The causal set approach to quantum gravity

Causal set theory (CST) is a discrete approach to quantum gravity which postulates that at the most fundamental level, spacetime is a locally finite partially ordered set. The partial order on the set is a proto-causality relation while local finiteness is a local discreteness requirement. In the continuum approximation the former corresponds to the spacetime causality relation and the latter ensures that finite volume regions in a spacetime contain only a finite number of"fundamental"spacetime events. CST is deeply rooted in the Lorentzian character of spacetime, where a primary role is played by the causal structure poset. Despite the assumption of fundamental discreteness, however, CST does not violate local Lorentz invariance in the continuum approximation. The combination of discreteness and Lorentz invariance on the other hand leads to a characteristic non-locality which distinguishes CST from most other approaches to quantum gravity. In this review we give a broad, semi-pedagogical introduction to CST, highlighting key results as well as some of the key open questions. This review is intended both for the beginner student in quantum gravity as well as more seasoned researchers in the field.


Overview
In this review, causal set theory (CST) refers to the specific proposal made by Bombelli, Lee, Meyer and Sorkin (BLMS) in their 1987 paper . In CST, the spacetime continuum is replaced by a locally finite partially ordered set, the causal set, which encodes the twin principles of causality and discreteness. These principles are realised via the continuum approximation of CST, in which the elements of the causal set correspond to spacetime events, such that (a) the order relation on the set is the causal ordering between the associated spacetime events and (b) the cardinality of a local region of the causal set corresponds to the local spacetime volume.
This review is intended as a semi-pedagogical introduction to CST. The aim is to give a broad survey of the main results and open questions and to direct the reader to some of the many interesting open research problems in CST, some of which are accessible even to the beginner.
We begin in Sect. 2 with a historical perspective on the ideas behind CST. The twin principles of discreteness and causality at the heart of CST have both been proposed -sometimes independently and sometimes together -starting with Riemann (1873); Robb (1914Robb ( , 1936 and later Zeeman (1964); Kronheimer and Penrose (1967);Finkelstein (1969); Hemion (1988) and Myrheim (1978), culminating in the CST proposal of BLMS . The continuum approximation of CST is an implementation of a deep result in Lorentzian geometry due to Hawking et al (1976) and its generalisation by Malament (1977), which states that the causal structure determines the conformal geometry of a future and past distinguishing causal spacetime. In following this history, the discussion will be necessarily somewhat technical. For those unfamiliar with the terminology of causal structure we point to standard texts (Hawking and Ellis 1973;Beem et al 1996;Wald 1984;Penrose 1972).
In Sect. 3, we state the CST proposal and describe its continuum approximation, in which the order relation is associated with causality and cardinality with local spacetime volumes. We will refer to causal sets which possess a continuum approximation to a spacetime as manifold-like. Important to CST is its Hauptvermutung or Fundamental Conjecture, which roughly states that a manifold-like causal set is equivalent to the continuum spacetime, modulo differences upto the discreteness scale. Much of the discussion on the Hauptvermutung is centered on the question of how to estimate the closeness of Lorentzian manifolds or more generally, causal sets. While there is no full proof of the conjecture, there is a lot of evidence in its favour as we will see in Sect. 4. An important outcome of CST discreteness in the continuum approximation is that it does not violate Lorentz invariance as shown in an elegant theorem by Bombelli et al (2009). Because of the centrality of this result we review this construction in some detail. The combination of discreteness and Lorentz invariance moreover gives rise to an inherent and characteristic non-locality, which distinguishes CST from other discrete approaches. Following Sorkin (1997), we discuss how these twin principles of CST force us to take certain "forks in the road" to quantum gravity.
We present some of the key developments in CST in Sects. 4, 5 and 6. We begin with the kinematical structure of theory and the program of "geometric reconstruction" in Sect. 4. Here, the aim is to reconstruct manifold invariants from "order invariants" in a manifold-like causal set. Finding appropriate manifold-like order invariants can be challenging, since there is little in the mathematics literature which correlates order theory to geometry using the CST continuum approximation. Thus, extracting such invariants requires new technical tools and insights some of which we will describe (Myrheim 1978;Brightwell and Gregory 1991;Meyer 1988;Bombelli and Meyer 1989;Bombelli 1987;Reid 2003;Major et al 2007;Rideout and Wallden 2009;Sorkin 2007b;Benincasa and Dowker 2010;Benincasa 2013;Benincasa et al 2011;Glaser and Surya 2013;Roy et al 2013;Buck et al 2015;Cunningham 2018a;Aghili et al 2018;Eichhorn et al 2018). The correlation between order invariants and manifold invariants in the continuum approximation provides support for the Hauptvermutung and proves weaker, observable-dependent versions of the conjecture.
Somewhere between dynamics and kinematics is the study of quantum fields on manifold-like causal sets, which we describe in Sect. 5. The simplest system is free scalar field theory on a causal set approximated by d-dimensional Minkowski spacetime M d . Because causal sets do not admit a natural Hamiltonian framework, a fully covariant construction is required to obtain the quantum field theory vacuum. A natural starting point is the advanced and retarded Green function for a free scalar field theory since it is defined using the causal structure of the spacetime. The explicit form for these were found for M d for d = 2, 4 (Daughton 1993;Johnston 2008Johnston , 2010 as well as de Sitter spacetime . In trying to find a quantisation scheme on the causal set without reference to the continuum, Johnston (2009) found a novel covariant definition of the scalar field vacuum. Sorkin (2011a) later showed that the construction of this Sorkin-Johnston (SJ) vacuum provides a new insight into quantum field theory even in the continuum. This has stimulated the interest of the algebraic field theory community (Fewster and Verch 2012;Brum and Fredenhagen 2014;Fewster 2018). The SJ vacuum has also been used to calculate Sorkin's spacetime entanglement entropy (SSEE) (Bombelli et al 1986;Sorkin 2014) in a causal set Sorkin and Yazdi 2018). The calculation in d = 2 is surprising since it gives rise to a volume law for the SSEE. It is only after a subtle double truncation procedure that the area law can be recovered (Sorkin and Yazdi 2018;Belenchia et al 2018).
In Sect. 6, we describe the CST approach to quantum dynamics, which roughly follows two directions. The first, is the "principled approach", where one starts with a general set of axioms which respect microscopic covariance and causality. An important class of such theories is the set of Markovian classical sequential growth models (CSG) of Rideout and Sorkin (Rideout andSorkin 2000a, 2001;Martin et al 2001;Rideout 2001;Varadarajan and Rideout 2006), which we will describe in some detail. The dynamical framework is best described in terms of measure theory, with the classical covariant observables represented by a covariant event algebra A over the sample space Ω g of past finite causal sets (Brightwell et al 2003;Dowker and Surya 2006). One of the main questions in CST dynamics is whether the overwhelming entropic presence of the Kleitmann-Rothschild (KR) posets in Ω g can be overcome by the dynamics (Kleitman and Rothschild 1975). These KR posets are very nonmanifold-like and "static", with just three "moments of time". Hence, their entropic contribution should be suppressed by the dynamics in the classical or continuum approximation of the theory. In the CSG models, the typical causal sets generated are very "tall" with countable rather than finite moments of time and, though not quite manifold-like, very unlike the entropically dominant KR posets or the subleading entropic contributions of Dhar (1978Dhar ( , 1980. Hence, the CSG dynamics does succeed in countering the dominating entropy of these very non-manifold-like posets. Moreover, it was shown that with every cosmological bounce, the dynamics renormalises the coupling constants of CSG which approach a fixed point (Martin et al 2001). The CSG models have sparked much interest in the mathematics community, and the use of new mathematical tools to study the asymptotic structure of the theory (Brightwell and Georgiou 2010;Brightwell and Luczak 2012, 2011. In CST, the appropriate route to quantisation is via the quantum measure or decoherence functional defined in the double-path integral formulation (Sorkin 1994(Sorkin , 1995(Sorkin , 2007d. In the quantum versions of the CSG (QSG) models the transition probabilities of CSG are replaced by the decoherence functional. While covariance can be easily imposed, a quantum version of microscopic causality is still missing (Henson 2005). Another indication of the non-triviality of quantisation comes from a prosaic generalisation of transitive percolation, which is the simplest of the CSG models. In this "complex percolation" (CP) dynamics the quantum measure does not extend to the full algebra of observables which is an impediment to the construction of covariant quantum observables (Dowker et al 2010c). This can be somewhat alleviated by taking a physically motivated approach to measure theory (Sorkin 2011b). An important future direction is to construct covariant observables in a wider class of quantum dynamics and look for a quantum version of coupling constant renormalisation.
Whatever the ultimate quantum dynamics however, little sense can be made of the theory without a fully developed quantum interpretation for closed systems, essential to quantum gravity. Sorkin's coevent interpretation (Sorkin 2007a;Dowker and Ghazi-Tabatabai 2008) provides a promising avenue based on the quantum measure approach. While a discussion of this is outside of the scope of the present work, one can use the broader "principle of preclusion", i.e., that sets of zero quantum measure do not occur (Sorkin 2007a;Dowker and Ghazi-Tabatabai 2008), to make a limited set of predictions in CP (Sorkin and Surya, work in progress).
The second approach to quantisation is more pragmatic, and uses the continuum inspired path integral formulation of quantum gravity for causal sets. Here, the path integral is replaced by a sum over the sample space Ω of causal sets, using the Benincasa-Dowker (BD) action, which limits to the Einstein-Hilbert action (Benincasa and Dowker 2010). This can be viewed as an effective, continuum-like dynamics, arising from the more fundamental dynamics described above. A recent analytic calculation in Loomis and Carlip (2018) showed that a sub-dominant class of non-manifold causal sets, the bilayer posets, are suppressed in the path integral when using the BD action, under certain dimension dependent conditions satified by the parameter space. This gives hope that an effective dynamics might be able to overcome the entropy of the non-manifold-like causal sets.
In Surya (2012), Glaser and Surya (2016), and Glaser et al (2018), Markov Chain Monte Carlo (MCMC) methods were used for a dimensionally restricted sample space Ω 2d of 2d orders, which corresponds to topologically trivial d = 2 quantum gravity. The quantum partition function over causal sets can be rendered into a statistical partition function via an analytic continuation of a "temperature" parameter, but which still retains the Lorentzian character of the theory. This theory exhibits a first order phase transition (Surya 2012;Glaser et al 2018) between a manifold-like phase and a layered, non-manifoldlike one. MCMC methods have also been used to examine the sample space Ω n of n-element causal sets and to estimate the onset of asymptotia, characterised by the dominance of the KR posets (Henson et al 2017). These techniques have recently been extended to topologically non-trivial d = 2 CST and in the future could be used to study higher dimensional CST. While this approach gives us expectation values of covariant observables which allows for a straightforward interpretation, establishing a link to the quantum partition function is nontrivial and an open problem.
In Sect. 7, we describe in brief some of the exciting phenomenology that comes out of the kinematical structure of causal sets. This includes the momentum space diffusion coming from CST discreteness ("swerves") (Dowker et al 2004) and the effects of non-locality on quantum field theory (Sorkin 2007b), which includes a recent proposal for dark matter (Saravani and Afshordi 2017). Of these, the most striking is the 1987 prediction of Sorkin for the value of the cosmological constant Λ (Sorkin 1991(Sorkin , 1997. While the original argument was a kinematic estimate, subsequently dynamical models of fluctuating Λ were examined (Ahmed et al 2004;Ahmed and Sorkin 2013;Zwane et al 2018) and have been compared with recent observations (Zwane et al 2018). This is an exciting future direction of research in CST which interfaces intimately with observation. We conclude with a brief outlook for CST in Sect. 8.
As is true of all other approaches to quantum gravity, CST is not as yet a complete theory. Some of the challenges faced are universal to quantum gravity as a whole, while others are specific to the approach. Although we have developed a reasonably good grasp of the kinematical structure of CST and some progress has been made in the construction of effective quantum dynamics, CST still lacks a satisfactory quantum dynamics built from first principles. Progress in this direction is therefore very important for the future of the program. From a broader perspective, it is the opinion of this author that a deeper understanding of CST will help provide key insights into the nature of quantum gravity from a fully Lorentzian, causal perspective, whatever ultimate shape the final theory takes.
It is not possible for this review to be truly complete. The hope is that the interested reader will use it as a springboard to the existing literature. Several older reviews also exist, with differing emphasis (Sorkin 2005b;Henson 2006bHenson , 2010Dowker 2005;Surya 2013;Wallden 2013). The focus of the current review is to provide as cohesive an account of the program as possible, so as to be useful to a starting researcher in the field.

A historical perspective
One of the most important conceptual realisations that arose from the special and general theories of relativity in the early part of the 20th century, was that space and time are part of a single construct, that of spacetime. At a fundamental level, one does not exist without the other. Unlike Riemannian spaces, spacetime has a Lorentzian signature (−, +, +, +) which gives rise to local lightcones and an associated global causal structure. The causal structure (M, ≺) of a causal spacetime (M, g) 1 is a partially ordered set or poset, with ≺ denoting the causal ordering on the event-set M .
Causal set theory (CST) as proposed in Bombelli et al (1987), takes the Lorentzian character of spacetime and the causal structure poset in particular, as a crucial starting point to quantisation. It is inspired by a long but sporadic history of investigations into Lorentzian geometry, in which the connections between (M, ≺) and the conformal geometry were eventually established. This history, while not a part of the standard narrative of General Relativity, is relevant to the sequence of ideas that led to CST. In looking for a quantum theory of spacetime, (M, ≺) has also been paired with discreteness, though the earliest ideas on discreteness go back to pre-quantum and pre-relativistic physics. We now give a broad review of this history.
The first few decades of the general theory of relativity were dedicated to understanding the physical implications of the theory and to finding solutions to the field equations. The approach to Lorentzian geometry was mostly practical: it was seen as a simple (though odd) generalisation of Riemannian geometry. 2 However, there were early attempts to use causality as a basis of Lorentzian geometry. Weyl and Lorentz (see Bell and Korté 2016) used light rays to attempt a reconstruction of d dimensional Minkowski spacetime M d , and Robb (1914Robb ( , 1936 suggested an axiomatic framework for spacetime where the causal precedence on the collection of events was seen to play a critical role. It was only several decades later, however, that the mathematical structure of Lorentzian geometry began to be explored more vigourously. In a seminal paper titled "Causality Implies the Lorentz Group", Zeeman (1964) identified the chronological poset (M d , ≺≺) in M d , where ≺≺ denotes the chronological relation on the event-set M d . Defining a chronological automorphism 3 f a of M d as the chronological poset-preserving bijection Zeeman showed that the group of chronological automorphisms G A is isomorphic to the group G Lor of inhomogeneous Lorentz transformations and dilations on M d when d > 2. While it is simple to see that the generators of G Lor preserve the chronological structure so that G Lor ⊆ G A , the converse is not obvious. In his proof Zeeman showed that every f a ∈ G A maps light rays to light rays, such that parallel light rays remain parallel and moreover that the map is linear. In Minkowski spacetime, moreover, every chronological automorphism is also a causal automorphism, so a Corollary to Zeeman's theorem is that the group of causal automorphisms is isomorphic to G Lor . This is a remarkable result, since it states that the physical invariants associated with M d follow naturally from its causal structure poset (M d , ≺) where ≺ denotes the causal relation on the event set M d . Kronheimer and Penrose (1967) subsequently generalised Zeeman's ideas to an arbitrary causal spacetime (M, g) where they identified both (M, ≺) and (M, ≺≺) with the event-set M devoid of the differential and topological structures associated with a spacetime. They defined an abstract causal space axiomatically, using both (M, ≺) and (M, ≺≺) along with a requirement of mixed transitivity between the relations ≺ and ≺≺, which mimics that in a causal spacetime.
Zeeman's result in M d was then generalised to a larger class of spacetimes by Hawking et al (1976) and Malament (1977). A chronological bijection generalises Zeeman's chronological automorphism between two spacetimes (M 1 , g 1 ) and (M 2 , g 2 ), and is a chronological order preserving bijection, where ≺≺ 1,2 refer to the chronology relations on M 1,2 , respectively. The existence of a chronological bijection between two strongly causal spacetimes was equated by Hawking et al (1976) to the existence of a conformal isometry, which is a bijection f : M 1 → M 2 such that f, f −1 are smooth (with respect to the manifold topology and differentiable structure) and f * g 1 = λ 2 g 2 for a real, smooth strictly positive function λ 2 on M 2 . Malament (1977) then generalised this result to the larger class of future and past distinguishing (FPD) 4 spacetimes. We refer to these results collectively as the Hawking-King-McCarthy-Malament theorem or HKMM theorem summarised as

Theorem 1 Hawking-King-McCarthy-Malament (HKMM)
If a chronological bijection f b exists between two d-dimensional spacetimes which are both future and past distinguishing, then these spacetimes are conformally isometric when d > 2.
It was shown by Levichev (1987) that a causal bijection implies a chronological bijection and hence the above theorem can be generalised by replacing "chronological" with "causal".
Thus, the causal structure poset (M, ≺) of an FPD spacetime is equivalent its conformal geometry. This means that (M, ≺) is equivalent to the spacetime, except for the local volume element encoded in the conformal factor λ 2 , which is a single scalar. As phrased by Finkelstein (1969), the causal structure in d = 4 is therefore (9/10) th of the metric! Enroute to a theory of quantum gravity one must pause to ask: what "natural" structure of spacetime should be quantised? Is it the metric or is it the causal structure poset? The former can be defined for all signatures, but the latter is an exclusive embodiment of a causal Lorentzian spacetime. In Fig. 2, we show a 3d projection of a non-Lorentzian and non-Riemannian d = 4 "space-time" with signature (−, −, +, +). The fact that a time-like direction can be continuously transformed into any other while still remaining timelike means that there is no order relation in the space and hence no causal structure poset. We can thus view the causal structure poset as an essential embodiment of Lorentzian spacetime.

Timelike
Spacelike Null Timelike Fig. 2 An example of a signature (−, −, +, +) spacetime with one spatial dimension suppressed. It is not possible to distinguish a past from a future timelike direction and hence order events, even locally.
Perhaps the first explicit statement of intent to quantise the causal structure of spacetime, rather than the spacetime geometry was by Kronheimer and Penrose (1967), who listed, as one of their motivations for axiomatising the causal structure: "To admit structures which can be very different from a manifold. The possibility arises, for example, of a locally countable or discrete event-space equipped with causal relations macroscopically similar to those of a space-time continuum." This brings to focus another historical thread of ideas important to CST, namely that of spacetime discreteness. The idea that the continuum is a mathematical construct which approximates an underlying physical discreteness was already present in the writings of Riemann as he ruminated on the physicality of the continuum (Riemann 1873): "Now it seems that the emperical notions on which the metric determinations of Space are based, the concept of a solid body and that of a light ray; lose their validity in the infinitely small; it is therefore quite definitely conceivable that the metric relations of Space in the infinitely small do not conform to the hypotheses of geometry; and in fact one ought to assume this as soon as it permits a simpler way of explaining phenomena." Many years later, in their explorations of spacetime and quantum theory, Einstein and Feynman each questioned the physicality of the continuum (Stachel 1986;Feynman 1944). These ideas were also expressed in Finkelstein's "spacetime code" (Finkelstein 1969), and most relevant to CST, in Hemion's use of local finiteness, to obtain discreteness in the causal structure poset (Hemion 1988). This last condition is the requirement there are only a finite number of fundamental spacetime elements in any finite volume Alexandrov interval A[p, q] ≡ I + (p) ∩ I − (q).
Though these ideas of spacetime discreteness resonate with the appearance of discreteness in quantum theory, the latter typically manifests itself as a discrete spectrum of a continuum observable. The discreteness proposed above is different: one is replacing the fundamental degrees of freedom, before quantisation, already at the kinematical level of the theory.
The most immediate motivation for discreteness however comes from the HKMM theorem itself. The missing (1/10) th of the d = 4 metric is the volume element. A discrete causal set can supply this volume element by substituting the continuum volume with cardinality. This idea was already present in Myrheim's remarkable (unpublished) CERN preprint (Myrheim 1978), which contains many of the main ideas of CST. Here he states: "It seems more natural to regard the metric as a statistical property of discrete spacetime. Instead we want to suggest that the concept of absolute time ordering, or causal ordering of, space-time points, events, might serve as the one and only fundamental concept of a discrete space-time geometry. In this view space-time is nothing but the causal ordering of events." The statistical nature of the poset is a key proposal that survives into CST with the spacetime continuum emerging via a random Poisson sprinkling. We will see this explicitly in Sect. 3. Another key concept which plays a role in the dynamics is that the order relation replaces coordinate time and any evolution of spacetime takes meaning only in this intrinsic sense (Sorkin 1997).
There are of course many other motivations for spacetime discreteness. One of the expectations from a theory of quantum gravity is that the Planck scale will introduce a natural cut-off which cures both the UV divergences of quantum field theory and regulates black hole entropy. The realisation of this hope lies in the details of a given discrete theory, and CST provides us a concrete way to study this question, as we will discuss in Sect. 5.
It has been 31 years since the original CST proposal of BLMS . The early work shed considerable light on key aspects of the theory Bombelli and Meyer 1989;Brightwell and Gregory 1991) and resulted in Sorkin's prediction of the cosmological constant Λ (Sorkin 1991). There was a seeming hiatus in the 1990s, which ended in the early 2000s with exciting results from the Rideout-Sorkin classical sequential growth models (Rideout andSorkin 2000b, 2001;Martin et al 2001;Rideout 2001). There have been several non-trivial results in CST in the intervening 19 odd years. In the following sections we will make a broad sketch of the theory and its key results, with this historical perspective in mind.

The causal set hypothesis
The main ideas in CST follow naturally from this historical perspective. We begin with the definition of a causal set: where |.| denotes the cardinality of the set and The acylic and transitive conditions together define a partially ordered set or poset, while the condition of local finiteness encodes discreteness.
The content of the HKMM theorem can be summarised in the statement: which lends itself to a discrete rendition, dubbed the "CST slogan": This captures the essence of the (yet to be specified) continuum approximation of a manifold-like causal set, which we denote by C ∼ (M, g). While the continuum causal structure gives the continuum conformal geometry by the HKMM theorem, the discrete causal structure represented by the underlying causal set is meant to approximate the entire spacetime geometry. Thus, discreteness supplies the missing conformal factor, or the missing (1/10) th of the metric, in d = 4. The transitivity condition x ≺ y, y ≺ z ⇒ x ≺ z is satisfied by the causality relation ≺ in any Lorentzian spacetime. Motivated thus, CST makes the following radical proposal : 1. Quantum gravity is a quantum theory of causal sets. 2. A continuum spacetime (M, g) is an approximation of an underlying causal set C, where (a) Order ∼ Causal Order In CST, the kinematical space of d = 4 continuum spacetime geometries or histories is replaced with a sample space Ω of causal sets. Thus, discreteness is viewed not only as a tool for regulating the continuum, but as a fundamental feature of quantum spacetime. Ω includes causal sets that have no continuum counterpart, i.e., they cannot be related via Conditions (2a) and (2b) to any continuum spacetime in any dimension. These non-manifold-like causal sets are expected to play an important role in the deep quantum regime. In order to make this precise we need to define what it means for a causal set to be manifold-like, i.e., to make precise the relation "C ∼ (M, g)".
Before doing so, it is important to understand the need for a continuum approximation at all. Without it, Condition (1) yields any quantum theory of locally finite posets: one then has the full freedom of choosing any poset calculus to construct a quantum dynamics, without need to connect with the continuum. Examples of such poset approaches to quantum gravity include those by Finkelstein (1969) andHemion (1988), and more recently Cortês and Smolin (2014). What distinguishes CST from these approaches is the critical role played by both causality and discrete covariance which informs the choice of the dynamics as well the physical observables. Condition (2) is the requirement that in the continuum approximation these observables should correspond to appropriate continuum topological and geometric covariant observables.
What do we mean by the continuum approximation Condition (2)? We begin to answer this by looking for the underlying causal set of a causal spacetime (M, g). A useful analogy to keep in mind is that of a macroscopic fluid, for example a glass of water. Here, there are a multitude of molecular-level configurations corresponding to the same macroscopic state. Similarly, we expect there to be a multitude of causal sets approximated by the same spacetime (M, g). And, just as the set of allowed microstates of the glass of water depends on the molecular size, the causal set microstate depends on the discreteness scale V c , which is a fundamental spacetime volume cut-off 5 .
Since the causal set C approximating (M, g) is locally finite, it represents a proper subset of the event-set M . An embedding is the injective map where ≺ C and ≺ M denote the order relations in C and M respectively. Because of our desire to link the cardinality of the causal set with the spacetime volume element, we are interested in embeddings that are uniform with respect to the spacetime volume measure of (M, g). A causal set is said to approximate a spacetime C ∼ (M, g) at density ρ c = V −1 c if there exists a faithful embedding The most obvious choice for Vc is the Planck volume, but for reasons of generality we will not impose it at this stage.

Fig. 5
The lightcone lattice in d = 2. The lattice on the left looks "regular" in a fixed frame but transforms into the "stretched" lattice on the right under a boost. The n ∼ ρcV correspondence cannot be implemented as seen from the example of the Alexandrov interval, which contains n = 7 lattice points in the lattice in the left but is empty after a boost.
Clearly, the map from C to (M, g) is not unique: if (M , g ) differs from (M, g) on scales much smaller than ρ −1 c , then C also faithfully embeds into (M , g ) 6 . The uniform distribution at density ρ c ensures that every finite spacetime volume V is represented by a finite number of elements n ∼ ρ c V in the causal set. It is natural to make these finite spacetime regions causally convex, so that they can be constructed from unions of Alexandrov intervals A(p, q) in (M, g). However, we must ensure covariance, since the goal is to be able to recover the approximate covariant spacetime geometry. For this Φ(C) must be uniformly distributed in (M, g) with respect to the spacetime volume measure. It is obvious that a "regular" lattice cannot suffice since such a lattice is not regular in all frames or coordinate systems and hence it is not possible to consistently assign n ∼ ρ c V (see Fig. 5). Another way of stating this is that regular lattices break not only continuum symmetries, but more generally, diffeomorphism invariance.
The issue of symmetry breaking is of course obvious even in Euclidean space. Any regular discretisation breaks the rotational and translational symmetry of the continuum. In the lattice calculations for QCD, these symmetries are restored only in the continuum limit, but are broken as long as the discreteness persists. In Christ et al (1982) it was suggested that symmetry can be restored in a randomly generated lattice where there lattice points are uniformly distributed via a Poisson process. This has the advantage of not picking any preferred direction and hence not explicitly breaking symmetry, at least on average. We will discuss this point in greater detail a few pages down.
Set in the context of spacetime, the Poisson distribution is a natural choice for Φ(C), with the probability of finding n elements in a spacetime region of 6 If we add in the requirement that (M, g) is "structureless" below scales ∼ ρ −1 c , i.e., Rρ −2 c << 1 where R is the spacetime curvature , then this source of non-uniqueness disappears.
volume v given by This means that on the average where n is the random variable associated with the random causal set Φ(C) 7 . This distribution then gives us the covariantly defined n ∼ ρ c V correspondence we seek. In a Poisson sprinkling into a spacetime (M, g) at density ρ c one selects points in (M, g) at random with respect to the Poisson distribution and imposes a partial ordering on these elements via the induced spacetime causality relation. Starting from (M, g), we can then obtain an ensemble of "microstates" or causal sets, which we denote by C(M, ρ c ), via the Poisson sprinkling 8 . We say that a causal set C is approximated by a spacetime (M, g) if C can be obtained from (M, g) via a high probability Poisson sprinkling. Conversely, for every C ∈ C(M, ρ c ) there is a natural embedding map where Φ(C) is a particular realisation in C(M, ρ c ). In Figure 6 we show a causal set obtained by Poisson sprinkling into d = 2 de Sitter spacetime. That there is a fundamental discrete randomness even kinematically is not always easy for a newcomer to CST to come to terms with. Not only does CST posit a fundamental discreteness, it also requires it to be probabilistic. Thus, even before coming to quantum probabilities, CST makes us work with a classical, stochastic discrete geometry.
Let us state some obvious, but important aspects of Eq. (7). Let Φ : C → (M, g) be a faithful embedding at density ρ c . While the set of all finite volume regions 9 v possess on average n = ρ c v elements of C 10 , the (Poisson) fluctuations in δn = √ n. Thus, it is possible that the region contains no elements at all, i.e., there is a "void". An important question to ask is how large a void does CST allow, since a sufficiently large void would have an obvious effect on our macroscopic perception of a continuum. If spacetime is unbounded, as it is in Minkowski spacetime, the probability for the existence of a void of any size is one. Can this be compatible at all with the idea of an emergent continuum in which the classical world can exist, unperturbed by the vagaries of quantum gravity?
The presence of a macroscopic void means that the continuum approximation is not realised in this region. A prediction of CST is then that the emergent 7 Since Φ(C) is a random causal set, any function of F : C → R is therefore a random variable.
8 C(M, ρc) explicitly depends on the spacetime metric g, which we have suppressed for brevity of notation. 9 We assume that these are always causally convex. 10 Henceforth we will identify Φ(C) with C, when Φ is a faithful embedding. continuum regions of spacetime are bounded both spatially and temporally, even if the underlying causal set is itself unbounded. Thus, a continuum universe is not viable forever. Of course, we must accept that the current phase of the observable universe, the big bang to the present does have a continuum realisation. Thus, CST is a viable theory only if the probability of occurance of a macroscopic void anywhere in our observable universe is truly negligible. In Dowker et al (2004) the probability for there to be at least one nuclear size void ∼ 10 −60 m 4 was calculated in a region of Minkowski spacetime which is the size of our present universe. Using general considerations they found that the probability is of order 10 84 × 10 168 × e −10 72 , which is an absurdly small number! Thus, CST poses no phenomenological inconsistency in this regard.
An example of a manifold-like causal set C which is obtained via a Poisson sprinkling into a 2d causal diamond is shown in Fig. 7. A striking feature of the resulting graph is that there is a high degree of connectivity. In the Hasse diagram of Fig. 7 only the nearest neighbour relations or links are depicted with the remaining relations following from transitivity. Two elements e, e ∈ C are said to be linked if e ≺ e and e ∈ C with such that e ≺ e ≺ e and e = e, e . In a causal set that is obtained from a Poisson sprinkling, the number of neighbours or the valency of any given element is typically very large. This is an important feature of continuum like causal sets and results from the fact that the elements of C are uniformly distributed in (M, g). For a given element e ∈ C, the probability of an event x e to be a link is equal to the probability that the Alexandrov interval A(e, x) does not contain any elements of C. Since the probability is significant only when V ∼ V c . As shown in Fig. 8, in M d , the set of events within a proper time ∝ (V ) 1/d to the future (or past) of a point p lies in the region between the future light cone and the hyperboloid −t 2 + Σ i x 2 i ∝ (V ) 2/d , with t > 0. Up to fluctuations, therefore, most of the future links to e lie within the hyperboloid with V = V c ± √ V c . This is a noncompact, infinite volume region and hence the number of future links to e is (almost surely) infinite. Since linked elements are the nearest neighbours of e, this means the valency of the graph C is infinite. It is this feature of manifoldlike causal sets which gives rise to a characteristic "non-locality", and plays a critical role in the continuum approximation of CST, time and again.
The Poisson distribution is not the only choice for a uniform distribution. A pertinent question is whether a different choice of distribution is possible, which would lead to a different manifestation of the continuum approximation. In Saravani and Aslanbeigi (2014), this question was addressed in some detail. Let C ∼ (M, g) at density ρ c . Consider k non-overlapping Alexandrov intervals of volume V in (M, g). Since C is uniformly distributed, n = ρ c V . The most optimal choice of distribution, is also one in which the fluctuations δn/ n = (n − n ) 2 / n are minimised. This ensures that C is as close to the continuum as possible. For the Poisson distribution δn/ n = 1/ n = 1/ √ ρ c V . Is this as good as it gets? It was shown that for d > 2, and under certain further assumptions, the Poisson distribution indeed does the best job. Strengthening these results is important as it can improve our understanding of the continuum approximation.

The Hauptvermutung or fundamental conjecture of CST
An important question is the uniqueness of the continuum approximation associated to a causal set C. Can a given C be faithfully embedded at density ρ c into two different spacetimes, (M, g) and (M , g )? As we have noted above, we expect that this is not the case if (M, g) and (M , g ) differ on scales smaller ρ c , or that they are close (M, g) ∼ (M , g ). Let us assume that a causal set can be identified with two macroscopically distinct spacetimes at the same density ρ c . Should this be interpreted as a hidden duality between these spacetimes as, for example isospectral manifolds or mirror manifolds in string theory (Greene and Plesser 1991)? The answer is clearly in the negative, since the aim of the CST continuum approximation is that C contains all the information in (M, g) at scales above ρ −1 c . Macroscopic non-uniqueness would therefore mean that the intent of the CST continuum approximation is not satisfied.
We thus state the fundamental conjecture of CST: The Hauptvermutung of CST: C can be faithfully embedded at density ρ c into two distinct spacetimes, (M, g) and (M , g ) iff they are approximately isometric.
By an approximate isometry at ρ c , (M, g) ∼ (M , g ) we mean that (M, g) and (M , g ) differ only at scales smaller than ρ c . Defining this rigourously has been challenging, but there have been concrete proposals Bombelli (2000); Noldus ( , 2002; Bombelli and Noldus (2004); Bombelli et al (2012). We will discuss these only very briefly in the next section, and instead use a more functional definition.
For now let us use the existence of this approximate isometry to flesh out a kinematic picture of CST. Let Ω be the set of all countable causal sets and H the set of all spacetimes in all dimensions. For a given ρ c , we can quotient H by the approximate isometry, H/ ∼. Then, the set of causal sets Ω cont in C(M, ρ c ) for any M ∈ [M ] ∈ H/ ∼ is a proper subset of Ω ⊃ Ω cont . In other words, Ω "contains" spacetimes of all dimensions! Thus, CST dynamics has the daunting task of not only obtaining manifold-like causal sets in the classical limit, but also ones that have dimension d = 4.
As mentioned in the introduction, the sample space of n element causal sets Ω n is dominated by the KR posets depicted in Fig. 9 and are hence very nonmanifold-like (Kleitman and Rothschild 1975;Dhar 1978Dhar , 1980. A KR poset has three "layers" (or abstract "moments of time"), with roughly n/4 elements in the bottom and top layer and such that each element in the bottom layer is related to roughly half those in the middle layer, and similarly each element in the top layer is related to roughly half those in the middle layer. These KR posets grow as 2 n 2 /4 and hence must play a role in the deep quantum regime. Since they are non-manifold-like they pose a challenge to the dynamics, which must overcome their entropic dominance in the classical limit of the theory.
Closely tied to the continuum approximation is the notion of coarse graining. Given (M, g) we can obtain C(M, ρ c ) at different ρ c . Given a causal set C which faithfully embeds into (M, g) at ρ c , density one can coarse grain it to a smaller subcausal set C ⊂ C which faithfully embeds into (M, g) at ρ c . A natural coarse graining would be via a random selection of elements in C such that for every n elements of C roughly n = ρ c ρc n elements are chosen. Even if C itself does not faithfully embed at a more fundamental discreteness scale ρ c , it is possible that such a coarse graining could itself be embeddable at a lower density ρ c . This would be in keeping with our sense in CST that the deep quantum regime is not manifold-like. One can also envisage manifold-like causal sets with a regular fixed lattice-like structure attached to each element similar to a "fibration" in the spirit of Kaluza-Klein theories. Instead of the coarse graining procedure, it would be more appropriate to take the quotient with respect to this fibre to obtain the continuum like causal set. Recently, the implications of coarse graining in CST, both dynamically and kinematically, were considered in Eichhorn (2018) using ideas based on renormalisation techniques.

Discreteness without Lorentz breaking (DWLB)
It is often assumed that a fundamental discreteness is incompatible with continuous symmetries. As was pointed out in Christ et al (1982), in the Euclidean context, symmetry can be preseved on average in a random lattice. In Bombelli et al (2009), it was shown that a causal set in C(M d , ρ c ) not only preserves Lorentz invariance on average, but in every realisation. Thus, in a very specific sense a manifold-like causal set does not break Lorentz invariance. In order to see the contrast between the Lorentzian and Euclidean cases we present the arguments of Bombelli et al (2009) starting with the easier Euclidean case.
Consider the Euclidean plane P = (R 2 , δ ab ), and let Φ : C(P, ρ c ) → P be the natural embedding map, where C(P, ρ c ) denotes the ensemble of Poisson sprinklings into P at density ρ c . A rotation r ∈ SO(2) about a point p ∈ P, induces a map r * : C(P, ρ c ) → C(P, ρ c ) where r * = Φ • r • Φ −1 and similarly a translation t in P induces the map t * : C(P, ρ c ) → C(P, ρ c ). The action of the Euclidean group is clearly not transitive on C(P, ρ c ) but has non-trivial orbits which provide a fibration of C(P, ρ c ). Thus the ensemble C(P, ρ c ) preserves the Euclidean group on average. This is the sense in which the discussion of Christ et al (1982) states that the random discretisation preserves the Euclidean group.
The situation is however different for a given realisation P ∈ C(P, ρ c ). Fixing an element e ∈ Φ(P ), we define a direction d ∈ S 1 , the space of unit vectors in P centered at e. Under a rotation r about e, d → r * (d) ∈ S 1 . In general, we want a rule that assigns a natural direction to every P ∈ C(P, ρ c ). One simple choice is to find the closest element to e in Φ(P ), which is well defined in this Euclidean context. Moreover, this element is almost surely unique, since the probability of two elements being at the same radius from e is zero in a Poisson distribution. Thus we can define a "direction map" D e : C(P, ρ c ) → S 1 for a fixed e ∈ Φ(P ) consistent with the rotation map, i.e., D e commutes with any r ∈ SO(2), or is equivaraint.
Associated with C(P, ρ c ), is a probability distribution µ arising from the Poisson sprinkling which associates with every measurable set α in C(P, ρ c ) a probability µ(α) ∈ [0, 1]. The Poisson distribution being volume preserving (Stoyan et al 1995), the measure on C(P, ρ c ) moreover must be independent of the action of the Euclidean group on C(P, ρ c ), i.e.: µ • r = µ.
In analogy with a continuous map, a measurable map is one whose preimage from a measurable set is itself a measurable set. The natural map D we have defined is a measurable map, and we can use it to define a measure on S 1 : µ D ≡ µ • D −1 . Using the invariance of µ under rotations and the equivariance of D under rotations we see that µ D is also invariant under rotations. Because S 1 is compact, this does not lead to a contradiction. In analogy with the construction used in Bombelli et al (2009) for the Lorentzian case, we choose a measurable set s ≡ (0, 2π/n) ∈ S 1 . A rotation by r(2π/n), takes s → s which is non-overlapping, so that after n successive rotations, r n (2π/n) • s = s. Since each rotation does not change µ D and µ D (S 1 ) = 1, this means that µ D (s) = 1/n. Thus, it is possible to assign a consistent direction for a given realisation P ∈ C(P, ρ c ) and hence break Euclidean symmetry. However, this is not the case for the space of sprinklings C(M d , ρ c ) into M d , where the hyperboloid H d−1 now denotes the space of future directed unit vectors and is invariant under the Lorentz group SO(n − 1, 1) about a fixed point p ∈ M d−1 . To begin with, there is no "natural" direction map. Let C ∈ C(M d , ρ c ). To find an element which is closest to some fixed e ∈ Φ(C), one has to take the infimum over J + (e) , or some suitable Lorentz invariant subset of it, which being non-compact, does not exist. Assume that some measurable direction map D : Ω M d → H d−1 , does exist. Then the above arguments imply that µ D must be invariant under Lorentz boosts. The action of successive Lorentz tranformations Λ can take a given measurable set h ∈ H d−1 to an infinite number of copies that are non-overlapping, and of the same measure. Since H d−1 is non-compact, this is not possible unless each set is of measure zero, but since this is true for any measurable set h and we require µ D (H d−1 ) = 1, this is a contradiction. This proves the following theorem : DWLB Theorem: In dimensions n > 1 there exists no equivariant measureable map D : In other words, even for a given sprinkling ω ∈ Ω M d it is not possible to consistently pick a direction in H d−1 . Consistency means that under a boost Λ : ω → Λ • w, and hence D(ω) → Λ • D(ω) ∈ H d−1 . Crucial to this argument is the use of the Poisson distribution 11 . Since causal set discreteness does not violate Lorentz invariance, this is also a prediction of the theory. Tests of Lorentz invariance over the last couple of decades have produced an ever-tightening bound, which is consistent with CST (Liberati and Mattingly 2016).

Forks in the road: What makes CST so "different"?
In many ways CST doesn't fit the standard paradigms adopted by other approaches to quantum gravity and it is worthwhile trying to understand the source of this difference. The program is minimalist but also rigidly constrained by its continuum approximation. The ensuing non-locality means that the apparatus of local physics is not readily availabl to CST. Sorkin (1991) describes the route to quantum gravity and the various forks at which one has to make choices. Different routes may lead to the same destination: for example (barring interpretational questions), simple quantum systems can be described equally well by the path integral and the canonical approach. However, this need not be the case in gravity: a set of consistent choices may lead you down a unique path, unreachable from another route. Starting from broad principles, Sorkin argued that certain choices at a fork are preferable to others for a theory quantum gravity. These include the choice of Lorentzian over Euclidean, the path integral over canonical quantisation and discreteness over the continuum. This set of choices leads to a CST-like theory, while choosing the Lorentzian-Hamiltonian-continuum route leads to a canonical approach like Loop Quantum Gravity.
Starting with CST as the final destination, we can work backward to retrace our steps to see what forks had to be taken and why other routes are impossible to take. The choice at the discreteness versus continuum fork and the Lorentzian versus Euclidean fork are obvious from our earlier discussions. As we explain below, the other essential fork that has to be taken in CST is the histories approach to quantisation.
One of the standard routes to quantisation is via the canonical approach. Starting with the phase space of a classical system, with or without constraints, quantisation rules give rise to the familiar apparatus of Hilbert spaces and self adjoint operators. In quantum gravity, apart from interpretational issues, this route has difficult technical hurdles, some of which have been partially overcome (Ashtekar and Pullin 2017). Essential to the canonical formulation is the 3+1 split of a spacetime M = Σ ×R, where Σ is a Cauchy hypersurface, on which are defined the canonical phase space variables which capture the instrinsic and extrinsic geometry of Σ.
The continuum approximation of CST however, does not allow a meaninful definition of a Cauchy hypersurface, because of the non-locality described above. The discrete analog of a spatial hypersurface is a set of maximally unrelated elements, an inextendible antichain A, which separates the set C into its future and past, so that we can express C = Fut(A) Past(A) A, where denotes disjoint union. However, an element in Past(A) can be linked to an element in Fut(A) thus "bypassing" A. An example of a "missing link" is depicted in Fig 11. Hence, unlike a Cauchy hypersurface, A is not a summary of its past (Major et al 2006), and hence a canonical decomposition is not viable. On the other hand, each causal set is a "history", and since the sample space of causal sets is countable, one can construct a "path integral" as a sum over histories. We will describe the dynamics of causal sets in more detail in Sect. 6.

Kinematics or geometric reconstruction
In this section we discuss the program of geometric reconstruction in which topological and geometric invariants of a continuum spacetime (M, g) are "reconstructed" from the underlying ensemble of causal sets. The assumption that such a reconstruction exists for any covariant observable in (M, g) comes from the Hauptvermutung of CST discussed in Sect. 3.
In the statement of the Hauptvermutung, we used the phrase "approximately isometric", with the promise of an explanation in this section. A rigourous definition requires the notion of closeness of two Lorentzian spacetimes. In Riemannian geometry, one has the Gromov-Hausdorff distance (Petersen 2006), but there is no simple extension to Lorentzian geometry, in part because of the indefinite signature. In Bombelli and Meyer (1989) a measure of closeness of two Lorentzian manifolds was given in terms of a pseudo distance func-e e' � Fig. 11 A "missing link" from e to e which "bypasses" the inextendible antichain A.
tion, which however is neither symmetric nor satisfies the triangle inequality. Subsequently, in a series of papers, a true distance function was defined on the space of Lorentzian geometries, dubbed the Lorentzian Gromov-Hausdorff distance (Bombelli 2000;Noldus , 2002Bombelli and Noldus 2004;Bombelli et al 2012). While this makes the statement of the Hauptvermutung precise, there is as yet no complete proof. Recently, a purely order theoretic criterion has been used to determine the closeness of causal sets and prove a version of the Hauptvermutung (Sorkin and Zwane, work in progress). Apart from these more formal constructions, as we will describe below, a large body of evidence has accumulated in favour of the Hauptvermutung. In the program of geometric reconstruction, we look for order invariants in continuum like causal sets which correspond to manifold (either topological or geometric) invariants of the spacetime. These manifold invariants include dimension, spatial topology, distance functions between fixed elements in the spacetime, scalar curvature, the discrete Einstein-Hilbert action, the Gibbons-Hawking-York boundary terms, Green functions for scalar fields, and the d'Alembertian operator for scalar fields. The identification of the order invariant O with the manifold invariant G then ensures that a causal set C that faithfully embeds into (M, g) cannot faithfully embed into a spacetime with a different manifold invariant 12 G . Thus, in this sense two manifolds can be defined to be close with respect to their specific manifold invariants. We can then state the limited, order-invariant version of the Hauptvermutung for each manifold invariant O: O-Hauptvermutung: If C faithfully embeds into (M, g) and (M , g ) then (M, g) and (M , g ) have the same manifold invariant G associated to O.
The longer our list of correspondences between order invariants and manifold invariants, the closer we are to proving the full Hauptvermutung.
In order to correlate a manifold invariant G with an order invariant O, we must recast geometry in purely order theoretic terms. Note that since locally finite posets appear in a wide range of contexts, the poset literature contains several order invariants, but these are typically not related to the manifold invariants of interest to us. The challenge is to choose the appropriate invariants that limit to manifold invariants. Guessing and verifying this using both analytic and numerical tools is the art of geometric reconstruction.
A labelling of a causal set C is an injective map: C → N, which is the analogue of a choice of coordinate system in the continuum. By an order invariant in a finite causal set C we mean a function O : C → R such that O is independent of the labelling of C. For a manifold-like causal set C ∈ C(M, ρ c ) 13 , associated to every order invariant O is the random variable O whose expectation value O in the ensemble C(M, ρ c ) is either equal to or limits (in the large ρ c limit) to a manifold invariant G of (M, g). We will typically restrict to compact regions of (M, g) in order to deal with finite values of O.
The first candidates for geometric order invariants were defined for C(A[p, q], ρ c ) where A[p, q] is an Alexandrov interval in M d . Some of these have been later generalised to Alexandrov intervals (or causal diamonds) in Riemman Normal Neighbourhoods (RNN) in curved spacetime. These manifold invariants are in this sense "local". In order to find spatial global invariants, the relevant spacetime region is a Gaussian Normal Neighbourhood (GNN) of a compact Cauchy hypersurface in a globally hyperbolic spacetime. As discussed in Sect. 3 compactness is necessary for manifold-likeness since otherwise there is a finite probability for there to be arbitrarily large voids which negates the discretecontinuum correspondence.
Before proceeding, we remind the reader that we are restricting ourselves to manifold-like causal sets in this section only because of the focus on CST kinematics and the continuum approximation. All the order invariants, however, can be calculated for any causal set, manifold-like or not. These order invariants give us an important class of covariant observables, essential to constructing a quantum theory of causal sets. As we will see in Sect. 6 they play an important role in the quantum dynamics.
The analytic results in this section are typically found in the continuum limit, ρ c → ∞. Strictly speaking, this limit is unphysical in CST because of the assumption of a fundamental discreteness. There are fluctuations at finite ρ c which give important deviations from the continuum with potential phenomenological consequences. These are however not always easy to calculate analytically and hence require simulations to assess the size of fluctuations at finite ρ c . As we will see below, CST kinematics therefore needs a combination of analytical and numerical tools.

Spacetime Dimension Estimators
The earliest result in CST is a dimension estimator for Minkowski spacetime due to Myrheim (1978) 14 and predates BLMS . A closely related dimension estimator was given by Meyer (1988), which is now collectively known as the Myrheim-Meyer dimension estimator.
The number of relations R in a finite n element causal set C is the number of ordered pairs e i , e j ∈ C such that e i ≺ e j . Since the maximum number of possible relations on n elements is n 2 , the ordering fraction is defined as It was shown by Myrheim (1978) that r is dependent only on the dimension when C faithfully embeds into M d .

Consider an Alexandrov interval
We are interested in calculating the expecation value of the random variable R associated with R for the ensemble C(A d , ρ c ). This is the probability that a pair of elements e 1 , e 2 ∈ A d are related. Given e 1 , the probability of there being an e 2 in its future is given by the volume of the region J + (e 1 ) ∩ J − (p) in units of the discreteness scale, while the probability to pick e 1 is given by the volume of A d . This joint probability can be calculated as follows. Wlog where T 1 is the proper time from x 1 to q. Evaluating the integral, one finds Using n = ρ c V , Meyer (1988) obtained a dimension estimator from R by noting that the ratio is only a function of d. In the large n limit, this is is half of Myrheim's ordering fraction r. As shown in Meyer (1988), the fluctuations in R are large and hence one should not expect to find the right dimension in a single realisation C ∈ C(A d , ρ c ); it is obtained only by averaging over the ensemble 15 . For 14 This remarkable preprint also contains the first expression, again without detailed proof, of the volume of a small causal diamond in an arbitrary spacetime. 15 For ρc large enough (or equivalent the n) any single C ∈ C(A d , ρc), one can sample smaller sub-causal sets in C. The expectation value R can be obtained from measurements in different sub-intervals A d (e i , e j ) ⊂ C of cardinality n ± √ n , n << n.
large enough ρ c , however, the relative fluctuations should become smaller, and allow one to distinguish causal sets obtained from sprinkling into different dimensional Alexandrov intervals. Such systematic tests have been carried out numerically using sprinklings into different spacetimes by Reid (2003) and show a general convergence as ρ c is taken to be large. How can we test this dimension estimator in practice? Let C be a causal set of sufficiently large cardinality n. If the dimension obtained from Eq. 16 is approximately an integer d, this means that C cannot be distinguished from a causal set that belongs to C(A d , ρ c ) using just the dimension estimator, for n ∼ ρ c vol(A d ). We denote this by C ∼ d A d . This also means that C cannot be a typical member of C(A d , 1) for integer d = d, so that C ∼ d A d . This is our first O-Hauptvermutung, with O = d, and provides a template for the remaining manifold-invariants in the next few subsections.
The equivalence C ∼ d A d itself does not of course imply that C ∼ A d or is even that C is manifold like. Rather, it is the limited statement that its dimension estimator is the same as that of a typical causal set in Using simulations Abajian and Carlip (2018) recently obtained the Myrheim-Myer dimension as function of interval size for nested intervals in a causal set in C(A d , ρ c ) for d = 3, 4, 5. As the interval size decreases, they found that the resulting causal sets are likely to be disconnected due to the large fluctuations at small volumes. In the extreme case, there is a single point with no relations and hence the Myrheim-Myer dimension goes to ∞ rather than 0. Using a criterion to discard such disconnected regions, it was shown that this dimension estimator gives a value of 2 at small volumes, even when d = 3, 4, 5, in support of the dimensional reduction conjecture in quantum gravity Carlip (2017) which we discuss briefly in Sect. 5.
Meyer's construction is in fact more general and yields a whole family of dimension estimators. If we think of the relation e 1 ≺ e 2 as a chain c 2 of two elements, then a k chain c k is the causal sequence e 1 ≺ e 2 . . . e k−1 ≺ e k (see Fig. 12). We denote the abundance or number of c k in C by C k . Its expectation value in C(A d [p, q], ρ c ) is given by a sequence of k nested integrals over a sequence of nested Alexandrov intervals, A d [p, q] ⊃ I(x 1 , q) ⊃ I(x 2 , q) . . . I(x k , q) which, as was shown by Meyer (1988), can be calculated inductively to give .
Thus for any k, k , the ratio of C k 1/k to C k 1/k only depends on the dimension.
Meyer's calculation of C k was generalised to a small causal diamond A d [p, q] that lies in an RNN of a general spacetime, i.e., one for which RT 2 << 1, where T is the proper time from p to q and R denotes components of the curvature at the center of the diamond (Roy et al 2013). In such a region the � �� � � 1 2 Fig. 12 Two different chains between x and x . One is a k = 4 chain and the other is a k = 7 chain. dimension satisfies the more complicated equation where f 0 (d) is given by Eq. (16). It is straightforward to show that the expression above reduces to the Myrheim-Meyer dimension estimator in M d . The calculation of Roy et al (2013) uses a result of Khetrapal and Surya (2013), which makes explicit earlier calculations of the volume of a causal diamond in an RNN (Myrheim 1978;Gibbons and Solodukhin 2007). The C k themselves are order invariants and hence are covariant observables for finite element causal sets. This class of dimension estimators is just one among several that have appeared in the literature, including the mid-point scaling estimator (Bombelli 1987;Reid 2003), and more recent ones (Glaser and Surya 2013;Aghili et al 2018). We refer the reader to the literature for more details.

Topological Invariants
The next step in our reconstruction is that of topology. There are several poset topologies described in the literature (see Stanley (2011) as well as Surya (2008) for a review). However, our interest is in finding one that most closely resembles the "coarse" continuum topology. It is clear that the full manifold topology cannot be reproduced in a causal set since it requires arbitrarily small open sets. However, according to the Hauptvermutung, topological invariants like the homology groups and the fundamental groups of (M, g) should be encoded in the causal set.
A natural choice for a topology in C based on the order relation is one generated by the order intervals I[e i , e j ] ≡ Fut(e i ) ∩ Past(e j ). Indeed, in the continuum the topology generated by their analogs, the Alexandrov intervals, can be shown to be equivalent to the manifold topology in strongly causal spacetimes (Penrose 1972). However, even for a causal set approximated by a finite region of M d , this order-interval topology can be seen to be roughly discrete or trivial. This is due to the fact that an order interval in C can be very "long and skinny", and thus contain only a single element in its interior. Hence the topology can typically distinguish any two elements in C, and possesses little structure. Some form of "localisation" therefore seems essential if wants to find the coarse continuum topology.
In Major et al (2007Major et al ( , 2009, localisation was provided by considering an inextendible antichain A ⊂ C (see Sect. 3.3), which is an (imperfect) analog of a Cauchy hypersurface. The natural topology on A is the discrete topology since there are no causal relations amongst the elements. In order to provide a topology on A, one needs to "borrow" information from a neighbourhood of A. The method devised was to consider elements to the future of A and "thicken" by a parameter v to some collar neighbourhood Here IFut and IPast denote the inclusive future and past respectively, where for any S ⊂ C, IFut(S) = Fut(S) ∪ S and IPast(S) = Past(S) ∪ S.

A topology can then be induced on
For a spacetime (M, g) with compact Cauchy hypersurface Σ, and for C ∈ C(M, ρ c ) it was shown in Major et al (2007Major et al ( , 2009) that there exists a range of values of v such that N v (A) is homological to Σ (upto the discreteness scale) as long as there is a sufficient separation between the discreteness scale c ≡ V 1/d c and K the scale of extrisinsic curvature of Σ.
A similar construction can be used to obtain an open covering of a region of a causal set sandwiched between two inextendible and non-overlapping antichains A 1 and A 2 . The resulting homology constructed from the nerve simplicial complex is then is associated with a spacetime region rather than just space, and hence includes topology change. There are also other possibilities for characterising the spatial homology using chain complexes which have only been partially investigatigated. An open question is to find the causal set analogues of other topological invariants 16 .

Geodesic distance: timelike, spacelike and spatial
In Minkowski spacetime, the proper time between two events is the longest path between them; the shortest path between two time-like separated events is of course any zig-zag null path, which has zero length. In a causal set C, if e i ≺ e f , a path from e i to e f is a chain c k+2 with k intervening elements, e i ≺ e 1 ≺ e 2 . . . ≺ e k ≺ e f and is said to be of length k. Thus, a natural choice for the timelike geodesic distance between e i and e f is to maximise the chain length between them, as was suggested by Myrheim (1978), i.e., the length of the longest chain l(e i , e j ). It was shown in Brightwell and Gregory (1991) that the expectation value of the associated random variable l in the ensemble For a finite ρ c , the fluctuations in l(e i , e j ) are very large (Meyer 1988;Bachmat 2007) and hence the correspondence becomes meaningful only when average over a large ensemble.
In Roy et al (2013), the proper time of a small causal diamond A d in an RNN of a d dimensional spacetime was calculated to leading order correction in terms of the random variables C k associated to the abundance of k-chains where where C k is the ensemble average in C(A d , ρ c ). This definition is not intrinsic to a single causal set but requires the full ensemble. Nevertheless, it is of interest to study the intrinsic version of the expression by replacing C k by the C k for each causal set and then taking the ensemble average to check for convergence. Spacelike distance is far less straightforward to compute from the poset, because events that are spacelike to each other have no natural relationship to each other. We saw this already in trying to find a topology on the inextendible antichain. Thus, the relationship must be "borrowed" from the elements in the causal past and future of the spacelike events. Brightwell and Gregory (1991) defined the following, naive spatial distance function in M d . For a given spacelike pair p, q ∈ M d , the common future and past are defined as J + (p, q) ≡ J + (p) ∩ J + (q) and J − (p, q) ≡ J − (p) ∩ J − (q) respectively. For every r ∈ J + (p, q) and s ∈ J − (p, q) let τ (s, r) be the timelike distance. Then the naive distance function is given by While this is a perfectly good continuum definition of the distance in M d , it fails for the causal set when d > 2 since the number of pairs (r, s) which minimize τ (r, s) lies in the region between the codimension 2 hyperboloid and the light cone τ = 0. In the causal set we can use the length of the maximal chain l(r, s) to obtain τ (r, s), but in d > 2 since there are an infinite number of proper time minimising pairs (r, s), there will almost surely be those for which l(r, s) is drastically underestimated. The minimisation in Eq. (23) will then always give 2 as the spatial distance! Rideout and Wallden (2009) later generalised the naive distance function using minimising pairs (r, s) such that either r or s is linked to both p and q. Instead of minimizing over these pairs (again infinite), the 2-link distance can be calculated by averaging over the pairs. Numerical simulations for the naive distance and the 2-link distance for sprinklings into a finite region of M 3 show that the latter stabilises as a function of ρ c . The former underestimates the spatial distance compared to the continuum, and the latter overestimates it. The spatial distance functions of both Brightwell and Gregory (1991) and Rideout and Wallden (2009) are however strictly "predistance" functions since they do not satisfy the triangle inequality.
Recently a one-parameter family of induced spatial distance functions was proposed for an inextendible antichain in a causal set by Eichhorn et al (2018). To begin with, a one parameter family of continuum induced distance functions d was constructed for a globally hyperbolic region (M, g) of spacetime with Cauchy hypersurface Σ using only the causal structure and the volume element. In M d with Σ an inertial slice, the volume of a past causal cone from p Σ has a simple relation to the diameter D of the base of the cone Since D is the distance between any two antipodal points on the S d−2 ⊂ Σ, this simple formula defines the induced distance on Σ. In a general spacetime with Cauchy hypersurface Σ, this formula can be used to extract an approximate induced distance function in a sufficiently small region of Σ. In order to define the distance function on all of Σ, a meso-scale cut-off was introduced, and the full distance function obtained by minimising over all segmented paths, such that each segment is bounded from above by . For << K , the scale of the extrinsic curvature of Σ, d was shown to converge to the induced spatial distance function d h on (Σ, h).
Since the d are constructed from the causal structure and volume element they are readily defined on an inextendible antichain on a causal set. For causal sets in C(M, ρ c ) with Σ ⊂ M the discrete distance function d was shown to significantly overestimate the continuum induced distance on Σ when the latter is close to the discreteness scale (V c ) 1/d . This is the "asymptotic silence" of Eichhorn et al (2017), which is an effect similar to the narrowing of light cones, and arises because fluctuations are large at the discreteness scale. At larger distances, on the other hand, d is a good approximation of the continuum induced distance when (V c ) 1/d << << K . It was shown moreover that the continuum induced distance is slightly underestimated for positive curvature and slightly overestimated for negative curvature, when restricted to small regions of Σ. This was confirmed by extensive numerical simulations in M d for d = 2, 3 (see Figure 13). This works paves the way to recovering more spatial geometric invariants from the causal set, and is currently in progress (Eichhorn, Surya and Versteegen).   Fig. 13 The error in the discrete spatial distance is plotted as a function of the continuum induced distance on Σ for causal sets in M 2 for Σ of constant negative and positive extrinsic curvature. The discrete spatial distance always overestimates the continuum distance around the discreteness scale, resulting in an effective "asymptotic silence". For larger distances, when there is a good separation of scales, the discrete distance gives a good approximation of the continuum induced distance.

The d'Alembertian for a scalar field
One of the very first questions that comes to mind in the continuum approximation of CST is whether a tangent space can be defined naturally on a causal set. To answer this (unfortunately in the negative), we need to examine the non-local nature of a manifold-like causal set in more detail. The nearest neighbours of an element e are those that it is linked to, both in its future and its past. In a causal set approximated by Minkowski spacetime for example, as discussed in Sect. 3, every element has an infinite number of nearest neighbours (see Fig. 8  from the set of links or next to nearest neighbours to e is not possible, since the valency of the graph is infinite. This means in particular that derivative operators cannot also be simpy defined. How then can we look for the effect of discreteness on the propagation of fields? We will discuss this in more detail in Sect. 5 but for now we notice that the best way forward is to look for scalar quantities, rather than more general tensorial ones, in making the discretecontinuum correspondence. A scalar field d'Alembertian is a good first step. In ( where L k (e) denotes the set of k th past nearest neighbours of e ∈ C. This is a highly non-local operator since it depends on the number of all the (possibly infinite) nearest k = 0, 1, 2, 3 neighbours. Notice the alternating sum whose precise coefficients turn out to be very important to the continuum limit. As was shown in M 2 by Sorkin (2007b) and in M 4 by Benincasa and Dowker (2010) (see also Benincasa (2013)), for φ of compact support, and in a frame in which it is slowly varying, there are miraculous cancellations that make the contributions far down the light cone negligible, thus making the operator effectively local. The expectation value of the random variable Bφ(x) associated where v ≡ vol(A(y, x)) and we have used the probability P n (v) for v to contain n elements, Eq. (7). We have also made the expression dimensionful, in order to be able to make a direct comparison with the continuum. In order to evaluate this integral, we first note that since φ is of compact support, the region of integration is compact. In a frame F φ in which φ is slowly varying, a small |y − x| expansion of φ(y) around φ(x) can be done. Following Sorkin (2007b); Benincasa and Dowker (2010); Benincasa (2013), the non-compact region of integration J − (x) can be split into 3 non-overlapping regions, W 1 , W 2 , W 3 in F φ : W 1 is a neighbourhood of x, W 2 is a neighbourhood of J − (x) but bounded away from the origin and W 3 is bounded away from ∂J − (x). The integral over W 3 was shown in Benincasa (2013) to be bounded from above by an integral that tends to zero faster than any power of ρ −1 c , while the integral over W 2 was shown to go to zero faster than ρ −3/2 c . The local contribution from W 1 dominates, and it was shown that Thus, B(φ) is "effectively local" since its dominant contribution comes from W 1 which is a local neighbourhood of x defined by the slowly varying frame F φ . In this frame, the contribution to Bφ(x) is dominated by the restrictions of L k to A(p, q) ∩ J − (x). Thus, while Bφ(x) is not determined just by the value of φ at x, it depends on φ only in an appropriately defined compact neighbourhood of x, rather than all of J − (x). This "restoration of locality" is an important subtlety in CST kinematics. How does a scalar field on a causal set evolve under this non-local d'Alembertian? There are indications that while the evolution in d = 2 is stable, it is unstable in d = 4 as suggested by Aslanbeigi et al (2014). Hence it is desirable to look for generalisations of the B κ operator. An infinite family of non-local d'Alembertians has been constructed by Aslanbeigi et al (2014) and shown to give the right continuum limit. It is still an open question whether there is a subfamily of these operators that lead to a stable evolution.
An interesting direction that has been explored by Yazdi and Kempf (2017) is to use the spectral information of the d'Alembertian operator to obtain all the information about the causal set. This was explored for for A 2 and it was shown that the spectrum of the d'Alembertian (or Feynman propagator) gives the link matrix (see Eqn. 54 below) or the set of all linked pairs in the causal set, and hence the full causal set. Extending these results to higher dimensions is an interesting open question.

The Ricci scalar and the Benincasa-Dowker action
Next we describe a very important development in CST : the construction of the discrete Einstein-Hilbert action or the Benincasa-Dowker (BD) action for a causal set (Benincasa and Dowker 2010;Dowker and Glaser 2013). The approach of Benincasa and Dowker (2010) was to generalise Bφ(x) to an RNN in curved spacetime in d = 2, d = 4. Again, the region of integration can be split into three as in the flat spacetime case. The contribution from W 3 i.e., away from a neighbourhood of ∂J − (x) can again be shown to be bounded from above by an integral that tends to zero faster than any power of ρ −1 c . In the limit, the contribution from the near region W 1 contained in an RNN is such that where R(x) is the Ricci scalar (Benincasa and Dowker 2010;Benincasa 2013). However, the calculation in region W 2 which is in the neighbourhood of ∂J − (x) but bounded away from the origin, is non-trivial, and needs a further set of assumptions to show that it does not contribute in the ρ c → ∞ limit. A painstaking calculation in Belenchia et al (2016a) using Fermi Normal Coordinates shows that this is indeed the case in an approximately flat region of a four dimensional spacetime. Generalising this calculation to arbitrary spacetimes is highly non-trivial but is an important open question in CST.
What is of course exciting about this form for the d'Alembertian is that it can be used to find the discrete Ricci curvature and hence the action. Assuming that holds in all spacetimes, and putting φ(x) = 1 18 Thus we can write the dimensionless discrete Ricci curvature at an element e ∈ C (Benincasa 2013) as where N k (e) = |L k (e)|. Summing over the n elements of a finite element causal set gives the dimensionless discrete action where N k is the total number of k-element order intervals in C. Benincasa and Dowker (2010) (see also Benincasa (2013)) showed that (under the assumption Eq. (29) ) the random variable S (4) associated with C(M, g) gives the Einstein-Hilbert action in the continuum limit upto (as yet unknown) boundary terms. Equation (33) is exactly true in an approximately flat region of a four dimensional spacetime as shown in Belenchia et al (2016a). Proving Eq. (29) in general is however non-trivial since there are caustics in a generic spacetime which complicate the calculation. On the other hand, numerical simulations suggest that again, upto boundary terms, the Benincasa-Dowker action S is the Einstein-Hilbert action (Benincasa 2013;Cunningham 2018b). We will discuss these boundary terms below.
Before doing so, we note that crucial to the validity of the causal set action are its fluctuations in a given causal set. These were shown in Sorkin (2007b) to be large for the operator B in M 2 . This can be traced to the fact that the elements in L k (e) for k = 0, 1, 2, 3 are very close to the discreteness scale and hence the d'Alembertian is susceptible to large Poisson fluctuations at small volumes. In order to "shield" the continuum from these fluctuations, a new mesoscale κ > c and its associated density ρ κ was introduced in Sorkin (2007b). Thus instead of a single discrete operator B, we have a one parameter family of operators: where ≡ ρ κ /ρ c , n(e, e ) = |I(e, e )| and This function smears out the contributions of the N k to B κ into four "layers" which appear with alternating sign, as shown in Fig. 15. Each layer is thus thickened from a single k to a range of k's corresponding to the intermediate mesoscale. When this mesoscale matches the discreteness scale, i.e., = 1, each layer collapses to a single value of k. This in turn gives us a one-parameter family of actions S κ (C). plays the role of a coupling constant and leads to an interesting phase structure in 2d CST as we will see in Sect. 6. The result for d = 2, 4 are due to Benincasa and Dowker (2010); Benincasa (2013) and generalised to arbitary dimensions by Dowker and Glaser (2013); Glaser (2014), using a dimension dependent smearing function f d (n, ).
There have been other attempts to obtain the action of a causal set. In Sverdlov and Bombelli (2009), the curvature at the center of an Alexandrov  Fig. 15 The function f (n, 0.05). There are 4 regions of alternating sign corresponding to 4 "smeared out" layers.
interval A d [p, q] in a RNN was obtained using the leading order corrections to the volume of a small causal diamond (Gibbons and Solodukhin 2007) where T is the proper time from p to q and V 0 is the flat spacetime volume. The expression obtained is in terms of the discrete volume and the length of the longest chain from p to q. Since R is approximately a constant in A d [p, q], this also gives the approximate action. Extending it to an action on the the full spacetime is however quite tricky since it is unclear how to localise the calculation.
The calculation for the abundance of k-chains C k in an RNN in Roy et al (2013) also gives rise to an expression for the curvature R(0) = −2(n + 2)(2n + 2)(3n + 2)2 3n+2 3n n 4 3n −1 . (37) where and While this expression is compact, it is not defined on a single causal, but rather, over the ensemble. Whether this can be expressed as a function on a single causal set or not is an interesting open question and under current investigation. As in the previous case, having obtained R(0), however, it is non-trivial to construct the action, without a localisation argument as in the BD action.

Boundary terms for the causal set action
Although the BD action gives the bulk Einstein Hilbert action in the continuum approximation, the role of boundary terms is less clear. As shown by Benincasa et al (2011) the expectation value for the BD action does not vanish for C(A 2 , ρ c ), where A 2 ⊂ M 2 as one might expect, but instead converges to a constant as ρ c → ∞ and is independent of vol(A 2 ). Buck et al (2015) showed more generally that for where J (d−2) ≡ ∂J + (p) ∩ ∂J − (q) is the codimension 2 "joint" of the causal diamond A d , which is a round sphere S d−2 . In d = 2 this is the volume of a zero sphere S 0 which is the constant found in (Benincasa et al 2011). This in turn corresponds to the Gibbons Hawking York (GHY) null boundary term of Lehner et al 2016) for a particular choice of the null affine parameter 19 . Extending this calculation to curved spacetime is challenging but would provide additional evidence that the BD action contains the null GHY term (Dhingra, Glaser and S. Surya, work in progress).
Simulations in d = 2, moreover give evidence for the fact that while the BD action appears to, in addition, contain timelike boundary terms, it does not contain spacelike boundary terms. Recent efforts by Cunningham (2018a) have been made to obtain time like boundaries in a causal set using numerical methods for d = 2, but it is an open question whether they admit a simple characterisation in arbitrary dimensions.
Unlike timelike boundaries, spacelike boundaries are naturally defined in a finite element causal set: a future/past spatial boundary is the futuremost/past-most inextendible antichain in the causal set, which we will denote as F 0 , P 0 respectively. GHY terms for spacelike boundaries play an important role in the additivity of the action in the continuum path integral, though such an additivity is far from guaranteed in a causal set because of non-locality.
The spatial causal set GHY terms were found by Buck et al (2015), and we will describe that construction here briefly. Let (M, g) be a spacetime with intial and final spatial boundaries (Σ ± , h ± ) . The GHY term on (Σ ± , h ± ) can be reexpressed as where ∂ ∂n is the normal derivative, and A Σ ± is the codimension 1 volume of Σ ± . Using the n ∼ ρ c v correspondence, this suggests that A Σ ± should be given by the cardinality F 0 ≡ |F 0 | or P 0 ≡ |P 0 | with the normal gradient represented by the change in the cardinality. But of course this is subtle, since apart from the future most F 0 or pastmost P 0 antichains, one needs another "close by" 19 It is an interesting question whether the choice of affine parameter along "almost" null directions can be obtained from the causal set.
antichain. Let us focus on (Σ + , h + ) wlog. There are two ways of finding this nearby antichain. To begin with if (M, g) ⊂ (M , g ) such that (Σ + , h + ) is not a boundary in (M , g ), then we can use this embedding to define the two antichains, in any C ∈ C(N, ρ c ): one to its immediate past F 0 (Σ + ) and one to its immediate future P 0 (Σ + ). Thus the GHY term should be proportional to the difference in the cardinality of these two antichains.
However, this partitioning is not intrisinsic to the causal set. Instead consider a partition C = C − ∪ C + , such that C + ∩ C − = ∅, and Fut(C − ) = C + , Past(C + ) = C − . Let F − 0 and P + 0 , be the future-most and past-most antichains of C − and C + respectively. We can then define the dimensionless causal set "boundary term" (Buck et al 2015) where and V d = (d + 1)π d+1 2 /Γ d+1 2 + 1 is the volume of the unit d-sphere. To make contact with the continuum, let (M, g) be a spacetime with compact Cauchy hypersurfaces. For a given Cauchy hypersurface (Σ, h) let M ± = J ± (Σ) and let C ± ∈ C(M ± , ρ c ). It was shown by Buck et al (2015) that in the limit ρ c → ∞ CBT is the associated random variable in (M, g). To obtain this expression, the volume of a half cone J + (p) ∩ J − (Σ) was calculated using a combination of RNCs and GNCs 20 for p ∈ J + (Σ) sufficiently close to Σ, where T is the proper time from p to Σ. As might be expected from dimensional considerations, the leading order correction to the flat spacetime volume of the half cone comes from the trace of the extrinsic curvature of Σ from which the GHY contribution can be obtained. If on the other hand, (Σ, h) is a future boundary of (M, g), then we require a second antichain in Past(F 0 ) for C ∈ C(M, ρ c ),. Define the antichain F 1 in C − to be the set of elements in C − such that ∀e ∈ F 1 , |Fut(e)∩C − | = 1 (where Fut(e) excludes the element e) 21 . The boundary term can then be expressed as which again yields the GHY term Eq. (44) in the limit. Indeed, a whole family of of boundary terms was obtained using the antichains F k [C − ] = {e ∈ C − ||Fut(e)| = k}, P k [C + ] = {e ∈ C + ||Past(e)| = k} 22 each of which gives the GHY term in the limit Eq. (44). A by product of the analysis of Buck et al (2015) is that for the partitioned causal set C = C − ∪ C + described above, the quantities Again, as for the boundary terms, one can construct a whole family of functions A d [C] each of which limit to the spatial volume of Σ as ρ c → ∞.

Localisation in a causal set
In these calculations generalisations are made to curved spacetime using an RNN which represent a local region of a spacetime. How are we to locate such local regions in a causal set using a purely order theoretic quantities? For a general causal set we can of course define a local region using the size of an interval, but for a manifold-like causal set, these will not necessarily correspond to regions in which the curvature is small. On the other hand, many of the order invariants we have obtained so far correspond to geometric invariants only in such RNN-type regions. A characterisation of intrinsic localisation was obtained by Glaser and Surya (2013) using the abundance of m element order intervals N d m for C ∈ C(A d , ρ c ). They found the following closed form expression for the associated expectation value The distribution of N d m with m has a characteristic form which depends on dimension, and as a by-product, can be used as a dimension estimator. However, it can also be used look for intervals in a manifold-like causal set which are approximately flat by comparing the interval abundances N d m to the above expression for N d m . While one might expect the fluctuations for a given causal set C to be large, numerical simulations show that there is typically a "self averaging" which results in relatively small fluctuations even for a given realisation. This makes it an ideal diagnostic tool for checking whether a neighbourhood in a manifold-like causal set is approximately flat or not. Once such local neighbourhoods have been found, a local check of geometric estimators can be made.
In Glaser and Surya (2013) the analytic curves were compared against simulations for a range of different causal sets including those that are not manifold-like . While curvature effects modified the large k behaviour of intervals, the distribution retains its characteristic form. Hence the dependence of the abundance of intervals with size also becomes a test for manifold-likeness. There are other ways of testing for manifold-likeness. In a similar approach, the distribution of maximal chains or linked paths of lenghth k in a finite element causal set C has been studied in M d , d = 2, 3, 4 and shown to have a characteristic dimension determined peak (Aghili et al 2018). (Bolognesi and Lamb 2016) suggest a novel way to test for manifold-likeness using the order invariant obtained from counting the number of elements with a fixed valency in a finite element causal set. Henson (2006a) proposed an algorithm to determine the embeddability of a causal set in M 2 , which again gives an intrinsic characterisation of manifold-likeness in d = 2. Extending and expanding on these studies will give us further useful tools in CST.

Kinematical Entropy
Since the classical continuum geometry itself is fundamentally statistical in CST, it is interesting to ask if a kinematic entropy can be assigned even classically to the continuum. In Dou and Sorkin (2003) a kinematic entropy was associated with a horizon H and a spatial or null hypersurface Σ in a dimensionally reduced d = 2 black hole spacetime by counting links beween elements in J − (Σ) ∩ J − (H) and those in J + (Σ) ∩ J + (H), with the additional requirement that the former is maximal and the latter minimal in their respective regions. A dimensionally reduced calculation showed that the number of links is proportional to the horizon area. Importantly, the calculation yields the same constant for a dimensionally reduced dynamical spacetime where a collapsing shell of null matter eventually forms a black hole. However, extending this calculation to higher dimensions is tricky. In Marr (2007) an entropy formula was proposed for higher dimensions by replacing links with other sub-causal sets. While they hold promise, these ideas have not been as yet fully explored.
In an analogy with Susskind's entropy bound, the maximum causal set entropy associated with a finite spherically symmetric spatial hypersurface Σ was defined by Rideout and Zohren (2006) as the number of maximal or future most elements in its future domain of dependence D + (Σ). It was shown that for several such examples this bound limits to the Susskind entropy bound in the continuum approximation. Again, extending this discussion to more general spacetimes is an interesting open question.
In Benincasa (2013) it a mutual information between different regions of a causal set was defined using the BD action. Dividing a causal set C into two (set-wise) disjoint regions C 1 and C 2 , we note that S BD (C) is non-additive since there can be order intervals between elements in C 1 and in C 2 that are not counted by either S BD (C 1 ) or S BD (C 2 ). The mutual information is then defined as and can be non-zero. In (Benincasa 2013) a spacetime region with a horizon H was considered to the past of a spacelike or null hypersurface Σ. Defining X = J + (H)∩J − (Σ) and Y = J − (H)∩J − (Σ) the mutual information between X and Y was calculated from a causal set obtained from sprinkling into X ∪Y . Equating this to the area of H ∩ Σ requires certain assumptions, and further work is required to prove it more generally. As we will see in the next section, the Sorkin spacetime entanglement entropy (SSEE) for a free scalar field provides a different avenue for exploring entropy.

Remarks
To conclude this section we note that several order invariants have been constructed on manifold-like causal sets whose expectation values limit to manifold invariants as ρ c → ∞. For finite ρ c there are important flucutations that distinguish the fundamental discreteness of causal sets from the continuum. Numerical simulations give us a handle on the relative importance of these fluctuations.
For each of these invariants, one has therefore proved a O-Hauptvermutung. While this set of invariants are not (even collectively) sufficient proof for the Hauptvermutung, they lend it strong support. In addition to these continuuminspired invariants, there are several other order theoretic invariants that we can construct, some of which may be important to the deep quantum regime but by themselves hold no direct continuum interpretation.

Matter on a continuum-like causal set
Before passing ont to the dynamics of CST, we look at a phenomenologically important question, namely how quantum fields behave on a fixed manifoldlike causal set. The simplest matter field to construct is the free scalar field on a causal set in M d . As we noted in the previous Section, this is the only class of matter fields that we know how to study, since at present no well defined representation of non-trivial tensorial fields on causal sets is known. However, as we will see, even this very simple class of matter fields yields both exciting new insights and interesting conundrums.

Causal set Green functions for a free scalar field
Consider the real scalar field φ : M d → R and its CST counterpart, φ : C → R where C ∈ C(M d , ρ c ). The Klein Gordon (KG) equation is replaced on the causal set by the B κ operator of Sect. 4, Eq. (34). In the continuum −1 gives the Green function, and we can do the same with B κ to obtain the discrete Green function B −1 κ . However, there are more direct ways of obtaining the Green function as was shown in (Daughton 1993;Salgado 2008;Johnston 2008;.
Define the causal matrix on a causal set C defined for every pair of elements e, e ∈ C. For C ∈ C(M d , ρ c ), C 0 (e, .) is therefore zero everywhere except within the past light cone of e at which it is 1. In d = 2, this is just half the massless retarded Green's function G 0 (x, x ) = 1 2 θ(t − t )θ(τ 2 (x, x )). Hence, we find the almost trivial relation C 0 (x, x ) = 2G (2) without having to take an expectation value, so that the dimensionless massless causal set retarded Green function is (Daughton 1993) To obtain the d = 4 massless causal set Green function define the link matrix For C ∈ C(M 4 , ρ c ) the expectation value of the associated random variable is where V (x, x ) = vol(J − (x) ∩ J + (x )) = π 24 τ 4 (x, x ). Since the exponential in the above expression is a Gaussian which, in the ρ c → ∞ limit ∝ δ(τ 2 ), we see that it resembles the massless retarded Green function in M 4 , G Hence we can write the dimensionless massless causal set scalar retarded Green function as (Johnston 2008(Johnston , 2010) In the continuum the massive Green function can be obtained from the massless Green function in M d via the formal expression (Dowker et al 2017) Using this as a template, with the discrete convolution operation given by matrix multiplication, A(e, e )B(e , e) , a candidate for the d = 2 dimensionless massive causal set Green function can be constructed as where M is dimensionless and we have used the relation C k (x, x ) = C k 0 (x, x ), where as before C k (x, x ) counts the number of k-element chains from x to x . For C ∈ C(M 2 , ρ c ) it can be shown that (Johnston 2008(Johnston , 2010) when M 2 = m 2 ρc . Similarly, a candidate for the d = 4 massive causal set Green function is where we have used the fact that the number of k-element linked paths L k = L k 0 . For C ∈ C(M 4 , ρ c ), when M 2 = m 2 √ ρc . These massive causal set Green function were first obtained by Johnston (2008Johnston ( , 2010 using an evocative analogy between Feynman paths and the kchains or k-linked paths (see Figure 17). "Amplitudes" a and b are assigned to a "hop" between two elements in the Feynman path, and to a "stop" at an intervening element, respectively. This gives a total "amplitude" a k+1 b k for each chain or linked path, so that the massive Green functions can be expressed as Finding causal set Green functions for other spacetimes is more challenging, but there have been some recent results  which show that the flat spacetime form of Johnston (2008,?) can be used in a wider context. These include (a) a causal diamond in an RNN of a d = 2 spacetime with M 2 = ρ c −1 (m 2 + ξR(0)), where R(0) is the Ricci scalar at the center of the diamond and ξ the non-minimal coupling (b) a causal diamond in an RNN of a d = 4 spacetime with M 2 = ρ c −1 (m 2 + ξR(0)) when R ab (0) ∝ g ab (0) (c) d = 4 de Sitter and anti de Sitter spacetimes with M 2 = ρ c −1 (m 2 + ξ). The de Sitter causal set Green function in particular allows us to explore cosmological consequences of discreteness, one of which we will describe below. It would be useful to extend this construction to other conformally flat spacetimes of cosmological relevance like the flat FRW spacetimes. Candidates for causal set Green functions in M 3 have also been obtained using both the volume of the causal interval and the length of the longest chain (Johnston 2010; Dowker et al 2017), but the comparisons with the continuum need further study.
As the attentive reader would have noticed, in d = 4 the causal set Green function matches the continuum only for ρ c → ∞, unlike in d = 2. At finite ρ c , however, there can be potentially observable differences with the continuum. Comparisons with observation can in turn put constraints on CST. Dowker et al (2010a) examined a model for the propagation of a classical massless scalar field from a source to a detector on a background causal set. In M d , an oscillating point source with scalar charge q, frequency ω and amplitude a, and a "head-on" rectangular shaped detector was considered, so that the field produced by the source is where P is the world line of the source and s the proper time along this world line. If D represents the spacetime volume swept out by the detector during its detection time T then the output of the detector is where R is the distance between the source and detector, ν is the component of the velocity along the displacement vector between the source and detector and v D is the spacetime volume of the detector region D. Here, R >> a and R >> ω −1 which in turn is much larger than the spatial and temporal extent of the detector region D. The causal set detector output can then be defined as whereD andP correspond to the detector and source subregions in the causal set and the causal set function L(e, e ) is equal to some normalisation constant κ when e and e are linked and is zero otherwise. For C ∈ C(M 4 , ρ c ) it was shown that, with the above constraints on R, ω, a and the dimensions of the detector, that F approximates to same continuum expression Eq. (67) when R >> ρ − 1 4 c . A detailed calculation gives an upper bound on the fluctuations, which, for a particular AGN model is one part in 10 12 for ρ c = ρ p . Hence the discreteness does not seem to mess with the coherence of waves from distant sources. As we will see in Sect. 7 there are other potential signatures of the discreteness that may have phenomenological consequences (Dowker et al 2004;Sorkin 1991Sorkin , 1997Ahmed et al 2004).

The Sorkin-Johnston (SJ) vacuum
Having obtained the classical Green function and the d'Alembertian operator in M 2 and M 4 , the obvious next step is to build a full quantum scalar field theory on the causal set. As we have mentioned earlier, the canonical route to quantisation is not an option for causal sets nor for fields on causal sets and hence there is a need to look at more covariant quantisation procedures. Johnston (2009Johnston ( , 2010 used the the covariantly defined Peierls' bracket as the starting point for quantisation, where is the Pauli Jordan function, and G R,A (x, x ) are the retarded and advanced Green's functions, respectively. As we have seen, these Green functions can be defined on some manifold-like causal set and hence provides a natural starting point for quantisation. However, even here, the standard route to quantisation involves the mode decomposition of the space of solutions of the Klein Gordan operator, ker( − m 2 ). In M d the space of solutions has a unique split into positive and negative frequency classes of modes with respect to which a vacuum can be defined. In his quest for a Feynman propagator, Johnston (2009) made a bold proposal, which as we will describe below, has led to a very interesting new direction in quantum field theory even in the continuum. This is the Sorkin-Johnston or SJ vacuum for a free quantum scalar field theory.
Noticing that the Pauli-Jordan function on a finite causal set C is a Hermitian operator, and that ∆(e, e ) itself is antisymmetric, Johnston used the fact that the eigenspectrum of i∆ This provides a natural split into a positive part and a negative part, without explicit reference to ker( − m 2 ) 23 . A spectral decomposition of i ∆ then gives i∆(e, e ) = λ k Defining the Wightmann function as the positive part of i∆, one can then construct the Feynman Green function. Importantly, a noninteracting theory with a Gaussian state, the Wightmann function is sufficient to describe the full theory and thus the vacuum state. Simulations in M 2 give a good agreement with the continuum, while those in M 4 are less clear since the causal set Green functions approach the continuum only for very large ρ c (Johnston 2009(Johnston , 2010. Sorkin (2011a) noticed that the construction on the causal set, which was born out of necessity, provides a new way of thinking of the quantum field theory vacuum. A well known feature of quantum field theory in a general curved spacetime is that the vacuum obtained from mode decomposition in ker( − m 2 ) is observer dependent and hence not unique. Since the SJ vacuum is intrinsically defined, at least in finite spacetime regions, one has a uniquely defined vacuum. The SJ state has therefore generated interest in the broader algebraic field theory community (Fewster and Verch 2012;Brum and Fredenhagen 2014;Fewster 2018). While not in itself Hadamard in general, it can be used to generate a new class of Hadamard states (Brum and Fredenhagen 2014).
In the continuum, the SJ vacuum was constructed for the massless scalar field in the d = 2 causal diamond (Afshordi et al 2012) and recently extended to the small mass case (Mathur and Surya, work in progress). It has also been obtained for the trousers topology and shown to produce a divergent energy along both the future and the past light cones associated with the Morse point singularity (Buck et al 2017). Numerical simulations of the SJ vacuum on causal sets are are approximated by de Sitter spacetime suggest that the causal set SJ state differs significantly from the Mottola-Allen α vacuua . This has potentially far reaching observational consequences which need further investigation.

Entanglement entropy
Using the Pauli Jordan operator i ∆ and the associated Wightman W , Sorkin (2014) defined spacetime entanglement entropy, Sorkins' Spacetime Entanglement Entropy (SSEE) where λ i are the generalised eigenvalues satisfying It was shown by  that for a smaller causal diamond sitting in the center of a larger one in M 2 that the SSEE has the expected behaviour in the limit that the size of the smaller diamond l is much smaller than that of the larger diamond, where l uv is the UV cut-off and b, c are constants that can be determined. One of the promises that discretisation holds is of curing the UV divergences of quantum field theory and in particular those coming from the calculation of the entanglement entropy of Bombelli et al (1986). As shown by Sorkin and Yazdi (2018) the causal set version of the above calculation does yield a finite answer, which however differs from the continuum. This is because the continuum spectrum of eigen values of Eq. (75) agrees with the discrete upto a "knee", beyond which the effects of discreteness begin to be seen, as shown in Fig. 18. Fig. 18 A log-log plot depicting the SJ spectra for causal sets in a causal diamond in M 2 . A comparison with the continuum (the straight black line) shows that the causal set SJ spectrum matches the continuum in the IR but has a characteristic "knee" in the UV after which it deviates significantly from the continuum. As the density of the causal set increases, this knee shifts to the UV.
Thus, the SSEE for the causal set is proportional to the volume rather than a constant. Sorkin and Yazdi (2018) truncated the spectrum twice -once in the larger diamond and once in the smaller one, in order to get the requisite area law. This raises very interesting and as yet unanswered puzzles about the nature of SSEE in the causal set. It is possible that in a fundamentally nonlocal theory like CST an area law is less natural than a volume law. Such a radical understanding could force us to rethink continuum inspired ideas about Black Hole entropy.
Extending the above calculation to actual black hole spacetimes is an important open problem. Ongoing simulations for causal sets obtained from sprinklings into 4d de Sitter spacetime show that this double truncation procedure gives the right de Sitter horizon entropy (Dowker, Surya, Sumati, X and Yazdi, work in progress), but one first needs to make an ansatz for locating the knee in the causal set i∆ spectrum.

Spectral dimensions
An interesting direction in causal set theory has been to calculate the spectral dimension of the causal set (Eichhorn and Mizera 2014;Belenchia et al 2016c;Carlip 2017). Carlip (2017) has argued that d = 2 is special in the UV limit, and that several theories of quantum gravity lead to such a dimensional reduction. In light of how we have presented CST, it seems that this continuum inspired description must be limited. However, it is nevertheless interesting to ask if causal sets that are manifold-like might exhibit such a behaviour around the discreteness scales at which the continuum approximation is known to break down. As we have seen, one such behaviour is discrete asymptotic silence (Eichhorn et al 2017). Eichhorn and Mizera (2014) calculated the spectral dimension on a causal set using a random walk on a finite element causal set. It was found that in contrast, the dimension at small scales goes up rather than down. On the other hand, citediondr showed that causal set inspired non-local d'Alembertians do give a spectral dimension of 2 in all dimensions. As we noted in Sect. 4, Abajian and Carlip (2018) showed that dimensional reduction of causal sets occurs for the Myrheim-Meyer dimension as one goes to smaller scales. Explorations in this direction are definitely interesting since they can yield insights into the continuum approximation.

Dynamics
Until now our focus has been on manifold-like causal sets, since the aim was to find useful manifold-like covariant observables as well as to make contact with phenomenology. However, as discussed in Sect. 3, the arena for CST is a sample space Ω of locally finite posets which replaces the space of 4-geometries, and contains non-manifold-like causal sets. A CST dynamics is given by the measure triple (Ω, A, µ) where A is an event algebra and µ is either a classical or a quantum measure. We will define these quantities later in this section.
To begin with, Ω itself can be chosen depending on the particular physical situation in mind. In the context of initial conditions for cosmology, for example, it is appropriate to restrict to the sample space of past finite countable causal sets Ω g , while for a unimodular type dynamics using the Einstein-Hilbert action, the natural restriction is to Ω n the sample space of causal sets of fixed cardinality n. We will see that dimensional restrictions on the sample space are also of interest and can lead to a closer comparison with other approaches to quantum gravity.
As discussed in Sect. 3 and 4, in the asymptotic n → ∞ limit the sample space Ω n is dominated by the non-manifold-like KR causal sets depicted in Fig. 9. This is the entropy problem of CST. These posets have approximately just three "moments" of time and hence should not be relevant in the classical or continuum approximation of the theory. In the classical limit, the quantum dynamics must be able to overcome this entropy.
For a quantum dynamics of CST we would like to start with a few basic axioms, including discrete general covariance and dynamical causality. A very important step in this direction was made by the Rideout and Sorkin (2000a) classical sequential growth models (CSG), which are Markovian growth models. These are also are required to satisfy Bell causality or spectator independence. The main challenge in CST is to build a viable quantum theory of CSG (QSG).
The appropriate framework in which to construct a QGS is using a quantum measure space which is a natural quantum generalisation of classical stochastic dynamics (Sorkin 1994(Sorkin , 1995(Sorkin , 2007d. This means replacing the classical probability measure P in the measure space triple (Ω, A, µ c ) with a quantum measure µ. The quantum measure is defined via a decoherence functional and can also be defined as a vector measure in a corresponding histories Hilbert space.
One can also construct a continuum inspired dynamics, where the discrete Einstein-Hilbert or BD action is used to give the measure for the discrete path integral or path sum. The quantum partition function can either be evaluated directly or converted into a statistical partition function over causal sets using an analytic continuation. This makes it amenable to Markov Chain Monte Carlo (MCMC) simulations as we will see below.
6.1 Classical sequential growth models The Rideout and Sorkin (2000a) classical sequential growth or CSG models are a class of stochastic dynamics in which causal sets a grown element by element, with the dynamics satisfying a few basic principles (Rideout andSorkin 2000a, 2001;Martin et al 2001;Rideout 2001;Varadarajan and Rideout 2006). The stochastic dynamics finds a natural expression in measure theory and allows for an explicit definition of covariant classical observables (Brightwell et al 2003;Dowker and Surya 2006). This measure theoretic structure provides an important template for the quantum theory, and hence we will first flesh it out in some detail before discussing quantum dynamics.
Let us start with a naive picture. Imagine living on a classical causal set universe, with our universe represented by a single causal set. Since causal sets are locally finite, the "passage of time" occurs with the addition of a new element. If we are to respect causality, this new element cannot be added so as to disturb the past, i.e., what has already happened. Instead it can be added to the future of some of the existing events or it can be unrelated to all of them. Every such "atomic change" in spacetime correponds to the causal set changing cardinality by one. Starting with a causal setc n of cardinality n, the passage of time means transitioning fromc n →c n+1 where the new element inc n+1 is to the future of some of the elements ofc n , but never in their past. In the infinite "time" limit, n → ∞, the dynamics, either deterministic, probabilistic or quantum, will take you fromc n to a countable causal set.
Working backwards, on the other hand, leads us to a "beginning", with n = 0. This gives the most natural initial condition for causal sets: begin with the empty set ∅. 24 The only way to go forward from here, is to make n = 1, i.e., we have a single element. For n = 2, the new element could either be to the future of the existing element or unrelated to it, as in Fig. 19.
Thus, one can build up the tree T of causal sets as n → ∞ as shown in Fig. 20. As n increases, the number of possibilities grows super exponentially as expected from the KR theorem (Kleitman and Rothschild 1975), and there is no easy enumeration of this space. The growth process generates a sample p q Fig. 19 The first two stages of a classical sequential growth(CSG) dynamics. The probability for a single element (red) to appear at coordinate time n = 1 is 1. Subsequently, the new element (blue) at n = 2 is added either to the future of the existing element with probability p or is unrelated to it with probability 1 − p.
spaceΩ g of causal sets which are are all past finite and labelled by the "time" at which each element is added. A causal setc inΩ g is said to be naturally labelled, i.e., there exists an injective map l :c → N (the natural numbers) which preserves the order relation inc, i.e., x ≺ y ⇒ l(x) < l(y). In the growth process, this label is the coordinate time.
p q Fig. 20 The CSG tree. There are three ways to get the 3-element unlabelled causal set whose natural labellings are given by the 3rd, 4th and 5th 3-element labelled causal sets in the figure. One path is via the 2-element chain and the other two are from the 2-element antichain. Covariance demands that the probability along each path is the same.
In the spirit of covariance, however, we cannot take the time label to be fundamental; the dynamics and the observables cannot depend on the order in which the elements are born. Thus, the probability to get a labelled causal setc n and any of its relabellings,c n must be the same. Identifying relabelled causal sets as the same object in the CST tree T gives us a non-trivial poset of causal sets or the "postcau" P of Rideout and Sorkin (2000a). On P, a covariant dynamics is thus path-independent: if there is more than one path from an unlabelled initial causal set c ni to an unlabelled final causal set c n f in P, then in order to satisfy covariance, the measure on both paths should be the same.
Apart from covariance, the dynamics should also satisfy an internal causality condition, dubbed Bell causality. Consider the transitionc n →c n+1 with proability α n where the new element e n+1 is added to the future of a "precursor" set p n ⊂c n , and is unrelated to a "specator set" s n ⊂c n . Causality suggests that the probability for the transition should not depend explicitly on the spectator set s n . For non-empty s n with |s n | < n, consider the causal setsc m =c n \s n andc m+1 =c n+1 \s n , where \ denotes set difference and m + |s n | = n. The transition probability α m forc m →c m+1 should then be proportional to α n . Ifc n →c n+1 is another transition fromc n , then defining p n , s n , α n , andc m+1 =c n+1 \s n , analogously, the condition of Bell causality is Though relatively easy to implement classically, a quantum version of Bell causality has been hard to find (Henson 2011). The triple requirements of (a) covariance, (b) Bell causality and (c) Markovian evolution define the classical sequential growth dynamics of Rideout and Sorkin (2000b). Starting from the empty set, a causal set is thus grown element by element, assigning probabilities to each transitionc n to ac n+1 , consistent with these requirements. Because of it being a Markovian evolution, the probability associated with any finite c n is given by the product of the transition probabilities along a path in P.
The dynamics was shown in Rideout and Sorkin (2000a) to be fully determined by the infinite set of coupling constants, t n , one for each stage of the growth. If q k denotes the transition probability from the k-element antichain to the k + 1-element antichain In general, each t n can be independent of the others. Including relations between the different t n simplifies the dynamics. The simplest example is that of transitive percolation determined by the probability (1−q) ≥ 0 of adding an element to the immediate future 25 of an existing element, and q of being unrelated to it. Thus, the probability of adding a new element to the immediate future of m elements of c n and of being unrelated to m others is (1 − q) m q m .
In terms of the general coupling constants, t n = t n ≡ 1−q q n . In Varadarajan and Rideout (2006) and Dowker and Surya (2006), a generalisation of the dynamics was explored, which allowed certain transition probabilities to vanish, consistent with (a) (b) and (c), and hence required a generalisation of the Bell causality condition. The resulting dynamics exhibits a certain "forgetfulness" when these transition proabilities vanish, but are otherwise very similar to the CSG models.
Since the generic dynamics consistent with (a),(b) and (c) does not by itself lead to constraints on the t n , this is an embarassment of riches. Does nature pick out one set over another? In Martin et al (2001), an evolutionary mechanism for doing so was suggested using cosmological bounces which give rise to new epochs which "renormalise" the coupling constants towards fixed points. A cosmological bounce in a causal set is naturally described by the appearance of a post which is an inextendible antichain of cardinality 1. Thus, every element in c either lies to its past or to its future. Moreover, because it is a single element maximal antichain, there are no missing links, and the post is indeed a summary of its past. The post is the causal set equivalent to a "bounce" but is non-singular in the causal set. We define the causal set between two posts as an "epoch", with the last epoch being the one after the last post. Let e be a post in c and let r = |past(e)|. Then a set of "effective" coupling constants in the epoch after e can be defined as (Martin et al 2001).
Denoting the set of effective couplings by 1 , . . .} with i = 0 being the original set of couplings, this corresponds to applying r copies of the transform M : In Martin et al (2001), it was shown that the fixed points of the map M give t n = t n (transitive percolation) for some t ≥ 0 and moreover M does not have any other cycles. Starting from any set T (0) for which lim n→∞ (t (0) n ) 1/n is finite, M r : T (0) → T (r) , is such that T (r) converges pointwise to t (r) n = t n for t = lim n→∞ (t (0) n ) 1/n . While this result does not guarantee that every T (0) will converge to transitive percolation, Martin et al (2001) examined several cases, and conjectured that the deviation from percolation-like values are "rare" and that typically, T (r) will be nearly transitive percolation like.
Such an evolutionary renormalisation thus brings the infinite dimensional coupling constant space to a one dimensional space, which is remarkable. Assuming that this is indeed the case in general, a sufficiently late epoch will likely have a transitive percolation dynamics.
What can one say about the causal sets generated from this dynamics? A very important result from transitive percolation is that the typical causal sets obtained are not KR like posets and hence the dynamics beats their entropic dominance. The question of whether there is a continuum-like limit for transitive percolation dynamics was explored in Rideout and Sorkin (2001), using a comparison criterion. The abundance of fixed small subcausal sets was examined as a function of the coupling, by fixing the density relations. Comparisons with Poisson sprinklings in flat spacetime showed a convergence, suggesting a continuum limit. In Ahmed and Rideout (2010), it was shown that the dynamics typically yields an exponentially expanding universe. Moreover, for (1 − q) 1 and n 1 1−q , after a post the universe enters a tree like phase and then a de Sitter-like phase, in which the cardinality of large causal diamonds are de Sitter like functions of the discrete proper time. In Glaser and Surya (2013), it was shown that despite this, the abundance of causal intervals is not de Sitter like, and thus, this is not strictly a manifold like phase. In Brightwell and Georgiou (2010) and Brightwell and Luczak (2015), moreover, it was shown explicitly that in the asymptotic limit n → ∞ the causal sets limit to "semi-orders" which, though temporally ordered, have no spatial structure at all, and are hence non-manifold-like. Nevertheless, the dominance of measure over entropy is important and the hope is that it will be reflected in the right quantum version of the dynamics.
Recently, Dowker and Zalel (2017) proposed a method for dealing with black hole singularies in CSG models. As in the case of cosmological bounces a new epoch is created beyond the singularity. Using similar construction as posts, it was demonstrated that a renormalisation of the coupling constants occurs in the new epoch.

Observables as beables
As mentioned in the introduction to this section, a dynamics for CST is given by the triple (Ω, A, µ). In CSG this is a probability measure space, where the sample spaceΩ g is the set of all past finite naturally labelled causal sets.
The event algebra A can be constructed from the sequential growth process as follows. We define a cylinder set cyl(c n ) ⊂Ω g as the set of all labelled causal sets inΩ g whose first n elements are the causal setc n . Figure 22 depicts an example of a cylinder set. 26 For every finite element causal setc n , cyl(c n ) ⊆ Ω g , and in the trivial n = 1 case, cyl(c 1 ) =Ω g . The cylinder sets in CSG satify a nesting property. Namely, if n > n and cyl(c n ) ∩ cyl(c n ) = ∅, then cyl(c n ) ⊂ cyl(c n ). Thus, non-trivial intersection of two different cylinder sets is only possible if one is strictly a subset of the other. Fig. 22 The cylinder set for the "V" poset consists of all countable causal sets inΩg whose first three elements are the labelled "V" poset. Examples of causal sets that lie in cyl(V) are depicted in the boxes.
The event algebraÃ is generated from the cylinder sets via finite unions, intersections and set differences. It is closed under finite set operations and contains the null set ∅ and alsoΩ g . In the growth process we assigned a probability µ(c n ) to every finite labelled causal setc n . By identifyingc n with its cylinder set cyl(c n ), we can assign the measure µ(cyl(c n )) ≡ µ(c n ). µ can then be defined on all the elements of A, which gives us the "pre-measure" space (Ω g ,Ã,μ ). An event is defined to be an element of A. We define a covariant event α as a measurable subset ofΩ g such that for everyc ∈ α, its relabellingc also belongs to α. Since a relabelling can happen arbitrarily far into the future, no event in A is covariant, since A is closed only under finite set operations. Take for example the covariant post event which is the set of all causal sets which have a post. This is a covariant event, and is the equivalent of the return event in the random walk. In both cases, the event cannot be defined using only countable set operations, and hence the post event does not belong to A.
One route to obtaining covariant events is to pass to the full sigma algebrã S generated byÃ, which is closed under countable set operations. For classical measure spaces, the Kolmogorov-Caratheodary-Hahn extension theorem allows us to extendμ toS and hence pass with ease to a full measure space (Ω g ,S,μ), whereμ|Ã =μ . Not every event in S is covariant, but we can restrict our attention to covariant events, i.e., sets that are invariant under relabellings. If ∼ denotes the equivalence upto relabellings one can define the quotient algebra S =S/ ∼ of covariant events. An element of S is measurable covariant set, or a covariant observable (or beable). Our example of the post event belongs to S. Another example of a covariant event is the set of originary causal sets, i.e., causal sets with a single initial element to the past of all other elements. Constructing more physically interesting covariant observables in S is important, since it tells us what covariant questions we can ask of quantum gravity.
A more covariant way to proceed is to generate the event algebra not via the cylinder sets inΩ g but using covariantly defined sets in Ω g , the sample space of unlabelled causal sets. Because causal sets are past finite we can use the analogue of past sets J − (X) to characterise causal sets in a covariant way. A finite unlabelled sub-causal set c n of c ∈ Ω g is said to be a partial stem if it contains its own past. A stem set stem(c n ) is then a subset of Ω g such that every c ∈ stem(c n ) contains the partial stemc n . Let S be the sigma algebra generated by the stem sets. Although S is a strictly smaller subalgebra of S, it differs on sets of measure zero for the CSG and extended CSG models as shown by Brightwell et al (2003) and Dowker and Surya (2006). Thus, one can characterise all the observables of CSG in terms of stem sets. This is a very non-trivial result and the hope is that some version of it will carry over to the quantum case.

A route to quantisation: The quantum measure
The generalisation of CSG to QSG is, at least formally, very straightforward. One "quantises" the classical covariant probability space (Ω g , S, µ c ), by simply replacing the classical probability µ c with a quantum measure µ : S → R + , where µ satisfies the quantum sum rule (Sorkin 1994(Sorkin , 1995Salgado 2002;Sorkin 2007d) for the mutually disjoint sets α, β, γ ∈ S. µ(.) is not in general a probability measure since it doesn't satisfy additivity µ(α∪β) = µ(α)+µ(β) for α∩β = ∅. As in the classical case, observables in this theory are simply the quantum measurable sets in S. The quantum measure µ(.) can be obtained from a decoherence functional D : S × S → C of quantum theory with where D satisfies Strong positivity.
In a QSG model the transition probabilities of CSG are replaced with the the decoherence functional D or quantum measure. Leaving aside Bell causality, the other principle of the growth dynamics are easy to implement. In Dowker et al (2010c), a simple complex percolation dynamics was studied, given by a product decoherence functionD(α, is obtained from the transition amplitudes q ∈ C, similar to transitive percolation. Thus, as in the case of CSG models, one starts with the labelled event algebra A generated by the cylinder sets, and a quantum pre-measurẽ D . Again, in order to obtain covariant observables one has to pass to the full sigma algebra S associated with A. However, unlike a classical measureD need not extend to a full sigma algebra. In Dowker et al (2010c), the quantum pre-measure was shown to be a vector pre-measureμ in the associated histories Hilbert space (Dowker et al 2010b). Extension ofμ to S is then possible provided certain convergence conditions are satisfied. 28 Although the vector measure is 1-dimensional in complex percolation dynamics, it was shown in Dowker et al (2010c) not to satisfy this convergence condition and hence one cannot pass to S to construct covariant observables. However a smaller algebra may be sufficient for answering physically interesting questions, which require far weaker convergence condition as suggested by Sorkin (2011b). This relaxation of conditions means that some simple measurable covariant observables can be constructed in complex percolation, including for the originary event (Sorkin and Surya, work in progress). Whether these results on extension are shared by all QSG models or not is of course an interesting question. Another possibility is that an extension of the measure in QSG could, for example, be a criterion for limiting the parameter space of QSG.
The space of QSG models is still very unexplored, partially for technical reasons, but is critical to a principled approach to the quantum dynamics in CST. The hope is that we will learn new tools in the coming years to be able to make significant progress in this direction.

A continuum-inspired dynamics
At a fundamental level, as we have seen, it is possible that the quantum dynamics of causal sets looks very different from that of a continuum theory of quantum gravity, even if the latter is formulated as a path integral. However, as one approaches the continuum approximation of the theory, the effective quantum dynamics must resemble the continuum inspired path integral dynamics. In CST, such a quantum partition function can be expressed as where S(c) is an action for causal sets, and the choice of sample space Ω is determined by the problem at hand. 29 The natural choice for S(c) is the d dimensional BD action S (d) BD (c) which limits to the Einstein-Hilbert action in the continuum. As discussed in Sect. 3, the sample space Ω n of causal sets of cardinality n is dominated by KR type causal sets. An important question is whether the action S (d) BD (c) can overcome the KR entropy in the large n limit.
Indeed, there is a hierarchy of sub-dominant causal sets which are non manifold-like (Dhar 1978(Dhar , 1980Kleitman and Rothschild 1975), with the set of bilayer posets B being the next subdominant class. A recent calculation by Loomis and Carlip (2018) shows that B is suppressed by the BD action when the mesoscale and dimension satisfy certain conditions. The only relations in a bilayer poset are links. Given that the maximum number of relations is n 2 the causal sets in B can be classified by the linking fraction p given by the ratio of the total number of links N 0 to the maximal possible number of links n 2 . Moreover, the action itself reduces to a simple sum of n and N 0 , which is just a function of p and n. In the limit of large n, Loomis and Carlip (2018) consider p to be a continuous variable using which the partition function Z B can be expressed as an integral over p, Z B = dp|B p,n |e iS(p)/ = e iµn dp|B p,n |e 1 2 iµλ0pn 2 +o(n 2 ) where B p,n denotes the class of n-element causal sets in B with linking fraction p and µ, λ 0 are related to the mesoscale and f d (n, ) that appear in S (d) BD (c). The challenge is then shifted to calculating |B p,n |. Using another parameter q which gives the cardinality of the upper layer as a further subclassification of B p,n , the leading order constribution to |B p,n | was found. The resulting partition function was then shown to be strictly suppressed when µλ 0 satisfy the condition 29 More appropriate to the quantum measure formulation, is using the decoherence func- is a causal set analog of the delta function associated with unitarity quantum theories.
This is an important analytic calculation and paves the way for a more rigourous understanding of the CST partition function. More than the partition function, however, is it the expectation value of observables or order invariants that is of physical significance. Evaluating this 30 for larger values of n is however a big challenge and we turn to numerical simulations to help us. One route could be to simply "perform" the sum above. However, given that |Ω n | grows super exponentially (to leading order it is ∼ 2 n 2 4 ), this is computationally challenging even for relatively small values of n. On the other hand, Markov Chain Monte Carlo (MCMC) methods for sampling the space Ω can be used if we can convert Z Ω into a statistical partition function.
In CST, there is no analogue of a Wick rotation: since the order relation derives from the causal structure, it cannot be "Euclideanised". On the other hand, there are other ways to analytically continue Z Ω (see Louko and Sorkin 1997 for a continuum example). One option, first explored in Surya (2012) is to introduce a new parameter β such that This allows us to analytically continue Z Ω,β from real to imaginary values of β, thus rendering the quantum partition function into a statistical partition function. We can then use standard tools in statistical physics, including MCMC methods, to find the expectation values of suitable observables (Surya 2012;Glaser and Surya 2016;Glaser et al 2018;Glaser 2018). In Henson et al (2017), MCMC methods were used to examine the sample space of naturally labelled posetsΩ n to determine the onset of the KR regime, using the uniform measure (β = 0). The Markov Chain was generated via a set of moves that sample Ω n . A mixture of two moves, the link move and the relation move, was used to obtain the quickest thermalisation. In the link move a pair of elements x, y are picked randomly and independently from the causal set c, and retained if l(x) < l(y). If x ≺ y and moreover the relation is a link, then the move is to "unlink" them. Those relations implied by this link via transitivity also need to be removed. These are relations between elements in IPast(x) and those in IFut(y) which are "mediated" solely either by x or y. On the other hand if x and y are not related, then one adds in a link between x and y, provided that there are no existing links between elements in IPast(x) and IFut(y), after which the transitive closure is taken. In the relation move, although the existence or non-existence of a link from x to y is also required, the move doesn't care about the sanctity of links, but is in other ways more restrictive. Thus, for both moves, picking of a pair of elements at random in c 30 We leave out interpretational questions! does not always lead to a possible move, let alone a probable one, and hence this MCMC model is slow to thermalise. Trying to find a more efficient move is however non-trivial precisely because of this issue with transitivity.
The simulations of Henson et al (2017) suggest that the onset of the asymptotic KR regime occurs for n as small as n ≈ 90. Ω n is very large even for n = 90 (∼ 2 90 2 !) and hence thermalisation starts to become a real issue. Recently, steps have been taken to incorporate the action (β = 0) into the measure, but again, because of thermalisation issues, the size of the posets are fairly small.
Instead of taking the full sample space, one can restrict Ω n to causal sets that capture some gross features of a class of spacetimes. As discussed above, for large enough n, Ω n contains causal sets that are approximated by spacetimes of arbitrary dimensions. It is thus of interest to restrict the sample space so that those causal sets that are manifold-like in the sample space are approximated only by spacetime regions of a given dimension. Such a restriction is typically hard to find, since it requires "tailoring" Ω using non-trivial order theoretic constraints determined by dimension estimators of the kind we have encountered in Sect. 4.
Somewhat fortuitously, this restriction is very natural in d = 2. Here, the sample space of "2d orders" Ω 2d is one in which the continuum dimension and the order theoretic dimension coincide (Brightwell et al 2008;El-Zahar and Sauer 1988;Winkler 1991). An n element 2d order is the intersection of two linear orders U = (u 1 , u 2 , . . . u n ) and V = (v 1 , v 2 , . . . v n ) where each u i and v i are valued on a set S n of n non-overlapping points in R. U and V can each be totally ordered by < in R. Their intersection is the poset Of special interest is the 2d order obtained from a Poisson sprinkling into a causal diamond, shown in Fig. 7 shows an example of a 2d order. It is obtained simply by choosing the entries of U and V from a fixed S n at random and independently. Interestingly, this random order in dominates Ω 2d in the large n limit as shown in El-Zahar and Sauer (1988); Winkler (1991), and grows as |Ω 2d | ∼ n!/2. Thus, unlike Ω n , the sample space is dominated by manifold-like causal sets, though it also contains causal sets that are distinctly non-manifold like. Hence this makes it an ideal starting point to study the (non-perturbative) quantum dynamics of causal sets. Moreover, as shown in Brightwell et al (2008), 2d orders also have trivial spatial homology in the sense of Major et al (2007) (see Sect. 4) and hence Ω 2d is the sample space of topologically trivial 2d causal set quantum gravity.
In Surya (2012), the partition function for Ω 2d was studied. Starting from where S 2d (c, ) is the BD action for d = 2, and non-locality parameter and taking β → iβ, Z 2d was studied using MCMC techniques. The MCMC move used by Surya (2012) in Ω 2d is very straighforward, unlike that in Ω n . A pair of elements is picked independently and at random in either U or V , and swapped. For example, if u i ↔ u j , then the elements (u i , v i ) and (u j , v j ) in U ∩ V are replaced by (u i = u j , v i = v i ) and (u j = u i , v j = v j ), hence changing the poset. Every move is possible, and hence one saves considerably on efficiency and thermalisation times.
The MCMC simulations of Surya (2012) give rise to a phase transition from a continuum phase at low β to a non-manifold phase at high β. This is very similar to the disordered to ordered phase in an Ising model. It was recently demonstrated by Glaser et al (2018) using finite size scaling arguments that that this is a first order phase transition. Varying with respect to the non-locality parameter = l 2 p /l 2 ∈ (0, 1] one finds a β 2 versus phase diagram which indicates that the continuum phase will survive the analytic continuation for any value of . The analysis of Glaser et al (2018) also suggests that the continuum phase corresponds to a spacetime with negative cosmological constant. This is an explicit example of a non-perturbative theory of quantum gravity in which the cosmological constant is generated via the dynamics.
The Hartle-Hawking wave function in 2d CST was constructed in Glaser and Surya (2016) using the natural no-boundary condition for causal sets, namely that that there is a single element e 0 to the past of all the other elements. ψ HH (A f ) is the wave function for a final antichain of cardinality |A f |, where one is summing over all causal sets that have an initial element e 0 and final boundary A f . In 2d CST this can be calculated explictly using MCMC methods to generate the expectation value of the action S 2d from which the partition function can be calculated and hence ψ HH (A f ), upto normalisation. The normalisation in turn was determined using a combination of analytic and numerical calculations. It was shown that ψ HH (A f ) peaks at low β on small antichains and this peak jumps at higher β to large A f . Indeed, the latter, low temperature phase is intriguing, since the dominant causal sets satisfy some rudimentary requirements of early universe cosmology. Namely, (a) the growth from a single element to a large antichain takes place rapidly and moreover (b) that each element in A f are all maximally causally related in their immediate past and hence A f is homogeneous. In other words, the "initial condition" of homogeneity is caused by past causal contact. However, this is a non manifold-like phase, and it is an open question how one exits this phase into a manifold like phase. If there is a mechanism to make β small, then this would imply the latter.
Will this analysis survive higher dimensions? One of the main issues at hand is that even for Ω 2d the space grows rapidly with n and hence thermalisation does become a major stumbling block. The finite sized scaling analysis of Glaser et al (2018) suggests that even for small n ∼ 80 one is in the asymptotic regime and hence the results of Surya (2012) are robust. Nevertheless, simulations in d = 4 will require far more extensive computational power to reach the asymptotic regime. Recently, using more sophisticated computational techniques (Cunningham 2018b), the algorithms of Surya (2012) have been updated and one can do MCMC simulations with n ∼ 300 in a reasonable time.
An important question, however is how to obtain a dimensionally restricted Ω n more generally. While 2d orders give rise to causal sets approximated by 2d spacetimes, this is not true in higher dimensions. A d-order is the intersection of d linear orders. However for d > 2 these correspond to "light-cubes" rather than lightcones, and are hence not particularly useful. On the other hand, a lattice inspired method is currently being used to generate sample spaces Ω L which are both dimensionally and topologically restricted and include manifold-like causal sets. In d = 2, the simplest example comes from causal sets obtained from sprinkling into the cylinder spacetime. Recent simulations (Cunningham and Surya, work in progress) suggest that the results of the topologically trivial case are largely unchanged, which is itself very interesting. The next step is to include topology change into the model, and hence bring it closer to a full 2d theory of quantum gravity.
Of course, 2d causal set quantum gravity without matter does not have a continuum counterpart, since in the continuum gravity is coupled to a scalar field, to give Liouville gravity. Studying 2d CST with matter is therefore an open interesting question. In Glaser (2018), Ising spins were coupled to the causal set by placing a spin s i = ±1 at every element e i and coupling spins along the links, i.e., where L ik is the link matrix and j the spin coupling constant. The phase structure of this model coupled to the BD action is substantially richer. In particular, the hope is that some of the resulting phase transitions are of higher order and hence comparisons with conformal field theories might be possible.
Further analysis of this model would definitely be useful and interesting. This is the first example of coupling causal set to matter in MCMC simulations.
In the MCMC simulations, labelled posets are used for practical reasons, since this is how they are stored on the computer. While a single poset admits many labellings the number of relabellings per poset depends on its automorphisms, i.e., relabellings that yield the same labelled poset. In Fig. 20, we have an example of a 3-element causal set with three distinct natural labellelings, while all others admit only one. Thus, the number of automorphisms varies from causal set to causal set. Enumerating the number of automorphisms quickly becomes impossible as n increases. A modification of the measure to accommodate this degeneracy due to relabellings is therefore highly nontrivial to construct. We therefore choose the most convenient measure: one that is uniform onΩ n , rather than on the unlabelled sample space Ω n . Thus, causal sets that admit more relabellings come with a higher natural weight than those that admit fewer relabellings. However, covariance is not compromised since the observables themselves are label independent and hence covariant. In the continuum path integral, it is a non-trivial exercise to find the "correct" measure especially in a gauge theory, with the measure being modified by the volume of the gauge orbits. In this discrete setting, the analogous gauge orbits correspond to the automorphisms and are not of the same cardinality for each c ∈Ω n . In this sense it is more natural to use the uniform measure, which naturally incorporates a weighted measure on the unlabelled space Ω n . Indeed, in the CSG models, the labelling is related to temporality and the choice of uniform measure onΩ g is a natural one.
While these numerical simulations have uncovered a wealth of information about the statistical thermodynamics of causal sets, one must pause to ask how this is related to the quantum dynamics. How are Z iβ and O iβ related to to Z β and O β . Should one assume analyticity in β to show that they are related? There is, for example no analogue of the Osterwalder-Schrader theorems to protect the real β results. Pursuing these questions further is important, though finding definitive and rigourous answers is perhaps beyond the scope of our present understanding of CST. A hope is that one can ultimately use the techniques developed to calculate decoherence functionals and hence the full quantum measure. While explicit summations in the path integral are difficult to carry out practically, it is possible that the MCMC thermalisation provides a clue at least about the quantum gravity ground state.

Phenomenology
While the deep realm of quantum gravity is extremely well shielded from experimental probes in the forseeable future, it is possible that certain properties of quantum gravity can "leak" into observationally accessible regimes. This is the reason for the push, in the last couple of decades, for exploring quantum gravity phenomenology. Without a full theory of quantum gravity, of course there is little hope that any phenomenology is entirely believable, since it requires assumptions about an incomplete theory. Nevertheless, quantum gravity phenomenology can be useful in setting realistic bounds on these leaked out properties, and hence constrain theories of quantum gravity, albeit weakly. Models of quantum gravity phenomenlogy moreover use distilled properties of the theory to build reasonable models that can be tested. Some of these properties are unique to a given approach.
In CST spacetime discreteness takes a special form and brings with it a certain non-locality that can affect observable physics. We have already encountered the possibility of voids in Sect. 3 as well as the propagation of scalar fields from distance sources in Sect. 5. The continuum approximation of CST is Lorentz invariant and consistent with stringent observational bounds as summarised in Liberati and Mattingly (2016). In addition, as suggested by Dowker et al (2004), there is the possibility of generating very high energies particles through long time diffusion in momentum space. This arises from the randomness of CST discreteness, which cause particles to "swerve", or suddenly change their momentum, as they traverse the causal set underlying our universe (Philpott et al 2009;Contaldi et al 2010). This spacetime Brownian motion was calculated in M d and can be constrained by observations (Kaloper and Mattingly 2006), but an open question is how to extend the calculation to our FRW universe.
There have been some very interesting recent ideas by Belenchia et al (2016b) for testing CST type non-locality via its effect on propagation in the continuum using the d'Alembertian operator. Belenchia et al (2015) have looked at the associated quantum field theory which contain critical instablities. These can be removed by modifying the d'Alembertian, but the relationship to CST is unclear. Saravani and Afshordi (2017) have proposed a candidate for dark matter as off-shell modes of the non-local CST d'Alembertian. This is an exciting proposal and needs to be investigated in more detail.
We will not review these very interesting ideas on CST phenomenology here, except one, namely the prediction of Λ.

The 1987 prediction for Λ
One of the most outstanding questions in theoretical physics is understanding the origin of "dark energy" which observationally has been seen to make up ∼ 70% of the total energy of the universe. The current observational value is ∼ 2.888 × 10 −122 in Planck units. Quantum field theory predictions for dark energy interpreted as the energy of vacuum fluctutions of quantum fields on the other hand gives a value ∼ 1 in Planck units. The gross conflict with observation obviously implies that this cannot be the source of Λ. 31 In light of this conundrum, the CST prediction for Λ due to Sorkin (1991) is startling in its simplicity and accuracy, especially since it was made several years before the 1998 observation. One begins with the framework of unimodular gravity (Sorkin 1997;Unruh and Wald 1989) in which the spacetime volume element is fixed. Λ then appears as a Lagrange multiplier in the action, with Λ dV = ΛV = constant, for any finite spacetime region of volume V . In a canonical formulation of the theory, therefore Λ and V are conjugate to each other, so that on quantisation there is an uncertainity relation ∆V ∆Λ ∼ 1.
Using the fact that ∆V is generated from Poisson fluctuations of the underlying causal set ensemble ∆V ∼ √ V .
Assuming Λ = 0, where H is the Hubble constant. If V is taken to be the volume of the visible universe Λ = ∆Λ ∼ 10 −120 , in Planck units. This is very close to the subsequently observed value of Λ! Importantly, the prediction also states that that under these assumptions, Λ always tracks the critical density. This argument is general and requires two important ingredients: (i) the assumption of unimodularity and hence the conjugacy between Λ and V , and (ii) that the fluctuation in V is Poisson, with V ∼ n, i.e., the number to volume correspondence. While (i) can be motivated by a wide range of theories of quantum gravity, (ii) is distinctive to causal set theory. No other discrete approach to quantum gravity makes the n ∼ V correspondence at a fundamental level and incorporates Poisson fluctuations kinematically in the continuum approximation. Quoting from Sorkin (1991), "Fluctuations in Λ arise as residual nonlocal quantum effects of spacetime discreteness". Interestingly, as shown by Sorkin (2005a), if spacetime admits large extra directions, then the contribution to V is very different and gives the wrong answer for ∆Λ.
Of course, an important question that arises in this quick calculation is why we should assume that Λ = 0. 32 The answer to this may well lie in the full and as yet unknown quantum dynamics. Nevertheless, phenomenologically this assumption leads to further predictions that can already be tested. The first conclusion is that a fluctuating Λ must violate conservation of the stress energy tensor, and hence the Einstein field equations.
In Ahmed et al (2004), a dynamical model for generating fluctuations of Λ was constructed, starting with the flat k = 0 FRW spacetime. In order to accommodate a fluctuating Λ, one of the two Friedmann equations must be dropped. In Ahmed et al (2004), the Friedmann equation was retained, 33 with and Λ modeled as a stochastic function of V , such that 32 In Samuel and Sinha (2006), a very striking analogy was made between a fluctuating Λ and the surface tension T of a fluid membrane. In addition, using the atomicity of the model, the mean value of T was shown to be zero, with a suggestion of how this might extend to CST. 33 Subsequently, more general "mixed equation" models were examined in Ahmed and Sorkin (2013), which indicate that the results of Ahmed et al (2004) are robust to these modifications.
More generally, Λ can be thought of as the action S per unit volume, which for causal sets means that Λ ∼ S/V . A very simple stochastic dynamics is then generated by assuming that every element contributes ± to S, so that where = √ κ). One then gets the integro-differential equations is the volume of the entire causal past of an event in the FRW spacetime. The stochastic equation is then generated as follows. At the i th step one has the variables a i (scale factor), N i , V i , S i and Λ i . The scale factor is updated using the discrete Friedmann equation a i+1 = a i + a i ρ+Λ 3 (τ i+1 − τ i ), from which V i = V (τ ) can be calculated and thence N i+1 = V i+1 / 4 . The action is then updated via S i+1 = S i + α ξ N i+1 − N i , where ξ is a Gaussian random variable, with ∆ξ = 1, and α is a tunable free parameter which controls the magnitude of the fluctuations. Finally, Λ i+1 = S i+1 /V i+1 , with S 0 = 0. It was shown in Ahmed et al (2004) that in order to be consistent with astrophysical observations, 0.01 < α < 0.02. The results of simulations moreover suggest that Λ is "everpresent" and tracks the energy density of the universe.
This model assumes spatial homogeneity and it is important to check how inhomogeneities affect these results. In Barrow (2007) and Zuntz (2008), inhomogeneities were modelled simplistically by taking Λ(x µ ), such that ∆Λ(x) is dependent only on Λ(y) for y ∈ J − (x). This would mean that well separated patches in the CMB sky would contain uncorrelated fluctuations in Ω Λ , which in turn are strongly constrained to < 10 −6 by observations and hence insufficient to account for Λ. In Ahmed et al (2004) and Zwane et al (2018), it was suggested that quantum Bell correlations may be a possible way to induce correlations in the CMB sky. However, incorporating inhomogeneities into the dynamics in a systematic way remains an important open question.
In Zwane et al (2018), a phenomenological model was adopted which uses the homogeneous temporal fluctuations in Λ coming from everpresent Λ to model a quintessence type spatially inhomogeneous scalar field with a potential term that varies from realization to realization. Using MCMC methods to sample the cosmological parameter space, as well as varying over stochastic realisations of everpresent Λ models, it was shown that Everpresent Λ agrees with the observations as well as ΛCDM models and in fact does better for the Baryonic Acoustic Oscillations (BAO) measurements. The very extensive and detailed analysis of Zwane et al (2018) sets the stage for direct comparisions with future observations and heralds an exciting phase of quantum gravity phenomenology.

Outlook
CST has come a long way in the last three decades, despite the fact that there are few dedicated practitioners. Over the last decade, in particular, there has been a growith of interest with inputs coming in from the wider quantum gravity community. This is heartening, since an extensive exploration of the theory is required in order to make significant progress. It is our hope that this review will spark the interest of the larger quantum gravity community.
We have in this review touched upon several open questions, many of which are challenging but some of which are straightforward to carry out. We will not summarise these but just pick two that are of utmost importance. One is the the pursuit of inhomogeneous models of everpresent Λ which can be tested against the most recent observations. The second, on the other side of the quantum gravity spectrum, is the construction from first principles of a viable quantum dynamics for causal sets. In between these two ends lie myriad interesting questions. We invite you to join us.