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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.

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(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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