In a recent paper [e-print quant-ph/0101012], Hardy has given a derivation of “quantum theory from five reasonable axioms.” Here we show that Hardy's first axiom, which identifies probability with limiting frequency in an ensemble, is not necessary for his derivation. By reformulating Hardy's assumptions, and modifying a part of his proof, in terms of Bayesian probabilities, we show that his work can be easily reconciled with a Bayesian interpretation of quantum probability.

We introduce a new “positive formalism” for encoding quantum theories in the general boundary formulation, somewhat analogous to the mixed state formalism of the standard formulation. This makes the probability interpretation more natural and elegant, eliminates operationally irrelevant structure and opens the general boundary formulation to quantum information theory.

It is an old idea, lately out of fashion but now experiencing a revival, that quantum mechanics may best be understood, not as a physical theory with a problematic probabilistic interpretation, but as something closer to a probability calculus per se. However, from this angle, the rather special C *-algebraic apparatus of quantum probability theory stands in need of further motivation. One would like to find additional principles, having clear physical and/or probabilistic content, on the basis (...) of which this apparatus can be reconstructed. In this paper, I explore one route to such a derivation of finite-dimensional quantum mechanics, by means of a set of strong, but probabilistically intelligible, axioms. Stated very informally, these require that systems appear completely classical as restricted to a single measurement, that different measurements, and likewise different pure states, be equivalent (up to the action of a compact group of symmetries), and that every state be the marginal of a bipartite non-signaling state perfectly correlating two measurements. This much yields a mathematical representation of (basic, discrete) measurements as orthonormal subsets of, and states, by vectors in, an ordered real Hilbert space – in the quantum case, the space of Hermitian operators, with its usual tracial inner product. One final postulate (a simple minimization principle, still in need of a clear interpretation) forces the positive cone of this space to be homogeneous and self-dual and hence, to be the state space of a formally real Jordan algebra. From here, the route to the standard framework of finite-dimensional quantum mechanics is quite short. (shrink)

The apparent nonlocality of quantum theory has been a persistent concern. Einstein et al. and Bell emphasized the apparent nonlocality arising from entanglement correlations. While some interpretations embrace this nonlocality, modern variations of the Everett-inspired many worlds interpretation try to circumvent it. In this paper, we review Bell's "no-go" theorem and explain how it rests on three axioms, local causality, no superdeterminism, and one world. Although Bell is often taken to have shown that local causality is ruled (...) out by the experimentally confirmed entanglement correlations, we make clear that it is the conjunction of the three axioms that is ruled out by these correlations. We then show that by assuming local causality and no superdeterminism, we can give a direct proof of many worlds. The remainder of the paper searches for a consistent, local, formulation of many worlds. We show that prominent formulations whose ontology is given by the wave function violate local causality, and we critically evaluate claims in the literature to the contrary. We ultimately identify a local many worlds interpretation that replaces the wave function with a separable Lorentz-invariant wave-field. We conclude with discussions of the Born rule, and other interpretations of quantum mechanics. (shrink)

Reconstructions of quantum theory are a novel research program in theoretical physics which aims to uncover the unique physical features of quantum theory via axiomatization. I focus on Hardy’s “Quantum Theory from Five Reasonable Axioms” (2001), arguing that reconstructions represent a modern usage of axiomatization with significant points of continuity to von Neumann’s axiomatizations in quantum mechanics. In particular, I show that Hardy and von Neumann share similar methodological ordering, have a common (...) operational framing, and insist on the empirical basis of axioms. In the reconstruction programme, interesting points of discontinuity with historical axiomatizations include the stipulation of a generalized space of theories represented by a framework and the stipulation of analytic machinery at two levels of generality (first by establishing a generalized mathematical framework and then by positing specific formulations of axioms). In light of the reconstruction programme, I show that we should understand axiomatization attempts as being context–dependent, context which is contingent upon the goals of inquiry and the maturity of both mathematical formalism and theoretical underpinnings within the area of inquiry. Drawing on Mitsch (2022)’s account of axiomatization, I conclude that reconstructions should best be understood as provisional, practical, representations of quantum theory that are well suited for theory development and exploration. However, I propose my context–dependent re–framing of axiomatization as a means of enriching Mitsch’s account. (shrink)

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. (shrink)

In this paper we use the framework of generalized probabilistic theories to present two sets of basic assumptions, called axioms, for which we show that they lead to the Hilbert space formulation of quantum mechanics. The key results in this derivation are the co-ordinatization of generalized geometries and a theorem of Solér which characterizes Hilbert spaces among the orthomodular spaces. A generalized Wigner theorem is applied to reduce some of the assumptions of Solér’s theorem to the theory (...) of symmetry in quantum mechanics. Since this reduction is only partial we also point out the remaining open questions. (shrink)

Quantum probability is used to provide a new organization of basic quantum theory in a logical, axiomatic way. The principal thesis is that there is one fundamental time evolution equation in quantum theory, and this is given by a new version of Born’s Rule, which now includes both consecutive and conditional probability as it must, since science is based on correlations. A major modification of one of the standard axioms of quantum theory (...) allows the implementation of various mathematically distinct models (commonly called ‘pictures’) of them. A natural definition of isomorphism of models is presented next. For a given Hamiltonian the standard ‘pictures’ of quantum theory (e.g., Schrödinger, Heisenberg, interaction) are isomorphic models, whose probabilities are nonetheless model independent. The Schrödinger equation remains valid in one model of the axioms, even though it is no longer the fundamental equation of time evolution. All the usual calculations of standard, textbook quantum theory remain valid, but they are put in a new light. For example, the ‘collapse’ of the state is now viewed in the Schrödinger model as _one_ step in a _two_ step algorithm for calculating _one_ conditional probability. In the isomorphic Heisenberg model, where states are constant in time, one has the same conditional probability given by the same formula. Also, entanglement is defined in a new, more general model independent way as an aspect of quantum probability without using ‘collapse’ or tensor products. (shrink)

The axioms for the density operator in quantum mechanics are discussed. A comparison is made with an axiomatization based on Gleason's theorem.

This axiomatization is based on the observation that ifG is the group of automorphisms of the states (induced, e.g., by suitable evolutions), then we can define a spherical function by mapping each element ofG to the matrix of its transition probabilities. Starting from five physically conservative axioms, we utilize the correspondence between spherical functions and representations to apply the structure theory for compact Lie groups and their orbits in representation spaces to arrive at the standard complex Hilbert space (...) structure of quantum mechanics. (shrink)

Exponential mappings into an imaginary space or number field for the axioms of a theory, which are in the form of propositional constants and variables, make possible: (a) an understanding of the meaning and differences between the Lorentz transformation constants, such that their product is still equal to one, but the axioms at each end of the transformations are logically inverse and separately consistent; (b) an interpretation of the psi function phase factor which is part of the (...) axiomE=hf; (c) the unification of the quantum-mechanical psi function and the electromagnetic wave function. Thus, those statements whose mechanisms are unknown (the axioms of the theory) are to be assigned the axiom propositional number symbol ϑ and are to be associated with the complex probability eiϑ, which is a uniform factor of the energy equations expressing the physical state. Such probabilistic axiom functions can be associated with both the special theory of relativity and the quantum-electromagnetic theory. (shrink)

The present paper is the first part of a work which follows up on H. Kummer: “A constructive approach to the foundations of quantum mechanics,”Found. Phys. 17, 1–63 (1987). In that paper we deduced the JB-algebra structure of the space of observables (=detector space) of quantum mechanics within an axiomatic theory which uses the concept of a filter as primitive under the restrictive assumption that the detector space is finite-dimensional. This additional hypothesis will be dropped in the (...) present paper.It turns out that the relevant mathematics for our approach to a quantum mechanical system with infinite-dimensional detector space is the noncommutative spectral theory of Alfsen and Shultz.We start off with the same situation as in the previous paper (cf. Sects. 1 and 2 of the present paper). By postulating four axioms (Axioms S, DP, R, and SP of Sec. 3), we arrive in a natural way at the mathematical setting of Alfsen and Shultz, which consists of a dual pair of real ordered linear spaces 〈Y, M〉: A base norm space, called the strong source space (which, however, in slight contrast to the setting of Alfsen and Shultz, is not 1-additive) and an order unit space, called the weak detector space, which is the norm and order dual space of Y. The last section of part I contains the guiding example suggested by orthodox quantum mechanics. We observe that our axioms are satisfied in this example. In the second part of this work (which will appear in the next issue of this journal) we shall postulate three further axioms and derive the JB-algebra structure of quantum mechanics. (shrink)

Algebraic quantum field theory (AQFT) puts forward three ``causal axioms'' that aim to characterize the theory as one that implements relativistic causation: the spectrum condition, microcausality, and primitive causality. In this paper, I aim to show, in a minimally technical way, that none of them fully explains the notion of causation appropriate for AQFT because they only capture some of the desiderata for relativistic causation I state or because it is often unclear how each axiom implements (...) its respective desideratum. After this diagnostic, I will show that a fourth condition, local primitive causality (LPC), fully characterizes relativistic causation in the sense of fulfilling all the relevant desiderata. However, it only encompasses the virtues of the other axioms because it is implied by them, as I will show from a construction by Haag and Schroer (1962). Since the conjunction of the three causal axioms implies LPC and other important results in QFT that LPC does not imply, and since LPC helps clarify some of the shortcomings of the three axioms, I advocate for a holistic interpretation of how the axioms characterize the causal structure of AQFT against the strategy in the literature to rivalize the axioms and privilege one among them. (shrink)

Conventional quantum mechanics with a complex Hilbert space and the Born Rule is derived from five axioms describing experimentally observable properties of probability distributions for the outcome of measurements. Axioms I, II, III are common to quantum mechanics and hidden variable theories. Axiom IV recognizes a phenomenon, first noted by von Neumann (in Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, 1955) and independently by Turing (Teuscher and Hofstadter, Alan Turing: Life and Legacy of (...) a Great Thinker, Springer, Berlin, 2004), in which the increase in entropy resulting from a measurement is reduced by a suitable intermediate measurement. This is shown to be impossible for local hidden variable theories. Axiom IV, together with the first three, almost suffice to deduce the conventional rules but allow some exotic, alternatives such as real or quaternionic quantum mechanics. Axiom V recognizes a property of the distribution of outcomes of random measurements on qubits which holds only in the complex Hilbert space model. It is then shown that the five axioms also imply the conventional rules for any finite dimension. (shrink)

We develop and defend the thesis that the Hilbert space formalism of quantum mechanics is a new theory of probability. The theory, like its classical counterpart, consists of an algebra of events, and the probability measures defined on it. The construction proceeds in the following steps: (a) Axioms for the algebra of events are introduced following Birkhoff and von Neumann. All axioms, except the one that expresses the uncertainty principle, are shared with the classical event (...) space. The only models for the set of axioms are lattices of subspaces of inner product spaces over a field K. (b) Another axiom due to Soler forces K to be the field of real, or complex numbers, or the quaternions. We suggest a probabilistic reading of Soler's axiom. (c) Gleason's theorem fully characterizes the probability measures on the algebra of events, so that Born's rule is derived. (d) Gleason's theorem is equivalent to the existence of a certain finite set of rays, with a particular orthogonality graph (Wondergraph). Consequently, all aspects of quantum probability can be derived from rational probability assignments to finite "quantum gambles". (e) All experimental aspects of entanglement- the violation of Bell's inequality in particular- are explained as natural outcomes of the probabilistic structure. (f) We hypothesize that even in the absence of decoherence macroscopic entanglement can very rarely be observed, and provide a precise conjecture to that effect .We also discuss the relation of the present approach to quantum logic, realism and truth, and the measurement problem. (shrink)

In this paper we review the general framework of operational probabilistic theories, along with the six axioms from which quantum theory can be derived. We argue that the OPT framework along with a relaxed version of five of the axioms, define a general information theory. We close the paper with considerations about the role of the observer in an OPT, and the interpretation of the von Neumann postulate and the Schrödinger-cat paradox.

Axioms in set theory typically have the form , where is a relation which links with in some way. In this paper we introduce a particular linkage relation and a single axiom based on from which all the axioms of (Zermelo set theory) can be derived as theorems. The single axiom is presented both in informal and formal versions. This calls for some discussion of pertinent features of formal and informal axiomatic method and some discussion of (...) pertinent features of the system of set theory to be erected on the single axiom. is shown to be somewhat stronger than , but much weaker than (Zermelo-Fraenkel set theory). (shrink)

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Although philosophers have considered some of the implications of the nature of quantum statistics of many-particle systems for the interpretation problem, e.g., Reichenbach, they have not produced a complete analysis of the relationship between aspects of quantum statistics and complications and/or possible solutions of the interpretation problem. While the present work by no means provides a complete account, it does explore some heretofore uncharted regions. One of the latter is an analysis of a situation that I call 'The (...) Paradox of Identical Particles ,' which arises in Bose-Einstein statistics. PIP refers to the fact that Elementary Quantum Mechanics seems to be committed to the thesis that systems of particles of the same species, "identical" particles, exhibit behavior differing from that of systems of particles of different species, simply in virtue of the fact that the former contain entities of the same species while the latter do not. Taken at face value this is tantamount to the Empedoclean "axiom" that like attracts like, while like and unlike repel, a somewhat embarassingly naive proposition for modern physicists to embrace, and one which, furthermore, poses a host of perplexing problems, e.g., how do the particles "know" whether they are of the same species or not? A result proven in the present work is that any "pluralistic" interpretation of quantum theory is unable to provide a satisfactory way of ridding quantum physicists of this appeal to what I call "species sensitivity." Arguing that Quantum Field Theory is most naturally and plausibly interpreted as a "monistic" theory--which means, inter alia, that for a system of n identical particles to exist is not for n distinct objects of any kind to exist, but rather for one unified underlying entity, a quantum field, to be in a certain kind of state or condition--I show that from this point of view PIP can be dissolved. On the basis of this and other considerations, I argue that the interpretation problem is more amenable to solution if we take the QFT point of view as the fundamental expression of quantum theory. (shrink)

The original development of the formalism of quantum mechanics involved the study of isolated quantum systems in pure states. Such systems fail to capture important aspects of the warm, wet, and noisy physical world which can better be modelled by quantum statistical mechanics and local quantum field theory using mixed states of continuous systems. In this context, we need to be able to compute quantum probabilities given only partial information. Specifically, suppose that B is (...) a set of operators. This set need not be a von Neumann algebra. Simple axioms are proposed which allow us to identify a function which can be interpreted as the probability, per unit trial of the information specified by B, of observing the (mixed) state of the world restricted to B to be σ when we are given ρ – the restriction to B of a prior state. This probability generalizes the idea of a mixed state (ρ) as being a sum of terms (σ) weighted by probabilities. The unique function satisfying the axioms can be defined in terms of the relative entropy. The analogous inference problem in classical probability would be a situation where we have some information about the prior distribution, but not enough to determine it uniquely. In such a situation in quantum theory, because only what we observe should be taken to be specified, it is not appropriate to assume the existence of a fixed, definite, unknown prior state, beyond the set B about which we have information. The theory was developed for the purposes of a fairly radical attack on the interpretation of quantum theory, involving many-worlds ideas and the abstract characterization of observers as finite information-processing structures, but deals with quantum inference problems of broad generality. (shrink)

This paper develops some ideas from previous work (coauthored, mostly with C.J.Isham). In that work, the main proposal is to assign as the value of a physical quantity in quantum theory (or classical physics), not a real number, but a certain kind of set (a sieve) of quantities that are functions of the given quantity. The motivation was in part physical---such a valuation illuminates the Kochen-Specker theorem; in part mathematical---the valuations arise naturally in the theory of presheaves; (...) and in part conceptual---the valuations arise from applying to propositions about the values of physical quantities some general axioms governing partial truth for any kind of proposition. In this paper, I give another conceptual motivation for the proposal. I develop (in Sections 2 and 3) the notion of a topos (of which presheaves give just one kind of example); and explain how this notion gives a satisfactory general framework for making sense of the idea of partial truth. Then I review (in Section 4) how our proposal applies this framework to the case of physical theories. (shrink)

A simple mathematical extension of quantum theory is presented. As well as opening the possibility of alternative methods of calculation, the additional formalism implies a new physical interpretation of the standard theory by providing a picture of an external reality. The new formalism, developed first for the single-particle case, has the advantage of generalizing immediately to quantum field theory and to the description of relativistic phenomena such as particle creation and annihilation.

This paper surveys the issue of relativistic causality within the framework of algebraic quantum field theory . In doing so, we distinguish various notions of causality formulated in the literature and study their relationships, and thereby we offer what we hope to be a useful taxonomy. We propose that the most direct expression of relativistic causality in AQFT is captured not by the spectrum condition but rather by the axiom of local primitive causality, in that it entails a (...) form of local determinism for quantum fields which generalizes the constraint of no superluminal propagation of classical field theories to relativistic quantum field theory. We discuss the status of the axiom of micro-causality by locating its place within a large family of separability/independence/locality conditions developed for AQFT and also by relating it to so-called no-signalling theorems. And we also provide a critical survey of attempts to understand the implications for relativistic causality of the distant correlations endemic to the states in models of AQFT satisfying the standard axioms, and we provide an assessment of attempts to employ Reichenbach's common cause principle in AQFT to defuse worries that these distant correlations implicate direct causal connections between relatively spacelike events. (shrink)

In their paper “A note on Misunderstandings of Piron's Axioms for Quantum Mechanics,” Foulis and Randall undertake a reply to our critique of Piron's “question-proposition system” (qp-s) which appeared in previous issues of this journal. In the present paper, we want briefly to refute the points of criticism raised by Foulis and Randall (FR). We argue that the “misunderstandings” are not ours, and we prove it.

The most majestic scientific achievement, of this century in mathematical beauty, axiomatic consistency, and experimental verifications has been special relativity with its unitary structure at the operator level, and canonical structure at the classical levels, which has turned out to be exactly valid for point particles moving in the homogenenous and isotropic vacuum (exterior dynamical problems). In recent decades a number of authors have studied nonunitary and noncanonical theories, here generally calleddeformations for the representation of broader conditions, such as extended (...) and deformable particles moving within inhomogeneous and anisotrophic physical media (interior dynamical problems). In this paper we show that nonunitary deformations, including, q-, k-, quatum-, Lie-isotopic, Lie-admissible, and other deformations, even thoughmathematically correct, have a number of problematic aspects ofphysical character when formulated on conventional spaces over conditional fields, such as lack of invariance of the basic space-time units, ambiguous applicability to measurements, loss of Hermiticity-observability in time, lack of invariant numerical predictions, loss of the axions of special relativity, and others. We then show that the classical noncanonical counterparts of the above nonunitary deformations are equally afflicted by corresponding problems of physical consistency. We also show that the contemporary formulation of gravity is afflicted by similar problematic aspects because Riemannian spaces are noncanonical deformations of Minkowskian spaces, thus having noninvariant space-time units. We then point out that new mathematical methods, calledisotopies, genotopies, hyperstructures and their isoduals, offer the possibilities of constructing a nonunitary theory, known asrelativistic hadronic mechanics which: (1) is as axiomatically consistent as relativistic quantum mechanics, (2) preserves the abstract axioms of special relativity, and (3) results in a completion of the conventional mechanics much along the celebrated Einstein-Podolski-Rosen argument. A number of novel applications are indicated, such as a geometric unification of the special and general relativity via the isominkowskian geometry in which the two relativities are differentiated via the invariant basic unit, while preserving conventional Riemannian metrics, Einstein's field equations, and related experimental verifications; a novel operator form of gravity verifying the axioms of relativistic quantum mechanics under the universal isopoincaré symmetry; a new structure model of hadrons with conventional massive particles as physical constituents which is compatile with composite quarks and with established unitary classifications; and other novels applications in nuclear physics, astrophysics, theoretical biology, and other fields. The paper ends with the proposal of a number of new experiments, some of which may imply new practical applications, such as conceivable new forms of recycling nuclear waste. The achievement of axiomatic consistency in the study of the above physical problems has been possible for the first time in this paper thanks to mathematical advances that recently appeared in a special issue of theRendiconti Circolo Matematico Palermo, and in other journals identified in the Acknowledgements. (shrink)

Piron's axioms for a realistically interpreted quantum mechanics are analyzed in detail within the context of a formal mathematical structure expressed in the conventional set-theoretic idiom of mathematics. As a result, some of the serious misconceptions that have encouraged recent criticisms of Piron's axioms are exposed.

Empirical adequacy, formal explanation and understanding are distinct goals of science. While no a priori criterion for understanding should be laid down, there may be inherent limitations on the way we are able to understand explanations of physical phenomena. I examine several recent contributions to the exercise of fashioning an explanatory discourse to mold the formal explanation provided by quantum mechanics to our modes of understanding. The question is whether we are capable of truly understanding (or comprehending) quantum (...) phenomena, as opposed to simply accepting the formalism and certain irreducible quantum correlations. The central issue is that of understanding versus merely redefining terms to paper over our ignorance. (shrink)

In the foundations of quantum mechanics Gleason’s theorem dictates the uniqueness of the state transition probability via the inner product of the corresponding state vectors in Hilbert space, independent of which measurement context induces this transition. We argue that the state transition probability should not be regarded as a secondary concept which can be derived from the structure on the set of states and properties, but instead should be regarded as a primitive concept for which measurement context is crucial. (...) Accordingly, we adopt an operational approach to quantum mechanics in which a physical entity is defined by the structure of its set of states, set of properties and the possible (measurement) contexts which can be applied to this entity. We put forward some elementary definitions to derive an operational theory from this State–COntext–Property (SCOP) formalism. We show that if the SCOP satisfies a Gleason-like condition, namely that the state transition probability is independent of which measurement context induces the change of state, then the lattice of properties is orthocomplemented, which is one of the ‘quantum axioms’ used in the Piron–Solèr representation theorem for quantum systems. In this sense we obtain a possible physical meaning for the orthocomplementation widely used in quantum structures. (shrink)

In 1981, Takeuti introduced quantum set theory as the quantum counterpart of Boolean valued models of set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed subspaces in a Hilbert space and showed that appropriate quantum counterparts of ZFC axioms hold in the model. Here, Takeuti's formulation is extended to construct a model of set theory based on the logic represented by the (...) lattice of projections in an arbitrary von Neumann algebra. A transfer principle is established that enables us to transfer theorems of ZFC to their quantum counterparts holding in the model. The set of real numbers in the model is shown to be in one-to-one correspondence with the set of self-adjoint operators affiliated with the von Neumann algebra generated by the logic. Despite the difficulty pointed out by Takeuti that equality axioms do not generally hold in quantum set theory, it is shown that equality axioms hold for any real numbers in the model. It is also shown that any observational proposition in quantum mechanics can be represented by a corresponding statement for real numbers in the model with the truth value consistent with the standard formulation of quantum mechanics, and that the equality relation between two real numbers in the model is equivalent with the notion of perfect correlation between corresponding observables (self-adjoint operators) in quantum mechanics. The paper is concluded with some remarks on the relevance to quantum set theory of the choice of the implication connective in quantum logic. (shrink)

When constrained by limited resources, how do we choose axioms of rationality? The target article relies on Bayesian reasoning that encounter serioustractabilityproblems. We propose another axiomatic foundation: quantum probability theory, which provides for less complex and more comprehensive descriptions. More generally, defining rationality in terms of axiomatic systems misses a key issue: rationality must be defined by humans facing vague information.

Having analyzed the formal aspects of Wallace’s proof of the Born rule, we now discuss the concepts and axioms upon which it is built. Justification for most axioms is shown to be problematic, and at times contradictory. Some of the problems are caused by ambiguities in the concepts used. We conclude the axioms are not reasonable enough to be taken as mandates of rationality in Everettian Quantum Mechanics. This invalidates the interpretation of Wallace’s result as meaning (...) it would be rational for Everettian agents to decide using the Born rule. (shrink)

We put forward a possible new interpretation and explanatory framework for quantum theory. The basic hypothesis underlying this new framework is that quantum particles are conceptual entities. More concretely, we propose that quantum particles interact with ordinary matter, nuclei, atoms, molecules, macroscopic material entities, measuring apparatuses, in a similar way to how human concepts interact with memory structures, human minds or artificial memories. We analyze the most characteristic aspects of quantum theory, i.e. entanglement and (...) non-locality, interference and superposition, identity and individuality in the light of this new interpretation, and we put forward a specific explanation and understanding of these aspects. The basic hypothesis of our framework gives rise in a natural way to a Heisenberg uncertainty principle which introduces an understanding of the general situation of ‘the one and the many’ in quantum physics. A specific view on macro and micro different from the common one follows from the basic hypothesis and leads to an analysis of Schrödinger’s Cat paradox and the measurement problem different from the existing ones. We reflect about the influence of this new quantum interpretation and explanatory framework on the global nature and evolutionary aspects of the world and human worldviews, and point out potential explanations for specific situations, such as the generation problem in particle physics, the confinement of quarks and the existence of dark matter. (shrink)

These are notes designed to bring the beginning student of the philosophy of quantum mechanics 'up to scratch' on the mathematical background needed to understand elementary finite-dimensional quantum theory. There are just three chapters: Ch. 1 'Vector Spaces'; Ch. 2 'Inner Product Spaces'; and Ch. 3 'Operators on Finite-Dimensional Complex Inner Product Spaces'. The notes are entirely self-contained and presuppose knowledge of only high school level algebra.

If $\mathcal{A}$ (V) is a net of local von Neumann algebras satisfying standard axioms of algebraic relativistic quantum field theory and V 1 and V 2 are spacelike separated spacetime regions, then the system ( $\mathcal{A}$ (V 1 ), $\mathcal{A}$ (V 2 ), φ) is said to satisfy the Weak Reichenbach's Common Cause Principle iff for every pair of projections A∈ $\mathcal{A}$ (V 1 ), B∈ $\mathcal{A}$ (V 2 ) correlated in the normal state φ there exists (...) a projection C belonging to a von Neumann algebra associated with a spacetime region V contained in the union of the backward light cones of V 1 and V 2 and disjoint from both V 1 and V 2 , a projection having the properties of a Reichenbachian common cause of the correlation between A and B. It is shown that if the net has the local primitive causality property then every local system ( $\mathcal{A}$ (V 1 ), $\mathcal{A}$ (V 2 ), φ) with a locally normal and locally faithful state φ and suitable bounded V 1 and V 2 satisfies the Weak Reichenbach's Common Cause Principle. (shrink)

The 1927 Solvay conference was perhaps the most important meeting in the history of quantum theory. Contrary to popular belief, the interpretation of quantum theory was not settled at this conference, and no consensus was reached. Instead, a range of sharply conflicting views were presented and extensively discussed, including de Broglie's pilot-wave theory, Born and Heisenberg's quantum mechanics, and Schrödinger's wave mechanics. Today, there is no longer an established or dominant interpretation of quantum (...) theory, so it is important to re-evaluate the historical sources and keep the interpretation debate open. This book contains a complete translation of the original proceedings, with background essays on the three main interpretations of quantum theory presented at the conference, and an extensive analysis of the lectures and discussions in the light of current research in the foundations of quantum theory. The proceedings contain much unexpected material, including extensive discussions of de Broglie's pilot-wave theory (which de Broglie presented for a many-body system), and a theory of 'quantum mechanics' apparently lacking in wave function collapse or fundamental time evolution. This book will be of interest to graduate students and researchers in physics and in the history and philosophy of quantum theory. (shrink)

In this work we attempt to confront the orthodox widespread claim, present in the philosophical and foundational debates about Quantum Mechanics (QM), that ‘superpositions are never actually observed in the lab’. In order to do so, we begin by providing a critical analysis of the famous measurement problem which, we will argue, was originated as a consequence of the strict application of the empirical-positivist requirements to subsume the quantum formalism under their specific understanding of a physical ‘theory’. (...) In particular, the ad hoc introduction of the projection postulate (or measurement rule) can be understood as the necessity of imposing a naive empiricist standpoint which presupposes that observations are self evident givens of “common sense” experience independent of metaphysical (or conceptual) presuppositions yet necessarily represented in binary terms. We then turn our attention to two “non-collapse” interpretations of QM—namely, modal and many worlds—which even though deny that the “collapse” is a real physical process retain anyhow the projection rule as a necessary axiom of the theory itself. In contraposition, following Einstein’s claim that “it is only the theory which decides what can be observed”, we propose a return to the realist representational understanding of physical theories in which ‘observation’ is not a “self evident” given of experience but something that can be only understood in the context of a theoretical (formal-conceptual) scheme. It is from this standpoint that we discuss a new non-classical conceptual representation which allows us to understand quantum phenomena in an intuitive (anschaulicht) manner. Leaving behind the projection postulate, we discuss the general physical conditions for measuring and observing quantum superpositions in the lab. (shrink)

Quantum information science is a source of task-related axioms whose consequences can be explored in general settings encompassing quantum mechanics, classical theory, and more. Quantum states are compendia of probabilities for the outcomes of possible operations we may perform on a system: ''operational states.'' I discuss general frameworks for ''operational theories'' (sets of possible operational states of a system), in which convexity plays key role. The main technical content of the paper is in a theorem (...) that any such theory naturally gives rise to a ''weak effect algebra'' when outcomes having the same probability in all states are identified and in the introduction of a notion of ''operation algebra'' that also takes account of sequential and conditional operations. Such frameworks are appropriate for investigating what things look like from an ''inside view,'' i.e., for describing perspectival information that one subsystem of the world can have about another. Understanding how such views can combine, and whether an overall ''geometric'' picture (''outside view'') coordinating them all can be had, even if this picture is very different in structure from the perspectives within it, is the key to whether we may be able to achieve a unified, ''objective'' physical view in which quantum mechanics is the appropriate description for certain perspectives, or whether quantum mechanics is truly telling us we must go beyond this ''geometric'' conception of physics. (shrink)

We motivate the possibility of using notions and methods derived from quantum physics, and more specifically from the research field known as ‘quantum cognition’, to optimally model different situations in the field of medicine, its decision-making processes and ensuing practices, particularly in relation to chronic and rare diseases. This also as a way to devise alternative approaches to the generally adopted double-blind gold standard.

Quantum theoretical developments in physical science challenge the foundational assumptions of both realist and constructivist social paradigms. Furthermore, when quantum metaphysics is coupled with biological, neuro-scientific discoveries that the brain regenerates and reprograms itself throughout life in response to environmental challenges and the force of attention and will, the result is a different picture of human nature and the social behavior that is possible, ethical, and scientifically plausible than that suggested by either social realists or constructivists. This article (...) explores the frontier of recent developments in the physical and biological sciences and considers how these findings might allow for a new foundation for social theory. (shrink)

This book explains, in simple but accurate terms, how orthodox quantum mechanics works. The author, a distinguished theoretical physicist, shows how this theory, realistically interpreted, assigns an important role to our conscious free choices. Stapp claims that mainstream biology and neuroscience, despite nearly a century of quantum physics, still stick essentially to failed classical precepts in which mental intentions have no effect upon our bodily actions. He shows how quantum mechanics provides a rational basis for a (...) better understanding of this connection, even allowing an explanation of certain phenomena currently held to be "paranormal". These ideas have major implications for our understanding of ourselves and our mental processes, and thus also for the meaningfulness of our lives. (shrink)

In this paper we investigate Accardi's claim that the "quantum paradoxes" have their roots in probability theory and that, in particular, they can be evaded by giving up Bayes' rule, concerning the relation between composite and conditional probabilities. We reach the conclusion that, although it may be possible to give up Bayes' rule and define conditional probabilities differently, this contributes nothing to solving the philosophical problems which surround quantum mechanics.

It is shown that the Hilbert space formalism of quantum mechanics can be derived as a corrected form of probability theory. These constructions yield the Schrödinger equation for a particle in an electromagnetic field and exhibit a relationship of this equation to Markov processes. The operator formalism for expectation values is shown to be related to anL 2 representation of marginal distributions and a relationship of the commutation rules for canonically conjugate observables to a topological relationship of two (...) manifolds is indicated. (shrink)