Causal Set Research Articles

Dark matter from spacetime nonlocality

We propose that dark matter is not yet another new particle in nature, but that it is a remnant of quantum gravitational effects on known fields. We arrive at this possibility in an indirect and surprising manner: by considering retarded, nonlocal, and Lorentzian evolution for quantum fields. This is inspired by recent developments in causal set theory, where such an evolution shows up as the continuum limit of scalar field propagation on a background causal set. Concretely, we study the quantum theory of a massless scalar field whose evolution is given not by the the d'Alembertian $\ensuremath{\square}$, but by an operator $\stackrel{\texttildelow{}}{\ensuremath{\square}}$ which is Lorentz invariant, reduces to $\ensuremath{\square}$ at low energies, and defines an explicitly retarded evolution: $(\stackrel{\texttildelow{}}{\ensuremath{\square}}\ensuremath{\phi})(x)$ only depends on $\ensuremath{\phi}(y)$, where $y$ is in the causal past of $x$. This modification results in the existence of a continuum of massive particles, in addition to the usual massless ones, in the free theory. When interactions are introduced, these massive or off-shell quanta can be produced by the scattering of massless particles, but once produced, they no longer interact, which makes them a natural candidate for dark matter.

Multiple SNP Set Analysis for Genome-Wide Association Studies Through Bayesian Latent Variable Selection.

The power of genome-wide association studies (GWAS) for mapping complex traits with single-SNP analysis (where SNP is single-nucleotide polymorphism) may be undermined by modest SNP effect sizes, unobserved causal SNPs, correlation among adjacent SNPs, and SNP-SNP interactions. Alternative approaches for testing the association between a single SNP set and individual phenotypes have been shown to be promising for improving the power of GWAS. We propose a Bayesian latent variable selection (BLVS) method to simultaneously model the joint association mapping between a large number of SNP sets and complex traits. Compared with single SNP set analysis, such joint association mapping not only accounts for the correlation among SNP sets but also is capable of detecting causal SNP sets that are marginally uncorrelated with traits. The spike-and-slab prior assigned to the effects of SNP sets can greatly reduce the dimension of effective SNP sets, while speeding up computation. An efficient Markov chain Monte Carlo algorithm is developed. Simulations demonstrate that BLVS outperforms several competing variable selection methods in some important scenarios.

Linear structures, causal sets and topology

Causal set theory and the theory of linear structures (which has recently been developed by Tim Maudlin as an alternative to standard topology) share some of their main motivations. In view of that, I raise and answer the question how these two theories are related to each other and to standard topology. I show that causal set theory can be embedded into Maudlin׳s more general framework and I characterise what Maudlin׳s topological concepts boil down to when applied to discrete linear structures that correspond to causal sets. Moreover, I show that all topological aspects of causal sets that can be described in Maudlin׳s theory can also be described in the framework of standard topology. Finally, I discuss why these results are relevant for evaluating Maudlin׳s theory. The value of this theory depends crucially on whether it is true that (a) its conceptual framework is as expressive as that of standard topology when it comes to describing well-known continuous as well as discrete models of spacetime and (b) it is even more expressive or fruitful when it comes to analysing topological aspects of discrete structures that are intended as models of spacetime. On one hand, my theorems support (a). The theory is rich enough to incorporate causal set theory and its definitions of topological notions yield a plausible outcome in the case of causal sets. On the other hand, the results undermine (b). Standard topology, too, has the conceptual resources to capture those topological aspects of causal sets that are analysable within Maudlin׳s framework. This fact poses a challenge for the proponents of Maudlin׳s theory to prove it fruitful.

Boundary terms for causal sets

We propose a family of boundary terms for the action of a causal set with a spacelike boundary. We show that in the continuum limit one recovers the Gibbons–Hawking–York boundary term in the mean. We also calculate the continuum limit of the mean causal set action for an Alexandrov interval in flat spacetime. We find that it is equal to the volume of the codimension-2 intersection of the two light-cone boundaries of the interval.

Comparison theorems for causal diamonds

We formulate certain inequalities for the geometric quantities characterizing causal diamonds in curved and Minkowski spacetimes. These inequalities involve the red-shift factor which, as we show explicitly in the spherically symmetric case, is monotonic in the radial direction and it takes its maximal value at the centre. As a byproduct of our discussion we re-derive Bishop's inequality without assuming the positivity of the spatial Ricci tensor. We then generalize our considerations to arbitrary, static and not necessarily spherically symmetric, asymptotically flat spacetimes. In the case of spacetimes with a horizon our generalization involves the so-called {\it domain of dependence}. The respective volume, expressed in terms of the duration measured by a distant observer compared with the volume of the domain in Minkowski spacetime, exhibits behaviours which differ if $d=4$ or $d>4$. This peculiarity of four dimensions is due to the logarithmic subleading term in the asymptotic expansion of the metric near infinity. In terms of the invariant duration measured by a co-moving observer associated with the diamond we establish an inequality which is universal for all $d$. We suggest some possible applications of our results including the comparison theorems for entanglement entropy, causal set theory and fundamental limits on computation.

Correction terms for propagators and d’Alembertians due to spacetime discreteness

The causal set approach to quantum gravity models spacetime as a discrete structure—a causal set. Recent research has led to causal set models for the retarded propagator for the Klein–Gordon equation and the d’Alembertian operator. These models can be compared to their continuum counterparts via a sprinkling process. It has been shown that the models agree exactly with the continuum quantities in the limit of an infinite sprinkling density—the continuum limit. This paper obtains the correction terms for these models for sprinkled causal sets with a finite sprinkling density. These correction terms are an important step towards testable differences between the continuum and discrete models that could provide evidence of spacetime discreteness.

Complex Quantum Network Manifolds in Dimension d > 2 are Scale-Free.

In quantum gravity, several approaches have been proposed until now for the quantum description of discrete geometries. These theoretical frameworks include loop quantum gravity, causal dynamical triangulations, causal sets, quantum graphity, and energetic spin networks. Most of these approaches describe discrete spaces as homogeneous network manifolds. Here we define Complex Quantum Network Manifolds (CQNM) describing the evolution of quantum network states, and constructed from growing simplicial complexes of dimension . We show that in d = 2 CQNM are homogeneous networks while for d > 2 they are scale-free i.e. they are characterized by large inhomogeneities of degrees like most complex networks. From the self-organized evolution of CQNM quantum statistics emerge spontaneously. Here we define the generalized degrees associated with the -faces of the -dimensional CQNMs, and we show that the statistics of these generalized degrees can either follow Fermi-Dirac, Boltzmann or Bose-Einstein distributions depending on the dimension of the -faces.

P00117 Explaining the survival disadvantage for men with non-small-cell lung cancer (NSCLC) compared with women in Osaka, Japan

Background Survival from non-small-cell lung cancer (NSCLC) is much higher for women than for men in Japan. We adapted methods from the causal inference setting using routine population-based cancer registry data to examine what proportion of the difference in survival can be explained by diagnostic and treatment factors. Methods 34 161 patients were diagnosed with NSCLC between 1993 and 2004 in Osaka, Japan, and followed up for at least 5 years. We regarded patients with localised cancer as being diagnosed early and with regional or distant spread as diagnosed late. G-computation procedures were used to estimate the proportion of the effect of sex on survival mediated by stage and by treatment, controlling for sociodemographic factors. Findings There were substantial sex differences in NSCLC net survival (at 1 year 47% for men versus 53% for women; 5 years, 17% versus 24%). Men were less likely to be diagnosed at an early stage (22% men diagnosed early versus 28% women, OR 0.75 [0.71–0.80], p Interpretation Late presentation and less curative treatment contribute to the survival disadvantage in men. Disentangling this disadvantage by treatment and stage informs us what could be done to most effectively reduce the inequality in outcome.

Life Satisfaction Moderates the Effectiveness of a Play-Based Parenting Intervention in Low-Income Mothers and Toddlers.

This multi-method multi-trait study examined moderators and mediators of change in the context of a parenting intervention. Low-income, diverse mothers of toddlers (average age 30months; N = 186, 90 girls) participated in a play-based intervention (Child-Oriented Play versus Play-as-Usual) aimed at increasing children's committed compliance and reducing opposition toward their mothers, observed in prohibition contexts, and at reducing mother-rated children's behavior problems 6months after the intervention. Mothers' subjective sense of life satisfaction and fulfillment during the intervention and objective ratings of psychosocial functioning by clinicians, obtained in a clinical interview were posed as moderators, and mothers' observed power-assertive discipline immediately following the intervention was modeled as a mediator of its impact. We tested moderated mediation using structural equation modeling, with all baseline scores (prior to randomization) controlled. Mothers' subjective sense of life satisfaction moderated the impact of the intervention, but clinicians' ratings did not. For mothers highly satisfied with their lives, participating in Child-Oriented Play group, compared to Play-as-Usual group, led to a reduction in power-assertive discipline which, in turn, led to children's increased compliance and decreased opposition and externalizing problems. There were no effects for mothers who reported low life satisfaction. The study elucidates the causal sequence set in motion by the intervention, demonstrates the moderating role of mothers' subjective life satisfaction, highlights limitations of clinicians' ratings, and informs future prevention and intervention efforts to promote adaptive parenting.

Using graph theory for automated electric circuit solving

Graph theory plays many important roles in modern physics and in many different contexts, spanning diverse topics such as the description of scale-free networks and the structure of the universe as a complex directed graph in causal set theory. Graph theory is also ideally suited to describe many concepts in computer science. Therefore it is increasingly important for physics students to master the basic concepts of graph theory. Here we describe a student project where we develop a computational approach to electric circuit solving which is based on graph theoretic concepts. This highly multidisciplinary approach combines abstract mathematics, linear algebra, the physics of circuits, and computer programming to reach the ambitious goal of implementing automated circuit solving.

A Model for the Universe that Begins to Resemble a Quantum Computer

This article presents a sequential growth model for the Universe that acts like a quantum computer. The basic constituents of the model are a special type of causal set (causet) called a c-causet. A c-causet is defined to be a causet that has a unique labeling. We characterize c-causets as those causets that form a multipartite graph or equivalently those causets whose elements are comparable whenever their heights are different. We show that a c-causet has precisely two c-causet offspring. It follows that there are 2n c-causets of cardinality n + 1. This enables us to classify c-causets of cardinality n + 1 in terms of n-bits. We then quantize the model by introducing a quantum sequential growth process. This is accomplished by replacing the n-bits by n-qubits and defining transition amplitudes for the growth transitions. We mainly consider two types of processes, called stationary and completely stationary. We show that for stationary processes, the probability operators are tensor products of positive rank-one qubit operators. Moreover, the converse of this result holds. Simplifications occur for completely stationary processes. We close with examples of precluded events.

Nonlocal scalar quantum field theory from causal sets

We study a non-local scalar quantum field theory in flat spacetime derived from the dynamics of a scalar field on a causal set. We show that this non-local QFT contains a continuum of massive modes in any dimension. In 2 dimensions the Hamiltonian is positive definite and therefore the quantum theory is well-defined. In 4-dimensions, we show that the unstable modes of the non-local d'Alembertian are propagated via the so called Wheeler propagator and hence do not appear in the asymptotic states. In the free case studied here the continuum of massive mode are shown to not propagate in the asymptotic states. However the Hamiltonian is not positive definite, therefore potential issues with the quantum theory remain. Finally, we conclude with hints toward what kind of phenomenology one might expect from such non-local QFTs.

Identifying Candidate Risk Factors for Prescription Drug Side Effects Using Causal Contrast Set Mining

Big longitudinal observational databases present the opportunity to extract new knowledge in a cost effective manner. Unfortunately, the ability of these databases to be used for causal inference is limited due to the passive way in which the data are collected resulting in various forms of bias. In this paper we investigate a method that can overcome these limitations and determine causal contrast set rules efficiently from big data. In particular, we present a new methodology for the purpose of identifying risk factors that increase a patients likelihood of experiencing the known rare side effect of renal failure after ingesting aminosalicylates. The results show that the methodology was able to identify previously researched risk factors such as being prescribed diuretics and highlighted that patients with a higher than average risk of renal failure may be even more susceptible to experiencing it as a side effect after ingesting aminosalicylates.

A new action for simplicial gravity in four dimensions

We develop a proposal for a theory of simplicial gravity with spinors as the fundamental configuration variables. The underlying action describes a mechanical system with finitely many degrees of freedom, the system has a Hamiltonian and local gauge symmetries. We will close with some comments on the resulting quantum theory, and explain the relation to loop quantum gravity and twisted geometries. The paper appears in parallel with an article by Cortês and Smolin (2014 arXiv:1407.0032), who study the relevance of the model for energetic causal sets and various other approaches to quantum gravity.

An Isometric Dynamics for a Causal Set Approach to Discrete Quantum Gravity

We consider a covariant causal set approach to discrete quantum gravity. We first review the microscopic picture of this approach. In this picture a universe grows one element at a time and its geometry is determined by a sequence of integers called the shell sequence. We next present the macroscopic picture which is described by a sequential growth process. We introduce a model in which the dynamics is governed by a quantum transition amplitude. The amplitude satisfies a stochastic and unitary condition and the resulting dynamics becomes isometric. We show that the dynamics preserves stochastic states. By "doubling down" on the dynamics we obtain a unitary group representation and a natural energy operator. These unitary operators are employed to define canonical position and momentum operators.

Emergent spacetime in stochastically evolving dimensions

Changing the dimensionality of the space–time at the smallest and largest distances has manifold theoretical advantages. If the space is lower dimensional in the high energy regime, then there are no ultraviolet divergencies in field theories, it is possible to quantize gravity, and the theory of matter plus gravity is free of divergencies or renormalizable. If the space is higher dimensional at cosmological scales, then some cosmological problems (including the cosmological constant problem) can be attacked from a completely new perspective. In this paper, we construct an explicit model of “evolving dimensions” in which the dimensions open up as the temperature of the universe drops. We adopt the string theory framework in which the dimensions are fields that live on the string worldsheet, and add temperature dependent mass terms for them. At the Big Bang, all the dimensions are very heavy and are not excited. As the universe cools down, dimensions open up one by one. Thus, the dimensionality of the space we live in depends on the energy or temperature that we are probing. In particular, we provide a kinematic Brandenberger–Vafa argument for how a discrete causal set, and eventually a continuum (3+1)-dim spacetime along with Einstein gravity emerges in the Infrared from the worldsheet action. The (3+1)-dim Planck mass and the string scale become directly related, without any compactification. Amongst other predictions, we argue that LHC might be blind to new physics even if it comes at the TeV scale. In contrast, cosmic ray experiments, especially those that can register the very beginning of the shower, and collisions with high multiplicity and density of particles, might be sensitive to the dimensional cross-over.

Spacetime entanglement entropy in 1 + 1 dimensions

Sorkin (Expressing entropy globally in terms of (4D) field-correlations arXiv:1205.2953) defines an entropy for a Gaussian scalar field ϕ in an arbitrary region of either a continuous spacetime or a causal set, given only the correlator within the region. The definition is global and independent of any choice of spacelike hypersurface. As a first application, we compute numerically the entanglement entropy in two cases where the asymptotic form is known or suspected from conformal field theory, finding excellent agreement when the required ultraviolet cutoff is implemented as a truncation on spacetime mode sums. We also show how the symmetry of entanglement entropy reflects the fact that RS and SR share the same eigenvalues, with R and S being arbitrary matrices.

On the causal set–continuum correspondence

We present two results that concern certain aspects of the question: when is a causal set well approximated by a Lorentzian manifold? The first result is a theorem that shows that the number–volume correspondence, if required to hold even for arbitrarily small regions, is best realized via Poisson sprinkling. The second result concerns a family of lattices in dimensional Minkowski space, known as Lorentzian lattices, which we show provide a much better number–volume correspondence than Poisson sprinkling for large volumes. We argue, however, that this feature should not persist in higher dimensions. We conclude by conjecturing a form of the aforementioned theorem that holds under weaker assumptions, namely that Poisson sprinkling provides the best number–volume correspondence in dimensions for spacetime regions with macroscopically large volumes.

The universe as a process of unique events

We describe a new class of models of quantum space-time based on energetic causal sets and show that under natural conditions space-time emerges from them. These are causal sets whose causal links are labelled by energy and momentum and conservation laws are applied at events. The models are motivated by principles we propose govern microscopic physics which posit a fundamental irreversibility of time. One consequence is that each event in the history of the universe has a distinct causal relationship to the rest; this requires a novel form of dynamics which an be applied to uniquely distinctive events. We hence introduce a new kind of deterministic dynamics for a causal set in which new events are generated from pairs of progenitor events by a rule which is based on extremizing the distinctions between causal past sets of events. This dynamics is asymmetric in time, but we find evidence from numerical simulations of a 1+1 dimensional model, that an effective dynamics emerges which restores approximate time reversal symmetry. Energetic causal set models differ from other spacetime-free causal set approaches, e.g. Ref. [1] proposed causal sets based on quantum information processing systems, and Ref. [2] proposed causal sets constructed out of standard model particles. Finally we also present a natural twistorial representation of energetic causal sets.

Surrogacy marker paradox measures in meta-analytic settings.

Because of the time and expense required to obtain clinical outcomes of interest, such as functional limitations or death, clinical trials often focus the effects of treatment on earlier and more easily obtained surrogate markers. Preliminary work to define surrogates focused on the fraction of a treatment effect "explained" by a marker in a regression model, but as notions of causality have been formalized in the statistical setting, formal definitions of high-quality surrogate markers have been developed in the causal inference framework, using either the "causal effect" or "causal association" settings. In the causal effect setting, high-quality surrogate markers have a large fraction of the total treatment effect explained by the effect of the treatment on the marker net of the treatment on the outcome. In the causal association setting, high-quality surrogate markers have large treatment effects on the outcome when there are large treatment effects on the marker, and small effects on the outcome when there are small effects on the marker. A particularly important feature of a surrogate marker is that the direction of a treatment effect be the same for both the marker and the outcome. Settings in which the marker and outcome are positively associated but the marker and outcome have beneficial and harmful or harmful and beneficial treatment effects, respectively, have been referred to as "surrogate paradoxes". If this outcome always occurs, it is not problematic; however, as correlations among the outcome, marker, and their treatment effects weaken, it may occur for some trials and not for others, leading to potentially incorrect conclusions, and real-life examples that shortened thousands of lives are unfortunately available. We propose measures for assessing the risk of the surrogate paradox using the meta-analytic causal association framework, which allows us to focus on the probability that a given treatment will yield treatment effect in different directions between the marker and the outcome, and to determine the size of a beneficial effect of the treatment on the marker required to minimize the risk of a harmful effect of the treatment on the outcome. We provide simulations and consider two applications.