The causal set approach to quantum gravity
Abstract
The causal set theory (CST) approach to quantum gravity postulates that at the most fundamental level, spacetime is discrete, with the spacetime continuum replaced by locally finite posets or “causal sets”. The partial order on a causal set represents a protocausality relation while local finiteness encodes an intrinsic discreteness. In the continuum approximation the former corresponds to the spacetime causality relation and the latter to a fundamental spacetime atomicity, so that finite volume regions in the continuum contain only a finite number of causal set elements. CST is deeply rooted in the Lorentzian character of spacetime, where a primary role is played by the causal structure poset. Importantly, the assumption of a fundamental discreteness in CST does not violate local Lorentz invariance in the continuum approximation. On the other hand, the combination of discreteness and Lorentz invariance gives rise to a characteristic nonlocality which distinguishes CST from most other approaches to quantum gravity. In this review we give a broad, semipedagogical 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.
Keywords
Causal set theory Quantum gravity Discreteness Causality Poset theory1 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 (Bombelli et al. 1987). In CST, the space of Lorentzian geometries is replaced by the set of locally finite posets, or causal sets. These causal sets encode the twin principles of causality and discreteness. In the continuum approximation of CST, where elements of the causal set set represent spacetime events, the order relation on the causal set corresponds to the spacetime causal order and the cardinality of an “order interval” to the spacetime volume of the associated causal interval.
This review is intended as a semipedagogical 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) and Robb (1914, 1936), and somewhat later by Zeeman (1964), Kronheimer and Penrose (1967), Finkelstein (1969), Hemion (1988) and Myrheim (1978), culminating in the CST proposal of BLMS (Bombelli et al. 1987). 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 spacetime causality is equivalent to the order relation and finite spacetime volumes to cardinality. Not all causal sets have a continuum approximation—in fact we will see that most do not. Those that do are referred to as manifoldlike. Important to CST is its “Hauptvermutung” or fundamental conjecture, which roughly states that a manifoldlike causal set is equivalent to the continuum spacetime, modulo differences up to 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 growing body 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 nonlocality, which distinguishes CST from other discrete approaches. Following Sorkin (1997), we then discuss how the twin principles behind 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 manifoldlike causal set. These are functions on the causal set that are independent of the labelling or ordering of the elements in the causal set. Finding the appropriate order invariants that correspond to manifold invariants can be challenging, since there is little in the mathematics literature which correlates order theory to Lorentzian geometry via the CST continuum approximation. Extracting such invariants requires new technical tools and insights, sometimes requiring a rethink of familiar aspects of continuum Lorentzian geometry. We will describe some of the progress made in this direction over the years (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. 2019; Eichhorn et al. 2019a). The correlation between order invariants and manifold invariants in the continuum approximation lends support for the Hauptvermutung and simultaneously establishes weaker, observabledependent versions of the conjecture.
Somewhere between dynamics and kinematics is the study of quantum fields on manifoldlike causal sets, which we describe in Sect. 5. The simplest system is free scalar field theory on a causal set approximated by ddimensional Minkowski spacetime \({\mathbb {M}}^d\). 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 functions for a free scalar field theory since it is defined using the causal structure of the spacetime. The explicit form for these Green functions were found for causal sets approximated by \({\mathbb {M}}^d\) for \(d=2,4\) (Daughton 1993; Johnston 2008, 2010) as well as de Sitter spacetime (Dowker et al. 2017). 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 discrete scalar field vacuum, starting from the covariantly defined Peierls’ bracket formulation of quantum field theory. Subsequently Sorkin (2011a) showed that the construction is also valid in the continuum, and can be used to give an alternative definition of the quantum field theory vacuum. This Sorkin–Johnston (SJ) vacuum provides a new insight into quantum field theory and 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 (Saravani et al. 2014; Sorkin and Yazdi 2018). The calculation in \(d=2\) is surprising since it gives rise to a volume law rather than an area law. What this means for causal set entanglement entropy is still an open question.
In Sect. 6, we describe the CST approach to quantum dynamics, which roughly follows two directions. The first, is based on “first principles”, 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 (CSG) models of Rideout and Sorkin (Rideout and Sorkin 2000a, 2001; Martin et al. 2001; Rideout 2001; Varadarajan and Rideout 2006), which we will describe in some detail. The dynamical framework finds a natural expression in terms of measure theory, with the classical covariant observables represented by a covariant event algebra \({\mathfrak {A}}\) over the sample space \(\varOmega _g\) 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 Kleitman–Rothschild (KR) posets in \(\varOmega _g\) can be overcome by the dynamics (Kleitman and Rothschild 1975). These KR posets are highly nonmanifoldlike and “static”, with just three “moments of time”. Hence, if the continuum approximation is to emerge in the classical limit of the theory, then the entropic contribution from the KR posets should be suppressed by the dynamics in this limit. In the CSG models, the typical causal sets generated are very “tall” with countable rather than finite moments of time and, though not quite manifoldlike, are very unlike the KR posets or even the subleading entropic contributions from nonmanifoldlike causal sets (Dhar 1978, 1980). The CSG models have generated some interest in the mathematics community, and new mathematical tools are now being used to study the asymptotic structure of the theory (Brightwell and Georgiou 2010; Brightwell and Luczak 2011, 2012, 2015).
In CST, the appropriate route to quantisation is via the quantum measure or decoherence functional defined in the doublepath integral formulation (Sorkin 1994, 1995, 2007d). In the quantum versions of the CSG (quantum sequential growth or 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 nontriviality of quantisation comes from a prosaic generalisation of transitive percolation, which is the simplest of the CSG models. In this “complex percolation” dynamics, however, 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), but the search is on to find a quantum dynamics for which the measure does extend. 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 GhaziTabatabai 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 GhaziTabatabai 2008), to make a limited set of predictions in complex percolation (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 \(\varOmega \) of causal sets, using the Benincasa–Dowker (BD) action, which goes over to the Einstein–Hilbert action (Benincasa and Dowker 2010) in the continuum limit. This can be viewed as an effective, continuumlike dynamics, arising from the more fundamental dynamics described above. A recent analytic calculation in Loomis and Carlip (2018) showed that a subdominant class of nonmanifoldlike causal sets, the bilayer posets, are suppressed in the path integral when using the BD action, under certain dimension dependent conditions satisfied by the parameter space. This gives hope that such an effective dynamics might be able to overcome the entropy of the nonmanifoldlike 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 \(\varOmega _{2d}\) of 2orders, which corresponds to topologically trivial \(d=2\) causal set 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, while retaining the Lorentzian character of the theory. This theory exhibits a first order phase transition (Surya 2012; Glaser et al. 2018) between a manifoldlike phase and a layered, nonmanifoldlike one. MCMC methods have also been used to examine the sample space \(\varOmega _n\) of nelement 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 nontrivial \(d=2\) and \(d=3\) CST (Cunningham and Surya 2019). While this approach gives us expectation values of covariant observables which allows for a straightforward interpretation, relating it to the complex or 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 nonlocality 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 \(\varLambda \) (Sorkin 1991, 1997). While the original argument was a kinematic estimate, dynamical models of fluctuating \(\varLambda \) were subsequently 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.
Finally, since this is an extensive review, in order to assist the reader we have made a list of some of the key definitions, as well as the abbreviations in Appendix A.
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 exist with differing emphasis (Sorkin 1991, 2005b; Henson 2006b, 2010; Dowker 2005; Sorkin 2009; Wallden 2013), some of which have an in depth discussion of the conceptual underpinnings of CST. 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. For more technical details, the reader is urged to look at the original references.
2 A historical perspective
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 \((M,\prec )\) 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, \((M,\prec )\) has also been paired with discreteness, though the earliest ideas on discreteness go back to prequantum and prerelativistic physics. We now give a broad review of this history.
The first few decades after the formulation of General Relativity were dedicated to understanding the physical implications of the theory and to finding solutions to the field equations. The attitude towards Lorentzian geometry was mostly practical: it was seen as a simple, though odd, generalisation of Riemannian geometry.^{2} There were however early attempts to understand this new geometry and to use causality as a starting point. Weyl and Lorentz (see Bell and Korté 2016) used light rays to attempt a reconstruction of d dimensional Minkowski spacetime \({\mathbb {M}}^d\), while Robb (1914, 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 vigorously.
Kronheimer and Penrose (1967) subsequently generalised Zeeman’s ideas to an arbitrary causal spacetime (M, g) where they identified both \((M,\prec )\) and \((M,{\prec \! \prec })\) with the eventset M, devoid of the differential and topological structures associated with a spacetime. They defined an abstract causal space axiomatically, using both \((M,\prec )\) and \((M,{\prec \! \prec })\) along with a mixed transitivity condition between the relations \(\prec \) and \({\prec \! \prec }\), which mimics that in a causal spacetime.
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”. Subsequently Parrikar and Surya (2011) showed that the causal structure poset \((M,\prec )\) of these spacetimes also contains information about the spacetime dimension.
Thus, the causal structure poset \((M,\prec )\) of a future and past distinguishing spacetime is equivalent its conformal geometry. This means that \((M,\prec )\) is equivalent to the spacetime, except for the local volume element encoded in the conformal factor \(\lambda \), which is a single scalar. As phrased by Finkelstein (1969), the causal structure in \(d=4\) is therefore \(\left( 9/10\right) {\mathrm {th}}\) of the metric!
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):To admit structures which can be very different from a manifold. The possibility arises, for example, of a locally countable or discrete eventspace equipped with causal relations macroscopically similar to those of a spacetime continuum.
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 \({\mathbf {A}}[p,q]\equiv I^+(p)\cap I^(q)\).Now it seems that the empirical 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.
Although 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 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).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, spacetime points, events, might serve as the one and only fundamental concept of a discrete spacetime geometry. In this view spacetime is nothing but the causal ordering of events.
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 cutoff 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 (Bombelli et al. 1987). The early work shed considerable light on key aspects of the theory (Bombelli et al. 1987; Bombelli and Meyer 1989; Brightwell and Gregory 1991) and resulted in Sorkin’s prediction of the cosmological constant \(\varLambda \) (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 and Sorkin 2000b, 2001; Martin et al. 2001; Rideout 2001). There have been several nontrivial 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.
3 The causal set hypothesis
We begin with the definition of a causal set:
Definition
 1.
Acyclic: \(x\prec y\) and \(y \prec x \) \(\Rightarrow x=y\), \(\forall x,y \in C\)
 2.
Transitive: \(x\prec y\) and \(y \prec z \) \(\Rightarrow x \prec z\), \(\forall x,y,z \in C\)
 3.
Locally finite: \(\forall x,y \in C\), \({\mathbf {I}}[x,y]< \infty \), where \({\mathbf {I}}[x,y]\equiv \mathrm {Fut}(x) \cap \mathrm {Past}(y)\) ,
 1.
Quantum gravity is a quantum theory of causal sets.
 2.A continuum spacetime (M, g) is an approximation of an underlying causal set \(C \sim (M,g)\), where
 (a)
Order \(\sim \) Causal Order
 (b)
Number \(\sim \) Spacetime Volume
 (a)
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) and Hemion (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. In particular, 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 molecularlevel 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 cutoff.^{7}
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 further on.
In a Poisson sprinkling into a spacetime (M, g) at density \(\rho _c\) one selects points in (M, g) uniformly at random 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 \({\mathcal {C}}(M,\rho _c)\), via the Poisson sprinkling.^{9} Each causal set thus obtained is a realisation, while any averaging is done over the whole ensemble.
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. (8). Let \(\varPhi : C \hookrightarrow (M,g)\) be a faithful embedding at density \(\rho _c\). While the set of all finite volume regions^{10} v possess on average \(\langle {{\mathbf {n}}} \rangle =\rho _c v\) elements of C,^{11} the Poisson fluctuations are given by \(\delta n =\sqrt{n}\). 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 is allowed, 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 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. We summarise this discussion. Let \(C \sim (M,g)\) at density \(\rho _c\). Consider k nonoverlapping Alexandrov intervals of volume V in (M, g). Since C is uniformly distributed, \(\langle {{\mathbf {n}}} \rangle = \rho _c V\). The most optimal choice of distribution, is also one in which the fluctuations \(\delta {\mathbf {n}}/\langle {{\mathbf {n}}} \rangle =\sqrt{\langle {({\mathbf {n}}\langle {{\mathbf {n}}} \rangle )^2} \rangle }/{\langle {{\mathbf {n}}} \rangle }\) are minimised. This ensures that C is as close to the continuum as possible. For the Poisson distribution \(\delta {\mathbf {n}}/\langle {{\mathbf {n}}} \rangle = 1/\sqrt{\langle {{\mathbf {n}}} \rangle } = 1/\sqrt{\rho _c V}\). Is this as good as it gets? It was shown by Saravani and Aslanbeigi (2014) that for \(d>2\), and under certain further technical assumptions, the Poisson distribution indeed does the best job. Strengthening these results is important as it can improve our understanding of the continuum approximation.
3.1 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 \(\rho _c\) into two different spacetimes, (M, g) and \((M',g')\)? We expect that this is the case if (M, g) and \( (M',g')\) differ on scales smaller than \(\rho _c\), or that they are, in an appropriate sense, “close” \((M,g) \sim (M',g')\). Let us assume that a causal set can be identified with two macroscopically distinct spacetimes at the same density \(\rho _c\). Should this be interpreted as a hidden duality between these spacetimes, as is the case for example for 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 to ensure that C contains all the information in (M, g) at scales above \(\rho _c^{1}\). Macroscopic nonuniqueness 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 \(\rho _c\) into two distinct spacetimes, (M, g) and \((M',g')\) iff they are approximately isometric.
Condition (1) tells us that the kinematic space of Lorentzian geometries must be replaced by a sample space \(\varOmega \) of causal sets. Let \(\varOmega \) be the set of all countable causal sets and \({\mathcal {H}}\) the set of all possible Lorentzian geometries, in all dimensions. If \(\, \sim \, \) denotes the approximate isometry at a given \(\rho _c\), as discussed above, the quotient space \({\mathcal {H}}/\!\!\sim \) corresponds to the set of all continuumlike causal sets \(\varOmega _{\mathrm{cont}}\subset \varOmega \) at that \(\rho _c\). Thus, causal sets in \(\varOmega \) correspond to Lorentzian geometries of all dimensions! Couched this way, we see that CST dynamics has the daunting task of not only obtaining manifoldlike 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 \(\varOmega _n\) is dominated by the KR posets depicted in Fig. 9 and are hence very nonmanifoldlike (Kleitman and Rothschild 1975). 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. The number of KR posets grows as \(\sim \, 2^{{n^2}/{4}}\) and hence must play a role in the deep quantum regime. Since they are nonmanifoldlike they pose a challenge to the dynamics, which must overcome their entropic dominance in the classical limit of the theory. Even if the entropy from these KR posets is suppressed by an appropriate choice of dynamics, however, there is a subdominant hierarchy of nonmanifoldlike posets (also layered) which also need to be reckoned with (Dhar 1978, 1980; Promel et al. 2001).
Closely tied to the continuum approximation is the notion of “coarse graining”. Given a spacetime (M, g) the set \({\mathcal {C}}(M,\rho _c)\) can be obtained for different values of \(\rho _c\). Given a causal set C which faithfully embeds into (M, g) at \(\rho _c\), one can then coarse grain it to a smaller subcausal set \(C' \subset C\) which faithfully embeds into (M, g) at \(\rho _c' <\rho _c\). A natural coarse graining would be via a random selection of elements in C such that for every n elements of C roughly \(n'=(\rho _c'/\rho _c) n\) elements are chosen. Even if C itself does not faithfully embed into (M, g) at \(\rho _c\), it is possible that a coarse graining of C can be faithfully embedded. This would be in keeping with our sense in CST that the deep quantum regime need not be manifoldlike. One can also envisage manifoldlike causal sets with a regular fixed latticelike 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) based on renormalisation techniques.
3.2 Discreteness without Lorentz breaking
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 preserved on average in a random lattice. In Bombelli et al. (2009), it was shown that a causal set in \({\mathcal {C}}({\mathbb {M}}^d,\rho _c)\) not only preserves Lorentz invariance on average, but in every realisation, with respect to the Poisson distribution. Thus, in a very specific sense a manifoldlike 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 \({\mathcal {P}}= ({\mathbb {R}}^2,\delta _{ab})\), and let \(\varPhi : {\mathcal {C}}({\mathcal {P}},\rho _c) \hookrightarrow {\mathcal {P}}\) be the natural embedding map, where \({\mathcal {C}}({\mathcal {P}},\rho _c)\) denotes the ensemble of Poisson sprinklings into \({\mathcal {P}}\) at density \(\rho _c\). A rotation \(r \in SO(2)\) about a point \(p \in {\mathcal {P}}\), induces a map \(r^* : {\mathcal {C}}({\mathcal {P}},\rho _c) \rightarrow {\mathcal {C}}({\mathcal {P}},\rho _c)\), where \(r^*=\varPhi ^{1}\circ r \circ \varPhi \) and similarly a translation t in \({\mathcal {P}}\) induces the map \(t^*: {\mathcal {C}}({\mathcal {P}},\rho _c) \rightarrow {\mathcal {C}}({\mathcal {P}},\rho _c)\). The action of the Euclidean group is clearly not transitive on \({\mathcal {C}}({\mathcal {P}},\rho _c)\) but has nontrivial orbits which provide a fibration of \({\mathcal {C}}({\mathcal {P}},\rho _c)\). Thus the ensemble \({\mathcal {C}}({\mathcal {P}},\rho _c)\) 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 \( P \in {\mathcal {C}}({\mathcal {P}},\rho _c)\). Fixing an element \(e \in \varPhi (P)\), we define a direction \(\mathbf {d}\in S^1\), the space of unit vectors in \({\mathcal {P}}\) centred at e. Under a rotation r about e, \(\mathbf {d}\rightarrow r^*(\mathbf {d})\in S^1\). In general, we want a rule that assigns a natural direction to every \(P \in {\mathcal {C}}({\mathcal {P}},\rho _c)\). One simple choice is to find the closest element to e in \(\varPhi (P)\), 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” \({\mathbf {D}}_e: {\mathcal {C}}({\mathcal {P}},\rho _c) \rightarrow S^1\) for a fixed \(e \in \varPhi (P)\) consistent with the rotation map, i.e., \({\mathbf {D}}_e\) commutes with any \(r\in SO(2)\), or is equivariant.
Associated with \({\mathcal {C}}({\mathcal {P}},\rho _c)\), is a probability distribution \(\mu \) arising from the Poisson sprinkling which associates with every measurable set \(\alpha \) in \({\mathcal {C}}({\mathcal {P}},\rho _c)\) a probability \(\mu (\alpha ) \in [0,1]\). The Poisson distribution being volume preserving (Stoyan et al. 1995), the measure on \({\mathcal {C}}({\mathcal {P}},\rho _c)\) moreover must be independent of the action of the Euclidean group on \({\mathcal {C}}({\mathcal {P}},\rho _c)\), i.e.: \(\mu \circ r =\mu \).
However, this is not the case for the space of sprinklings \({\mathcal {C}}({{\mathbb {M}}^d},\rho _c)\) into \({\mathbb {M}}^d\), where the hyperboloid \({\mathcal {H}}^{d1}\) now denotes the space of future directed unit vectors and is invariant under the Lorentz group \(SO(n1,1)\) about a fixed point \(p\in {\mathbb {M}}^{d1}\) (see Fig. 10). To begin with, there is no “natural” direction map. Let \(C \in {\mathcal {C}}({\mathbb {M}}^d,\rho _c)\). To find an element which is closest to some fixed \(e \in \varPhi (C)\), one has to take the infimum over \(J^+(e)\), or some suitable Lorentz invariant subset of it, which being noncompact, does not exist. Assume that some measurable direction map \(D: \varOmega _{{\mathbb {M}}^d} \rightarrow {\mathcal {H}}^{d1}\), does exist. Then the above arguments imply that \(\mu _D\) must be invariant under Lorentz boosts. The action of successive Lorentz transformations \(\varLambda \) can take a given measurable set \(h \in {\mathcal {H}}^{d1}\) to an infinite number of copies that are nonoverlapping, and of the same measure. Since \({\mathcal {H}}^{d1}\) is noncompact, this is not possible unless each set is of measure zero, but since this is true for any measurable set h and we require \(\mu _D({\mathcal {H}}^{d1})=1\), this is a contradiction. This proves the following theorem (Bombelli et al. 2009):
Theorem 2
In other words, even for a given sprinkling \(\omega \in \varOmega _{{\mathbb {M}}^d}\) it is not possible to consistently pick a direction in \({\mathcal {H}}^{d1}\). Consistency means that under a boost \(\varLambda : \omega \rightarrow \varLambda \circ w\), and hence \(D(\omega ) \rightarrow \varLambda \circ D(\omega ) \in {\mathcal {H}}^{d1}\). Crucial to this argument is the use of the Poisson distribution.^{12} Thus, an important prediction of CST is local Lorentz invariance. Tests of Lorentz invariance over the last couple of decades have produced an evertightening bound, which is therefore consistent with CST (Liberati and Mattingly 2016).
3.3 Forks in the road: what makes CST so “different”?
In many ways CST does not 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 nonlocality means that the apparatus of local physics is not readily available 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 CSTlike theory, while choosing the Lorentzian–Hamiltoniancontinuum 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 \(M=\varSigma \times {\mathbb {R}}\), where \(\varSigma \) is a Cauchy hypersurface, on which are defined the canonical phase space variables which capture the intrinsic and extrinsic geometry of \(\varSigma \).
Before moving on, we comment on the condition of local finiteness which, as we have pointed out, provides an intrinsic definition of spacetime discreteness, which does not need a continuum approximation. An alternative definition would be for the causal set to be countable, which along with the continuum approximation is sufficient to ensure the number to volume correspondence. This includes causal sets with order intervals of infinite cardinality, and allows us to extend causal set discretisation to more general spacetimes, like anti de Sitter spacetimes, where there exist events p, q in the spacetime for which \(\mathrm {vol}({\mathbf {A}}[p,q])\) is not finite. However, what is ultimately of interest is the dynamics, and in particular, the sample space \(\varOmega \) of causal sets. In the growth models we will encounter in Sects. 6.1, 6.2 and 6.3 the sample space consists of past finite posets, while in the continuuminspired dynamics of Sect. 6.4 it consists of finite element posets. Thus, while countable posets may be relevant to a broader framework in which to study the dynamics of causal sets, it suffices for the present to focus on locally finite posets.
4 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 rigorous 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 function, 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 2002, 2004; Bombelli 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 \({\mathcal {O}}\) with the manifold invariant \({\mathcal {G}}\) then ensures that a causal set C that faithfully embeds into (M, g) cannot faithfully embed into a spacetime with a different manifold invariant \({\mathcal {G}}'\).^{13} 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, orderinvariant version of the Hauptvermutung:
\({\mathcal {O}}\) Hauptvermutung: If C faithfully embeds into (M, g) and \((M',g')\) then (M, g) and \((M',g')\) have the same manifold invariant \({\mathcal {G}}\) associated with \({\mathcal {O}}\).
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 \({\mathcal {G}}\) with an order invariant \({\mathcal {O}}\), 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 correspond 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: \(C \rightarrow {\mathbb {N}}\), 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 \({\mathcal {O}}: C \rightarrow {\mathbb {R}}\) such that \({\mathcal {O}}\) is independent of the labelling of C. For a manifoldlike causal set^{14} \(C \in {\mathcal {C}}(M,\rho _c)\), associated to every order invariant \({\mathcal {O}}\) is the random variable \({\mathbf {O}}\) whose expectation value \(\langle {{\mathbf {O}}} \rangle \) in the ensemble \({\mathcal {C}}(M,\rho _c)\) is either equal to or limits (in the large \(\rho _c\) limit) to a manifold invariant \({\mathcal {G}}\) of (M, g). We will typically restrict to compact regions of (M, g) in order to deal with finite values of \({\mathbf {O}}\).
The first candidates for geometric order invariants were defined for \({\mathcal {C}}({\mathbf {A}}[p,q],\rho _c)\) where \({\mathbf {A}}[p,q]\) is an Alexandrov interval in \({\mathbb {M}}^d\). Some of these have been later generalised to Alexandrov intervals (or causal diamonds) in Riemann 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 manifoldlikeness 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 manifoldlike 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, manifoldlike 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, \(\rho _c\rightarrow \infty \). Strictly speaking, this limit is unphysical in CST because of the assumption of a fundamental discreteness. There are fluctuations at finite \(\rho _c\) 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 \(\rho _c\). As we will see below, CST kinematics therefore needs a combination of analytical and numerical tools.
4.1 Spacetime dimension estimators
The earliest result in CST is a dimension estimator for Minkowski spacetime due to Myrheim (1978)^{15} and predates BLMS (Bombelli et al. 1987). A closely related dimension estimator was given by Meyer (1988), which is now collectively known as the Myrheim–Meyer dimension estimator.
We now describe the construction of a closely related dimension estimator by Meyer (1988). Consider an Alexandrov interval \({\mathbf {A}}_d[p,q] \subset {\mathbb {M}}^d\) of volume \(V>> \rho _c^{1}\). We are interested in calculating the expectation value of the random variable \({\mathbf {R}}\) associated with R for the ensemble \({\mathcal {C}}({\mathbf {A}}_d,\rho _c)\). This is the probability that a pair of elements \(e_1,e_2 \in {\mathbf {A}}_d[p,q]\) are related. Given \(e_1\), the probability of there being an \(e_2\) in its future is given by the volume of the region \(J^+(e_1) \cap J^(p)\) in units of the discreteness scale, while the probability to pick \(e_1\) is given by the volume of \({\mathbf {A}}_d[p,q]\). This joint probability can be calculated as follows.
However, the fluctuations in \({\mathbf {R}}\) are large and hence the right dimension cannot be obtained from a single realisation \(C \in {\mathcal {C}}({\mathbf {A}}_d,\rho _c)\), but rather by averaging over the ensemble. For large enough \(\rho _c\), 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 \(\rho _c\) is taken to be large, or equivalently the interval size is taken to be large.
How can we use this dimension estimator in practice? Let C be a causal set of sufficiently large cardinality n. If the dimension obtained from Eq. (18) is approximately an integer d, this means that C cannot be distinguished from a causal set that belongs to \({\mathcal {C}}({\mathbf {A}}_d,\rho _c)\) using just the dimension estimator, for \(n \sim \rho _c\mathrm {vol}({\mathbf {A}}_d)\). We denote this by \(C \sim _d {\mathbf {A}}_d\). This also means that C cannot be a typical member of \({\mathcal {C}}({\mathbf {A}}_{d'},1)\) for dimension \(d'\ne d\), so that \(C \not \sim _{d'} {\mathbf {A}}_{d'}\). The equivalence \(C\sim _d {\mathbf {A}}_d\) itself does not of course imply that \(C \sim {\mathbf {A}}_d\) or even that C is manifoldlike. Rather, it is the limited statement that its dimension estimator is the same as that of a typical causal set in \( {\mathcal {C}}({\mathbf {A}}_d,\rho _c)\) for \(n \sim \rho _c\mathrm {vol}({\mathbf {A}}_d)\).
This is our first example of a \({\mathcal {O}}\)Hauptvermutung, where the order invariant \({\mathcal {O}}\) is the ordering fraction r and the spacetime dimension d is the corresponding manifold invariant \({\mathcal {G}}\). This example provides a useful template in the search for manifoldlike order invariants some of which we will describe in the next few subsections.
This class of dimension estimators is just one among several that have appeared in the literature, including the midpoint scaling estimator (Bombelli 1987; Reid 2003), and more recent ones (Glaser and Surya 2013; Aghili et al. 2019). We refer the reader to the literature for more details.
4.2 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 \({\mathbf {I}}[e_i,e_j] \equiv \mathrm {Fut}(e_i) \cap \mathrm {Past}(e_j)\). 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 \({\mathbb {M}}^d\), this orderinterval topology is roughly discrete or trivial. This is because the intersection of any two intervals in the continuum can be of order the discreteness scale and hence contain just a single element of the causal set, thus trivialising the topology. A way forward is to use the causal structure to obtain a locally finite open covering of C and construct the associated “nerve simplicial complex” (see Munkres 1984).
In Major et al. (2007, 2009), a “spatial” homology of C was obtained in this manner by considering an inextendible antichain \({\mathcal {A}}\subset C\) (see Sect. 3.3), which is an (imperfect) analog of a Cauchy hypersurface. The natural topology on \({\mathcal {A}}\) is the discrete topology since there are no causal relations amongst the elements. In order to provide a topology on \({\mathcal {A}}\), one needs to “borrow” information from a neighbourhood of \({\mathcal {A}}\). The method devised was to consider elements to the future of \({\mathcal {A}}\) and “thicken” by a parameter v to some collar neighbourhood \({\mathcal {T}}_v({\mathcal {A}}) \equiv \{e  \, \mathrm {IFut}({\mathcal {A}})\cap \mathrm {IPast}(e) \le v \}\). Here \(\mathrm {IFut}\) and \(\mathrm {IPast}\) denote the inclusive future and past respectively, where for any \(S \subset C\), \(\mathrm {IFut}(S)=\mathrm {Fut}(S) \cup S\) and \(\mathrm {IPast}(S)=\mathrm {Past}(S) \cup S\).
A topology can then be induced on \({\mathcal {A}}\) from \({\mathcal {T}}_v({\mathcal {A}})\) by considering the open cover \(\{{\mathcal {O}}_v \equiv \mathrm {Past}(e) \cap {\mathcal {A}}\}\) of \({\mathcal {A}}\), for \(e \in {{\mathcal {M}}}_v({\mathcal {A}})\), the set of future most elements of \({\mathcal {T}}_v({\mathcal {A}})\). The “nerve” simplicial complex \({\mathcal {N}}_v({\mathcal {A}})\) can be constructed from \(\{{\mathcal {O}}_v\}\) for every v. For a spacetime (M, g) with compact Cauchy hypersurface \(\varSigma \), and for \(C \in {\mathcal {C}}(M,\rho _c)\) it was shown in Major et al. (2007, 2009) that there exists a range of values of v such that \({\mathcal {N}}_v({\mathcal {A}})\) is homological to \(\varSigma \) (up to the discreteness scale) as long as there is a sufficient separation between the discreteness scale \(\ell _c\equiv V_c^{{1}/{d}}\) and \(\ell _K\) the scale of extrinsic curvature of \(\varSigma \).
One might also imagine a similar construction on C using the nerve simplicial complex of causal intervals of a given minimal cardinality v which cover C. However, in the continuum the intersection of such intervals may not only be of order the discreteness scale, but also such that they “straddle” each other. As an example consider the equal volume intervals \({\mathbf {A}}[p_1,q_1], {\mathbf {A}}[p_2,q_2]\) in \({\mathbb {M}}^2\) where \(p_1,q_1\) are at \(x=0\) in a frame (x, t), with the xcoordinate of \(p_2\) being \(<0\) and that of \(q_2\) being \(>0\). These two intervals not only intersect, but straddle each other, i.e., the set difference \({\mathbf {A}}[p_1,q_1] \backslash {\mathbf {A}}[p_2,q_2]\) is disconnected as is \({\mathbf {A}}[p_2,q_2]\backslash {\mathbf {A}}[p_1,q_1]\). By choosing \(p_2,q_2\) appropriately, the intersection region can be made very “thin”, pushing most of the volume of \({\mathbf {A}}[p_2,q_2]\) out of \({\mathbf {A}}[p_1,q_1]\). Thus, while they intersect in \({\mathbb {M}}^2\) these intervals would not intersect in the corresponding causal set C. This results in a nontrivial cycle in the associated nerve simplicial complex for C, which is absent in the continuum. Such a construction can be therefore made to work only in a sufficiently localised region within C.
An example of a localised subset of C is the region sandwiched between two inextendible and nonoverlapping antichains \({\mathcal {A}}_1\) and \({\mathcal {A}}_2\). The resulting homology constructed from the nerve simplicial complex of the order intervals of volume \(\sim v\) is then is associated with a spacetime region rather than just space, and hence includes topology change. While preliminary investigations along these lines have been started, there is much that remains to be understood. Another possibility for characterising the spatial homology uses chain complexes but this has only been partially investigated. A further open direction is to obtain the causal set analogues of other topological invariants.
4.3 Geodesic distance: timelike, spacelike and spatial
Rideout and Wallden (2009) 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 minimising over these pairs (again infinite), the 2link distance can be calculated by averaging over the pairs. Numerical simulations for the naive distance and the 2link distance for sprinklings into a finite region of \({\mathbb {M}}^3\) show that the latter stabilises as a function of \(\rho _c\). 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.
4.4 The d’Alembertian for a scalar field
It is already clear from the picture that emerges in \({\mathbb {M}}^d\) that, unlike a regular lattice, a simple construction of a locally defined tangent space 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 simply 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.
How does a scalar field on a causal set evolve under this nonlocal 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 \(B_\kappa \) operator. An infinite family of nonlocal 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 \({\mathbf {A}}_2[p,q] \subset {\mathbb {M}}^2\) and it was shown that the spectrum of the d’Alembertian (or Feynman propagator) gives the link matrix (see Eq. (56) below), i.e., the matrix of all linked pairs using which the entire causal set can be reconstructed via transitivity. Extending these results to higher dimensions is an interesting open question.
4.5 The Ricci scalar and the Benincasa–Dowker action
Equation (35) is exactly true in an approximately flat region of a four dimensional spacetime as shown in Belenchia et al. (2016a). Proving Eq. (31) in general is however nontrivial since there are caustics in a generic spacetime which complicate the calculation. On the other hand, numerical simulations suggest that again, up to boundary terms, the Benincasa–Dowker action S is the Einstein–Hilbert action (Benincasa 2013; Cunningham 2018b). We will discuss these boundary terms below.
The result for \(d=2,4\) are due to Benincasa and Dowker (2010) and Benincasa (2013) and were generalised to arbitrary dimensions by Dowker and Glaser (2013) and Glaser (2014), using a dimension dependent smearing function \(f_d(n,\epsilon )\).
4.6 Boundary terms for the causal set action
Simulations of causal sets corresponding to different regions of \({\mathbb {M}}^2\) moreover suggest that while the BD action contains 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/pastmost inextendible antichain in the causal set, which we denote as \({\mathcal {F}}_0,{\mathcal {P}}_0\) 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 nonlocality).
4.7 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 find such local regions in a causal set using a purely order theoretic quantities? For a causal set a natural definition of a local region is given by the size of an interval, but for a manifoldlike causal set, this 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 RNNtype regions.
There are other ways of testing for manifoldlikeness. In a similar approach, the distribution of the longest chains or linked paths of length k in a finite element causal set C has been studied in \({\mathbb {M}}^d\), \(d=2,3,4\) and shown to have a dimensiondependent peak (Aghili et al. 2019). In Bolognesi and Lamb (2016), a novel way to test for manifoldlikeness was given, using the order invariant obtained from counting the number of elements with a fixed valency in a finite element causal set. In Henson (2006a), an algorithm for determining the embeddability of a causal set in \({\mathbb {M}}^2\) was given, which again gives an intrinsic characterisation of manifoldlikeness in \(d=2\). Extending and expanding on these studies using causal sets obtained from sprinklings into different types of spacetimes would be a straightforward but useful exercise.
4.8 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 \(\varSigma \) in a dimensionally reduced \(d=2\) black hole spacetime by counting links between elements in \(J^(\varSigma ) \cap J^(H)\) and those in \(J^+(\varSigma ) \cap J^+(H)\), with the additional requirement that the former is futuremost and the latter pastmost 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 proves to be tricky. In Marr (2007), an entropy formula was proposed for higher dimensions by replacing links with other subcausal sets. While these ideas hold promise, they have not as yet been fully explored.
In analogy with Susskind’s entropy bound, the maximum causal set entropy associated with a finite spherically symmetric spatial hypersurface \(\varSigma \) was defined by Rideout and Zohren (2006) as the number of maximal or future most elements in its future domain of dependence \(D^+(\varSigma )\). 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.
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.
4.9 Remarks
To conclude this section we note that several order invariants have been constructed on manifoldlike causal sets whose expectation values limit to manifold invariants as \(\rho _c\rightarrow \infty \). At finite \(\rho _c\) there are fluctuations that serve to distinguish the fundamental discreteness of causal sets from the continuum, and these have potential phenomenological consequences. Numerical simulations are often important in assessing the relative importance of these fluctuations.
For each of these invariants, one has therefore proved an \({\mathcal {O}}\)Hauptvermutung. While this collection of order invariants is not sufficient to prove the full Hauptvermutung, they lend it strong support. These order invariants are moreover important observables for the full theory. In addition to these manifoldlike order invariants, there are several other order invariants that can be constructed, some of which may be important to the deep quantum regime but by themselves hold no direct continuum interpretation.
5 Matter on a continuumlike causal set
Before passing on 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 is the free scalar field on a causal set in \({\mathbb {M}}^d\). 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 nontrivial tensorial fields on causal sets is known. However, as we will see, even this very simple class of matter fields brings with it both exciting new insights and interesting conundrums.
5.1 Causal set Green functions for a free scalar field
Consider the real scalar field \(\phi : {\mathbb {M}}^d \rightarrow {\mathbb {R}}\) and its CST counterpart, \(\phi :C \rightarrow {\mathbb {R}}\) where \(C \in {\mathcal {C}}({\mathbb {M}}^d,\rho _c)\). The Klein Gordon operator of the continuum is replaced on the causal set by the \(B_\kappa \) operator of Sect. 4, Eq. (36). In the continuum \(\Box ^{1}\) gives the Green function, and we can do the same with \(B_\kappa \) to obtain the discrete Green function \(B_\kappa ^{1}\).
Finding causal set Green functions for other spacetimes is more challenging, but there have been some recent results (Dowker et al. 2017) which show that the flat spacetime form of Johnston (2008, 2010) can be used in a wider context. These include (a) a causal diamond in an RNN of a \(d=2\) spacetime with \(M^2={\rho _c}^{1}({m^2+\xi R(0)})\), where R(0) is the Ricci scalar at the centre of the diamond and \(\xi \) is the nonminimal coupling, (b) a causal diamond in an RNN of a \(d=4\) spacetime with \(R_{ab}(0) \propto g_{ab}(0)\) and \(M^2={\rho _c}^{1}({m^2+\xi R(0)})\) when (c) \(d=4\) de Sitter and anti de Sitter spacetimes with \(M^2={\rho _c}^{1}({m^2+\xi })\).
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 \({\mathbb {M}}^3\) 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.
5.2 The Sorkin–Johnston (SJ) vacuum
Having obtained the classical Green function and the d’Alembertian operator in \({\mathbb {M}}^2\) and \({\mathbb {M}}^4\), 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.
However, even here, the standard route to quantisation involves the mode decomposition of the space of solutions of the Klein Gordan operator, \(\ker (\Box m^2)\). In \({\mathbb {M}}^d\) 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.
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 \(\ker ({\widehat{\Box }}m^2)\) 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. As a result, the SJ state has generated some interest in the broader algebraic field theory community (Fewster and Verch 2012; Brum and Fredenhagen 2014; Fewster 2018). For example, while not in itself Hadamard in general, the SJ vacuum 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 2019). 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 \(\alpha \) vacuua (Surya et al. 2019). This has potentially far reaching observational consequences which need further investigation.
5.3 Entanglement 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, X and Yazdi, work in progress), but one first needs to make an ansatz for locating the knee in the causal set \(i \varDelta \) spectrum.
5.4 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. It is nevertheless interesting to ask if causal sets that are manifoldlike might exhibit such a behaviour around the discreteness scales at which the continuum approximation is known to break down. As we have seen earlier (Sect. 4.3), 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, Belenchia et al. (2016c) showed that causal set inspired nonlocal 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. Recently in Eichhorn et al. (2019b), the spectral dimension was calculated on a maximal antichain for a causal set obtained from sprinklings into \({\mathbb {M}}^d\), \(d=2,3\) using the induced distance function of Eichhorn et al. (2019a). It was seen to decrease at small scales, thus bringing the results closer to that conjectured by Carlip (2017).
6 Dynamics
Until now our focus has been on manifoldlike causal sets, since the aim was to find useful manifoldlike covariant observables as well as to make contact with phenomenology. However, as discussed in Sect. 3, the arena for CST is a sample space \(\varOmega \) of locally finite posets which replaces the space of 4geometries, and contains nonmanifoldlike causal sets. A CST dynamics is given by the measure triple \((\varOmega , {\mathfrak {A}}, \mu )\) where \({\mathfrak {A}}\) is an event algebra and \(\mu \) is either a classical or a quantum measure. We will define these quantities later in this section.
To begin with, \(\varOmega \) 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 \(\varOmega _{g}\), while for a unimodular type dynamics using the Einstein–Hilbert action, the natural restriction is to \(\varOmega _n\) 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 Sects. 3 and 4, in the asymptotic \(n \rightarrow \infty \) limit the sample space \(\varOmega _n\) is dominated by the nonmanifoldlike 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 play a role in the classical or continuum approximation of the theory.
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 classical sequential growth models (CSG)(Rideout and Sorkin 2000a), which are Markovian growth models. We will describe these in Sects. 6.1 and 6.2.
One of the main challenges in CST is to build a viable quantum sequential growth(QSG) dynamics. The appropriate framework for the dynamics is as a quantum measure space which is a natural quantum generalisation of classical stochastic dynamics (Sorkin 1994, 1995, 2007d). This means replacing the classical probability measure P in the measure space triple \((\varOmega , {\mathcal {A}}, \mu _c)\) with a quantum measure \(\mu \). The quantum measure is defined via a decoherence functional and can also be defined as a vector measure in a corresponding histories Hilbert space. We will discuss this in Sect. 6.3.
It is also of interest to construct an effective continuuminspired 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 in Sect. 6.4.
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 are grown element by element, with the dynamics satisfying a few basic principles (Rideout and Sorkin 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. 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 corresponds to the causal set changing cardinality or “growing” by one. Starting with a causal set \({{\tilde{c}}}_n\) of cardinality n, the passage of time means transitioning from \({{\tilde{c}}}_n\rightarrow {{\tilde{c}}}_{n+1}\) where the new element in \({{\tilde{c}}}_{n+1}\) is to the future of some of the elements of \({{\tilde{c}}}_n\), but never in their past. In the infinite “time” limit, \(n\rightarrow \infty \), the dynamics, either deterministic, probabilistic or quantum, will take you from \({{\tilde{c}}}_n\) to a countable causal set.
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 set \({{\tilde{c}}}_n\) and any of its relabellings, \({{\tilde{c}}}_n'\) must be the same. Identifying relabelled causal sets as the same object in the CST tree \({\mathcal {T}}\) gives us a nontrivial poset of causal sets or the “postcau” \({\mathcal {P}}\) of Rideout and Sorkin (2000a). On \({\mathcal {P}}\), a covariant dynamics is thus pathindependent: if there is more than one path from an unlabelled initial causal set \(c_{n_{i}}\) to an unlabelled final causal set \(c_{n_{f}}\) in \({\mathcal {P}}\), then in order to satisfy covariance, the measure on both paths should be the same.
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 transition \({{\tilde{c}}}_n\) to a \({{\tilde{c}}}_{n+1}\), 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 \({\mathcal {P}}\).
In Varadarajan and Rideout (2006) and Dowker and Surya (2006), a generalisation of the dynamics was explored, where some of the transition probabilities were allowed to vanish, consistent with (a) (b) and (c). This requires a generalisation of the Bell causality condition. The resulting dynamics exhibits a certain “forgetfulness” when these transition probabilities vanish, but are otherwise very similar to the CSG models.
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 continuumlike 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, suggestive of a continuum limit. In Ahmed and Rideout (2010), it was shown that the dynamics typically yields an exponentially expanding universe. Moreover, for \((1q) \ll 1 \) and \(n \gg \frac{1}{1q}\), after a post the universe enters a tree like phase and then a de Sitterlike 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 manifoldlike phase. In Brightwell and Georgiou (2010) and Brightwell and Luczak (2015), moreover, it was shown explicitly that in the asymptotic limit \(n \rightarrow \infty \) the causal sets limit to “semiorders” which, though temporally ordered, have no spatial structure at all, and are hence nonmanifoldlike. 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 singularities in CSG models. As in the case of cosmological bounces a new epoch is created beyond the singularity. Using “breaks” which are multielement versions of a post, they demonstrated that a renormalisation of the coupling constants occurs in the new epoch.
6.2 Observables as beables
As mentioned in the introduction to this section, a dynamics for CST is given by the triple \((\varOmega , {\mathfrak {A}}, \mu )\). In CSG this is a probability measure space, where the sample space \({\tilde{\varOmega _g}} \) is the set of all past finite naturally labelled causal sets.
The event algebra \({{\tilde{{\mathfrak {A}}}}}\) 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 \(\emptyset \) as well as \({\tilde{\varOmega _g}} \). In the growth process we assign a probability \(\mu ({{\tilde{c}}}_n)\) to every finite labelled causal set \({{\tilde{c}}}_n\). By identifying \({{\tilde{c}}}_n\) with its cylinder set \(\mathrm {cyl}({{\tilde{c}}}_n)\), we define the measure \(\mu (\mathrm {cyl}({{\tilde{c}}}_n))\equiv \mu ({{\tilde{c}}}_n)\) and hence on all elements of \({{\tilde{{\mathfrak {A}}}}}\), since \(\mu \) is finitely additive. This makes \(({\tilde{\varOmega _g}} , {{\tilde{{\mathfrak {A}}}}}, {{\tilde{\mu }}}')\) a “premeasure” space.
An event \(\alpha \) is an element of \({\mathfrak {A}}\) deemed to be covariant as a measurable subset \(\alpha \subset {\tilde{\varOmega _g}} \) if for every \({{\tilde{c}}}\in \alpha \), its relabelling \({{\tilde{c}}}'\) also belongs to \(\alpha \). Since a relabelling can happen arbitrarily far into the future, no event in \({\mathfrak {A}}\) is covariant, since \({\mathfrak {A}}\) 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 \({\mathfrak {A}}\).
One route to obtaining covariant events is to pass to the full sigma algebra \({{\tilde{{\mathfrak {S}}}}}\) generated by \({{\tilde{{\mathfrak {A}}}}}\), which is closed under countable set operations. For classical measure spaces, the Kolmogorov–Caratheodory–Hahn extension theorem allows us to extend \({{\tilde{\mu }}}'\) to \({{\tilde{{\mathfrak {S}}}}}\) and hence pass with ease to a full measure space \(({\tilde{\varOmega _g}} , {{\tilde{{\mathfrak {S}}}}}, {{\tilde{\mu }}})\), where \({{\tilde{\mu }}}_{{{\tilde{{\mathfrak {A}}}}}} = {{\tilde{\mu }}}'\). Not every event in \({\mathfrak {S}}\) is covariant, but we can restrict our attention to covariant events, i.e., sets that are invariant under relabellings. If \(\sim \) denotes the equivalence up to relabellings one can define the quotient algebra \({\mathfrak {S}}={{\tilde{{\mathfrak {S}}}}}/\sim \) of covariant events. An element of \({\mathfrak {S}}\) is measurable covariant set, or a covariant observable (or beable). Our example of the post event belongs to \({\mathfrak {S}}\). 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 \({\mathfrak {S}}\) is important, since it tells us what covariant questions we can ask of causal set quantum gravity.
A more covariant way to proceed is to generate the event algebra not via the cylinder sets in \({\tilde{\varOmega _g}} \) but by using covariantly defined sets in \(\varOmega _g\), 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 subcausal set \(c_n\) of \( c \in \varOmega _g\) is said to be a partial stem if it contains its own past. A stem set \(\mathrm {stem}(c_n)\) is then a subset of \(\varOmega _g\) such that every \(c \in \mathrm {stem}(c_n)\) contains the partial stem \(\tilde{c_n}\). Let \(\mathcal S\) be the sigma algebra generated by the stem sets. Although \({\mathcal {S}} \) is a strictly smaller subalgebra of \({\mathfrak {S}}\), 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 nontrivial result and the hope is that some version of it will carry over to the quantum case.
6.3 A route to quantisation: the quantum measure

Hermiticity: \(D(\alpha ,\beta )=D^*(\beta ,\alpha )\)

Countable biadditivity: \(D(\alpha , \sqcup _i \beta _i)=\sum _i D(\alpha , \beta _i)\) and \(D(\sqcup _i\alpha _i, \beta )=\sum _i D(\alpha _i, \beta )\)

Normalisation: \(D(\varOmega ,\varOmega )=1\)

Strong positivity: \(M_{ij}\equiv D(\alpha _i, \alpha _j)\) for any finite collection \(\{\alpha _i\} \) is positive semidefinite
Although the vector measure is 1dimensional in complex percolation dynamics, it was shown in Dowker et al. (2010c) not to satisfy this convergence condition and hence one cannot pass to \({\mathfrak {S}}\) 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. Very recently a class of QSG dynamics that does admit an extension has been found (Surya and Zalel, work in progress).
The space of QSG models is largely unexplored. It is however critical to study it extensively in order to find the right CST quantum dynamics based on first principles.
6.4 A continuuminspired dynamics
The natural choice for S(c) is the d dimensional BD action \(S_{\mathrm{BD}}^{(d)}(c)\) which limits to the Einstein–Hilbert action in the continuum. As discussed in Sect. 3, the sample space \(\varOmega _n\) of causal sets of cardinality n is dominated by KR type causal sets. An important question is whether the action \(S_{\mathrm{BD}}^{(d)}(c)\) can overcome the KR entropy in the large n limit.
One route could be to simply “perform” the sum above. However, given that \(\varOmega _n\) grows superexponentially (to leading order it is \(\sim \,2^{\frac{n^2}{4}}\)), 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 \(\varOmega \) can be used if we can convert \(Z_{\varOmega }\) into a statistical partition function.
In Henson et al. (2017), MCMC methods were used to examine the sample space of naturally labelled posets \({\tilde{\varOmega }}_n\) to determine the onset of the KR regime, using the uniform measure (\(\beta =0\)). The Markov Chain was generated via a set of moves that sample \(\varOmega _n\). A mixture of two moves, the link move and the relation move, was used to obtain the quickest thermalisation.
To illustrate the complexity of these moves we describe in detail the link move. A pair of elements \(e,e'\) are picked randomly and independently from the causal set c, and retained if \(L(e) < L(e')\), where L is the natural labelling defined in Sect. 6.1. If \(e \prec e'\) 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 \(\mathrm {IPast}(e)\) and those in \(\mathrm {IFut}(e')\) which are “mediated” solely either by e or \(e'\). On the other hand if e and \(e'\) are not related, then one adds in a link between e and \(e'\), provided that there are no existing links between elements in \( \mathrm {IPast}(e)\) and \(\mathrm {IFut}(e')\), after which the transitive closure is taken. In the relation move, although the existence or nonexistence of a link from e to \(e'\) 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 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 nontrivial precisely because of transitivity.
The simulations of Henson et al. (2017) suggest that the onset of the asymptotic KR regime occurs for n as small as \(n \approx 90\). \(\varOmega _n\) is very large even for \(n=90\) (\(\sim \,2^{90^2}\) !) and hence thermalisation becomes a problem very quickly. Recently, steps have been taken to incorporate the action (\(\beta \ne 0\)) 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 \(\varOmega _n\) to causal sets that capture some gross features of a class of spacetimes. As discussed above, for large enough n, \(\varOmega _n\) 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 manifoldlike 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” \(\varOmega \) using nontrivial order theoretic constraints determined by dimension estimators of the kind we have encountered in Sect. 4.
For \(d=2\), the total orders U, V can be thought of as the set of lightcone coordinates of a causal set obtained from an embedding (not necessarily faithful) into a causal diamond in \({\mathbb {M}}^2\). Of special interest is the 2order obtained from a Poisson sprinkling, an example of which is shown in Fig. 7. As shown in Brightwell et al. (2008) this is equivalent to choosing the entries of U and V from a fixed \(S_n\) at random and independently. Importantly, this random order dominates \(\varOmega _{\mathrm {2d}}\) in the large n limit as shown in ElZahar and Sauer (1988) and Winkler (1991), and grows as \(\varOmega _{\mathrm {2d}} \sim \,n!/2 \). Thus, unlike \(\varOmega _n\), the sample space is dominated by manifoldlike causal sets, though it also contains causal sets that are distinctly nonmanifoldlike. This makes it an ideal starting point to study the nonperturbative quantum dynamics of causal sets. Moreover, as shown in Brightwell et al. (2008), 2orders also have trivial spatial homology in the sense of Major et al. (2007) (see Sect. 4) and hence \(\varOmega _{\mathrm {2d}}\) is the sample space of topologically trivial \(\mathrm {2d}\) causal set quantum gravity.
Importantly, the MCMC simulations of Surya (2012) give rise to a phase transition from a continuum phase at low \(\beta \) to a nonmanifoldlike phase at high \(\beta \). This is very similar to the disordered to ordered phase transition in an Ising model. The \(\beta ^2\) versus \(\epsilon \) phase diagram moreover indicates that the continuum phase survives the analytic continuation for any value of \(\epsilon \).
It was recently demonstrated by Glaser et al. (2018) using finite size scaling arguments that that this is a first order phase transition. The analysis moreover suggests that the continuum phase corresponds to a spacetime with negative cosmological constant. This is an explicit example of a nonperturbative theory of quantum gravity in which the cosmological constant is generated via the dynamics.
This simple system also allows us to examine other physically relevant questions. Of particular interest is the Hartle–Hawking wave function using the noboundary proposal. In 2d CST, this was constructed by Glaser and Surya (2016) using a natural noboundary condition for causal sets, namely requiring the existence of an “initial” element \(e_0\) to the past of all the other elements. \(\psi _{\mathrm{HH}} ({\mathcal {A}}_f)\) is the wave function for a final antichain of cardinality \({\mathcal {A}}_f\), where one is summing over all causal sets that have an initial element \(e_0\) and final boundary \({\mathcal {A}}_f\).
The MCMC simulations give the expectation value of the action \(S_{\mathrm {2d}}\) from which the partition function can be calculated by numerical integration, up to normalisation. The normalisation itself was determined in Glaser and Surya (2016) using a combination of analytic and numerical calculations. The results of the extensive analysis was that the Hartle–Hawking wave function \(\psi _{\mathrm{HH}}({\mathcal {A}}_f)\) peaks at low \(\beta \) on antichains of small cardinality, with the peak jumping at higher \(\beta \) to antichains with cardinality \(\sim n/2\). Interestingly, in the latter, high \(\beta \) (low temperature) phase, the dominant causal sets satisfy some of the rudimentary features of early universe cosmology: (a) the growth from a single element to a large antichain takes place rapidly and (b) each element in \({\mathcal {A}}_f\) is causally related to all the elements in its immediate past which makes \({\mathcal {A}}_f\) “homogeneous”. However, this is a non manifoldlike phase, and it is an open question how one exits this phase into a manifoldlike phase. If there is a dynamical mechanism that makes \(\beta \) small, then this would be a promising new mechanism for generating cosmologically relevant initial conditions for the universe.
Will this analysis survive higher dimensions? One of the issues at hand is that even for 2orders the cardinality of \(\varOmega _{2d}\) grows rapidly with n and hence thermalisation can become a major stumbling block. However, the finite sized scaling analysis of Glaser et al. (2018) and the techniques used therein, tell us that it suffices to be in the asymptotic regime. For 2orders, this is already true around \(n \sim \,80 \) and hence the results of Surya (2012) and Glaser and Surya (2016) are at least qualitatively robust. Nevertheless, to get to the asymptotic regime in \(d=4\) will require far more extensive computational power. Recently, using new sophisticated computational techniques (Cunningham 2018b), the algorithms of Surya (2012) have been updated, so that \(n \sim \,300 \) simulations can be done in a reasonable time.
An important question, however is how to obtain a dimensionally restricted \(\varOmega _n\) more generally. While 2orders are a good representation of \(\mathrm {2d}\) (topologically trivial) causal set quantum gravity, this is not true for higher order theoretic dimension. For \(d>2\) a dorder is an embedding into a space with “lightcubes” rather than lightcones. Though potentially interesting, this does not serve our more narrowly defined goal of obtaining a continuuminspired dimensionally reduced sample space.
Recently, a lattice inspired method has been investigated to generate sample spaces which are both dimensionally and topologically restricted. These are obtained as embeddings (not necessarily faithful) into a fixed spacetime, and thus include manifoldlike causal sets. In \(d=2\), the simplest example comes from causal sets obtained from sprinkling into the flat cylinder spacetime \(ds^2=  dt^2+d\theta ^2\), \(\theta \in [0,2\pi ]\). Recently it has been demonstrated that topologically nontrivial lattice inspired models in \(d = 2\) and \(d = 3\) also exhibit phase transitions similar to 2orders (Cunningham and Surya 2019). The next step is to include a wider class of embeddings as well as topology change into the model, and hence bring it closer to a full 2d theory of quantum gravity.
In the MCMC simulations discussed above, labelled posets are used for practical reasons, since this is how they are stored on the computer. A single unlabelled poset admits many relabellings or “automorphisms”, but the number of relabellings varies from poset to poset even for the same cardinality. For example, in the list of coloured or labelled 3element causal sets in Fig. 20, we see that there is only one 3element causal set with three distinct natural labellings, while all the others admit only one natural labelling. Enumerating the number of automorphisms for a given causal set quickly becomes very difficult as n increases.
In the continuum path integral, the “correct” measure in a gauge theory involves the volume of the gauge orbits. In this discrete setting, as we have discussed above, the analogous gauge orbits corresponding to to the automorphisms, are not of the same cardinality for each \(c\in {\tilde{\varOmega }}_n\).
Indeed, the choice of measure is not obvious in CST since it is not merely a discretisation of the continuum theory, with the path sum Eq. (84) including causal sets that are nonmanifoldlike. There is no underlying order theoretic reason to pick the specific BD action; we have done so, “inspired” by the continuum. For continuum like causal sets of a fixed dimension the number of relabellings is approximately the same, so that they appear roughly with the same weight in the path integral. However, it is the relative weight compared the noncontinuumlike causal sets that depends on the relabellings. In the classical sequential growth model described above, the labelling is related to temporality and hence the choice of a uniform measure on the set of labelled causal sets \({\tilde{\varOmega _g}} \) is a natural one. In the MCMC simulations, therefore we pick a measure that is uniform on \({\tilde{\varOmega }}_n\), rather than on the unlabelled sample space \(\varOmega _n\). Causal sets that admit more relabellings come with a higher natural weight than those that admit fewer relabellings. However, discrete covariance or label invariance is not compromised since the observables themselves are label independent.
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, as \(\beta \rightarrow i \beta \). There is for example no analogue of the Osterwalder–Schrader theorems to protect the results we have obtained in the MCMC simulations. Pursuing these questions further is important, though finding definitive and rigorous answers is perhaps beyond the scope of our present understanding of CST.
7 Phenomenology
While the deep realm of quantum gravity is extremely well shielded from experimental probes in the foreseeable 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 underlying 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 special type of nonlocality 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 \({\mathbb {M}}^d\) 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 nonlocality 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 instabilities. 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 offshell modes of the nonlocal CST d’Alembertian. This is an exciting proposal and should be investigated in more detail.
We will not review these very interesting ideas on CST phenomenology here, except one, namely the prediction of \(\varLambda \).
7.1 The 1987 prediction for \(\varLambda \)
One of the most outstanding questions in theoretical physics is understanding the origin of “dark energy” which observationally has been seen to make up \(\sim \) 70% of the total energy of the universe. The current observational value is \(\sim \,2.888 \times 10^{122}\) in Planck units. Quantum field theory predictions for dark energy interpreted as the energy of vacuum fluctuations of quantum fields on the other hand gives a huge value, perhaps as large as \(\sim \, 1\) in Planck units. The gross conflict with observation obviously implies that this cannot be the source of \(\varLambda \).^{30}
This argument is general and requires three important ingredients: (i) the assumption of unimodularity and hence the conjugacy between \(\varLambda \) and V, (ii) the number to volume correspondence \(V \sim n\) and (iii) that there are fluctuations in V which are Poisson, with \(\delta V = \sqrt{V} \sim \sqrt{n}\). While (i) can be motivated by a wide range of theories of quantum gravity, (ii) and (iii) are both distinctive to causal set theory. No other discrete approach to quantum gravity makes the \(n \sim V\) correspondence at a fundamental level and also incorporates Poisson fluctuations kinematically in the continuum approximation. Quoting from Sorkin (1991), “Fluctuations in \(\varLambda \) 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 \(\varDelta \varLambda \).
Of course, an important question that arises in this quick calculation is why we should assume that \(\langle {\varLambda } \rangle =0\).^{31} 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 \(\varLambda \) must violate conservation of the stress energy tensor, and hence the Einstein field equations.
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 by taking \(\varLambda (x^\mu )\), such that \(\varDelta \varLambda (x)\) is dependent only on \(\varLambda (y)\) for \(y \in J^(x)\). This would mean that well separated patches in the CMB sky would contain uncorrelated fluctuations in \(\varOmega _\varLambda \), which in turn are strongly constrained to \(< 10^{6}\) by observations and hence insufficient to account for \(\varLambda \). 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 \(\varLambda \) to model a quintessence type spatially inhomogeneous scalar field with a potential term that varies from realisation to realisation. Using MCMC methods to sample the cosmological parameter space, and generate different stochastic realisations, it was shown that these CST inspired models agrees with the observations as well as \(\varLambda CDM\) 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 comparisons with future observations and heralds an exciting phase of quantum gravity phenomenology.
8 Outlook
CST has come a long way in the last three decades, despite the fact that there are only a few practitioners who have been able to dedicate their time to it. Over the last decade, in particular, there has been a growth of interest with inputs 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, and continue what has been a productive dialogue.
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 CSTinspired inhomogeneous models of fluctuating \(\varLambda \) 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. Between these two ends lie myriad interesting questions. We invite you to join us.
Footnotes
 1.
Henceforth, we will assume that spacetime is causal, i.e., without any closed causal curves.
 2.
Hence the term “pseudoRiemannian”.
 3.
 4.
A point p in a spacetime is said to be strongly causal if every neighbourhood of p contains a subneighbourhood such that no causal curve intersects it more than once. All the events in a strongly causal spacetime are strongly causal.
 5.
These are spacetimes in which the chronological past and future \(I^\pm (p)\) of each event p is unique, i.e., \(I^\pm (p)=I^\pm (q) \Rightarrow p=q\).
 6.
These are the exclusive future and past sets since they do not include the element itself.
 7.
The most obvious choice for \(V_c\) is the Planck volume, but we will not require it at this stage.
 8.
Since \(\varPhi (C)\) is a random causal set, any function of \({{\mathbf {F}}}: C \rightarrow {\mathbb {R}}\) is therefore a random variable.
 9.
\({\mathcal {C}}(M,\rho _c)\) explicitly depends on the spacetime metric g, which we have suppressed for brevity of notation.
 10.
We assume that these are always causally convex.
 11.
Henceforth we will identify \(\varPhi (C)\) with C, whenever \(\varPhi \) is a faithful embedding.
 12.
It is interesting to ask if other choices of uniform distribution satisfy the above theorem. If so, then our criterion for a uniform distribution could not only include ones that minimise the fluctuations but also those that respect Lorentz invariance.
 13.
This is in the sense of an ensemble, since the faithful embedding is defined statistically.
 14.
We remind the reader that the ensemble depends on the spacetime (M, g) but we suppress the dependence on g for the sake of brevity.
 15.
This remarkable preprint also contains the first expression, again without detailed proof, of the volume of a small causal diamond in an arbitrary spacetime.
 16.
Note that this is the exclusive interval and hence there exists exactly one element \(e''\) such that \(e\prec e'' \prec e'\).
 17.
By doing so, we violate the condition that \(\phi \) is of compact support. However, given that the regions \({\mathcal {W}}_3\) and by assumption \({\mathcal {W}}_2\) contribute negligibly, we can always ensure this by only requiring constancy of \(\phi \) in a neighbourhood of \({\mathcal {W}}_1\).
 18.
\(\epsilon \) is a new free parameter in the theory, whose value should ultimately be decided by the fundamental dynamics.
 19.
It is an interesting question whether the choice of affine parameter along “almost” null directions can be obtained from the causal set.
 20.
This calculation has later been extended by Jubb (2017) to higher orders to obtain more information about the spatial geometry.
 21.
Note that while \({\mathcal {F}}_1\cap {\mathcal {F}}_0=\emptyset \), \({\mathcal {F}}_1\) is not necessarily an inextendible antichain.
 22.
The expression in Buck et al. (2015) holds for any two subsets of C not just those we consider here.
 23.
The identification of \(\ker (\Box m^2)\) with \(Im(i\varDelta )\) is in fact well known (Wald 1994) when the latter is restricted to functions of compact support.
 24.
Of course, we could insist that there is no beginning, in which case n is never finite.
 25.
By this we mean that the new element is “linked” to an existing one, not just related to it.
 26.
A useful example to keep in mind is the 1d random walk. Let \(\gamma ^T\) be a finite element path in the \(tx\) plane from \(t=0\) to \(t=T\). A cylinder set \(\mathrm {cyl}(\gamma ^T)\) is then the set of all infinite time paths, which coincide with \(\gamma ^T\) from \(t=0\) to \(t=T\).
 27.
We will not discuss the very rich and interesting literature on the coevent interpretation of the quantum measure, which though incomplete, contains essential features that one would seek for a theory of quantum gravity (Sorkin 2007c).
 28.
In general, these are given by the conditions in the Kolmogorov–Caratheodory–Hahn–Kluvanek theorem (Diestel and Uhl 1977).
 29.
We leave out interpretational questions!
 30.
On the other hand, it would be interesting to understand why the back of the envelope quantum field theory calculation is not observationally relevant. Interesting insights into this question could come from a better understanding of the SJ vacuum in de Sitter spacetime.
 31.
In Samuel and Sinha (2006), a very striking analogy was made between a fluctuating \(\varLambda \) 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.
 32.
Notes
Acknowledgements
I am indebted to Rafael Sorkin for his deep insights and vast knowledge, that have directly and indirectly shaped this review. I am also deeply indebted to Fay Dowker for our interactions and collaborations over the past 25 years, which have helped enrich my understanding of quantum gravity. I am grateful to my other collaborators, including David Rideout, Joe Henson, Graham Brightwell, Petros Wallden, Lisa Glaser, Denjoe O’Connor, Ian Jubb, Yasaman Yazdi and my students Nomaan X and Abhishek Mathur, for their active and continuous engagement with the questions in CST, which have led to fruitful discussions, arguments, disagreements and debates over the years. Finally, I would like to thank Yasaman Yazdi and Stav Zalel for a careful reading through the first draft of the manuscript and giving me useful feedback. This research was supported in part by the Emmy Noether Fellowship (2017–2018) and also by a Visiting Fellowship (2019–2022) at the Perimeter Institute of Theoretical Physics.
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