Entanglement and nonlocality are studied in the framework of pre-/postselected ensembles with the aid of weak measurements and the two-state-vector formalism. In addition to the EPR–Bohm experiment, we revisit the Hardy and Cheshire Cat experiments, whose entangled pre- or postselected states give rise to curious phenomena. We then turn to even more peculiar phenomenon suggesting “emerging correlations” between independent preand postselected ensembles of particles. This can be viewed as a quantum violation of the classical “pigeonhole principle.”

Introduction

It is seldom acknowledged that seven years before the celebrated Bell paper [1], Bohm and Aharonov [2] published an analysis of the EPR paradox [3]. They suggested an experimental setup, based on Compton scattering, for testing nonlocal correlations between the polarizations of two annihilation photons. In 1964, Bell proposed his general inequality, thereby excluding local realism. During the same time, Aharonov et al. constructed the foundations of a time-symmetric formalism of quantum mechanics [4]. While Bell's proof utilizes entanglement to demonstrate nonlocal correlations,wewill describe inwhat follows the emergence of nonlocal correlations between product states. For this purpose, however, we shall invoke weak measurements of pre- and postselected ensembles.

In classical mechanics, initial conditions of position and velocity for every particle fully determine the time evolution of the system. Therefore, trying to impose an additional final condition would lead either to redundancy or to inconsistency with the initial conditions. This is radically different in quantum mechanics. Because of the uncertainty principle, an initial state vector does not fully determine, in general, the outcome of a future measurement. However, adding a final (backward-evolving) state vector results in a more complete description of the quantum system in between these two boundary conditions, which has a bearing on the determination of measurement outcomes.

The basis for this time-symmetric formulation of quantum mechanics was laid down by Aharonov, Bergman, and Lebowitz (ABL), who derived a symmetric probability rule concerning measurements performed on systems, while taking into account the final state of the system, in addition to the usual initial state [4].

Abstract

I begin with some reminiscences of my delayed appreciation of the significance of John Bell'swork. Preparatory remarks on my general perspective concerning the interpretation of quantum mechanics are presented. I argue against the conflation of hyperplane dependence with frame dependence, which occurs occasionally; and that the ‘elements of reality’ of Hardy's famous gedankenexperiment can retain their Lorentz invariance, i.e., their frame independence, if one recognizes the hyperplane dependence of their localization, which follows. Finally, I criticize a view of the nature of Lorentz transformations presented by Asher Peres and co-workers which conflicts with the view employed here.

Initial Reactions to Bell's Work

Like many others in the physics community, I was late to come to an appreciation of the significance of John Bell's papers on the foundations of quantum mechanics (QM). But unlike many of those, my indifference to Bell was not due to a dismissive attitude to foundational studies per se or to an intransigent commitment to some version of Copenhagenism. In 1971, for example, I was very much involved in the international conference on QM foundations at my home institution, where Bell presented the paper “On the hypothesis that the Schrödinger equation is exact,” published later as [1], but I paid minimal attention to his presentation. The cause of this poor judgment (as I eventually came to realize it was) was that I was already convinced that the Schrödinger equation was not exact in the sense Bell meant, but must be augmented with primordial state reductions. I was among those who, in the words of Bob Wald, as reported by Roger Penrose [2], were inclined to “take it seriously,” the QM state that is, and, consequently, could not “really believe in it,” i.e., really believe that purely unitary QM is a complete theory. I was delighted when the experimental tests of Bell's inequalities upheld the QM predictions. I was, furthermore, uninterested in hidden variable reconstructions of QM, such as Bohmian mechanics [3] and Many Worlds interpretations [4, 5], that were, in principle, not susceptible to empirical tests of their novel details.

Abstract

I present the background of the Bohm approach that led John Bell to a study of quantum nonlocality, from which his famous inequalities emerged. I recall the early experiments done at Birkbeck with the aim of exploring the possibility of ‘spontaneous collapse’, a way suggested by Schrödinger to avoid the conclusion that quantum mechanics is grossly nonlocal. I also review some of the work that John did which directly impinged on my own investigations into the foundations of quantum mechanics and report some new investigations towards a more fundamental theory.

Introduction

My first encounter with quantum nonlocality was in discussions with David Bohm when I joined him as an assistant lecturer at Birkbeck in the sixties. Although my PhD thesis had been in condensed matter physics under the supervision of Cyril Domb, I had always been fascinated and puzzled by quantum phenomena, and when the opportunity to study the subject with Bohm came up, I took it. I would listen to the discussions between Bohm and Roger Penrose, who was at Birkbeck at the time, and it soon became clear that the prescribed interpretation of quantum mechanics that I was taught as an undergraduate left too many questions unanswered.

At that time there was a very strange atmosphere in the physics community. I was often informed that there were no problems with the interpretation of the quantum formalism. If I did not follow the well-prescribed rules of the quantum algorithm, I would be ‘wasting my time’. There was no alternative, just do it! But Louis de Broglie and David Bohm had shown there was another way. However, their views were strongly opposed by many, and as far as I could judge their reasons did not seem to be based on mathematics or on logic, but on a preconceived notion that the formalism was, in principle, the best we could do. The experimental data were bizarre when looked upon from the standpoint of classical physics, but the uncertainty principle somehow prevented us from examining the actual process in detail. Reality was somehow veiled (see [1]).Were we condemned to use a mere algorithm or was there an underlying ontology?

Abstract

John Bell was a staunch supporter of the dynamical wave function collapse approach to making a well-defined quantum theory. Through letters from him, I reminisce on my handful of interactions with him, all of which were memorable tome. Then I discuss nonlocality, violation of Bell's inequality and some further implications within the framework of the CSL (continuous spontaneous localization) model of dynamical collapse.

Interactions

I was an instructor at Harvard with a newly minted Ph.D. from MIT, earned with an indifferent thesis on particle theory. It took me two years to come to terms with, dare I say it, the fact that I did not care about the S-matrix. I was going to be out of a job anyway in 1966 so, in the fall of 1965, I put my full energy into writing my very first paper, about what I did care: that, to my eyes there was something deeply wrong with the quantum theory I had been taught; that the rules of how to use it were inadequate. The title of the paper [1] was “Elimination of the reduction postulate from quantum theory and a framework for hidden variable theories.” It was a long title, because it was really two rather separate papers.

The first was based upon the perceived inadequacy that the collapse (reduction) postulate of the theory is ill-defined.

The second stated some postulates I thought a good hidden variable theory ought to obey, and gave a model for their satisfaction in a two-dimensional (i.e., spin-1/2) Hilbert space. Try as I might, I was not able to generalize my model to a higher-dimensional (i.e., spin-1) Hilbert space, but I put it out there for someone more clever than I to achieve. I sent the paper to a very few people who I thought might be interested. One was John Bell, and he replied (see Figure 23.1)!

So someone more clever than I pointed out that generalizing my model could not be achieved. I therefore excised the model and, after rewriting and retitling, published my very first paper [2]!

After a three-year stint at Case Institute of Technology, which graduated to become Case Western Reserve University while I was there, in 1969 I became ensconced at Hamilton College.

John Bell lived in Ireland for only 21 years, but throughout his life he remembered his Irish upbringing with fond memories, pride and gratitude.

Irish Physics and the Irish Tradition

Ireland has a very respectable tradition in physics, and particularly mathematical physics [1]. As early as the eighteenth century, Robert Boyle is usually given credit for establishing the experimental tradition in physics [2, 3]. Through the nineteenth century, luminaries such as William Rowan Hamilton [4], James MacCullagh [5], Thomas Andrews [6], George Francis FitzGerald [7, 8] and Joseph Larmor [9] all made very important contributions to the establishment of physics as an intellectual discipline in its own right, while it should not be forgotten that although the academic careers of George Gabriel Stokes [10],William Thomson [Lord Kelvin] [11] and John Tyndall [12] were spent in England or Scotland, all had Irish origins which they never forgot.

An important aspect of Irish mathematical physics in the nineteenth century was the existence of the so-called Irish tradition, which consisted of studying the wave theory of light and the ether. It was a tradition that was to be important for John Bell and it was to transcend what may be described as the most important event in nineteenth century physics – James Clerk Maxwell's theory of electromagnetism, and his conclusion that light was an electromagnetic wave [13]. The work of MacCullagh and Hamilton was performed in the 1830s and 1840s, while Maxwell's mature work was not to emerge until the 1860s.

MacCullagh improved the theory of Christian Huygens and Augustine Fresnel, being able first to derive the laws of reflection and refraction of light at the surfaces of crystals and metals, and then to write down equations for a light-bearing ether that justified his previous work. He was also able to handle the phenomenon of total internal reflection, and his model of the ether involved an effect that he called ‘rotational elasticity’, which was easy to describe mathematically but difficult to picture physically [14].

MacCullagh is probably not very well known even among physicists. One physicist, though, who was well aware of his achievements was Richard Feynman.

Abstract

I briefly recall intersections of my research interests with those of John Bell. I then argue that the noise needed in theories of objective state vector reduction most likely comes from a fluctuating complex part in the classical spacetime metric; that is, state vector reduction is driven by complex-number-valued “space–time foam.”

Introduction

My research interests have intersected those of John Bell three times. The first was when I found the forward lepton theorem for high-energy neutrino reactions, showing that for forward lepton kinematics, a conserved vector current (CVC) and a partially conserved axial vector current (PCAC) imply that the neutrino cross section can be related to a pion scattering cross section [1]. This led to an exchange of letters and discussions with Bell during 1964–5, which are described in the Commentaries for my selected papers [2]. The second was in the course of my work on the axial-vector anomaly [3], when I had further correspondence with John Bell, as described both in the Commentaries [4] and in the volume of essays on Yang–Mills theories assembled by 't Hooft [5]. The third time was a few years after Bell's death in 1990, when I became interested in the foundations of quantum theory and the quantum measurement problem, in the course of writing my book on quaternionic quantum mechanics [6]. Foundational issues in standard, complex quantum theory had preoccupied Bell for much of his career and led to his best known work. However, my correspondence with Bell in the 1960s never touched on quantum foundations, and I only read his seminal writings on the subject much later on. In this article I focus on this third area of shared interests.

Objective Reduction Models

There is now a well-defined phenomenology of state vector reduction, pioneered by the work of Ghirardi, Rimini, andWeber (GRW) and of Philip Pearle, and worked on by many others. John Bell was interested in this program from the outset.

Introduction

Bell's theorem establishes a contradiction between locality (and a few other assumptions) and certain predictions of quantum mechanics [2]. There have been two main issues surrounding the deep implications of this celebrated theorem. The first issue is basic, and it concerns whether the theorem really establishes that nonlocality is a necessary feature of any empirically viable theory and hence a feature of nature itself. One aspect of the issue concerns exactly what the underlying assumptions of Bell's theorem are. For example, some argue that these assumptions include the assumption of counterfactual definiteness [3–5], while others disagree [6–8]. The other aspect of the issue concerns which assumption should be dropped due to the resulting contradiction. Some believe that standard quantum mechanics may avoid nonlocality by denying counterfactual definiteness [3–5].

The second issue is much deeper, and it concerns how to make sense of the strange nonlocality when assuming that Bell's theorem establishes that our world is nonlocal. In particular, it has yet to be determined whether such quantum nonlocality is compatible with the theory of relativity, e.g., whether the nonlocality requires the existence of a preferred Lorentz frame [7, 9, 10]. It seems that “there is a preferred frame of reference, and in this preferred frame of reference things do go faster than light … Behind the apparent Lorentz invariance of the phenomena, there is a deeper level which is not Lorentz invariant” [11]. But it also seems possible that the existence of a preferred Lorentz frame is an illusion, since it cannot be detected according to standard quantum mechanics. A more subtle issue is whether quantum nonlocality permits superluminal signaling. It is widely thought that the answers to these questions will lead to a deeper understanding of Bell's theorem and quantum nonlocality.

In this paper, we will present a new analysis of quantum nonlocality and its apparent incompatibility with relativity.

I am very pleased to see, in this volume, the papers written in John's memory. He had many interests and it is good to see such a variety of authors. I thank all of them for their work. I am sure that John would have enjoyed the book.

Abstract

Between 1964 and 1990, the notion of nonlocality in Bell's papers underwent a profound change as his nonlocality theorem gradually became detached from quantum mechanics, and referred to wider probabilistic theories involving correlations between separated beables. The proposition that standard quantum mechanics is itself nonlocal (more precisely, that it violates ‘local causality’) became divorced from the Bell theorem per se from 1976 on, although this important point is widely overlooked in the literature. In 1990, the year of his death, Bell would express serious misgivings about themathematical form of the local causality condition and leave ill-defined the issue of the consistency between special relativity and violation of the Bell-type inequality. In our view, the significance of the Bell theorem, in both its deterministic and stochastic forms, can only be fully understood by taking into account the fact that a fully Lorentz covariant version of quantum theory, free of action at a distance, can be articulated in the Everett interpretation.

Introduction

John S. Bell's last word on his celebrated nonlocality theorem and its interpretation appeared in his 1990 paper ‘La nouvelle cuisine’, first published in the year of his untimely death. Bell was careful here to distinguish between the issue of ‘no superluminal signalling’ in quantum theory (both quantum field theory and quantum mechanics) and a principle he first introduced explicitly in 1976 and called ‘local causality’ [1]. In relation to the former, Bell expressed concerns that amplify doubts he had already expressed in 1976. These concerns touch on what is now widely known as the no-signalling theorem in quantum mechanics, and ultimately have to do with Bell's distaste for what he saw as an anthropocentric element in orthodox quantum thinking. In relation to local causality, Bell emphasised that his famous factorizability (no-correlations) condition is not to be seen ‘as the formulation of local causality, but as a consequence thereof’ and stressed how difficult he found it to articulate this consequence. He left the question of any strict inconsistency between violation of factorizability and special relativity theory unresolved, a not insignificant shift from his thinking up to the early 1980s.

Introduction

This contribution to the book in honour of J. S. Bell will probably differ from the remaining ones, in particular since only a part of it will be devoted to specific technical arguments. In fact I have considered it appropriate to share with the community of physicists interested in the foundational problems of our best theory the repeated interactions I had with him in the last four years of his life, the deep discussions in which we have been involved in particular in connection with the elaboration of collapse theories and their interpretation, and the contributions he gave to the development of this approach at a formal level, as well as championing it on repeated occasions. In brief, I intend to play here the role of one of those lucky persons who became acquainted with him personally, who has exchanged important views with him, who has learned a lot from his deep insight and conceptual lucidity, and, last but not least, whose scientific work has been appreciated by him.

Moreover, due to the fact that this book intends to celebrate the 50th anniversary of the derivation of the fundamental inequality which bears his name, I will also devote a small part of the text to recalling his clearcut views about the locality issue, views that I believe have not been grasped correctly by a remarkable part of the scientific community. I will analyze this problem in quite general terms at the end of the paper.

Some of Bell's Scientific Achievements

Bell received bachelor's degrees in experimental physics and in mathematical physics at Queen'sUniversity of Belfast in the years 1948 and 1949, respectively, and a PhD in physics at the University of Birmingham. We all know very well that already at that time he was absolutely unsatisfied with the conceptual structure of quantum mechanics and with the way in which it was taught. This is significantly expressed by the statement he made during an interview to Jeremy Bernstein: I remember arguing with one of my professors, a Doctor Sloane, about that. I was getting very heated and accusing him, more or less, of dishonesty. He was getting very heated too and said, ‘You're going too far.’

Spooky Action at a Distance

In the context of correlation experiments involving pairs of experiments performed at very nearly the same time in very far-apart experimental regions, Einstein famously said [1],

But on one supposition we should in my opinion hold absolutely fast: “The real factual situation of the system S2 is independent of what is done with system S1 which is spatially separated from the former.”

This demand is incompatible with the basic ideas of standard (Copenhagen/orthodox) quantum mechanics, which makes two relevant claims:

1. (1) Experimenters in the two labs make “local free choices” that determine which experiments will be performed in their respective labs. These choices are free in the sense of not being predetermined by the prior history of the physically described aspects of the universe, and they are localized in the sense that the physical effects of these free choices are inserted into the physically described aspects of the universe only within the laboratory in which the associated experiment is being performed.

2. (2) These choices of what is done with the system being measured in one lab can (because of a measurement-induced global collapse of the quantum state) influence the outcome of the experiment performed at very close to the same time in the very faraway lab.

This influence of what is done with the system being measured in one region upon the outcome appearing at very nearly the same time in a very faraway lab was called “spooky action at a distance” by Einstein, and was rejected by him as a possible feature of “reality.”

John Bell's Quasi-classical Statistical Theory

Responding to the seeming existence in the quantum world of “spooky actions,” John Bell [2] proposed a possible alternative to the standard approach that might conceivably be able to reconcile quantum spookiness with “reality.” This alternative approach rests on the fact that quantum mechanics is a statistical theory.We already have in physics a statistical theory called “classical statistical mechanics.”

In 1964, John Stewart Bell published an important result that was later called Bell's theorem [1]. It states that certain predictions of quantum mechanics cannot be accounted for by any local realistic theory. Bell's theorem has been called “the most profound discovery of science” [2]. By introducing a notion of quantum nonlocality, it not only transforms the study of the foundations of quantum mechanics, but also paves the way to many quantum technologies that have been developed in the last decades. It can be expected that Bell's work will play a more important role in the physics of the future. Admittedly, there are still controversies on the underlying assumptions and deep implications of Bell's theorem. This also poses a challenge to us in understanding quantum theory, as well as the physical world at the most fundamental level.

This book is an anthology celebrating the 50th anniversary of Bell's theorem. It contains 26 original essays written by physicists and philosophers of physics, reflecting the latest thoughts of leading experts on the subject. The content includes recollections of John Bell, an introduction to Bell's nonlocality theorem, a review of its experimental tests, analyses of its meaning and implications, investigations of the nature of quantum nonlocality, discussions of possible ways to avoid nonlocality, and last but not least, analyses of various nonlocal realistic theories. The book is accessible to graduate students in physics. It will be of value to students and researchers with an interest in the philosophy of physics and especially to physicists and philosophers working on the foundations of quantum mechanics.

This book is arranged in four parts. The first part introduces John Bell, the great Northern Irish physicist, and it also contains a few physicists’ treasured recollections of him. In Chapter 1, Andrew Whitaker introduces the Irish tradition of physics, Bell's Belfast background, and his studies of physics at Queen's University Belfast. Whitaker argues that there may exist a connection between Bell's work and the Irish tradition, especially concerning his views on the apparent incompatibility between quantum nonlocality and special relativity.

Abstract

J.S. Bell's remarkable 1964 theorem showed that any theory sharing the empirical predictions of orthodox quantum mechanics would have to exhibit a surprising – and, from the point of view of relativity theory, very troubling – kind of nonlocality. Unfortunately, even still on this 50th anniversary, many commentators and textbook authors continue to misrepresent Bell's theorem. In particular, one continues to hear the claim that Bell's result leaves open the option of concluding either nonlocality or the failure of some unorthodox “hidden variable” (or “determinism” or “realism”) premise. This mistaken claim is often based on a failure to appreciate the role of the earlier 1935 argument of Einstein, Podolsky, and Rosen in Bell's reasoning. After briefly reviewing this situation, I turn to two alternative versions of quantum theory – the “many worlds” theory of Everett and the quantum Bayesian interpretation of Fuchs, Schack, Caves, and Mermin – that purport to provide actual counterexamples to Bell's claim that nonlocality is required to account for the empirically verified quantum predictions. After analyzing each theory's grounds for claiming to explain the EPR–Bell correlations locally, however, one can see that (despite a number of fundamental differences) the two theories share a common for-all-practical-purposes (FAPP) solipsistic character. This dramatically undermines such theories’ claims to provide a local explanation of the correlations and thus, by concretizing the ridiculous philosophical lengths to which one must go to elude Bell's own conclusion, reinforces the assertion that nonlocality really is required to coherently explain the empirical data.

Introduction

Fifty years ago, in 1964 [1], John Stewart Bell first proved the theorem that has become widely known as “Bell's Theorem” but that Bell himself instead referred to as the “locality inequality theorem” [2]. In Bell's own view, the theorem showed that the empirical predictions of local theories will be constrained by Bell's inequality (or as Bell himself preferred to call it, the “locality inequality”). Hence, nonlocality is a necessary feature of any theory that shares the empirical predictions of standard quantum mechanics.

Abstract

Various interpretations of quantum mechanics, favored (or neglected) by John Bell in the context of his nonlocality theorem, are compared and discussed.

Varenna 1970

I met John Bell for the first time at the Varenna conference of 1970 [1]. I had been invited on the suggestion of Eugene P. Wigner, who had already helped me to publish my first paper on the concept that was later called decoherence – to appear in the first issue of Foundations of Physics a few months after the conference [2]. This concept arose from my conviction, based on many applications of quantum mechanics to composite systems under various conditions, that Schrödinger's wave function in configuration space, or more generally the superposition principle, is universally valid and applicable. In particular, stable narrow wavepackets can represent classical configurations, while their superpositions define novel individual properties – such as “momentum,” defined as a plane wave superposition of different positions. Superpositions of macroscopically different properties, on the other hand, are regularly irreversibly “dislocalized” (distributed over many degrees of freedom) by means of interactions described by the Schrödinger equation. The corresponding disappearance of certain local superpositions (“decoherence”) seems to explain the phenomenon of a classical world as well as the apparent occurrence of quantum jumps or stochastic “events” – see Sect. 20.4 or [3] for a historical overview of the conceptual development of quantum theory. So I had never felt any motivation to think of “hidden variables” or any other physics behind the successful wave function.

Therefore, I was very surprised on my arrival in Varenna to hear everybody discuss Bell's inequality. It had been published a few years before the conference, but I had either not noticed it or not regarded it as particularly remarkable until then. As this inequality demonstrates that the predictions of quantum theory would require any conceivable reality possibly underlying the nonlocal wave function to be nonlocal itself, I simply found my conviction that the latter describes individual reality confirmed. For example, I had often discussed the conservation of total spin or angular momentum in an individual decay process, which requires nonlocal entanglement between the fragments at any distance in a form that was later called a “Bell state”.

Introduction

Annus mirabilis 1964

1964 was a wonderful year for physics. In this annus mirabilis, summarising and simplifying a little, the quark model was proposed by Gell-Mann and Zweig and charm and colour properties were introduced, Higgs published his work on the scalar boson, the _ boson and CP violation were discovered, cosmic background radiation was identified and, finally, Bell inequalities were suggested.

Most of these works were later subjects of a Nobel prize. Unluckily John Bell left us too early for this achievement, but, at least in my opinion, his work [1] is probably the most fruitful among all these huge advances, since now we can recognise that it was the root of a fundamental change in our vision of the physical world, providing, on one hand, the possibility of a test of quantum mechanics against a whole class of theories (substantially all the theories that we could define as “classical”) and, on the other hand, introducing a clear notion of quantum nonlocality that represents not only a challenge in understanding quantum mechanics, but also one of the most important resources paving the way to quantum technologies that were developed in the last decades based on the quantum correlations discussed in the Bell paper.

In the following we will present the experimental progress in testing Bell inequalities. Rather amazingly even 50 years after Bell's paper, a conclusive test of his proposal is still missing [2, 3].

Nevertheless, huge progress has been achieved with respect to the first experiments.We will discuss these developments and the remaining problems for a conclusive test of Bell inequalities in detail.

The Bell Inequalities, Their Hypotheses and the Experimental Loopholes

In order to understand the experimental difficulties in achieving a final test of Bell inequalities, let us analyse a little how these inequalities are derived and the explicit and implicit hypotheses they contain.

The proof of Bell inequalities usually starts by considering two separated subsystems of an entangled state, which are addressed to two independent measurement apparatuses measuring the expectation values of two dichotomic observables, Aa and Bb. a, b are two parameters describing the settings of the measuring apparatus A and B, respectively.

Abstract

In 1964, John Stewart Bell famously demonstrated that the laws of standard quantum mechanics demand a physical world that cannot be described entirely according to local laws. The present article argues that this nonlocality must be gravitationally related, as it comes about only with quantum state reduction, this being claimed to be a gravitational effect. A new formalism for curved space–times, palatial twistor theory, is outlined, which appears now to be able to accommodate gravitation fully, providing a nonlocal description of the physical world.

Nonlocality and Quantum State Reduction

Although quantum entanglement is a normal consequence of the unitary (Schrödinger) evolution U of a multiparticle quantum system, this evolution is nevertheless local in the sense that it is described as the continuous (local) evolution in the relevant configuration space. What the EPR situations considered by John Stewart Bell [1] demonstrated was that when widely separated quantum measurements, of appropriate kinds, are performed on such entangled states (a current record for distance separation being 143 km, for entangled photon pairs [2]), the results of such measurements (which may be probabilistic or of a yes/no character) cannot be mimicked by any local realistic model. Thus, in trying to propose a mathematical model of what is going on realistically in the physical world, one would need to face up to the actual physical process involved in quantum state reduction R, which is an essential feature of quantum measurement.

For many years, my own position on the state-reduction issue has been that R would have to involve a fundamental mathematical extension of current quantum theory, giving a description of something objectively taking place ‘out there’ in the physical world (OR: objective reduction), rather than R being the effect of some kind of ‘interpretation’ of the standard unitary quantum formalism. Moreover, the extended theory should be able to describe the one physical world that we all experience, rather than some sort of coexisting superposition of vast numbers of alternatives. To be more specific about my own viewpoint, it has long been my position that such necessary deviation from unitarity U in this OR would result from a correct melding of the quantum formalism with that of general relativity.