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43 References# A 2D model of Causal Set Quantum Gravity: The emergence of the continuum

Abstract

Non-perturbative theories of quantum gravity inevitably include configurations 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.

arXiv:0706.0375v3 [gr-qc] 17 Sep 2008

A 2D model of Causal Set Quantum Gravity:

The emergence of the continuum.

Graham Brightwell

1

, Joe Henson

2

and Sumati Surya

3

1

London School of Economics, London, UK,

2

Perimeter Insitute, Waterloo, Canada

& University of U trecht, Utr echt, Netherlands,

3

Raman Research Instit ute, 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 prob lem 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 f r om non-manifoldlike conﬁgurations. It has been argued that

1

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. Namely, for a ny

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

c

, where V

c

is the discreteness scale. This means

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 point s of S in R is a Poisson random variable

with mean V/V

c

, and (b) the order relation ≺ in C is in 1-1 correspo ndence with

the induced causal order on Φ(C), i.e., x ≺ y ⇔ Φ(x) is to the causal past of Φ(y)

[4, 5].

While a quantum version of a classical sequential growth dynamics for causets

may eventually provide a more na tura l 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

1

This can be replaced by the condition that x ≺ y, y ≺ x ⇒ x = y. For both choices one avoids

causal “loops”.

2

[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 or ders. 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 topolo gy. Causal set theory does not in principle

assume a ﬁxed spacetime dimension, and hence our 2D model is indeed a restriction

of the f ull 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 topo lo gy is routinely adopted in other approaches to quantum grav-

ity, the hope is that our model can ultimately b e generalised to include a sum over

3

all 2D topologies.

Although our wor k does not lead directly t o 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 confo rmally ﬂat spacetime

intervals (I, g). Using a ﬁducial ﬂat metric, η

ab

dx

a

dx

b

= −dudv in light cone co-

ordinates (u, v), these geometries are represented by diﬀeomorphism classes of the

metrics

g

ab

dx

a

dx

b

= −Ω

2

(u, v) dudv, (1)

with Ω

2

(u, v) the conformal factor. Quantisation of this set of spacetimes on I can

be thought of as a Lo rentzian 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 I

0

of

2

M and a bijection Ψ : ∂I → ∂I

0

. 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 o f a time parameter [13]. Fixing t he time coordinate is thus

given a covar ia nt meaning, corresponding to t he volume constraint

V =

Z

Ω(u, v) dudv = constant. (2)

This constraint places restrictions on the map Ψ; starting with an “initial” event

p

0

= (u

0

, v

0

) in the ﬁducial metric η

ab

, the “ﬁnal” event p

f

= (u

f

, v

f

) of the interval

I

0

= [(u

f

, v

f

), (u

0

, v

0

)] in (R

2

, η

ab

) is determined (upto boosts) by the condition

R

u

f

u

0

R

v

f

v

0

Ω(u, v) dudv = V.

The Einstein action on an interval includes a term o n 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 ≪ V

c

. Since the boundary of I

′

is piecewise spacelike, the

4

Einstein action takes the form

S =

1

16πG

Z

I

′

RdV −

1

8πG

Z

∂I

′

kdS −

X

j

1

8πG

θ

j

−

1

8πG

ΛV

I

′

, (3)

where θ

j

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 =

1

16πG

(2πi − 2ΛV

I

′

), (4)

which is a constant over the entire class of spacetimes under consideration.

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 characterisatio n 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 = {e

1

, ...e

n

} 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, e

i

≺ e

j

or e

j

≺ e

i

in X. We use the

notation Q = (e

π(1)

, e

π(2)

. . . e

π(i)

, . . . e

π(n)

) to denote a linear order on S, where π is a

permutation on n elements, so that e

π(i)

≺ e

π(i+1)

for all i. A linearly ordered subset

of a causet is known as a cha in. Similarly, a totally unordered causet is one such

that e

i

⊀ e

j

for all i, j, and a totally unordered subset is known as an antichain. For

causets Q

1

, Q

2

, . . . , Q

k

on the same set S, the intersection P =

T

k

i=1

Q

i

is deﬁned

by setting e

i

≺ e

j

in P if and o nly if e

i

≺ e

j

in all of the Q

i

. For example, if

Q

1

= (e

1

, e

2

) and Q

2

= (e

2

, e

1

), then since e

1

≺ e

2

in Q

1

and e

2

≺ e

1

in Q

2

, they

are unrelated the intersection Q

1

∩ Q

2

which is therefore a two element antichain.

The “dimension” of a causet P is the minimum k such t hat 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 a s the intersection of two linear orders, but are not

themselves linear orders

2

(see Fig 1).

In the rest o f 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

2

Note tha t ca us e t dimension in this context is not apriori related to the space time dimension.

5

1

3

4

2

5

2

3 4

(a)

(b)

(c)

1

2 3 4

1

Figure 1: E xamples of labelled 2D orders, obtained from the intersections of the following

linear orders: (a) L = (e

1

, e

3

, e

2

, e

4

, e

5

) and M = (e

2

, e

4

, e

1

, e

3

, e

5

) (b) L = (e

1

, e

2

, e

3

, e

4

)

and M = (e

2

, e

1

, e

4

, e

3

) and (c) L = (e

1

, e

2

, e

3

, e

4

) and M = (e

4

, e

3

, e

2

, e

1

).

precisely by Bombelli as fo llows [5]. Let Φ : C → M be an embedding of a causet C

of cardinality V /V

c

into a spacetime of ﬁnite volume V . Consider sampling intervals

of volume V

c

< V

0

< V . Then

P

V

0

(n) ≡

1

n!

e

−

V

0

V

c

V

0

V

c

!

n

(5)

is the probability of ﬁnding n < N elements of Φ(C) in a region of volume V

0

for

a Poisson embedding. Deﬁne the indicator function F

n

=

R

χ

n

(I)dI/

R

dI, for the

embedded causet Φ(C), where χ

n

(I) = 1, or 0 depending on whether the interval

I (o f volume V

0

) has n points in it or not, and the integral is over all possible

intervals I of volume V

0

in (M, g). Then, if |F

n

− P

V

0

| < δ, Φ will be said to be a δ-

faithful embedding with respect to V

0

. We will henceforth use the phrase “faithfully

embeddable” to imply in the (δ, V

0

) sense. Speciﬁcally, we will require that V

c

≪

V

0

≪ V and 0 < δ ≪ 1. For suitable choices of δ and V

0

, a causet generated

by a Poisson sprinkling into M with density V

−1

c

will be, with high probability,

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.

To see t hat 2D orders are appropriate for o ur 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:

(u

1

, v

1

) (u

2

, v

2

) ⇔ u

1

≤ u

2

and v

1

≤ v

2

, (6)

6

where (u

1,2

, v

1,2

) 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 o nly if it is the intersection of the two co-ordinate linear

orders, i.e., if a nd only if its “dimension” is at most 2 [18].

Although every 2D order can be embedded int o a conformally ﬂat 2D spacetime,

not all of them can be faithfully embedded. For example, the intersection of the lin-

ear orders L = (e

1

, e

2

, e

3

, e

4

, . . . , e

N

) and M = (e

2

, e

1

, e

3

, e

4

, . . . , e

N

) has an ant ichain

{e

1

, e

2

}, while all other e

i

are to the future of bo t h e

1

and e

2

, and linearly ordered.

Thus L ∩ M is almost a chain, except for the past-most two elements {e

1

, e

2

}, 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 = (u

1

, v

1

), q = (u

2

, v

2

) such that p ≺ q in (I, η), but p ⊀ q in (I − R, η),

so that u

1

< u

2

, v

1

< v

2

does not imply that (u

1

, v

1

) ≺ (u

2

, v

2

). 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 o f 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)

2

η which “compensates” for the varying embedding density of E(C)

in (I, η), so that Φ : C → (I, g) is a faithful embedding (see F ig 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 t ime. From this

perspective, the extension of I to I

′

seems reasonable, since I

′

has bo th initial and

7

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 s uitable change in the d en s ity of th e embedding pushes the elements

of C in regions A, B, F, G of (a) into a portion of the spacetime (I, η) without ch anging

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, Ω

2

(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 β

1

. β

1

is 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 β

1

6= 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 A

1

= (e

1

, e

2

, . . . e

m

),

A

2

= (e

′

1

, e

′

2

, . . . e

′

m

) be two non-intersecting antichains in C, whose elements are

related to each other by e

i

≺ e

′

i

, e

′

i+1

and e

′

i

≻ e

i−1

, e

i

, where we are treating indices

modulo m (see Fig 3 for an example). We note that:

Claim 1 A 2D order cannot contain a crown poset with m > 2.

Proof Suppose the crown C

m

is the intersection of linear orders L and M. Let

8

e

′

i

be the lowest primed vertex in L. Then e

i

and e

i−1

both appear below all the

primed vertices in L. Now suppose wlog that e

i

appears above e

i−1

in M. Then e

′

i+1

appears above e

i

in M, and hence is above e

i−1

in M as well as in L. As e

′

i+1

is not

above e

i−1

in the crown C

m

, this contradicts the assertion that L ∩ M = C

m

. 2

3 1 2 4

1’

2’

3’ 4’

Figure 3: An eight element crown poset, constructed from A

1

= (e

1

, e

2

, e

3

, e

4

) and A

2

=

(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 to pology 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 t his sense that the class of 2 D 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 Ω

2D

of unlabelled 2D orders

e

Z = (phase) ×

X

Ω

2D

1. (7)

The appearance of a uniform weight in the partition function comes from the triv-

9

iality of the continuum theory, and at ﬁrst glance suggests that any semi-classical

regime will be impossible in t he model. In path integral quantum mecha nics, for

example, the set of all paths is dominated by non-classical, non-diﬀerentiable paths.

The inclusion of the non-trivial weight exp(i

S(γ)

~

) is crucial in obtaining the correct

classical limit.

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 r andom 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 la belled random 2D orders P (N) ≡ L ∩ M which are random

variables deﬁned by choo sing L and M randomly and independently from the N!

linear orderings of {e

1

, ..., e

N

}. 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 r andom 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 pro cess (o r 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 (u

a

, v

a

) and (u

b

, v

b

) respectively. Thus I is the rectangle

consisting of all po ints with u-coordinate in [u

a

, u

b

] and v-coordinate in [v

a

, v

b

].

10

Now let C be the causet, with elements {e

1

, . . . , e

N

}, obtained by choosing points

{Φ(e

1

), ..., Φ(e

N

)} independently uniformly at random from I, and taking C to be

the induced order: e

i

≺ e

j

in C if Φ(e

i

) < Φ(e

j

) in the causal o r der on the manifold.

Let (u

i

, v

i

) denote the coordinates of the sprinkled element Φ(e

i

). With probability 1 ,

all the values u

i

and v

i

are 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 (u

i

, v

i

) is

chosen uniformly over the rectangle I, so t he coordinates u

i

and v

i

are 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 NV

c

. For

a spacetime with non-trivial confo r mal 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 fro m ﬂat spacetime at scales much

larger than the cut-o ﬀ , are not equivalent to the random 2D orders P (N).

The following result was ﬁrst proved by El-Zahar and Sa uer [23], and was stated

in this fo rm 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 t hem 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 f ollows that as N → ∞, t he partition function (7) is dom-

inated by causets which faithfully embed into an interval of Minkowski spacetime

of volume V = NV

c

. This emergence of manifoldlike causets in an apparently fea-

tureless partition function is surprising, to say the least. Dominance of a class of

11

conﬁgurations in the partition function has a standard interpretation in quantum

theory, which translates in our case t o 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 V

c

is 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 S

1

× 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 fo r the disc t opology. It would

be o f great interest to check if a continuum spacetime is also emergent for this class

of causets. Such a mo del 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-manifo ldlike causets a r e 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 o f 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

12

Minkowski space, and a sk how these are related. It is probably too much to expect

that every statement a bout 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 manifo ld, in the sense considered here. If

so, t his bodes well for the entropy problem in CSQG.

References

[1] J. Ambjorn, J. Jurkiewicz and R. Loll, “Quantum gravity, or the art of build-

ing spacetime,” in D. Oriti (ed), “Approaches to Quantum Gravity - Toward

a new understanding of space and time”, Cambridge University Press, 2006;

L. Smolin, “The case for background independence,”, in D. Rickles et al.

(ed.), “ The structural foundations of quantum gravity”, Oxford University

Press, USA (2007), 196-239. ; F. Dowker, “Causal sets and the deep struc-

ture of spacetime,” in A. Ashtekar (ed), “100 Years of Relativity. Space-time

Structure: Einstein and Beyond”, World Scientiﬁc, (2005).

[2] L. Bombelli, J. Henson and R. D. Sorkin, arXiv:gr-qc/0605006.

[3] R. D. Sorkin, Int. J. Theor. Phys. 36, 2759 (1997); M. Ahmed, S. Dodelson,

P. B. Greene and R. Sorkin, Phys. Rev. D 69, 103523 (2004).

[4] L. Bombelli, J. Lee, D. Meyer and R. D. Sorkin, Phys. Rev. Lett. 59, 521

(1987).

[5] L. Bombelli. Talk at “Causet 2006: A Topical Schoo l funded by the European

Network on Random Geometry”, Imperial College, London, U.K., September

18-22, 20 06.

[6] D. P. Rideout and R. D. Sorkin, Phys. Rev. D 61, 024 002 (2000).

[7] J. W. Barrett and L. Crane, Class. Quant. Grav. 17, 3101 (2000) ; J. Ambjorn,

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[8] T. Regge, Nuovo Cimento, 19, 558, (1 961); R. D. Sorkin, Phys. Rev. D 12,

385 (1975).

13

[9] J. Henson, Stud. Hist. Philos. Mod. Phys. 36, 519 (2005).

[10] R. D. Sorkin, “Does locality fail at intermediate length scales?”, in D. Oriti

(ed), “Approaches to Quantum Gravity - Toward a new understanding of space

and time”, Cambridge University Press, 2 006; J. Henson, “The causal set ap-

proach to quantum gravity,” in D. Oriti (ed), “Approaches to Quantum Grav-

ity - Toward a new understanding of space and time”, Cambridge University

Press, 20 06.

[11] F. Markopoulou and L. Smolin, Nucl. Phys. B 508, 409 (1997).

[12] F. David, arXiv:hep-th/9303127; F. David (Saclay), Lectures at 8th Jerusalem

Winter School for Theoretical Physics, Two-Dimensional Gravity and Random

Surfaces, Jerusalem, Israel, Dec 27 - Jan 4, 1991. Published in Jerusalem

Gravity 1990:0125-141 (QC178:J4:1990).

[13] W. G. Unruh, Phys. Rev. D 40, 1048 (1 989). R. D. Sorkin, Int. J. Theor.

Phys. 33, 523 (1994).

[14] G. Hayward, Phys. Rev. D 47, 3275, (1993).

[15] J. B. Hartle and R. Sorkin, Gen. Rel. Grav. 13, 541 (1981).

[16] G. S. Birman and K. Nomizu, Mich. Math. J. 31, 77-81, (1984).

[17] P. R. Law, Rocky Mountain Journal of Mathematics, 22, 1365, (1992).

[18] William T. Trotter, “Combinatorics and Part ia lly Ordered Sets”, The Johns

Hopkins University Press, 1992.

[19] S. Major, D. Rideout, S. Surya, Journal of Math. Phys. 48, 032501 (2007).

[20] D. Kleitman and B. L. Rothschild, Trans. Am. Math. Soc. 205, 205 (1975).

[21] Peter Winkler, Order 1, 317, (1985).

[22] D. Stoyan, W.S. Kendall, J. Mecke, “Stochastic Geometry and Its Applica-

tions”, John Wiley & Sons, 1996.

[23] M.H. El-Zahar and N.W. Sauer, Order 5, 2 39, (1988).

14

[24] P. Winkler, Order 7, 329, (1991).

15

- CitationsCitations11
- ReferencesReferences43

- In the unlabelled case, in either model, almost every element of D n has a unique representation as an intersection of two linear orders, and, if Φ is any isomorphism-invariant statement about partial orders with a limiting probability in either model, a limiting probability exists in the other case as well and the two limits are equal. The implications of this result for causal sets were explored by Brightwell, Henson and Surya [21]. The result of El-Zahar and Sauer implies that a 2-dimensional partial order chosen uniformly at random is (effectively) distributed as the partial order induced on a Poisson process in [0, 1] 2 (or, equivalently, an interval in M 2 ), and so such a random partial order can be embedded faithfully in an interval in M 2 .

[Show abstract] [Hide abstract]**ABSTRACT:**The causal set approach to quantum gravity is based on the hypothesis that the underlying structure of spacetime is that of a random partial order. We survey some of the interesting mathematics that has arisen in connection with the causal set hypothesis, and describe how the mathematical theory can be translated to the application area. We highlight a number of open problems of interest to those working in causal set theory.- . The uniform distribution over Ω 2d without restrictions on N f is dominated by random 2d orders which are approximated by 2d Minkowski spacetime [11] and we will see in what follows that fixing N f changes the associated partition function as a function of N f .

[Show abstract] [Hide abstract]**ABSTRACT:**We define the Hartle-Hawking no-boundary wave function for causal set quantum gravity over the discrete analogs of spacelike hypersurfaces. Using Markov Chain Monte Carlo and numerical integration methods we analyse this wave function in non perturbative 2d causal set quantum gravity. Our results provide new insights into the role of quantum gravity in the observable universe. We find that non-manifold contributions to the Hartle-Hawking wave function can play a significant role. These discrete geometries exhibit a rapid spatial expansion with respect to the proper time and also possess a spatial homogeneity consistent with our current understanding of the observable universe.- @BULLET For the universe in which we live, spacetime is almost flat at the largest scales. @BULLET In a model of full 2-d quantum gravity of causal sets [16], the main contribution from the sum over histories comes from causets which correspond to an interval in M 2 . While this is a toy model, it provides another reason to be interested in causal sets which are approximated by Minkowski spacetime.

[Show abstract] [Hide abstract]**ABSTRACT:**Any discrete approach to quantum gravity must provide some prescription as to how to deduce continuum properties from the discrete substructure. In the causal set approach it is straightforward to deduce timelike distances, but surprisingly difficult to extract spacelike distances, because of the unique combination of discreteness with local Lorentz invariance in that approach. We propose a number of methods to overcome this difficulty, one of which reproduces the spatial distance between two points in a finite region of Minkowski space. We provide numerical evidence that this definition can be used to define a `spatial nearest neighbor' relation on a causal set, and conjecture that this can be exploited to define the length of `continuous curves' in causal sets which are approximated by curved spacetime. This provides evidence in support of the ``Hauptvermutung'' of causal sets. Comment: 32 pages, 16 figures, revtex4; journal version- [Show abstract] [Hide abstract]
**ABSTRACT:**Why are "analogue spacetimes'' interesting? For the purposes of this workshop the answer is simple: Analogue spacetimes provide one with physically well-defined and physically well-understood concrete models of many of the phenomena that seem to be part of the yet incomplete theory of "quantum gravity'', or more accessibly, "quantum gravity phenomenology''. Indeed "analogue spacetimes'' provide one with concrete models of "emergence'' (whereby the effective low-energy theory can be radically different from the high-energy microphysics). They also provide many concrete and controlled models of "Lorentz symmetry breaking'', and extensions of the usual notions of pseudo-Riemannian geometry such as "rainbow spacetimes'', and pseudo-Finsler geometries, and more. I will provide an overview of the key items of "unusual physics'' that arise in analogue spacetimes, and argue that they provide us with hints of what we should be looking for in any putative theory of "quantum gravity''. For example: The dispersion relations that naturally arise in the known emergent/analogue spacetimes typically violate analogue Lorentz invariance at high energy, but do not do so in completely arbitrary manner. This suggests that a search for arbitrary violations of Lorentz invariance is possibly overkill: There are a number of natural and physically well-motivated restrictions one can put on emergent/ analogue dispersion relations, considerably reducing the plausible parameter space. - [Show abstract] [Hide abstract]
**ABSTRACT:**There are several indications (from different approaches) that Spacetime at the Plank Scale could be discrete. One approach to Quantum Gravity that takes this most seriously is the Causal Sets Approach. In this approach spacetime is fundamentally a discrete, random, partially ordered set (where the partial order is the causal relation). In this contribution, we examine how timelike and spacelike distances arise from a causal set (in the case that the causal set is approximated by Minkowski spacetime), and how one can use this to obtain geometrical information (such as lengths of curves) for the general case, where the causal set could be approximated by some curved spacetime. Comment: 8 pages, 2 figures, based on talk by P. Wallden at the NEB XIII conference - [Show abstract] [Hide abstract]
**ABSTRACT:**We present a computational tool that can be used to obtain the "spatial" homology groups of a causal set. Localisation in the causal set is seeded by an inextendible antichain, which is the analog of a spacelike hypersurface, and a one parameter family of nerve simplicial complexes is constructed by "thickening" this antichain. The associated homology groups can then be calculated using existing homology software, and their behaviour studied as a function of the thickening parameter. Earlier analytical work showed that for an inextendible antichain in a causal set which can be approximated by a globally hyperbolic spacetime region, there is a one parameter sub-family of these simplicial complexes which are homological to the continuum, provided the antichain satisfies certain conditions. Using causal sets that are approximated by a set of 2d spacetimes our numerical analysis suggests that these conditions are generically satisfied by inextendible antichains. In both 2d and 3d simulations, as the thickening parameter is increased, the continuum homology groups tend to appear as the first region in which the homology is constant, or "stable" above the discreteness scale. Below this scale, the homology groups fluctuate rapidly as a function of the thickening parameter. This provides a necessary though not sufficient criterion to test for manifoldlikeness of a causal set. Comment: Latex, 46 pages, 43 .eps figures, v2 numerous changes to content and presentation

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