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We describe a Markov-Chain-Monte-Carlo algorithm which can be used to generate naturally labeled n-element posets at random with a probability distribution of one's choice. Implementing this algorithm for the uniform distribution, we explore the approach to the asymptotic regime in which almost every poset takes on the three-layer structure described by Kleitman and Rothschild (KR). By tracking the n-dependence of several order-invariants, among them the height of the poset, we observe an oscillatory behavior which is very unlike a monotonic approach to the KR regime. Only around n=40 or so does this "finite size dance" appear to give way to a gradual crossover to asymptopia which lasts until n=85, the largest n we have simulated., Comment: 28 pages, 16 figures

Quantum mechanics allows only certain sets of experimental results (or "probabilistic models") for Bell-type quantum non-locality experiments. A derivation of this set from simple physical or information theoretic principles would represent an important step forward in our understanding of quantum mechanics, and this problem has been intensely investigated in recent years. "Macroscopic locality," which requires the recovery of locality in the limit of large numbers of trials, is one of several principles discussed in the literature that place a bound on the set of quantum probabilistic models. A similar question can also be asked about probabilistic models for the more general class of quantum contextuality experiments. Here, we extend the Macroscopic Locality principle to this more general setting, using the hypergraph approach of Ac\'in, Fritz, Leverrier and Sainz [Comm. Math. Phys. 334(2), 533-628 (2015)], which provides a framework to study both phenomena of nonlocality and contextuality in a unified manner. We find that the set of probabilistic models allowed by our Macroscopic Non-Contextuality principle is equivalent to an important and previously studied set in this formalism, which is slightly larger than the quantum set. In the particular case of Bell Scenarios, this set is equivalent to the set of "Almost Quantum" models, which is of particular interest since the latter was recently shown to satisfy all but one of the principles that have been proposed to bound quantum probabilistic models, without being implied by any of them (or even their conjunction). Our condition is the first characterisation of the almost quantum set from a simple physical principle., Comment: 21 pages (main text 12 pages); v2 minor typos corrected, extended presentation and discussion, published version

A spacetime condensation phenomenon underlies the emergence of a macroscopic universe in causal dynamical triangulations, where the time extension of the condensate is strictly smaller than the total time. It has been known for some time that the volumes of spatial slices in the bulk of the macroscopic universe follow a time evolution which resembles that of a sphere, and their effective dynamics is well described by a minisuperspace reduction of the general relativistic action. More recently, it has been suggested that the same minusuperspace model can also provide an understanding of the condensation phenomenon itself, thus explaining the presence of an extended droplet of spacetime connected to a stalk of minimal spatial extension. We show here that a minisuperspace model based on the general relativistic action fails in that respect for the (2+1)-dimensional case, while a successful condensation is obtained from a minisuperspace model of Horava-Lifshitz gravity., Comment: 32 pages, 5 figures; three new references and several minor improvements

Recently many simple principles have been proposed that can explain quantum limitations on possible sets of experimental probabilities in nonlocality and contextuality experiments. However, few implications between these principles are known. Here it is shown that lack of irreducible third-order interference (a generalisation of the idea that no probabilistic interference remains unaccounted for once we have taken into account interference between pairs of slits in a $n$-sit experiment) implies the principle known as the E principle or Consistent Exclusivity (that, if each pair of a set of experimental outcomes are exclusive alternatives in some measurement, then their probabilities are consistent with the existence of a further measurement in which they are all exclusive). This is a step towards a more unified understanding of quantum nonlocality and contextuality, which promises to allow derivations of important results from minimal, easily grasped assumptions. As one example, this result implies that lack of third order interference bounds violation of the CHSH-Bell inequality to 2.883., Comment: Explanatory comments on proofs, citations and comparisons with other results have been added, and other edits have been made. 5 pages

Bayesian networks provide a powerful tool for reasoning about probabilistic causation, used in many areas of science. They are, however, intrinsically classical. In particular, Bayesian networks naturally yield the Bell inequalities. Inspired by this connection, we generalise the formalism of classical Bayesian networks in order to investigate non-classical correlations in arbitrary causal structures. Our framework of `generalised Bayesian networks' replaces latent variables with the resources of any generalised probabilistic theory, most importantly quantum theory, but also, for example, Popescu-Rohrlich boxes. We obtain three main sets of results. Firstly, we prove that all of the observable conditional independences required by the classical theory also hold in our generalisation; to obtain this, we extend the classical $d$-separation theorem to our setting. Secondly, we find that the theory-independent constraints on probabilities can go beyond these conditional independences. For example we find that no probabilistic theory predicts perfect correlation between three parties using only bipartite common causes. Finally, we begin a classification of those causal structures, such as the Bell scenario, that may yield a separation between classical, quantum and general-probabilistic correlations., Comment: 26+14 pages, many figures, code at http://www.mattpusey.uk/gdag/ v2: Various minor improvements, esp. to appendix A

We introduce a framework for studying non-locality and contextuality inspired by the path integral formulation of quantum theory. We prove that the existence of a strongly positive joint quantum measure -- the quantum analogue of a joint probability measure -- on a set of experimental probabilities implies the Navascues-Pironio-Acin (NPA) condition $Q^1$ and is implied by the stronger NPA condition $Q^{1+AB}$. A related condition is shown to be equivalent to $Q^{1+AB}$., Comment: 28 pages

This paper addresses arguments that "separability" is an assumption of Bell's theorem, and that abandoning this assumption in our interpretation of quantum mechanics (a position sometimes referred to as "holism") will allow us to restore a satisfying locality principle. Separability here means that all events associated to the union of some set of disjoint regions are combinations of events associated to each region taken separately. In this article, it is shown that: (a) localised events can be consistently defined without implying separability; (b) the definition of Bell's locality condition does not rely on separability in any way; (c) the proof of Bell's theorem does not use separability as an assumption. If, inspired by considerations of non-separability, the assumptions of Bell's theorem are weakened, what remains no longer embodies the locality principle. Teller's argument for "relational holism" and Howard's arguments concerning separability are criticised in the light of these results. Howard's claim that Einstein grounded his arguments on the incompleteness of QM with a separability assumption is also challenged. Instead, Einstein is better interpreted as referring merely to the existence of localised events. Finally, it is argued that Bell rejected the idea that separability is an assumption of his theorem., Comment: 26 pages main body, 33 total. v2: several references and comments added, many minor revisions made, appendix B significantly revised

In a recent article entitled "A simple explanation of the quantum violation of a fundamental inequality," Cabello proposes a condition on a class of probabilistic models that, he claims, gives the same bound on contextuality for the KCBS inequality as quantum mechanics, and also rules out PR-box nonlocality. He conjectures that the condition will also reproduce quantum limits on contextuality in other scenarios. Here we show that the proposed principle is actually too weak to derive these results. Cabello has implicitly assumed in the proofs that if all pairs in a set of events are pairwise exclusive (so that their probabilities must sum to less than 1), the set can itself be considered exclusive. Perhaps surprisingly, this is not in general true in the general probabilistic framework under discussion. We call this hidden assumption "Consistent Exclusivity" or CE. With this extra assumption Cabello's proofs are sound. Furthermore, it is established that CE holds in quantum mechanics, providing a reasonable and simple new principle that may characterise quantum non-contextuality in many scenarios., Comment: 9 pages, 2 figures. Version 2: note added to acknowledge the previously overlooked work of Fritz, Sainz, Augusiak, Brask, Chaves, Leverrier, and Ac\'in, which gives results and arguments essentially the same as some of those given here. Some references added. v3: comment altered for various reasons, particularly to correct a mistaken labeling of a result

R\'edei and san Pedro discuss my "Comparing Causality Principles," their main aim being to distinguish reasonable weakened versions of two causality principles presented there, "SO1" and "SO2". They also argue that the proof that SO1 implies SO2 contains a flaw. Here, a reply is made to a number of points raised in their paper. It is argued that the "intuition" that SO1 should be stronger than SO2 is implicitly based on a false premise. It is pointed out that a similar weakening of SO2 was already considered in the original paper. The technical definition of the new conditions is shown to be defective. The argument against the stronger versions of SO1 and SO2 given by R\'edei and san Pedro is criticised. The flaw in the original proof is shown to be an easily corrected mistake in the wording. Finally, it is argued that some cited results on causal conditions in AQFT have little relevance to these issues, and are, in any case, highly problematic in themselves., Comment: 6 pages, 1 figure. To appear in Stud. Hist. Philos. Mod. Phys. (some minor modifications made here)

Bell's theorem shows that the reasonable relativistic causal principle known as "local causality" is not compatible with the predictions of quantum mechanics. It is not possible maintain a satisfying causal principle of this type while dropping any of the better-known assumptions of Bell's theorem. However, another assumption of Bell's theorem is the use of classical logic. One part of this assumption is the principle of "ontic definiteness", that is, that it must in principle be possible to assign definite truth values to all propositions treated in the theory. Once the logical setting is clarified somewhat, it can be seen that rejecting this principle does not in any way undermine the type of causal principle used by Bell. Without ontic definiteness, the deterministic causal condition known as Einstein Locality succeeds in banning superluminal influence (including signalling) whilst allowing correlations that violate Bell's inequalities. Objections to altering logic, and the consequences for operational and realistic viewpoints, are also addressed., Comment: 40 pages (26 main text), 3 figures

We model the classical transmission of a massless scalar field from a source to a detector on a background causal set. The predictions do not differ significantly from those of the continuum. Thus, introducing an intrinsic inexactitude to lengths and durations - or more specifically, replacing the Lorentzian manifold with an underlying discrete structure - need not disrupt the usual dynamics of propagation., Comment: 16 pages, 1 figure. Version 2: reference added

This paper presents an brief review of some recent work on the causal set approach to quantum gravity. Causal sets are a discretisation of spacetime that allow the symmetries of GR to be preserved in the continuum approximation. One proposed application of causal sets is to use them as the histories in a quantum sum-over-histories, i.e. to construct a quantum theory of spacetime. It is expected by many that quantum gravity will introduce some kind of "fuzziness", uncertainty and perhaps discreteness into spacetime, and generic effects of this fuzziness are currently being sought. Applied as a model of discrete spacetime, causal sets can be used to construct simple phenomenological models which allow us to understand some of the consequences of this general expectation., Comment: 24 pages, 4 figures. Based on a proceedings article for the "Foundations of Space and Time" conference, Cape Town, August 2009, in honour of George Ellis' 70th birthday.

Employing standard results from spectral geometry, we provide strong evidence that in the classical limit the ground state of three-dimensional causal dynamical triangulations is de Sitter spacetime. This result is obtained by measuring the expectation value of the spectral dimension on the ensemble of geometries defined by these models, and comparing its large scale behaviour to that of a sphere (Euclidean de Sitter). From the same measurement we are also able to confirm the phenomenon of dynamical dimensional reduction observed in this and other approaches to quantum gravity -- the first time this has been done for three-dimensional causal dynamical triangulations. In this case, the value for the short-scale limit of the spectral dimension that we find is approximately 2. We comment on the relevance of these results for the comparison to asymptotic safety and Horava-Lifshitz gravity, among other approaches to quantum gravity., Comment: 25 pages, 6 figures. Version 2: references to figures added, acknowledgment added.

A new procedure for coarse-graining dynamical triangulations is presented. The procedure provides a meaning for the relevant value of observables when "probing at large scales", e.g. the average scalar curvature. The scheme may also be useful as a starting point for a new type of renormalisation procedure, suitable for dynamically triangulated quantum gravity. Random Delaunay triangulations have previously been used to produce discretisations of continuous Euclidean manifolds, and the coarse-graining scheme is an extension of this idea, using random simplicial complexes produced from a dynamical triangulation. In order for a coarse-graining process to be useful, it should preserve the properties of the original dynamical triangulation that are relevant when probing at large scales. Some general discussion of this point is given, along with some arguments in favour of the proposed scheme., Comment: 19 pages (14 main body), 3 figures. Accepted for publication in Class. Quant. Grav

This paper reviews the histories approach to quantum mechanics. This discussion is then applied to theories of quantum gravity. It is argued that some of the quantum histories must approximate (in a suitable sense) to classical histories, if the correct classical regime is to be recovered. This observation has significance for the formulation of new theories (such as quantum gravity theories) as it puts a constraint on the kinematics, if the quantum/classical correspondence principle is to be preserved. Consequences for quantum gravity, particularly for Lorentz symmetry and the idea of "emergent geometry", are discussed., Comment: 35 pages (29 pages main body), two figures

We introduce a new matrix model that describes Causal Dynamical Triangulations (CDT) in two dimensions. In order to do so, we introduce a new, simpler definition of 2D CDT and show it to be equivalent to the old one. The model makes use of ideas from dually weighted matrix models, combined with multi-matrix models, and can be studied by the method of character expansion., Comment: 9 pages, 2 figures; slightly extended version with more details on the character expansion

Non-perturbative theories of quantum gravity inevitably include configurations that fail to resemble physically reasonable spacetimes at large scales. Often, these configurations are entropically dominant and pose an obstacle to obtaining the desired classical limit. We examine this "entropy problem" in a model of causal set quantum gravity corresponding to a discretisation of 2D spacetimes. Using results from the theory of partial orders we show that, in the large volume or continuum limit, its partition function is dominated by causal sets which approximate to a region of 2D Minkowski space. This model of causal set quantum gravity thus overcomes the entropy problem and predicts the emergence of a physically reasonable geometry., Comment: Corrections and clarifications. Conclusions unchanged

This paper concerns sprinklings into Minkowski space (Poisson processes). It proves that there exists no equivariant measurable map from sprinklings to spacetime directions (even locally). Therefore, if a discrete structure is associated to a sprinkling in an intrinsic manner, then the structure will not pick out a preferred frame, locally or globally. This implies that the discreteness of a sprinkled causal set will not give rise to ``Lorentz breaking'' effects like modified dispersion relations. Another consequence is that there is no way to associate a finite-valency graph to a sprinkling consistently with Lorentz invariance., Comment: 7 pages, laTeX

One obtains Bell's inequalities if one posits a hypothetical joint probability distribution, or {\it measure}, whose marginals yield the probabilities produced by the spin measurements in question. The existence of a joint measure is in turn equivalent to a certain causality condition known as ``screening off''. We show that if one assumes, more generally, a joint {\it quantal measure}, or ``decoherence functional'', one obtains instead an analogous inequality weaker by a factor of $\sqrt{2}$. The proof of this ``Tsirel'son inequality'' is geometrical and rests on the possibility of associating a Hilbert space to any strongly positive quantal measure. These results lead both to a {\it question}: ``Does a joint measure follow from some quantal analog of `screening off'?'', and to the {\it observation} that non-contextual hidden variables are viable in histories-based quantum mechanics, even if they are excluded classically., Comment: 38 pages, TeX. Several changes and added comments to bring out the meaning more clearly. Minor rewording and extra acknowledgements, now closer to published version

This article concerns the fate of local Lorentz invariance in quantum gravity, particularly for approaches in which a discrete structure replaces continuum spacetime. Some features of standard quantum mechanics, presented in a sum-over-histories formulation, are reviewed, and their consequences for such theories are discussed. It is argued that, if the individual histories of a theory give bad approximations to macroscopic continuum properties in some frames, then it is inevitable that the theory violates Lorentz symmetry., Comment: 15 pages, 1 figure, laTeX

The ideas of spacetime discreteness and causality are important in several of the popular approaches to quantum gravity. But if discreteness is accepted as an initial assumption, conflict with Lorentz invariance can be a consequence. The causal set is a discrete structure which avoids this problem and provides a possible history space on which to build a ``path integral'' type quantum gravity theory. Motivation, results and open problems are discussed and some comparisons to other approaches are made. Some recent progress on recovering locality in causal sets is recounted., Comment: 22 pages, 4 figures, Latex. Extended version of a review to be published in "Approaches to Quantum Gravity - Towards a new understanding of space and time" (ed. D. Oriti), Cambridge University Press, 2006. Ref added. Dedicated to Rafael Sorkin on the occasion of his 60th birthday

A criticism sometimes made of the causal set quantum gravity program is that there is no practical scheme for identifying manifoldlike causal sets and finding embeddings of them into manifolds. A computational method for constructing an approximate embedding of a small manifoldlike causal set into Minkowski space (or any spacetime that is approximately flat at short scales) is given, and tested in the 2 dimensional case. This method can also be used to determine how manifoldlike a causal set is, and conversely to define scales of manifoldlikeness., Comment: latex; 8 pages, one figure; accepted for publication in Classical and Quantum Gravity Letters

The principle of common cause is discussed as a possible fundamental principle of physics. Some revisions of Reichenbach's formulation of the principle are given, which lead to a version given by Bell. Various similar forms are compared and some equivalence results proved. The further problems of causality in a quantal system, and indeterministic causal structure, are addressed, with a view to defining a causality principle applicable to quantum gravity., Comment: 27 pages, 1 figure. Some mistakes corrected, discussions expanded, and a reference added. In new version, some notation and wording changed, another new reference added; now closer to published version

Contrary to what is often stated, a fundamental spacetime discreteness need not contradict Lorentz invariance. A causal set's discreteness is in fact locally Lorentz invariant, and we recall the reasons why. For illustration, we introduce a phenomenological model of massive particles propagating in a Minkowski spacetime which arises from an underlying causal set. The particles undergo a Lorentz invariant diffusion in phase space, and we speculate on whether this could have any bearing on the origin of high energy cosmic rays., Comment: 13 pages. Replaced version with corrected fundamental solution, missing m's (mass) and c's (speed of light) added and reference on diffusion on the three sphere changed. Note with additional references added and addresses updated, as in published version

The ``generic'' family of classical sequential growth dynamics for causal sets provides cosmological models of causal sets which are a testing ground for ideas about the, as yet unknown, quantum theory. In particular we can investigate how general covariance manifests itself and address the problem of identifying and interpreting covariant ``observables'' in quantum gravity. The problem becomes, in this setting, that of identifying measurable covariant collections of causal sets, to each of which corresponds the question: ``Does the causal set that occurs belong to this collection?'' It has for answer the probability measure of the collection. Answerable covariant questions, then, correspond to measurable collections of causal sets which are independent of the labelings of the causal sets. However, what the transition probabilities of the classical sequential growth dynamics provide directly is a measure on the space of {\it labeled} causal sets and the physical interpretation of the covariant measurable collections is consequently obscured. We show that there is a physically meaningful characterisation of the class of measurable covariant sets as unions and differences of ``stem sets''., Comment: LaTeX, 1+19 pages, 2 figures

We present a spontaneous collapse model of a field theory on a 1+1 null lattice, in which the causal structure of the lattice plays a central role. Issues such as ``locality,'' ``non-locality'' and superluminal signaling are addressed in the context of the model which has the virtue of extreme simplicity. The formalism of the model is related to that of the consistent histories approach to quantum mechanics., Comment: 12 pages, 1 figure. Journal version

This essay discusses the idea that a Theory of Everything would not be complete without a theory of consciousness as one of its parts, and the suggestion that new physics may be needed to describe consciousness. I argue that the motivations behind searching for such a theory arise as the result of misunderstandings over the use of language when talking about consciousness., Comment: This paper has been withdrawn by the author, due to large numbers of crank responses

Identifying an appropriate set of ``observables'' is a nontrivial task for most approaches to quantum gravity. We describe how it may be accomplished in the context of a recently proposed family of stochastic (but classical) dynamical laws for causal sets. The underlying idea should work equally well in the quantum case., Comment: plainTeX, 17 pages on 8.5x11 inch paper. To appear in the proceedings of the Alternative Natural Philosophy Association meeting, held August 16-21, 2001, Cambridge, England