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arXiv:0706.0375v3 [gr-qc] 17 Sep 2008
A 2D model of Causal Set Quantum Gravity:
The emergence of the continuum.
Graham Brightwell1, Joe Henson2and Sumati Surya3
1London School of Economics, London, UK,
2Perimeter Insitute, Waterloo, Canada
& University of Utrecht, Utrecht, Netherlands,
3Raman Research Institute, Bangalore, India
September 17, 2008
Abstract
Non-perturbative theories of quantum gravity inevitably include configu-
rations that fail to resemble physically reasonable spacetimes at large scales.
Often, these configurations are entropically dominant and pose an obstacle
to obtaining the desired classical limit. We examine this “entropy problem”
in a model of causal set quantum gravity corresponding to a discretisation of
2D spacetimes. Using results from the theory of partial orders we show that,
in the large volume or continuum limit, its partition function is dominated
by causal sets which approximate to a region of 2D Minkowski space. This
model of causal set quantum gravity thus overcomes the entropy problem and
predicts the emergence of a physically reasonable geometry.
In approaches to quantum gravity where the continuum is replaced by a more
primitive entity, manifoldlikeness is typically a feature of only a small proportion
of the configurations. In order to obtain the correct continuum limit, this small
set of configurations needs to be dynamically favoured over the often overwhelming
entropic contribution from non-manifoldlike configurations. It has been argued that
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some form of this “entropy problem” is of critical importance in dynamical trian-
gulations, graph-based approaches and in causal set quantum gravity(CSQG) [1].
This present work shows how the problem is overcome in a simplified 2D model of
CSQG.
In CSQG, continuum spacetime arises as an approximation to a fundamentally
discrete structure, the causal set. Here, order and number correspond to the con-
tinuum notions of causal order and spacetime volume. Despite being discrete, local
Lorentz invariance in the continuum approximation is restored by using a random
lattice [2]. These basic features of the theory led to an early prediction for the
cosmological constant, confirmed several years later by observation [3]. The con-
struction of a model of CSQG with physically realistic predictions is therefore of
considerable interest.
The fundamental entity that replaces spacetime in CSQG is a causal set, or
causet, (C,≺), which is a locally finite partially ordered set.
x,y,z ∈ C (i) x ⊀ x (irreflexive)1, (ii) x ≺ y, y ≺ z ⇒ x ≺ z (transitive) and
(iii) Cardinality(Future(x) ∩ Past(y)) is finite (locally finite), where Future(x) =
{z|x ≺ z}, and Past(x) = {z|z ≺ x}. Local finiteness means that discreteness is
taken to be fundamental, and not simply a tool for regularisation. The continuum
(M,g) arises as an approximation of a causet C if there exists a “faithful embed-
ding” Φ : C → (M,g) at density V−1
that (a) the distribution of Φ(C) ⊂ M is indistinguishable from that obtained via
a Poisson sprinkling into (M,g), i.e., a random discrete set S such that, for any
region R of volume V , the number of points of S in R is a Poisson random variable
with mean V/Vc, and (b) the order relation ≺ in C is in 1-1 correspondence with
the induced causal order on Φ(C), i.e., x ≺ y ⇔ Φ(x) is to the causal past of Φ(y)
[4, 5].
Namely, for any
c , where Vcis the discreteness scale. This means
While a quantum version of a classical sequential growth dynamics for causets
may eventually provide a more natural framework for quantisation [6], it is useful
to consider the standard path-integral paradigm. As in other discrete approaches
[7], the path integral is replaced by a sum, which in CSQG is over causets, with an
appropriate “causet action” providing the measure. The Regge action is an obvious
choice for discrete theories of quantum gravity based on triangulations of spacetimes
1This can be replaced by the condition that x ≺ y, y ≺ x ⇒ x = y. For both choices one avoids
causal “loops”.
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[8]. However, in CSQG, because of the intrinsic non-locality of causets, an action
defined as a sum of strictly local quantities is likely to fail. The construction of a
causet action is deeply intertwined with the question of locality and the associated
problems in constructing a consistent quantum dynamics [9, 10]. While prescriptions
for a localised D’alembertian [10] may eventually lead to an approximately local
action for causets, it is a worthwhile exercise to sidestep this question by considering
simplified models.
One possible approach is to make a precise definition of the class of “manifold-
like” causets, and restrict the history space to this class. Such causets have a natural
corresponding continuum action which can be used to define the partition function.
While manifoldlikeness is a trivial prediction of such a model, it may nevertheless
display features that yield interesting insights into CSQG. Without further restric-
tions, however, such a model is not obviously tractable. One such restriction is by
spacetime dimension, the simplest choice being dimension 2.
The specific model of 2D CSQG we present here is constructed via a restriction
to the class of so-called 2D orders. This class contains not only all causets that
approximate to conformally flat 2D spacetime intervals, but also some that are non-
manifoldlike. Moreover, all causets in this class share a certain topological triviality.
This allows us to meaningfully address the entropy problem and the question of
manifoldlikeness within the model. We find that the entropy problem is tamed in
an unexpected way, and that it is possible to characterise its physical consequences
with results that may be surprising.
Such a model can be regarded as a restriction of the full theory of CSQG, anal-
ogous to mini and midi-superspace models in canonical approaches to quantum
gravity – the hope would be to gain insights into the full theory by understanding
details of the simplified model. Indeed, our model is more fully dynamical than such
reduced models, because rather than freezing local degrees of freedom, one is simply
restricting to a class of causal sets that are naturally associated with discretisations
of 2D spacetime, with a fixed topology. Causal set theory does not in principle
assume a fixed spacetime dimension, and hence our 2D model is indeed a restriction
of the full theory. However, this simply brings it on par with the starting point of
other routes to quantisation which must assume a fixed spacetime dimension. While
a restriction to topology is routinely adopted in other approaches to quantum grav-
ity, the hope is that our model can ultimately be generalised to include a sum over
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all 2D topologies.
Although our work does not lead directly to a 4D theory, it is an example of
how the continuum can be recovered from a quantum causet model and hence may
prompt more optimism on the general approach presented above. Moreover, it is an
explicit demonstration that the causet approach is rich enough to allow formulations
with physically sensible outcomes, without the addition of extra variables [11].
We consider a causet “discretisation” of the set of 2D conformally flat spacetime
intervals (I,g). Using a fiducial flat metric, ηabdxadxb= −dudv in light cone co-
ordinates (u,v), these geometries are represented by diffeomorphism classes of the
metrics
gabdxadxb= −Ω2(u,v)dudv,(1)
with Ω2(u,v) the conformal factor. Quantisation of this set of spacetimes on I can
be thought of as a Lorentzian analog of Euclidean 2D quantum gravity on a disc
[12]. As a topological space, I is simply homeomorphic to a disc, with the boundary
condition that there exists an interval I0of2M and a bijection Ψ : ∂I → ∂I0. In 2D
the conformal factor encodes all geometric degrees of freedom so that all Lorentzian
metrics on this manifold have the form (1).
We will also adopt the unimodular modification of gravity, in which spacetime
volume plays the role of a time parameter [13]. Fixing the time coordinate is thus
given a covariant meaning, corresponding to the volume constraint
?
This constraint places restrictions on the map Ψ; starting with an “initial” event
p0= (u0,v0) in the fiducial metric ηab, the “final” event pf= (uf,vf) of the interval
I0 = [(uf,vf),(u0,v0)] in (R2,ηab) is determined (upto boosts) by the condition
?uf
The Einstein action on an interval includes a term on the null boundary. In
order to simplify the action, we take the interval I to be enlarged ever so slightly,
to a region I′⊃ I with spacelike boundary components. Because of the nature of
the manifold approximation, a causet discretisation is insensitive to modifications of
the boundary on scales much smaller than the discretisation scale, i.e. if the volume
of the region I′− I is ≪ Vc. Since the boundary of I′is piecewise spacelike, the
V =Ω(u,v) dudv = constant. (2)
u0
?vf
v0Ω(u,v)dudv = V.
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Einstein action takes the form
S =
1
16πG
?
I′RdV −
1
8πG
?
∂I′kdS −
?
j
1
8πGθj−
1
8πGΛVI′, (3)
where θj are the four boost parameters corresponding to the four “joints” in ∂I′
[14, 15]. The Lorentzian Gauss Bonnet theorem [16, 17] then simplifies the action
to
S =
16πG(2πi − 2ΛVI′),
which is a constant over the entire class of spacetimes under consideration.
1
(4)
The first step in constructing the causet discretisation of this continuum theory
is to characterise the class of causets of finite cardinality which embed faithfully
into conformally flat 2D intervals. While such a characterisation appears daunting
in general, for our model these causets lie in the set of “2D orders”, a well-studied
class of partially ordered sets.
To define this class, some nomenclature is necessary. Consider a set of elements
S = {e1,...en} and a partial order ≺ on this set. A causet X on the underlying set
S is a linear order if and only if, for all i,j, ei≺ ej or ej≺ eiin X. We use the
notation Q = (eπ(1),eπ(2)...eπ(i),...eπ(n)) to denote a linear order on S, where π is a
permutation on n elements, so that eπ(i)≺ eπ(i+1)for all i. A linearly ordered subset
of a causet is known as a chain. Similarly, a totally unordered causet is one such
that ei⊀ ejfor all i,j, and a totally unordered subset is known as an antichain. For
causets Q1,Q2,...,Qkon the same set S, the intersection P =?k
Q1= (e1,e2) and Q2= (e2,e1), then since e1≺ e2in Q1and e2≺ e1in Q2, they
are unrelated the intersection Q1∩ Q2which is therefore a two element antichain.
The “dimension” of a causet P is the minimum k such that P can be written as
the intersection of k linear orders. Our main interest is in “2-dimensional” or 2D
orders: ones that can be written as the intersection of two linear orders, but are not
themselves linear orders2(see Fig 1).
i=1Qiis defined
by setting ei ≺ ej in P if and only if ei ≺ ej in all of the Qi. For example, if
In the rest of this paper we will say that a causet C “corresponds” to a given
spacetime (M,g) if there exists a faithful embedding Φ : C → (M,g), defined
2Note that causet dimension in this context is not apriori related to the spacetime dimension.
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1
3
4
2
5
2
34
(a)
(b)
(c)
1
2 34
1
Figure 1: Examples of labelled 2D orders, obtained from the intersections of the following
linear orders: (a) L = (e1,e3,e2,e4,e5) and M = (e2,e4,e1,e3,e5) (b) L = (e1,e2,e3,e4)
and M = (e2,e1,e4,e3) and (c) L = (e1,e2,e3,e4) and M = (e4,e3,e2,e1).
precisely by Bombelli as follows [5]. Let Φ : C → M be an embedding of a causet C
of cardinality V/Vcinto a spacetime of finite volume V . Consider sampling intervals
of volume Vc< V0< V . Then
PV0(n) ≡1
n!e−V0
Vc
?
V0
Vc
?n
(5)
is the probability of finding n < N elements of Φ(C) in a region of volume V0for
a Poisson embedding. Define the indicator function Fn=?χn(I)dI/?dI, for the
I (of volume V0) has n points in it or not, and the integral is over all possible
intervals I of volume V0in (M,g). Then, if |Fn−PV0| < δ, Φ will be said to be a δ-
faithful embedding with respect to V0. We will henceforth use the phrase “faithfully
embeddable” to imply in the (δ,V0) sense. Specifically, we will require that Vc≪
V0 ≪ V and 0 < δ ≪ 1. For suitable choices of δ and V0, a causet generated
by a Poisson sprinkling into M with density V−1
faithfully embedded in M. On the other hand, regular discrete lattices tend not to
be faithfully embedded: the regular structure leaves large intervals void of points.
embedded causet Φ(C), where χn(I) = 1, or 0 depending on whether the interval
c
will be, with high probability,
To see that 2D orders are appropriate for our purposes, consider a conformally
flat 2D spacetime (M,g). The causal order ? between events p and q in such a
spacetime can be encoded in the statement:
(u1,v1) ? (u2,v2)⇔u1≤ u2
andv1≤ v2,(6)
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where (u1,2,v1,2) are light cone coordinates of p and q, respectively. For conformally
flat spacetimes any choice of light cone coordinates is such that the ordering on each
co-ordinate u or v is a linear order. This means exactly that a finite causet can be
embedded in (M,g) if and only if it is the intersection of the two co-ordinate linear
orders, i.e., if and only if its “dimension” is at most 2 [18].
Although every 2D order can be embedded into a conformally flat 2D spacetime,
not all of them can be faithfully embedded. For example, the intersection of the lin-
ear orders L = (e1,e2,e3,e4,...,eN) and M = (e2,e1,e3,e4,...,eN) has an antichain
{e1,e2}, while all other eiare to the future of both e1and e2, and linearly ordered.
Thus L ∩ M is almost a chain, except for the past-most two elements {e1,e2}, and
so clearly does not faithfully embed into a 2D spacetime, at least for N sufficiently
large. Thus, in this sense, not every 2D order corresponds to a 2D spacetime.
But can a 2D order faithfully embed into a spacetime of a different topology than
the interval? Consider, for example, a flat 2D interval (I,η) with a large region R cut
out of it (see Fig 2 (a)). (I−R,η) is not a causally convex subset of (I,η) and hence
its intrinsic causal order differs from that of (I,η). In particular, it contains pairs
of events p = (u1,v1),q = (u2,v2) such that p ≺ q in (I,η), but p ⊀ q in (I − R,η),
so that u1< u2, v1< v2does not imply that (u1,v1) ≺ (u2,v2). This means that a
causal set C that faithfully embeds, via some Φ, into (I − R,η) cannot be realised
as an intersection of the lightcone coordinates of Φ(C), for R large. However, it
is always possible to make appropriate changes in the embedding density in order
to construct an embedding E : C → (I,η): E will not be faithful, but C can be
realised as the intersection of the lightcone coordinates of E(C) and is hence a 2D
order. It would thus appear that the class of 2D orders includes faithful embeddings
into intervals with regions cut out of it, i.e., topologies different from the disc. On
the other hand, it is always possible to choose an interval spacetime (I,g), with
a g = Ω(u,v)2η which “compensates” for the varying embedding density of E(C)
in (I,η), so that Φ : C → (I,g) is a faithful embedding (see Fig 2 (b)) Relevant
to our model, is the resulting statement that a 2D order which embeds into an
interval spacetime with holes can equivalently be obtained as a discretisation of a
conformally flat interval spacetime.
The class of continuum manifolds of typical interest in 2D quantum gravity are
ones with spacelike boundaries representing an initial and a final time. From this
perspective, the extension of I to I′seems reasonable, since I′has both initial and
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u
v
B
p
q
A
B
(a)
(b)
F
G
A
F
G
Figure 2: (a) If Φ : C → (I − R,η) is faithful, Φ(C) uniformly populates the regions
A,B,F,G. (b) A suitable change in the density of the embedding pushes the elements
of C in regions A,B,F,G of (a) into a portion of the spacetime (I,η) without changing
the order-causality correspondence. By choosing a conformal factor Ω(u,v) which is ap-
proximately one in the regions A,B,F,G and vanishingly small elsewhere, Φ(C) can be
equivalently thought of as a faithful embeddedding into (I,Ω2(u,v)η).
final spacelike boundaries. Now, except spacetimes on the interval topology, all
2D spacetimes of finite volume which satisfy this boundary requirement have non-
contractible loops and non-vanishing first Betti numbers β1. β1is also non-vanishing
for any spatial slice in these spacetimes. In [19] it was shown that causal sets C that
faithfully embed into a globally hyperbolic region of a spacetime contain sufficient
structure to reproduce the spatial continuum homology with high probability. The
construction in [19] uses the idea of a thickened antichain from which a nerve sim-
plicial complex is constructed. In particular, it can be shown that if β1?= 0 for this
nerve simplex, it implies the existence of a “crown” sub-poset in C. A crown poset is
defined as follows. Let C have cardinality, 2m, m > 2, and let A1= (e1,e2,...em),
A2 = (e′
related to each other by ei≺ e′
modulo m (see Fig 3 for an example). We note that:
1,e′
2,...e′
m) be two non-intersecting antichains in C, whose elements are
i,e′
i+1and e′
i≻ ei−1,ei, where we are treating indices
Claim 1 A 2D order cannot contain a crown poset with m > 2.
Proof
Suppose the crown Cmis the intersection of linear orders L and M. Let
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e′
primed vertices in L. Now suppose wlog that eiappears above ei−1in M. Then e′
appears above eiin M, and hence is above ei−1in M as well as in L. As e′
above ei−1in the crown Cm, this contradicts the assertion that L ∩ M = Cm.
ibe the lowest primed vertex in L. Then eiand ei−1both appear below all the
i+1
i+1is not
2
3 1 2 4
1’
2’
3’4’
Figure 3: An eight element crown poset, constructed from A1= (e1,e2,e3,e4) and A2=
(e′
1,e′
2,e′
3,e′
4).
To see this result within the wider context of the theory of poset dimension,
the reader should consult [18]. What one would like to deduce from this is that
2D orders are exclusively associated with the interval topology. Of course, if the
scale of the topology is of order the discretisation scale, then this is no longer the
case. However, such continuum structure is considered irrelevant from the causet
perspective, and hence discretisation of such spacetimes is not pertinent. Thus,
within these limitations we may conclude that if a 2D order approximates to a
2 dimensional spacetime, then the latter belongs to the class of conformally flat
interval spts. It is in this sense that the class of 2D orders distinguishes the topology
of the interval from all the others relevant to 2D quantum gravity. The set of all
2D orders is thus a meaningful causet discretisation of the class of 2D interval
spacetimes.
We are now in a position to write down the causet partition function. For 2D
orders that do have a continuum approximation, our discretisation gives us a uniform
measure coming from the continuum action (3). Moreover, since the set of all 2D
orders includes all causets corresponding to 2D intervals, but none corresponding to
other 2D spacetimes, it is natural to extend this uniform measure on manifoldlike 2D
orders to all 2D orders. The partition function for our model is thus the unweighted
sum over the set Ω2Dof unlabelled 2D orders
?Z = (phase) ×
?
Ω2D
1. (7)
The appearance of a uniform weight in the partition function comes from the triv-
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iality of the continuum theory, and at first glance suggests that any semi-classical
regime will be impossible in the model. In path integral quantum mechanics, for
example, the set of all paths is dominated by non-classical, non-differentiable paths.
The inclusion of the non-trivial weight exp(iS(γ)
classical limit.
?) is crucial in obtaining the correct
Indeed, as shown in [20], a uniform measure over the set of all N element causets,
not just those which are 2D orders, is completely dominated in the large N limit by
the Kleitman-Rothschild three-level (or three “moments-of-time”) causets, which are
most non-manifoldlike. It is hoped that a suitable action for causets would repair this
entropic problem and yield the correct continuum approximation or classical limit.
In our model, however, since the action is trivial, the partition function is determined
solely by entropic effects. Nevertheless, because our measure vanishes on all N
element causets which are not 2D orders, a meaningful continuum approximation
does indeed emerge from this theory as we will discuss below.
As labels are the discrete analogues of coordinates they are considered unphysi-
cal in causet theory. Our interest therefore lies with isomorphism classes of labelled
2D orders, i.e., with unlabelled 2D orders. The random variable U(N) on the iso-
morphism classes of labelled 2D orders each taken with equal probability therefore
matches the normalised partition function for our model (7). We will also be in-
terested in so-called labelled random 2D orders P(N) ≡ L ∩ M which are random
variables defined by choosing L and M randomly and independently from the N!
linear orderings of {e1,...,eN}. The study of random k-dimensional orders was ini-
tiated in the 1980s by Winkler [21]. The case k = 2 has been of particular interest,
because of its connection to random permutations, and to Young tableaux. The typ-
ical structure of a random 2-dimensional order is now reasonably well understood.
From the perspective of CSQG, this model of random orders plays a crucial role.
Indeed, a random order from this model can equivalently be generated by taking
a sequence of N independent random points in a fixed interval I of 2D Minkowski
spacetime, according to the volume measure [21]. This in turn is equivalent to the
Poisson process (or sprinkling) in the interval, conditioned on the number of points
being N [22]. The equivalence of the two models can be seen as follows.
Let I be the interval of 2D Minkowski spacetime between two points a and b,
with lightcone coordinates (ua,va) and (ub,vb) respectively. Thus I is the rectangle
consisting of all points with u-coordinate in [ua,ub] and v-coordinate in [va,vb].
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Now let C be the causet, with elements {e1,...,eN}, obtained by choosing points
{Φ(e1),...,Φ(eN)} independently uniformly at random from I, and taking C to be
the induced order: ei≺ ejin C if Φ(ei) < Φ(ej) in the causal order on the manifold.
Let (ui,vi) denote the coordinates of the sprinkled element Φ(ei). With probability 1,
all the values uiand viare different. As described above, C is the intersection of the
linear orders U, V obtained from these u and v values. Each of the pairs (ui,vi) is
chosen uniformly over the rectangle I, so the coordinates uiand viare independent
of each other, and of all other choices. Thus no permutation of the elements of C
can be more likely to occur as the order U than any other, i.e., the random linear
order U is distributed uniformly over all linear orders of elements of C. The order
V is also uniform over the set of all linear orders, and is independent of U. The
process of taking a sprinkling and deriving a (labelled) causet from it is therefore
equivalent to taking a random causet according to P(N).
This means that a “typical” random order from P(N) corresponds (in the sense
of a faithful embedding) to an interval of 2D Minkowski space of volume NVc. For
a spacetime with non-trivial conformal factor, while the process of sprinkling is still
a random process, there will in general be correlations in the u and v values. Hence,
sprinklings into such spacetimes which differ from flat spacetime at scales much
larger than the cut-off, are not equivalent to the random 2D orders P(N).
The following result was first proved by El-Zahar and Sauer [23], and was stated
in this form by Winkler [24], who gave an alternative proof and considered the (more
complicated) labelled case as well.
Theorem 1 Let Φ be an isomorphism-invariant statement about 2D orders which
has a limiting probability either in P(N) or in U(N). Then a limiting probability
exists in the other case as well and the two probabilities are equal.
The proofs of El-Zahar and Sauer, and of Winkler, also give that the number of
N-element 2D orders is N!/2(1 + o(1)), and that almost all of them have a unique
representation, up to isomorphism, as an intersection of two linear orders. Here,
“limiting probability” refers to the probability in the N → ∞ limit. From our
discussion above, it then follows that as N → ∞, the partition function (7) is dom-
inated by causets which faithfully embed into an interval of Minkowski spacetime
of volume V = NVc. This emergence of manifoldlike causets in an apparently fea-
tureless partition function is surprising, to say the least. Dominance of a class of
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configurations in the partition function has a standard interpretation in quantum
theory, which translates in our case to the statement that 2D Minkowski spacetime
is a prediction of our theory.
The large N limit taken above can be interpreted as a large volume, if the dis-
creteness scale Vcis held constant, or a continuum limit if, instead, the total volume
V is held constant. We see in the above model that, in the continuum approxima-
tion, fluctuations die out altogether, with flat space dominating. Thus, despite the
possibility of having no classical limit at all in 2D, the continuum approximation is
actually a classical limit. This is not a feature to be found in other 2D quantum
gravity models [12]. However, it is something that is desirable in a model of 4D
CSQG, since the discreteness scale is of order the quantum gravity scale. These
results are therefore interesting from this point of view. The size of quantum fluctu-
ations at given N remains to be calculated and will require numerical analysis. We
leave this for future investigations.
A similar model may be constructed for the cylinder topology S1× R, the other
class of 2 dimensional spacetimes with fixed topology. Causets on the cylinder can
be partly characterised by the existence of the crown sub-posets described above,
resulting in a non-vanishing first Betti number (which means that they are not
2D orders). However, we know of no definitive characterisation of such “cylinder”
posets analogous to the 2D orders discussed above for the disc topology. It would
be of great interest to check if a continuum spacetime is also emergent for this class
of causets. Such a model would help in a more straightforward comparison with
existing 2D quantum gravity models – would the radius of the cylinder fluctuate in
the continuum limit as in other models, or would a classical limit be obtained?
In this model, as in other lower-dimensional models, many of the problems that
exist in the 4D case are avoided, but not always in a way that immediately suggests
answers to the 4D problems. Nonetheless, there are some lessons from 2D to be
learned. It is an encouraging and non-trivial fact that, in the set of 2D orders,
sprinklings of flat space naturally dominate. Once the restriction to 2D orders is
made, non-manifoldlike causets are no longer entropically preferred. In 4D, causets
that can be embedded into intervals of Minkowski space are known as “4D sphere
orders” [18]. It would be of great interest to know whether any analog of the
El-Zahar/Sauer result holds here: we can define the probability spaces U(N) and
P(N) as in the 2D case, where P(N) now refers to sprinkling into an interval of 4D
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Minkowski space, and ask how these are related. It is probably too much to expect
that every statement about 4D sphere orders has the same limiting probability in
the two models, but nevertheless it may well be true that a causet drawn from U(N)
typically corresponds to an interval in the manifold, in the sense considered here. If
so, this bodes well for the entropy problem in CSQG.
References
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