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Causal Sets Approach to Quantum Gravity
Mir Emad Aghili1 ∗
1
Department of Physics and Astronomy, University of Mississippi, 38677
Introduction
The fact that singularities happen in general relativity is only one of the reasons for us to
look for a new theory for gravity. There are many alternative theories for general relativity,
among those are the theories that are trying to use quantum field theory and define force
carriers and etc. Also there are ones that try to quantize general relativity itself since
quantum theories have been the most successful theory of the nature. Loop quantum gravity
is one of the most famous quantum theories which quantizes the gravitational action using
the Hamiltonian procedure and its constraints. Causal set theory is a different approach
that will be discussed in this paper.
Kinematics and Topology
Causal set theory was first started by idea of ’t Hooft [1] and Myrheim [2] and the pro-
gram was established by R. Sorkin and L. Bombelli [3]. It is the most minimalistic approach
to quantum gravity that only uses a set of points with their causal relation to explain the
spacetime. In causal set theory the spacetime does not exist is smaller scales and it is and
emergent of causal sets. spacetime is considered to be a locally finite partially ordered set
that is:
(1) x x (reflexive)
(2) y x , x y → x = y (antisymmetric)
(3) x y , y z → x z (transitive)
(4) for any pair of elements x and z of set α, the set {y|x y z} is finite (locally finite)
where x y means that x comes before (precedes) y. 2n
subsets of real numbers 1 to n
ordered by inclusion is one of the simplest partially ordered sets. Each element of causal
set is an event in spacetime that can be connected to the events that are causally (through
a timelike or null path) related to them and disconnected from the spacelike events. The
average density of the points in causal set is ρ and a given volume element can be measured
by number of elements that it contains (V = N/ρ). A chain is a set of connected events
and antichain is a set of spacelike events. The volume contained between the chronological1
∗maghili@go.olemiss.edu
1Chronological past/future of a point x (usually denoted by I±(x)) consists of all of the points that are
timelike related to it.
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future of a point p and chronological future of point q is called and Alexandrov set.
Figure 1: An Alexandrov interval of p, q
Figure 2: Another view of an Alexandrov
interval
It was shown in [2] that one can encode all the properties of a manifold and the metric
up to a conformal coefficient in the causal relation of points in a manifold
Lorentzian Metric = Local Causal Structure + Volume Element.
This is one of the evidences that causal set can be a good alternative for a new theory for
gravity. There are two different methods to reconstruct a manifold form causal sets. In first
method one initially starts with a causal set. In fact most of the causal sets existing can
not be embedded in a manifold. Knowing that, from the ones that can be embedded in the
manifold one should be able to extract information such as the dimension of the manifold.
There are some work done on the dimension of a causal set by D. Meyer [4] which is based
on the relation of the volume to the longest chain (discrete version of proper time2
) and it
works well for flat manifolds.
The embedded causal set should be faithful to the original causal set i.e. the causal
relation of the causal set should remain valid once their are embedded. A causal set might
be embedded in different manifolds but these manifolds should be reasonably close. This
leads us to the Hauptfermütüng conjecture
Hauptvermütüng: If a causal set C faithfully embeds at the same density into two dis-
tinct spacetimes (M1, g1) and (M2, g2) then these spacetimes are related by an approximate
2proper time is the time that all the observers agree on that and is the longest path between two points
separated by a time like distance
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Figure 3: A simple causal set known as
“crown poset” which can not be embedded
to a 2 dimensional manifold.
Figure 4: A B4 = (x + 1)4
binomial causal
set. Length of each antichain is equal to
value coefficient of each term and number of
links is equal to the corresponding power.
isometry.
Although this conjecture has not been proved yet the causal set physicists are very close
to the prove and we take it as to be true. To prove this one needs to define a notion of
distance between the different Lorentzian manifolds that can be constructed over the same
causal set [5].
Second way to obtain a causal set is to sprinkle the points randomly in an already existing
manifold which produces a causal set that is already embedded in manifold .But one has
to be careful when they sprinkle the points in the manifold. Sprinkling should be in a way
that it is covariant. It is shown in [6] that using Poisson process for sprinkling does satisfy
that property. In Poisson sprinkling the probability of having i points in volume V is
(ρV )i
i!
exp(−ρV ). (1)
If we take the average distance between the points in the sprinklings to be 1 then the each
volume is equal to the number of points it is containing. The difficulty in causal sets is that
the nearest neighbors can be arbitrarily far as long as they are close to the light cone, this
means that one has to give up locality and also the volume of nearest neighbors is infinite
because nearest neighbors are not defined with a ball, rather they are defined by hyperbola.
Dynamics
To study dynamics of causal sets one needs to know about the action on causal sets and
propagation of different fields on causal sets. If we want to reproduce general relativity in
continuum limit, the action should reproduce Einstein-Hilbert action. For that one needs
to know the notion of scalar curvature in causal sets. Different methods have been used
to find scalar curvature. One of the ways is to write scalar curvature in terms of relevant
parameters in causal set such as volume and proper time [7]
V = Vflat 1 −
d
24(d + 1)(d + 2)
Rτ2
+
d
24(d + 1)
R00τ2
+ · · · (2)
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Using that and another covariant equation with the same parameters and solve for scalar
curvature [8]
R =
1
D
V (τ)
kdτd
− 1 (Id,0 + Id,1 +
1
4
kd)τd
−
α(p,q)
V (p, x)
kdτd(p, x)
− 1 dd
x (3)
ad =
d
24(d + 1)(d + 2)
, bd =
d
24(d + 1)
D = (add + bdId,1)τd+2
Id,0 =
cd−1
2dd(d + 1)(d + 2)
, Id,1 =
cd−2Jd+1
2d+1d(d + 2)
where kd = cd−1
2d−1d
, d is dimension, τ is the proper time, V is the volume of an Alexandrov
set and
cd =
2πd/2
d Γ(d/2)
, Jd =
π/2
−π/2
(cosθ)d
dθ.
If one discretizes these equation and write them in terms of number of points and find the
discrete scalar curvature and consequently the action.
Another solution to this problem is to first find the nonlocal d’Alembertian on causal set
B(d)
φ(x) =
1
l2
αdφ(x) + βd
nd
i=1
C
(d)
i
y∈Li
φ(y) (4)
where B is causal set d’Alembertian. In continuum limit it should reproduce the regular
local d’Alembertian
lim
ρ→0
Bφ(x) = ( + R(x))φ(x) (5)
where R is the scalar curvature and the constants are to be determined using this condition.
Letting this operator act on a constant scalar field of compact support one can get a result
for scalar curvature. There has been some effort in this direction for 2-D [9] and later it was
extended to 4 and d-dimensions [10] [11].
Recently there has been some work along reproducing the Gibbons-Hawking boundary term
in causal sets [12]. Gibbons-Hawking boundary term is
Sboundary =
∂M
dd−1
x
√
hK (6)
where K is extrinsic curvature of the boundary hypersurface, h is the determinant of metric
on boundary hypersurface and is 1 when the boundary is timelike and −1 when the
boundary is spacelike. This term is necessary in general relativity for spacetimes with
boundary. One can produce a candidate for boundary term using a few layers of the future
and past of the boundary. Number of layers in calculations depends on the dimension.
Conclusion
Although in the case of quantum gravity it is hard to see the effects, one reason is
smallness of gravitational coupling constant G and the other reason is that one can not do
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any measurements in such small scales that quantum gravity shows its effects.
There has been some heuristic predictions of cosmological constant using causal sets [13],
but theory of quantum gravity is still not experimentally measurable. many alternatives
have failed but the causal set theory for gravity is still one of the promising theories.
References
[1] G. ’t Hooft, “Quantum Gravity: a fundamental problem and some radical ideas,” in
Recent Developments in Gravitation. Cargèse 1978, M. Lévi and S. Deser, eds. (Plenum
1979).
[2] J. Myrheim, “Statistical Geometry,” 1978 preprint, Ref. TH 2538-CERN, unpublished.
[3] L. Bombelli, J. Lee, D. A. Meyer and R. D. Sorkin, “Spacetime as a Causal Set,” Phys.
Rev. Lett. 59, 521-524 (1987).
[4] D. Meyer, The Dimension of Causal Sets, PhD thesis, M.I.T. 1988.
[5] L. Bombelli, “Statistical Lorentzian Geometry and the Closeness of Lorentzian Mani-
folds,” arXiv:gr-qc/0002053v2 (2000).
[6] L. Bombelli, J. Henson and R. Sorkin, “Discreteness without Symmetry Breaking: a
Theorem,” arXiv:gr-gc/060500v1 (2006).
[7] G. Gibbons and S. Solodukhin, “The Geometry of Small Causal Diamonds,” arXiv:hep-
th/0703098v3 (2015).
[8] R. Sverdlov and L. Bombelli, “Gravity and Matter is Causal Set Theory,”
arXiv:0801.0240v2 (2008).
[9] D. Benincasa and F. Dowker, “The Scalar Curvature of a Causal Set,” arXiv:1001.2725v4
(2011).
[10] F. Dowker and L. Glaser, “Causal Set d’Alembertians for Various Dimensions,”
arXiv:1305.2588v2 (2013).
[11] A. Belenchia, D. Benincasa and F. Dowker, “The Continuum Limit of a 4-dimensional
Causal Set Scalar d’Alembertian,” arXiv:1510.04656v1 (2015).
[12] M. Buck, F. Dowker, I. Jubb and S. Surya, “Boundary Terms for Causal Sets,”
arXiv:1502.05388 (2015).
[13] R. Sorkin, Paper presented to the conference on The History of Modern Gauge Theories,
held Logan, Utah, July 1987: Int. J. Theor. Phys. 33 : 523-534 (1994).
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