We describe numerical and analytical investigations of causal sets sprinkled into spacetime manifolds. The first part of the paper is a numerical study of finite causal sets sprinkled into Alexandrov subsets of Minkowski spacetime of dimensions $1 + 1$, $1 + 2$ and $1 + 3$. In particular we consider the rank 2 past of sprinkled causet events, which is the set of events that are two links to the past. Assigning one of the rank 2 past events as `preferred past' for each event yields a `preferred past structure', which was recently proposed as the basis for a causal set d'Alembertian. We test six criteria for selecting rank 2 past subsets. One criterion performs particularly well at uniquely selecting -- with very high probability -- a preferred past satisfying desirable properties. The second part of the paper concerns (infinite) sprinkled causal sets for general spacetime manifolds. After reviewing the construction of the sprinkling process with the Poisson measure, we consider various specific applications. Among other things, we compute the probability of obtaining a sprinkled causal set of a given isomorphism class by combinatorial means, using a correspondence between causal sets in Alexandrov subsets of $1 + 1$ dimensional Minkowski spacetime and 2D-orders. These methods are also used to compute the expected size of the past infinity as a proportion of the total size of a sprinkled causal set.
Evidence is provided for a conjecture that, in the continuum limit, the mean of the causal set action of a causal set sprinkled into a globally hyperbolic Lorentzian spacetime, M, of finite volume equals the Einstein Hilbert action of M plus the volume of the co-dimension 2 intersection of the future boundary with the past boundary. We give the heuristic argument for this conjecture and analyse some examples in 2 dimensions and one example in 4 dimensions.
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 correct…
We study the continuum limit of the Benincasa-Dowker-Glaser causal set action on a causally convex compact region. In particular, we compute the action of a causal set randomly sprinkled on a small causal diamond in the presence of arbitrary curvature in various spacetime dimensions. In the continuum limit, we show that the action admits a finite limit. More importantly, the limit is composed by an Einstein-Hilbert bulk term as predicted by the Benincasa-Dowker-Glaser action, and a boundary term exactly proportional to the codimension-two joint volume. Our calculation provides strong evidence …
The causal set approach to quantum gravity embodies the concepts of causality and discreteness. This article explores some foundational and conceptual issues within causal set theory.
While it is possible to build causal sets that approximate spacetime manifolds, most causal sets are not at all manifold-like. We show that a Lorentzian path integral with the Einstein-Hilbert action has a phase in which one large class of non-manifold-like causal sets is strongly suppressed, and suggest a direction for generalization to other classes. While we cannot yet show our argument holds for all non-manifold-like sets, our results make it plausible that the path integral might lead to emergent manifold-like behavior with no need for further conditions.
The Causal Set hypothesis asserts that spacetime, ultimately, is discrete and its underlying structure is that of a locally finite partial ordered set, and macroscopic causality reflects a deeper notion of order in terms of which all the geometrical structure of spacetime must find their ultimate expression. After reviewing the main aspects of Causal Sets Kinematics, and the recently developed Stochastic Dynamics. We concentrate on possible implications in the fields of cosmology and black holes. In the context of black hole, we propose a possible interpretation of the entropy as the number of…
We investigate the spectral dimension obtained from non-local continuum d'Alembertians derived from causal sets. We find a universal dimensional reduction to 2 dimensions, in all dimensions. We conclude by discussing the validity and relevance of our results within the broader context of quantum field theories based on these nonlocal dynamics.
Starting from certain causality conditions and a discrete form of general covariance, we derive a very general family of classically stochastic, sequential growth dynamics for causal sets. The resulting theories provide a relatively accessible ``half way house'' to full quantum gravity that possibly contains the latter's classical limit (general relativity). Because they can be expressed in terms of state models for an assembly of Ising spins living on the relations of the causal set, these theories also illustrate how non-gravitational matter can arise dynamically from the causal set without …
We investigate the behavior of small subsets of causal sets that approximate Minkowski space in three, four, and five dimensions, and show that their effective dimension decreases smoothly at small distances. The details of the short distance behavior depend on a choice of dimensional estimator, but for a reasonable version of the Myrheim-Meyer dimension, the minimum dimension is $d \approx 2$, reproducing results that have been seen in other approaches to quantum gravity.
We present a new method for embedding a causal set into an interval of Minkowski spacetime. The method uses spacetime volumes for causally related elements to define causal set analogs of Minkowski inner products. These are used to construct matrices of inner products which are then factored using the singular value decomposition to give coordinates in Minkowski spacetime. Results are presented showing good quality embeddings into Minkowski spacetime for dimensions d=2,3,4. The method applies in any dimension and does not require spacelike distances to be used as an input. It offers a new way …
We propose a manifestly covariant framework for causal set dynamics. The framework is based on a structure, dubbed covtree, which is a partial order on certain sets of finite, unlabeled causal sets. We show that every infinite path in covtree corresponds to at least one infinite, unlabeled causal set. We show that transition probabilities for a classical random walk on covtree induce a classical measure on the $σ$-algebra generated by the stem sets.
We examine the validity and scope of Johnston's models for scalar field retarded Green functions on causal sets in 2 and 4 dimensions. As in the continuum, the massive Green function can be obtained from the massless one, and hence the key task in causal set theory is to first identify the massless Green function. We propose that the 2-d model provides a Green function for the massive scalar field on causal sets approximated by any topologically trivial 2 dimensional spacetime. We explicitly demonstrate that this is indeed the case in a Riemann normal neighbourhood. In 4-d the model can again …
A one parameter family of retarded linear operators on scalar fields on causal sets is introduced. When the causal set is well-approximated by 4 dimensional Minkowski spacetime, the operators are Lorentz invariant but nonlocal, are parametrised by the scale of the nonlocality and approximate the continuum scalar D'Alembertian, $\Box$, when acting on fields that vary slowly on the nonlocality scale. The same operators can be applied to scalar fields on causal sets which are well-approximated by curved spacetimes in which case they approximate $\Box - {1/2}R$ where $R$ is the Ricci scalar curvat…
Causal set theory provides a model of discrete spacetime in which spacetime events are represented by elements of a causal set---a locally finite, partially ordered set in which the partial order represents the causal relationships between events. The work presented here describes a model for matter on a causal set, specifically a theory of quantum scalar fields on a causal set spacetime background. The work starts with a discrete path integral model for particles on a causal set. Here quantum mechanical amplitudes are assigned to trajectories within the causal set. By summing these over all t…
We describe an algebraic way to code the causal information of a discrete spacetime. The causal set C is transformed to a description in terms of the causal pasts of the events in C. This is done by an evolving set, a functor which to each event of C assigns its causal past. Evolving sets obey a Heyting algebra which is characterised by a non-standard notion of complement. Conclusions about the causal structure of the causal set can be drawn by calculating the complement of the evolving set. A causal quantum theory can be based on the quantum version of evolving sets, which we briefly discuss.
In this paper we will explore two different proposals for the action for causal sets: the Benincasa-Dowker action and a modified version of the chain action. We propose a variational principle for two-dimensional causal sets and use it for both actions to determine which causal sets at least on average satisfy a discrete version of the Einstein equation. Specifically, we test this method on causal sets embedded in 2d Minkowski, de Sitter, and anti-de Sitter spacetimes and compare these results to the most prominent nonmanifoldlike causal sets, Kleitman-Rothschild 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 asymp…
We evaluate the spectral dimension in causal set quantum gravity by simulating random walks on causal sets. In contrast to other approaches to quantum gravity, we find an increasing spectral dimension at small scales. This observation can be connected to the non-locality of causal set theory that is deeply rooted in its fundamentally Lorentzian nature. Based on its large-scale behaviour, we conjecture that the spectral dimension can serve as a tool to distinguish causal sets that approximate manifolds from those that do not. As a new tool to probe quantum spacetime in different quantum gravity…
After sketching a context in which to seek observable signals of spatio-temporal discreteness, I briefly review the status of the causal set program for quantum gravity, concluding with a simple model for the field produced by a moving charge in a background causal set.
A scheme for an algebraic quantization of the causal sets of Sorkin et al. is presented. The suggested scenario is along the lines of a similar algebraization and quantum interpretation of finitary topological spaces due to Zapatrin and this author. To be able to apply the latter procedure to causal sets Sorkin's `semantic switch' from `partially ordered sets as finitary topological spaces' to `partially ordered sets as locally finite causal sets' is employed. The result is the definition of `quantum causal sets'. Such a procedure and its resulting definition is physically justified by a prope…
Causal sets are particular partially ordered sets which have been proposed as a basic model for discrete space-time in quantum gravity. We show that the class C of all countable past-finite causal sets contains a unique causal set (U,<) which is universal (i.e., any member of C can be embedded into (U,<)) and homogeneous (i.e., (U,<) has maximal degree of symmetry). Moreover, (U,<) can be constructed both probabilistically and explicitly. In contrast, the larger class of all countable causal sets does not contain a universal object.
We explore whether the growth dynamics paradigm of Causal Set Theory is compatible with past-infinite causal sets. We modify the Classical Sequential Growth dynamics of Rideout and Sorkin to accommodate growth "into the past" and discuss what form physical constraints such as causality could take in this new framework. We propose convex-suborders as the "observables" or "physical properties" in a theory in which causal sets can be past-infinite and use this proposal to construct a manifestly covariant framework for dynamical models of growth for past-infinite causal sets.
Causal set quantum gravity is a Lorentzian approach to quantum gravity, based on the causal structure of spacetime. It models each spacetime configuration as a discrete causal network of spacetime points. As such, key questions of the approach include whether and how a reconstruction of a sufficiently coarse-grained spacetime geometry is possible from a causal set. As an example for the recovery of spatial geometry from discrete causal structure, the construction of a spatial distance function for causal sets is reviewed. Secondly, it is an open question whether the path sum over all causal se…
These are lecture notes on causal set theory prepared in Jan. 2002 for a Summer School in Valdivia, Chile. In some places, they are more complete, in others much less so, regrettably. An extensive set of references and a glossary of terms can be found at the end of the notes.
We construct a Dirac theory on causal sets; a key element in the construction being that the causet must be regarded as emergent in an appropriate sense too. We further notice that mixed norm spaces appear in the construction allowing for negative norm particles and "ghosts".
A large class of the dynamical laws for causal sets described by a classical process of sequential growth yield a cyclic universe, whose cycles of expansion and contraction are punctuated by single `origin elements' of the causal set. We present evidence that the effective dynamics of the immediate future of one of these origin elements, within the context of the sequential growth dynamics, yields an initial period of de Sitter like exponential expansion, and argue that the resulting picture has many attractive features as a model of the early universe, with the potential to solve some of the …
Dual structures on causal sets called timelets are introduced, being discrete analogs of global time coordinates. Algebraic and geometrical features of the set of timelets on a causal set are studied. A characterization of timelets in terms of incidence matrix of causal set is given. The connection between timelets and preclusive coevents is established, it is shown that any timelet has a unique decomposition over preclusive coevents. The equivalence classes of timelets with respect to reascaling are shown to form a simplicial complex.
In order to understand the detailed mechanism by which a fundamental discreteness can provide a finite entanglement entropy, we consider the entanglement entropy of two classes of free massless scalar fields on causal sets that are well approximated by causal diamonds in Minkowski spacetime of dimensions 2,3 and 4. The first class is defined from discretised versions of the continuum retarded Green functions, while the second uses the causal set's retarded nonlocal d'Alembertians parametrised by a length scale $l_k$. In both cases we provide numerical evidence that the area law is recovered wh…
We study the utility of chains defined on causal sets in estimating continuum properties like the curvature, the proper time and the spacetime dimension through a numerical analysis. In particular, we show that in $\text{dS}_2$ and $\text{FLRW}_3$ spacetimes the formalism of arXiv:1212.0631 with slight modifications gives the right continuum properties. We also discuss a possible test of manifoldlikeness using this formalism by considering two models of non-manifoldlike causal sets. This is a part of a broader idea of the geometrical reconstruction of continuum properties given a discrete sub …
A new approach to quantum gravity is described which joins the loop representation formulation of the canonical theory to the causal set formulation of the path integral. The theory assigns quantum amplitudes to special classes of causal sets, which consist of spin networks representing quantum states of the gravitational field joined together by labeled null edges. The theory exists in 3+1, 2+1 and 1+1 dimensional versions, and may also be interepreted as a theory of labeled timelike surfaces. The dynamics is specified by a choice of functions of the labelings of d+1 dimensional simplices,whi…
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 fo…
This is is a copy of dissertation that I have submitted in defense of my ph.d. thesis, with some minor changes that I have made since then. The goal of the project is to generalize matter fields and their Lagrangians from regular space time to causal sets.
The model is a particular case of causal set. This is a discrete model of spacetime in a microscopic level. In paper the most general properties of the model are investigated without any reference to a dynamics. The dynamics of the model is introduced in [arXiv: 1004.5077]. These two papers introduce a consistent description of the model.
We propose an approach to quantum theory based on the energetic causal sets, introduced in Cortês and Smolin (2013). Fundamental processes are causal sets whose events carry momentum and energy, which are transmitted along causal links and conserved at each event. Energetic causal set models differ from other spacetime-free causal set approaches, e.g. Ref. [2] proposed causal sets based on quantum information processing systems, and Ref. [3] proposed causal sets constructed out of standard model particles. Fundamentally there are amplitudes for causal processes in energetic causal sets, but no…
Energetic causal sets are causal sets endowed by a flow of energy-momentum between causally related events. These incorporate a novel mechanism for the emergence of space-time from causal relations. Here we construct a spin foam model which is also an energetic causal set model. This model is closely related to the model introduced in parallel by Wolfgang Wieland in arXiv:1407.0025, and this construction makes use of results used there. What makes a spin foam model also an energetic causal set is Wieland's identification of new momenta, conserved at events (or four-simplices), whose norms are …
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.
We show that the Feynman propagator (or the d'Alembertian) of a causal set contains the complete information about the causal set. Intuitively, this is because the Feynman propagator, being a correlator that decays with distance, provides a measure for the invariant distance between pairs of events. Further, we show that even the spectra alone (of the self-adjoint and anti-self-adjoint parts) of the propagator(s) and d'Alembertian already carry large amounts of geometric information about their causal set. This geometric information is basis independent and also gauge invariant in the sense th…
In order to construct a quantum theory of gravity, we may have to abandon certain assumptions we were making. In particular, the concept of spacetime as a continuum substratum is questioned. Causal Sets is an attempt to construct a quantum theory of gravity starting with a fundamentally discrete spacetime. In this contribution we review the whole approach, focusing on some recent developments in the kinematics and dynamics of the approach.
We study the distribution of maximal-chain lengths between two elements of a causal set and its relationship with the embeddability of the causal set in a region of flat spacetime. We start with causal sets obtained from uniformly distributed points in Minkowski space. After some general considerations we focus on the 2-dimensional case and derive a recursion relation for the expected number of maximal chains $n_k$ as a function of their length $k$ and the total number of points $N$ between the maximal and minimal elements. By studying these theoretical distributions as well as ones generated …
We consider Lorentzian manifolds as examples of partially ordered measure spaces, sets endowed with compatible partial order relations and measures, in this case given by the causal structure and the volume element defined by each Lorentzian metric. This places the structure normally used to describe spacetime in geometrical theories of gravity in a more general context, which includes the locally finite partially ordered sets of the causal set approach to quantum gravity. We then introduce a function characterizing the closeness between any two partially ordered measure spaces and show that, …
A classical precursor to a full quantum dynamics for causal sets has been forumlated in terms of a stochastic sequential growth process in which the elements of the causal set arise in a sort of accretion process. The transition probabilities of the Markov growth process satisfy certain physical requirements of causality and general covariance, and the generic solution with all transition probabilities non-zero has been found. Here we remove the assumption of non-zero probabilities, define a reasonable extension of the physical requirements to cover the case of vanishing probabilities, and fin…
Many of the most important astrophysical tests of Lorentz symmetry also assume that energy-momentum of the observed particles is exactly conserved. In the causal set approach to quantum gravity a particular kind of Lorentz symmetry holds but energy-momentum conservation may be violated. We show that incorrectly assuming exact conservation can give rise to a spurious signal of Lorentz symmetry violation for a causal set. However, the size of this spurious signal is much smaller than can be currently detected and hence astrophysical Lorentz symmetry tests as currently performed are safe from cau…
The framework of perturbative algebraic quantum field theory (pAQFT) is used to construct QFT models on causal sets. We discuss various discretised wave operators, including a new proposal based on the idea of a `preferred past', which we also introduce, and show how they may be used to construct classical free and interacting field theory models on a fixed causal set; additionally, we describe how the sensitivity of observables to changes in the background causal set may be encapsulated in a relative Cauchy evolution. These structures are used as the basis of a deformation quantization, using…
