogy of the Feynman and many-world interpretation − psi-function as physical quantity – Things in the world and instances in time – The generation of the physi-cal by mathematical – The generalized notion of observer – Subjective or objective probability – Energy as the change of probability per the unite of time – The generalized principle of least action from a new view-point – The exception of two dimensions and Fermat’s last theorem. (shrink)

A suitable unified statistical formulation of quantum and classical mechanics in a *-algebraic setting leads us to conclude that information itself is noncommutative in quantum mechanics. Specifically we refer here to an observer's information regarding a physical system. This is seen as the main difference from classical mechanics, where an observer's information regarding a physical system obeys classical probability theory. Quantum mechanics is then viewed purely as a mathematical framework for the probabilistic description (...) of noncommutative information, with the projection postulate being a noncommutative generalization of conditional probability. This view clarifies many problems surrounding the interpretation of quantum mechanics, particularly problems relating to the measuring process. (shrink)

There is a large literature on the issue of the lack of properties (i.e. accidents) in quantum mechanics (the problem of “hidden variables”) and also on the indistinguishability of particles. Both issues were discussed as far back as the late 1920’s. However, the implications of these challenges to classical ontology were taken up rather late, in part in the ‘quantum set theory’ of Takeuti (Curr Issues Quant Logic 303–322, 1981), Finkelstein (in Beltrametti EG, Van Fraassen BC (eds) (...) Current issues in quantum logic. Plenum, New York, 1981) and the work of Décio Krause (1992)—and subsequent publications). But the problems created by quantum mechanics go far beyond set theory or the identity of indiscernibles (another subject that has been often discussed)—it extends, I argue, to our accounts of truth. To solve this problem, i.e. to have an approach to truth that facilitates a transition from a classical to a quantum ontology one must have a unified framework for them both. This is done within the context of a pluralist view of truthmaking, where the truthmakers are unified in having a monoidal structure. The structure of the paper is as follows. After a brief introduction, the idea of a monoid is outlined (in Sect. 1) followed by a standard set of axioms that govern the truthmaker relation from elements of the monoid to the set of propositions. This is followed, in Sect. 2, by a discussion of how to have truthmakers for two kinds of necessities: tautologies and analytic truths. The next Sect. 3, then applies these ideas to quantum mechanics. It gives an account of quantum states and shows how these form a monoid. The final section then argues that quantum logic does not, despite what one might initially suspect, stand in the way of an account of quantum truth. (shrink)

The quantum harmonic oscillator is described in terms of two basic sets of coordinates: linear coordinates x, px and angular coordinates eiφ, Pφ (action-angle variables). The angular “coordinate” eiφ is assumed unitary, the conjugate momentum pφ is assumed Hermitian, and eiφ and pφ are assumed to be a canonical pair. Two transformations are defined connecting the angular coordinates to the linear coordinates. It is found that x, px can be physical, i.e., Hermitian and canonical, only under constraints on the (...) pφ eigenvalue spectrum. The conclusion is that eiφ can be a unitary operator. A parallel analysis of the classical harmonic oscillator is done with equivalent results. (shrink)

It is argued that the measurement problem reduces to the problem of modeling quasi-classical systems in a modified quantum mechanics with superselection rules. A measurement theorem is proved, demonstrating, on the basis of a principle for selecting the quantities of a system that are determinate (i.e., have values) in a given state, that after a suitable interaction between a systemS and a quasi-classical systemM, essentially only the quantity measured in the interaction and the indicator quantity ofM are determinate. (...) The theorem justifies interpreting the noncommutative algebra of observables of a quantum mechanical system as an algebra of “beables,” in Bell's sense. (shrink)

The particles of quantum mechanics (QM) are discrete undulatory entities which are described in terms of the complex state vectors of the theory in full agreement with experiment. The wave-particle paradox stems from the fact that undulation and discreteness are inconsistent within the classical theory which was historically the point of departure for the canonical foundation. The author describes his prolonged efforts of anchoring the state vector of QM in experiment rather in obsolete theory.

We show that it is impossible in quantum mechanics to describe two separated physical systems. This is due to the mathematical structure of quantum mechanics. It is possible to give a description of two separated systems in a theory which is a generalization of quantum mechanics and of classical mechanics, in the sense that this theory contains both theories as special cases. We identify the axioms of quantum mechanics that make it (...) impossible to describe separated systems. One of these axioms is equivalent to the superposition principle. We show how these findings throw a different light on the paradox of Einstein, Podolsky, and Rosen. (shrink)

The properties of classical and quantum systems are characterized by different algebraic structures. We know that the properties of a quantum mechanical system form a partial Boolean algebra not embeddable into a Boolean algebra, and so cannot all be co-determinate. We also know that maximal Boolean subalgebras of properties can be (separately) co-determinate. Are there larger subsets of properties that can be co-determinate without contradiction? Following an analysis of Bohrs response to the Einstein-Podolsky-Rosen objection to the complementarity interpretation (...) of quantum mechanics, a principled argument is developed justifying the selection of particular subsets of properties as co-determinate for a quantum system in particular physical contexts. These subsets are generated by sets of maximal Boolean subalgebras, defined in each case by the relation between the quantum state and a measurement (possibly, but not necessarily, the measurement in terms of which we seek to establish whether or not a particular property of the system in question obtains). If we are required to interpret quantum mechanics in this way, then predication for quantum systems is quite unlike the corresponding notion for classical systems. (shrink)

The classical Hamiltonian in generalized coordinates is given asH=1/2 Σ i.k p i g ik p k . We show that there is no operator of the formP i= −iA(qi) (∂/∂qi)+Gi(qi) (note that the Hermitian momentum operatorP i H is of this form) such that the quantum Hamiltonian operatorH Q is given asH Q =1/2 Σ i,k P i g ik P k or1/2 Σ i,k g ik P i P k , etc. In order to maintain a direct (...) transition of this sort from classical to quantum theory, using the classical Hamiltonian as a starting point, we must rely on our previous prescriptions, writing the quantum Hamiltonian asH Q =1/2 Σ i,k P i + g ik P k , whereP i + denotes the adjoint of the operatorP i=−ih ∂/∂qi. (shrink)

Classical physics and quantum physics suggest two meta-physical types of reality: the classical notion of a objectively definite reality with properties "all the way down," and the quantum notion of an objectively indefinite type of reality. The problem of interpreting quantum mechanics is essentially the problem of making sense out of an objectively indefinite reality. These two types of reality can be respectively associated with the two mathematical concepts of subsets and quotient sets which are category-theoretically (...) dual to one another and which are developed in two dual mathematical logics, the usual Boolean logic of subsets and the more recent logic of partitions. Our sense-making strategy is "follow the math" by showing how the mathematics of set partitions can be transported in a natural way to complex vector spaces where it yields the mathematical machinery of QM. And then we show how the machinery of QM can be transported the other way down to set-like vector spaces over ℤ₂ yielding a rather fulsome "toy" or pedagogical model of "quantum mechanics over sets." In this way, we try to make sense out of objective indefiniteness and thus to interpret quantum mechanics. (shrink)

In his entry on "Quantum Logic and Probability Theory" in the Stanford Encyclopedia of Philosophy, Alexander Wilce (2012) writes that "it is uncontroversial (though remarkable) the formal apparatus quantum mechanics reduces neatly to a generalization of classical probability in which the role played by a Boolean algebra of events in the latter is taken over the 'quantum logic' of projection operators on a Hilbert space." For a long time, Patrick Suppes has opposed this view (see, for (...) example, the paper collected in Suppes and Zanotti (1996). Instead of changing the logic and moving from a Boolean algebra to a non-Boolean algebra, one can also 'save the phenomena' by weakening the axioms of probability theory and work instead with upper and lower probabilities. However, it is fair to say that despite Suppes' efforts upper and lower probabilities are not particularly popular in physics as well as in the foundations of physics, at least so far. Instead, quantum logic is booming again, especially since quantum information and computation became hot topics. Interestingly, however, imprecise probabilities are becoming more and more popular in formal epistemology as recent work by authors such as James Joye (2010) and Roger White (2010) demonstrates. (shrink)

This article concerns a phenomenon of elementary quantum mechanics that is quite counter-intuitive, very non-classical, and apparently not widely known: a quantum particle can get reflected at a potential step downwards. In contrast, classical particles get reflected only at upward steps. As a consequence, a quantum particle can be trapped for a long time (though not forever) in a region surrounded by downward potential steps, that is, on a plateau. Said succinctly, a quantum particle tends (...) not to fall off a table. The conditions for this effect are that the wave length is much greater than the width of the potential step and the kinetic energy of the particle is much smaller than the depth of the potential step. We point out how the topic is accessible with elementary methods, but also with mathematical rigor and numerically. (shrink)

This paper describes a long-standing, though little-known, debate between Paul Dirac and Werner Heisenberg over the nature of scientific methodology, theory change, and intertheoretic relations. Following Heisenberg’s terminology, their disagreements can be summarized as a debate over whether the classical and quantum theories are “open” or “closed.” A close examination of this debate sheds new light on the philosophical views of two of the great founders of quantum theory.

The paper discusses this year’s Nobel Prize in physics for experiments of entanglement “establishing the violation of Bell inequalities and pioneering quantum information science” in a much wider, including philosophical context legitimizing by the authority of the Nobel Prize a new scientific area out of “classical” quantum mechanics relevant to Pauli’s “particle” paradigm of energy conservation and thus to the Standard model obeying it. One justifies the eventual future theory of quantum gravitation as belonging to the (...) newly established quantum information science. Entanglement, involving non-Hermitian operators for its rigorous description, non-unitarity as well as nonlocal and superluminal physical signals “spookily” (by Einstein’s flowery epithet) synchronizing and transferring some nonzero action at a distance, can be considered to be quantum gravity so that its local counterpart to be Einstein’s gravitation according to general relativity therefore pioneering an alternative pathway to quantum gravitation different from the “secondary quantization” of the Standard model. So, the experiments of entanglement once they have been awarded by the Nobel Prize launch particularly the relevant theory of quantum gravitation grounded on “quantum information science” thus granted to be nonclassical quantum mechanics in the shared framework of the generalized quantum mechanics obeying rather quantum-information conservation than only energy conservation. The concept of “dark phase” of the universe naturally linked to the very well confirmed “dark matter” and “dark energy” and opposed to its “light phase” inherent to classical quantum mechanics and the Standard model obeys quantum-information conservation, after which reversible causality or the mutual transformation of energy and information are valid. The mythical Big Bang after which energy conservation holds universally is to be replaced by an omnipresent and omnitemporal medium of decoherence of the dark and nonlocal phase into the light and local phase. The former is only an integral image of the latter and borrowed in fact rather from religion than from science. Physical, methodological and proper philosophical conclusions follow from that paradigm shift heralded by this year’s Nobel Prize in physics. For example, the scientific theory of thinking should originate from the dark phase of the universe, as well: probably only approximately modeled by neural networks physically belonging to the light phase thoroughly. A few crucial philosophical sequences follow from the break of Pauli’s paradigm: (1) the establishment of the “dark” phase of the universe as opposed to its “light” phase, only to which the Cartesian dichotomy of “body” and “mind” is valid; (2) quantum information conservation as relevant to the dark phase, furthermore generalizing energy conservation as to its light phase, productively allowing for physical entities to appear “ex nihilo”, i.e., from the dark phase, in which energy and time are yet inseparable from each other; (3) reversible causality as inherent to the dark phase; (4) the interpretation of gravitation only mathematically: as an interpretation of the incompleteness of finiteness to infinity, for example, following the Gödel dichotomy (“either contradiction or incompleteness”) about the relation of arithmetic to set theory; (5) the restriction of the concept of hierarchy only to the light phase; (6) the commensurability of both physical extremes of a quantum and the universe as a whole in the dark phase obeying quantum information conservation and akin to Nicholas of Cusa’s philosophical and theological worldview. (shrink)

Determinismus und Indeterminismus in der modernen Physik is one of Cassirer’s least known and studied works, despite his own assessment as “one of his most important achievements”. A prominent theme locates quantum mechanics as a yet further step of the tendency within physical theory towards the purely functional theory of the concept and functional characterization of objectivity. In this respect DI can be considered an “update”, like the earlier monograph Zur Einsteinschen Relativitätstheorie: Erkenntnistheoretische Betrachtungen, to Substanzbegriff und Funktionsbegriff, (...) a seminal work considering only classical and pre-relativistic physics. But how does DI cohere with the three volumes of The Philosophy of Symbolic Forms providing a systematic survey of symbolic meanings in diverse aspects of culture, each with its own mode of “objectification” via self-created signs and images? Cassirer’s “phenomenology of cognition” via distinct types of symbolic form locates Dirac’s Principles of Quantum Mechanics as an exemplar within physical theory of purely symbolic thought, a realm of pure relations and their correlated meanings. In particular, Dirac’s characterization of the new notion of “physical state” in quantum mechanics by a “symbolic algebra of observables” severed from particular representations points to a limiting pole of the Bedeutungsfunktion, the third and highest mode of symbolic formation. (shrink)

Recently, Brukner and Zeilinger 3354) have claimed that the Shannon information is not well defined as a measure of information in quantum mechanics, adducing arguments that seek to show that it is inextricably tied to classical notions of measurement. It is shown here that these arguments do not succeed: the Shannon information does not have problematic ties to classical concepts. In a further argument, Brukner and Zeilinger compare the Shannon information unfavourably to their preferred information measure, I , (...) with regard to the definition of a notion of “total information content.” This argument is found unconvincing and the relationship between individual measures of information and notions of “total information content” investigated. We close by considering the prospects of Zeilinger's Foundational Principle as a foundational principle for quantum mechanics. (shrink)

The explicit history of the “hidden variables” problem is well-known and established. The main events of its chronology are traced. An implicit context of that history is suggested. It links the problem with the “conservation of energy conservation” in quantum mechanics. Bohr, Kramers, and Slaters (1924) admitted its violation being due to the “fourth Heisenberg uncertainty”, that of energy in relation to time. Wolfgang Pauli rejected the conjecture and even forecast the existence of a new and unknown then (...) elementary particle, neutrino, on the ground of energy conservation in quantum mechanics, afterwards confirmed experimentally. Bohr recognized his defeat and Pauli’s truth: the paradigm of elementary particles (furthermore underlying the Standard model) dominates nowadays. However, the reason of energy conservation in quantum mechanics is quite different from that in classical mechanics (the Lie group of all translations in time). Even more, if the reason was the latter, Bohr, Cramers, and Slatters’s argument would be valid. The link between the “conservation of energy conservation” and the problem of hidden variables is the following: the former is equivalent to their absence. The same can be verified historically by the unification of Heisenberg’s matrix mechanics and Schrödinger’s wave mechanics in the contemporary quantum mechanics by means of the separable complex Hilbert space. The Heisenberg version relies on the vector interpretation of Hilbert space, and the Schrödinger one, on the wave-function interpretation. However the both are equivalent to each other only under the additional condition that a certain well-ordering is equivalent to the corresponding ordinal number (as in Neumann’s definition of “ordinal number”). The same condition interpreted in the proper terms of quantum mechanics means its “unitarity”, therefore the “conservation of energy conservation”. In other words, the “conservation of energy conservation” is postulated in the foundations of quantum mechanics by means of the concept of the separable complex Hilbert space, which furthermore is equivalent to postulating the absence of hidden variables in quantum mechanics (directly deducible from the properties of that Hilbert space). Further, the lesson of that unification (of Heisenberg’s approach and Schrödinger’s version) can be directly interpreted in terms of the unification of general relativity and quantum mechanics in the cherished “quantum gravity” as well as a “manual” of how one can do this considering them as isomorphic to each other in a new mathematical structure corresponding to quantum information. Even more, the condition of the unification is analogical to that in the historical precedent of the unifying mathematical structure (namely the separable complex Hilbert space of quantum mechanics) and consists in the class of equivalence of any smooth deformations of the pseudo-Riemannian space of general relativity: each element of that class is a wave function and vice versa as well. Thus, quantum mechanics can be considered as a “thermodynamic version” of general relativity, after which the universe is observed as if “outside” (similarly to a phenomenological thermodynamic system observable only “outside” as a whole). The statistical approach to that “phenomenological thermodynamics” of quantum mechanics implies Gibbs classes of equivalence of all states of the universe, furthermore re-presentable in Boltzmann’s manner implying general relativity properly … The meta-lesson is that the historical lesson can serve for future discoveries. (shrink)

A motivation is given for expressing classical mechanics in terms of diagonal projection matrices and diagonal density matrices. Then quantum mechanics is seen to be a simple generalization in which one replaces the diagonal real matrices with suitable Hermitian matrices.