The Born rule may be stated mathematically as the rule that probabilities in quantum theory are expectation values of a complete orthogonal set of projection operators. This rule works for single laboratory settings in which the observer can distinguish all the different possible outcomes corresponding to the projection operators. However, theories of inflation suggest that the universe may be so large that any laboratory, no matter how precisely it is defined by its internal state, may exist (...) in a large number of very distantly separated copies throughout the vast universe. In this case, no observer within the universe can distinguish all possible outcomes for all copies of the laboratory. Then normalized probabilities for the local outcomes that can be locally distinguished cannot be given by the expectation values of any projection operators. Thus the Born rule dies and must be replaced by another rule for observational probabilities in cosmology. The freedom of what this new rule is to be is the measure problem in cosmology. A particular volume-averaged form is proposed. (shrink)

In the libertarian ``agent causation'' view of free will, free choices are attributable only to the choosing agent, as opposed to a specific cause or causes outside the agent. An often-repeated claim in the philosophical literature on free will is that agent causation necessarily implies lawlessness, and is therefore ``antiscientific." That claim is critiqued and it is argued, on the contrary, that the volitional powers of a free agent need not be viewed as anomic, specifically with regard to the quantum (...) statistical law. Assumptions about the role and nature of causation, taken as bearing on volitional agency, are examined and found inadequate to the task. Finally, it is suggested that quantum theory may constitute precisely the sort of theory required for a nomic grounding of libertarian free will. (shrink)

We have recently developed a new understanding of probability in quantum gravity. In this paper we provide an overview of this new approach and its implications. Adopting the de Broglie–Bohm pilot-wave formulation of quantum physics, we argue that there is no Born rule at the fundamental level of quantum gravity with a non-normalisable Wheeler–DeWitt wave functional \(\Psi\). Instead the universe is in a perpetual state of quantum nonequilibrium with a probability density \(P\ne \left| \Psi \right| ^{2}\). Dynamical relaxation (...) to the Born rule can occur only after the early universe has emerged into a semiclassical or Schrödinger approximation, with a time-dependent and normalisable wave functional \(\psi\), for non-gravitational systems on a classical spacetime background. In that regime the probability density \(\rho\) can relax towards \(\left| \psi \right| ^{2}\) (on a coarse-grained level). Thus the pilot-wave theory of gravitation supports the hypothesis of primordial quantum nonequilibrium, with relaxation to the Born rule taking place soon after the big bang. We also show that quantum-gravitational corrections to the Schrödinger approximation allow quantum nonequilibrium \(\rho \ne \left| \psi \right| ^{2}\) to be created from a prior equilibrium ( \(\rho =\left| \psi \right| ^{2}\) ) state. Such effects are very tiny and difficult to observe in practice. (shrink)

We clarify the role of the Born rule in the Copenhagen Interpretation of quantum mechanics by deriving it from Bohr's doctrine of classical concepts, translated into the following mathematical statement: a quantum system described by a noncommutative C*-algebra of observables is empirically accessible only through associated commutative C*-algebras. The Born probabilities emerge as the relative frequencies of outcomes in long runs of measurements on a quantum system; it is not necessary to adopt the frequency interpretation of single-case (...) probabilities (which will be the subject of a sequel paper). Our derivation of the Born rule uses ideas from a program begun by Finkelstein (1965) and Hartle (1968), intending to remove the Born rule as a separate postulate of quantum mechanics. Mathematically speaking, our approach refines previous elaborations of this program - notably the one due to Farhi, Goldstone, and Gutmann (1989) as completed by Van Wesep (2006) - in replacing infinite tensor products of Hilbert spaces by continuous fields of C*-algebras. In combination with our interpretational context, this technical improvement circumvents valid criticisms that earlier derivations of the Born rule have provoked, especially to the effect that such derivations were mathematically flawed as well as circular. Furthermore, instead of relying on the controversial eigenvector-eigenvalue link in quantum theory, our derivation just assumes that pure states in classical physics have the usual interpretation as truthmakers that assign sharp values to observables. (shrink)

It was repeatedly underlined in literature that quantum mechanics cannot be considered a closed theory if the Born Rule is postulated rather than derived from the first principles. In this work the Born Rule is derived from the time-reversal symmetry of quantum equations of motion. The derivation is based on a simple functional equation that takes into account properties of probability, as well as the linearity and time-reversal symmetry of quantum equations of motion. The derivation presented (...) in this work also allows to determine certain limits to applicability of the Born Rule. (shrink)

We show that there is a hidden freedom in quantum many-body theory associated with overcompleteness of the time evolution through the single-particle subspace of a many-body system. To fix the freedom, an additional constraint is necessary. We argue that the appropriate constraint on the time evolution through the subspace is to quantize the propagation of entangled pairs of particles, represented by the single-particle spectral function, instead of individual particles. This solution method creates a surface that indicates the multiplicity of every (...) solution to the inverse problem defined by matching the freedom to the constraint. Upon measurement, the system collapses nonlocally onto a single quantized solution. In addition to a combinatoric multiplicity, each solution acquires a multiplicity due to its stability when subject to a small variation in the microscopic degrees of freedom. Numerical calculations for a two-level system show that our theory improves upon standard theory in the description of non-quasiparticle spectral features. Our reinterpretation of quantum many-body theory is not based on the Born rule and offers a more faithful representation of experiments than current theory by modeling individual, quantized events with an explicit collapse model. (shrink)

Despite the tremendous empirical success of quantum theory there is still widespread disagreement about what it can tell us about the nature of the world. A central question is whether the theory is about our knowledge of reality, or a direct statement about reality itself. Current interpretations of quantum theory, regardless of their stance on this question, regard the Born rule as fundamental and add an independent state update (or ‘collapse’) rule to describe how quantum states change (...) upon measurement. In this paper we present an alternative perspective and derive a unified probability rule that subsumes both the Born rule and the collapse rule. We show that this more fundamental probability rule can provide a rigorous foundation for informational, or ‘knowledge-based’, interpretations of quantum theory. Our result requires an assumption of instrument non- contextuality, a key notion that generalises previous approaches to non-contextuality. Therefore, the framework also permits one to consider non-contextuality in scenarios with arbitrary causal structure. (shrink)

The Born rule is derived from operational assumptions, together with assumptions of quantum mechanics that concern only the deterministic development of the state. Unlike Gleason’s theorem, the argument applies even if probabilities are de…ned for only a single resolution of the identity, so it applies to a variety of foundational approaches to quantum mechanics. It also provides a probability rule for state spaces that are not Hilbert spaces.

It is shown that violation of the Born Rule leads to a breakdown of the correspondence between the quantum electromagnetic field and its classical counterpart. Specifically, the relationship of the quantum coherent state to the classical electromagnetic field turns out to imply that if the Born Rule were violated, this could result in apparent deviations from the energy conservation law applying to the field and its sources. The result, which is fully general and independent of interpretations (...) of quantum theory, suggests that the Born Rule is just as fundamental a law of Nature as are the field conservation laws. (shrink)

I develop the decision-theoretic approach to quantum probability, originally proposed by David Deutsch, into a mathematically rigorous proof of the Born rule in (Everett-interpreted) quantum mechanics. I sketch the argument informally, then prove it formally, and lastly consider a number of proposed ``counter-examples'' to show exactly which premises of the argument they violate. (This is a preliminary version of a chapter to appear --- under the title ``How to prove the Born Rule'' --- in Saunders, Barrett, (...) Kent and Wallace, "Many worlds? Everett, quantum theory and reality", forthcoming from Oxford University Press.). (shrink)

Born’s quantum probability rule is traditionally included among the quantum postulates as being given by the squared amplitude projection of a measured state over a prepared state, or else as a trace formula for density operators. Both Gleason’s theorem and Busch’s theorem derive the quantum probability rule starting from very general assumptions about probability measures. Remarkably, Gleason’s theorem holds only under the physically unsound restriction that the dimension of the underlying Hilbert space \ must be larger than (...) two. Busch’s theorem lifted this restriction, thereby including qubits in its domain of validity. However, while Gleason assumed that observables are given by complete sets of orthogonal projectors, Busch made the mathematically stronger assumption that observables are given by positive operator-valued measures. The theorem we present here applies, similarly to the quantum postulate, without restricting the dimension of \ and for observables given by complete sets of orthogonal projectors. We also show that the Born rule applies beyond the quantum domain, thereby exhibiting the common root shared by some quantum and classical phenomena. (shrink)

The centered Everett interpretation solves a problem that various approaches to quantum theory face. In this paper, I continue developing the theory underlying that solution. In particular, I defend the centered Everett interpretation against a few objections, and I provide additional motivation for some of its key features.

We provide a derivation of the Born Rule in the context of the Everett (Many-Worlds) approach to quantum mechanics. Our argument is based on the idea of self-locating uncertainty: in the period between the wave function branching via decoherence and an observer registering the outcome of the measurement, that observer can know the state of the universe precisely without knowing which branch they are on. We show that there is a uniquely rational way to apportion credence in such (...) cases, which leads directly to the Born Rule. Our analysis generalizes straightforwardly to cases of combined classical and quantum self-locating uncertainty, as in the cosmological multiverse. (shrink)

In Everettian quantum mechanics, justifications for the Born rule appeal to self-locating uncertainty or decision theory. Such justifications have focused exclusively on a pure-state Everettian multiverse, represented by a wave function. Recent works in quantum foundations suggest that it is viable to consider a mixed-state Everettian multiverse, represented by a (mixed-state) density matrix. Here, we develop the conceptual foundations for decoherence and branching in a mixed-state multiverse, and extend arguments for the Born rule to this setting. (...) This extended framework provides a unification of 'classical' and 'quantum' probabilities, and additional theoretical benefits, for the Everettian picture. (shrink)

Recently, W. H. Zurek presented a novel derivation of the Born rule based on a mechanism termed environment-assisted invariance, or “envariance” [W. H. Zurek, Phys. Rev. Lett. 90(2), 120404 (2003)]. We review this approach and identify fundamental assumptions that have implicitly entered into it, emphasizing issues that any such derivation is likely to face.

Nelson’s stochastic quantum mechanics provides an ideal arena to test how the Born rule is established from an initial probability distribution that is not identical to the square modulus of the wavefunction. Here, we investigate numerically this problem for three relevant cases: a double-slit interference setup, a harmonic oscillator, and a quantum particle in a uniform gravitational field. For all cases, Nelson’s stochastic trajectories are initially localized at a definite position, thereby violating the Born rule. For (...) the double slit and harmonic oscillator, typical quantum phenomena, such as interferences, always occur well after the establishment of the Born rule. In contrast, for the case of quantum particles free-falling in the gravity field of the Earth, an interference pattern is observed before the completion of the quantum relaxation. This finding may pave the way to experiments able to discriminate standard quantum mechanics, where the Born rule is always satisfied, from Nelson’s theory, for which an early subquantum dynamics may be present before full quantum relaxation has occurred. Although the mechanism through which a quantum particle might violate the Born rule remains unknown to date, we speculate that this may occur during fundamental processes, such as beta decay or particle-antiparticle pair production. (shrink)

To solve the probability problem of the Many Worlds Interpretation of Quantum Mechanics, D. Wallace has presented a formal proof of the Born rule via decision theory, as proposed by D. Deutsch. The idea is to get subjective probabilities from rational decisions related to quantum measurements, showing the non-probabilistic parts of the quantum formalism, plus some rational constraints, ensure the squared modulus of quantum amplitudes play the role of such probabilities. We provide a new presentation of Wallace’s proof, (...) reorganized to simplify some arguments, and analyze it from a formal perspective. Similarities with classical decision theory are made explicit, to clarify its structure and main ideas. A simpler notation is used, and details are filled in, making it easier to follow and verify. Some problems have been identified, and we suggest possible corrections. (shrink)

Everett’s Relative State Interpretation has gained increasing interest due to the progress of understanding the role of decoherence. In order to fulfill its promise as a realistic description of the physical world, two postulates are formulated. In short they are for a system with continuous coordinates \, discrete variable j, and state \\), the density \=|\psi _j|^2\) gives the distribution of the location of the system with the respect to the variables \ and j; an equation of motion for the (...) state \. The first postulate connects the mathematical description to the physical reality, which has been missing in previous versions. The contents of the standard postulates are derived, including the appearance of Hilbert space and the Born rule. The approach to probabilities earlier proposed by Greaves replaces the classical probability concept in the Born rule. The new quantum probability concept, earlier advocated by Deutsch and Wallace, is void of the requirement of uncertainty. (shrink)

Abstract: This paper develops and advocates a rule for assigning self-locating credences in quantum branching scenarios, called Indexed Branch-Counting. It is argued that Indexed Branch-Counting can be justified on both accuracy-theoretic grounds and on the grounds that it satisfies a requirement of exchangeability for probability assignments. Since Indexed Branch-Counting diverges from the Born Rule, this poses trouble for Everettian approaches to probability. The paper also addresses a common argument against branch-counting, namely that the rule is incoherent (...) in light of putative vagueness in the number of branches. Finally, the paper addresses a recent proposal from Simon Saunders that aims to reconcile branch-counting with the Born Rule, arguing that the proposal faces challenges. (shrink)

Considering the “Born-Alive” Rule and Possession of Sperm Following Death Content Type Journal Article Category Recent Developments Pages 323-327 DOI 10.1007/s11673-011-9324-0 Authors Bernadette Richards, Law School, The University of Adelaide, South Australia, Australia Bill Madden, School of Law, University of Western Sydney, Sydney, NSW, Australia Tina Cockburn, School of Law, Queensland University of Technology, Brisbane, Qld, Australia Journal Journal of Bioethical Inquiry Online ISSN 1872-4353 Print ISSN 1176-7529 Journal Volume Volume 8 Journal Issue Volume 8, Number 4.

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)

English law is unambiguous that legal personality, and with it all legal rights and protections, is assigned at birth. This rule is regarded as a bright line that is easily and consistently applied. The time has come, however, for the rule to be revisited. This article demonstrates that advances in fetal surgery and (anticipated) artificial wombs do not marry with traditional conceptions of birth and being alive in law. These technologies introduce the possibility of ex utero gestation, and/or (...) temporary existence ex utero, and consequently developing human beings that are novel to the law. Importantly, therefore, the concepts of birth and born alive no longer distinguish between human beings deserving of legal protection in the way originally intended. Thus, there is a need for reform, for a new approach to determining the legal significance of birth and what being legally alive actually encompasses. Investigating the law of birth is of crucial importance, because of the implications of affording or denying the subjects of new reproductive technologies rights and protections. A determination of the legal status of the subject of fetal surgery or an artificial womb will determine what can and cannot be done to each entity. Moreover, the status afforded to these entities will drastically impact on the freedoms of pregnant women. (shrink)

Rule systems are used every day to share experience, pre-existing knowledge, beliefs and ethical rules, and to provide instructions for future action. This article expands and builds upon an approach pioneered by Julius M. Moravcsik to argue that ethics cannot be completely codified into a rigid set of rules, because any such set lacks a misapplication-correcting sensibility. Thus, an ethics that is transferred purely by means of a rule-set is incomplete and thus cannot be used reliably to guide (...) future action. The balancing sensibility is formed out of pre-existing expert knowledge and takes account of the limitations of rule-sets in specific contexts. Moravcsik’s approach is then expanded by incorporating it into a holistic framework for the analysis, guidance and development of actions to be taken to support the emergence, selection and implementation of solutions for a sustainable future. This approach has profound implications for managers and organisers in new or critical situations. (shrink)

In previous articles we presented a simple set of axioms named “Contexts, Systems and Modalities”, where the structure of quantum mechanics appears as a result of the interplay between the quantized number of modalities accessible to a quantum system, and the continuum of contexts that are required to define these modalities. In the present article we discuss further how to obtain Born’s rule within this framework. Our approach is compared with other former and recent derivations, and its strong (...) links with Gleason’s theorem are particularly emphasized. (shrink)

We present a new experimental approach using a three-path interferometer and find a tighter empirical upper bound on possible violations of Born’s Rule. A deviation from Born’s rule would result in multi-order interference. Among the potential systematic errors that could lead to an apparent violation we specifically study the nonlinear response of our detectors and present ways to calibrate this error in order to obtain an even better bound.

The Born’s rule to interpret the square of wave function as the probability to get a specific value in measurement has been accepted as a postulate in foundations of quantum mechanics. Although there have been so many attempts at deriving this rule theoretically using different approaches such as frequency operator approach, many-world theory, Bayesian probability and envariance, literature shows that arguments in each of these methods are circular. In view of absence of a convincing theoretical proof, recently (...) some researchers have carried out experiments to validate the rule up-to maximum possible accuracy using multi-order interference (Sinha et al, Science, 329, 418 [2010]). But, a convincing analytical proof of Born’s rule will make us understand the basic process responsible for exact square dependency of probability on wave function. In this paper, by generalizing the method of calculating probability in common experience into quantum mechanics, we prove the Born’s rule for statistical interpretation of wave function. (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)

Born in China in 551 B.C., Confucius rose from poverty to the heights of his country's ruling class. But then he quit his high post for the life of an itinerant philosopher. "The Analects" collects his teachings on education and government, the definition of nobility, the equality of man, and the right way and purpose of living - ideas that eventually spread to the West and influenced the great thinkers of the Enlightenment. And five centuries before Christ, Confucius set (...) forth his own Golden Rule: "Do not impose on others what you do not wish for yourself.". (shrink)

The aim of the famous Born and Jordan 1925 paper was to put Heisenberg’s matrix mechanics on a firm mathematical basis. Born and Jordan showed that if one wants to ensure energy conservation in Heisenberg’s theory it is necessary and sufficient to quantize observables following a certain ordering rule. One apparently unnoticed consequence of this fact is that Schrödinger’s wave mechanics cannot be equivalent to Heisenberg’s more physically motivated matrix mechanics unless its observables are quantized using this (...) rule, and not the more symmetric prescription proposed by Weyl in 1926, which has become the standard procedure in quantum mechanics. This observation confirms the superiority of Born–Jordan quantization, as already suggested by Kauffmann. We also show how to explicitly determine the Born–Jordan quantization of arbitrary classical variables, and discuss the conceptual advantages in using this quantization scheme. We finally suggest that it might be possible to determine the correct quantization scheme by using the results of weak measurement experiments. (shrink)

Successfully navigating the norms of a society is a complex task that involves recognizing diverse kinds of rules as well as the relative weight attached to them. In the United States, different kinds of rules—federal statutes and regulations, scientific norms, and professional ideals—guide the work of researchers. Penalties for violating these different kinds of rules and norms can range from the displeasure of peers to criminal sanctions. We proposed that it would be more difficult for researchers working in the U.S. (...) who were born in other nations to distinguish the seriousness of violating rules across diverse domains. We administered a new measure, the evaluating rules in science task, to National Institutes of Health-funded investigators. The ERST assessed perceptions of the seriousness of violating research regulations, norms, and ideals, and allowed us to calculate the degree to which researchers distinguished between the seriousness of each rule category. The ERST also assessed researchers’ predictions of the seriousness that research integrity officers would assign to the rules. We compared researchers’ predictions to the seriousness ratings of 112 RIOs working at U.S. research-intensive universities. U.S.-born researchers were significantly better at distinguishing between the seriousness of violating federal research regulations and violating ideals of science, and they were more accurate in their predictions of the views of RIOs. Acculturation to the U.S. moderated the effects of nationality on accuracy. We discuss the implications of these findings in terms of future research and education. (shrink)

Probabilities in quantum theory are traditionally given by Born’s rule as the expectation values of projection operators. Here it is shown that Born’s rule is insufficient in universes so large that they contain identical multiple copies of observers, because one does not have definite projection operators to apply. Possible replacements for Born’s rule include using the expectation value of various operators that are not projection operators, or using vari-.

A geometric approach to quantum mechanics with unitary evolution and non-unitary collapse processes is developed. In this approach the Schrödinger evolution of a quantum system is a geodesic motion on the space of states of the system furnished with an appropriate Riemannian metric. The measuring device is modeled by a perturbation of the metric. The process of measurement is identified with a geodesic motion of state of the system in the perturbed metric. Under the assumption of random fluctuations of the (...) perturbed metric, the Born rule for probabilities of collapse is derived. The approach is applied to a two-level quantum system to obtain a simple geometric interpretation of quantum commutators, the uncertainty principle and Planck’s constant. In light of this, a lucid analysis of the double-slit experiment with collapse and an experiment on a pair of entangled particles is presented. (shrink)

Everettian quantum mechanics (EQM) results in ‘multiple, emergent, branching quasi-classical realities’ (Wallace [2012]). The possible outcomes of measurement as per ‘orthodox’ quantum mechanics are, in EQM, all instantiated. Given this metaphysics, Everettians face the ‘probability problem’—how to make sense of probabilities and recover the Born rule. To solve the probability problem, Wallace, following Deutsch ([1999]), has derived a quantum representation theorem. I argue that Wallace’s solution to the probability problem is unsuccessful, as follows. First, I examine one of (...) the axioms of rationality used to derive the theorem, ‘branching indifference’ (BI). I argue that Wallace is not successful in showing that BI is rational. While I think it is correct to put the burden of proof on Wallace to motivate BI as an axiom of rationality, it does not follow from his failing to do so that BI is not rational. Thus, second, I show that there is an alternative strategy for setting one’s credences in the face of branching which is rational and which violates BI. This is ‘branch counting’ (BC). Wallace is aware of BC and has proffered various arguments against it. However, third, I argue that Wallace’s arguments against BC are unpersuasive. I conclude that the probability problem in EQM persists. 1 Introduction2 Branching Indifference 2.1 The positive argument for branching indifference 2.2 The negative argument for branching indifference 2.3 Branching indifference section summary3 Branch Counting 3.1 Branch counting is rational under the subjective-uncertainty viewpoint 3.2 Branch counting is rational under the objective-determinism viewpoint 3.3 Branch counting section summary4 Number of Branches 4.1 Veracity of the framework 4.2 No such thing as the number of branches 4.2.1 The number of branches is indeterminate 4.2.2 The number of branches is indeterminable 4.3 Rationality and weight5 Conclusion. (shrink)

Overruling of law said to establish the “right not to be born”The French government has given in to public pressure and overturned a controversial legal ruling which recognised the right of a disabled chld to seek damaages. Most notably, the ruling, widely described as establishing a child's right “not to be born”, had provoked “outrage” amongst groups defending the rights of the disabled and led to a ban on prenatal scans by French gynaecologists. Once again, only parents will (...) be able to seek damages but some people think the ruling has been misinterpreted.The Perruche judgment, a controversial ruling which has been said to establish the “right not to be born”, has now been overruled by a majority of the National Assembly of the French Parliament.1, 2 A bill has been put forward which states “no one can sue for damages for the sole fact of their birth”. It was considered by the senate until January 22, 2002.2In 1982, Josette Perruche discovered red spots covering her four year old daughter. The child was diagnosed with German measles. Mrs Perruche told her …. (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)

These lecture notes present a concise and introductory, yet as far as possible coherent, view of the main formalizations of quantum mechanics and of quantum field theories, their interrelations and their theoretical foundations. The "standard" formulation of quantum mechanics (involving the Hilbert space of pure states, self-adjoint operators as physical observables, and the probabilistic interpretation given by the Born rule) on one hand, and the path integral and functional integral representations of probabilities amplitudes on the other, are the (...) standard tools used in most applications of quantum theory in physics and chemistry. Yet, other mathematical representations of quantum mechanics sometimes allow better comprehension and justification of quantum theory. This text focuses on two of such representations: the algebraic formulation of quantum mechanics and the "quantum logic" approach. Last but not least, some emphasis will also be put on understanding the relation between quantum physics and special relativity through their common roots - causality, locality and reversibility, as well as on the relation between quantum theory, information theory, correlations and measurements, and quantum gravity. Quantum mechanics is probably the most successful physical theory ever proposed and despite huge experimental and technical progresses in over almost a century, it has never been seriously challenged by experiments. In addition, quantum information science ha s become an important and very active field in recent decades, further enriching the many facets of quantum physics. Yet, there is a strong revival of the discussions about the principles of quantum mechanics and its seemingly paradoxical aspects: sometimes the theory is portrayed as the unchallenged and dominant paradigm of modern physical sciences and technologies while sometimes it is considered a still mysterious and poorly understood theory, waiting for a revolution. This volume, addressing graduate students and seasoned researchers alike, aims to contribute to the reconciliation of these two facets of quantum mechanics. (shrink)

Entanglement was initially thought by some to be an oddity restricted to the realm of thought experiments. However, Bell’s inequality delimiting local - behavior and the experimental demonstration of its violation more than 25 years ago made it entirely clear that non-local properties of pure quantum states are more than an intellectual curiosity. Entanglement and non-locality are now understood to figure prominently in the microphysical world, a realm into which technology is rapidly hurtling. Information theory is also increasingly recognized by (...) physicists and philosophers as intimately related to the foundations of mechanics. The clearest indicator of this relationship is that between quantum information and entanglement. To some degree, a deep relationship between information and mechanics in the quantum context was already there to be seen upon the introduction by Max Born and Wolfgang Pauli of the idea that the essence of pure quantum states lies in their provision of probabilities regarding the behavior of quantum systems, via what has come to be known as the Born rule. The significance of the relationship between mechanics and information became even clearer with Leo Szilard’s analysis of James Clerk Maxwell’s infamous demon thought experiment. Here, in addition to examining both entanglement and quantum information and their relationship, I endeavor to critically assess the influence of the study of these subjects on the interpretation of quantum theory. (shrink)

This paper compares and contrasts relational quantum mechanics with a pragmatist view of quantum theory. I first explain important points of agreement. Then I point to two problems faced by RQM and sketch DP?s solutions to analogous problems. Since both RQM and DP have taken the Born rule to require relative facts I next say what these might be. My main objection to RQM as originally conceived is that its ontology of relative facts is incompatible with scientific objectivity (...) and undercuts the evidential base of quantum theory. In contrast DP?s relative facts have all the objectivity we need to accept quantum theory as scientific knowledge. But a very recent modification to RQM has successfully addressed my main objection,bringing the two views into even closer alignment. (shrink)

We show that the quantum mechanical rules for manipulating probabilities follow naturally from standard probability theory. We do this by generalizing a result of Khinchin regarding characteristic functions. From standard probability theory we obtain the methods usually associated with quantum theory; that is, the operator method, eigenvalues, the Born rule, and the fact that only the eigenvalues of the operator have nonzero probability. We discuss the general question as to why quantum mechanics seemingly necessitates different methods than standard (...) probability theory and argue that the quantum mechanical method is much richer in its ability to generate a wide variety of probability distributions which are inaccessibe by way of standard probability theory. (shrink)

Recently quantum probability theory started to be actively used in studies of human decision-making, in particular for the resolution of paradoxes (such as the Allais, Ellsberg, and Machina paradoxes). Previous studies were based on a cognitive metaphor of the quantum double-slit experiment—the basic quantum interference experiment. In this paper, we report on an economics experiment based on a triple-slit experiment design, where the slits are menus of alternatives from which one can choose. The test of nonclassicality is based on the (...) Sorkin equality (which was only recently tested in quantum physics). Each alternative is a voucher to buy products in one or more stores. The alternatives are obtained from all disjunctions including one, two or three stores. The participants have to reveal the amount for which they are willing to sell the chosen voucher. Interference terms are computed by comparing the willingness to sell a voucher built as a disjunction of stores and the willingness to sell the vouchers corresponding to the singleton stores. These willingness to sell amounts are used to estimate probabilities and to test both the law of total probabilities and the Born Rule. Results reject neither classical nor quantum probability. We discuss this initial experiment and our results and provide guidelines for future studies. (shrink)

In classical game theory the idea that players randomize between their actions according to a particular optimal probability distribution has always been viewed as puzzling. In this paper, we establish a fundamental connection between n-person normal form games and quantum mechanics, which eliminates the conceptual problems of these random strategies. While the two theories have been regarded as distinct, our main theorem proves that if we do not give any other piece of information to a player in a game, than (...) the payoff matrix—the axiom of “no-supplementary data” holds—then the state of mind of a rational player is algebraically isomorphic to a pure quantum state. The “no supplementary data” axiom is captured in a Lukasiewicz’s three-valued Kripke semantics wherein statements about whether a strategy or a belief of a player is rational are initially indeterminate i.e. neither true, nor false. As a corollary, we show that in a mixed Nash equilibrium, the knowledge structure of a player implies that probabilities must verify the standard “Born rule” postulate of QM. The puzzling “indifference condition” wherein each player must be rationally indifferent between all the pure actions of the support of his equilibrium strategy is resolved by his state of mind being described by a “quantum superposition” prior a player is asked to make a definite choice in a “measurement”. Finally, these results demonstrate that there is an intrinsic limitation to the predictions of game theory, on a par with the “irreducible randomness” of quantum physics. (shrink)

The subjective Everettian approach to quantum mechanics presented by Deutsch and Wallace fails to constitute an empirically viable theory of quantum phenomena. The decision theoretic implementation of the Born rule realized in this approach provides no basis for rejecting Everettian quantum mechanics in the face of empirical data that contradicts the Born rule. The approach of Greaves and Myrvold, which provides a subjective implementation of the Born rule as well but derives it from empirical (...) data rather than decision theoretic arguments, avoids the problem faced by Deutsch and Wallace and is empirically viable. However, there is good reason to cast doubts on its scientific value. (shrink)

Recently a new attempt to go beyond quantum mechanics (QM) was presented in the form of so called prequantum classical statistical field theory (PCSFT). Its main experimental prediction is violation of Born’s rule which provides only an approximative description of real probabilities. We expect that it will be possible to design numerous experiments demonstrating violation of Born’s rule. Moreover, recently the first experimental evidence of violation was found in the triple slit interference experiment, see Sinha, et (...) al. (Foundations of Probability and Physics-5. American Institute of Physics, Ser. Conference Proceedings, vol. 1101, pp. 200–207, 2009). Although this experimental test was motivated by another prequantum model, it can be definitely considered as at least preliminary confirmation of the main prediction of PCSFT. In our approach quantum particles are just symbolic representations of “prequantum random fields,” e.g., “electron-field” or “neutron-field”; photon is associated with classical random electromagnetic field. Such prequantum fields fluctuate on time and space scales which are essentially finer than scales of QM, cf. ’t Hooft’s attempt to go beyond QM (see ’t Hooft arXiv:hep-th/0105105, 2001; arXiv:quant-ph/0212095, 2002; arXiv:quant-ph/0701097, 2007). In this paper we elaborate a detection model in the PCSFT-framework. In this model classical random fields (corresponding to “quantum particles”) interact with detectors inducing probabilities which match with Born’s rule only approximately. Thus QM arises from PCSFT as an approximative theory. New tests of violation of Born’s rule are proposed. (shrink)

A simple proof is given that the probabilities of observations in a large universe are not given directly by Born’s rule as the expectation values of projection operators in a global quantum state of the entire universe. An alternative procedure is proposed for constructing an averaged density matrix for a random small region of the universe and then calculating observational probabilities indirectly by Born’s rule as conditional probabilities, conditioned upon the existence of an observation.

I defend the Deutsch-Wallace (DW) theorem against a dilemma presented by Dawid and Thébault (2014), and endorsed in part by Read (2018), and Brown and Porath (2020), according to which the theorem is either redundant or in conflict with general frequency-to-chance inferences. I argue that neither horn of the dilemma is well-posed. On the one hand, the DW theorem is not in conflict with general frequency-to-chance inferences on the most natural way of stating the theorem. On the other hand, the (...) DW theorem is crucial for establishing the Born rule as a prediction of Everettian quantum mechanics (EQM), and so cannot be redundant within the theory. (shrink)

The paper takes up Bell's “Everett theory” and develops it further. The resulting theory is about the system of all particles in the universe, each located in ordinary, 3-dimensional space. This many-particle system as a whole performs random jumps through 3N-dimensional configuration space – hence “Tychistic Bohmian Mechanics”. The distribution of its spontaneous localisations in configuration space is given by the Born Rule probability measure for the universal wavefunction. Contra Bell, the theory is argued to satisfy the minimal (...) desiderata for a Bohmian theory within the Primitive Ontology framework. TBM's formalism is that of ordinary Bohmian Mechanics, without the postulate of continuous particle trajectories and their deterministic dynamics. This “rump formalism” receives, however, a different interpretation. We defend TBM as an empirically adequate and coherent quantum theory. Objections voiced by Bell and Maudlin are rebutted. The “for all practical purposes”-classical, Everettian worlds exist sequentially in TBM. In a temporally coarse-grained sense, they quasi-persist. By contrast, the individual particles themselves cease to persist. (shrink)