Introduction to causal sets: an alternate view of spacetime structure

David D. Reid
Department of Physics and Astronomy
Eastern Michigan University, Ypsilanti, MI 48197

II. THE PROBLEM OF QUANTUM GRAVITY

Abstract. This paper provides a thorough introduction
to the causal set hypothesis aimed at students, and other in-
A. What is quantum gravity?
terested persons, with some knowledge of general relativity
and nonrelativistic quantum mechanics. I elucidate the argu-
ments for why the causal set structure might be the appro- The question of what one means by “quantum grav-
priate structure for a theory of quantum gravity. The logical ity” is not a simple question to answer for the obvious
and formal development of a causal set theory as well as a reason that we do not yet have a complete understanding
arXiv:gr-qc/9909075v1 22 Sep 1999

few illuminating examples are also provided. of quantum gravity. Hence, the answers to this question
are both short and long and perhaps as numerous as the
number of approaches attempting to solve the problem.
I. INTRODUCTION Most physicists agree that by “gravity” we mean Ein-
stein’s theory of general relativity (and possibly a few
When studying general relativity, students often find modified versions of it). General relativity most popu-
that two of the most compelling topics, cosmology and larly interprets gravitation as a result of the geometrical
black holes, lead directly to the need for a theory of quan- structure of spacetime. The geometrical interpretation
tum gravity. However, not much is said about quantum fits because the theory is formally cast in terms of met-
gravity at this level. Those who search for more informa- rical structure gµν on a manifold M .
tion will find that most discussions center around the two There is somewhat less agreement on the meaning of
best know approaches: canonical quantization [1] and su- “quantum.” At first glance, it seems odd that there
perstring theories [2]. This paper seeks to introduce the would be less agreement on the aspect with which we
problem of quantum gravity in the context of a third have much more experience. On the other hand, how-
view, causal sets, which has emerged as an important ever, the fact that we have only been able to perform
concept in the pursuit of quantum gravity. weak-field experimental tests of general relativity leaves
The causal set idea is an hypothesis for the structure of us with much less information to debate. Our experience
spacetime. This structure is expected to become appar- with quantum mechanics tells us that the deviations from
ent for extremely tiny lengths and extremely short times. classical physics it describes are important when dealing
This hypothesis, in its current form, has grown out of an with size scales on the order of magnitude of an atom
attempt to find an appropriate structure for a physical and smaller. Is there a natural size scale at which we ex-
theory of quantum gravity. There is a long tradition of pect the predictions of general relativity to be inaccurate
the importance of causality in relativity. Many of the requiring a new more fundamental theory?
issues faced when confronting the problem of quantum The scale at which theories become important is set
gravity bring considerations of time and causality to the by the values of the fundamental parameters related to
forefront. the processes being described. For example, the speed of
There are many approaches to quantum gravity. Usu- light c is the fundamental constant that determines the
ally, these approaches go through cycles of rapid progress, velocity scale for which relativistic effects (special rela-
during which times an approach will appear very promis- tivity) are appreciable. Likewise, Planck’s constant ℏ,
ing, followed by (sometimes long) periods of slow, or no, among others, sets the scale for systems that must be
progress. The causal set approach has gone through these described by quantum mechanics. The fundamental con-
cycles as well, although to a lesser extent than some, with stants that are relevant to a theory of quantum gravity
early work by Finkelstein [3], Myrhiem [4], ’t Hooft [5], are the speed of light, Planck’s constant, and the univer-
and Sorkin [6]. The recent upswing of interest in causal sal gravitation constant G. These three quantities com-
sets was ignited by a paper written in the late 1980s [7]. bine to form the length and time scales at which classical
Since the hope is that causal sets will lead to a working general relativity break down:
model for quantum gravity, it seems appropriate to begin
1/2
by describing the problem of quantum gravity in general. ℓP = Gℏ/c3 ∼ 10−35 m (1)
The basic ideas behind the causal set approach and some tP = ℓP /c ∼ 10 −44
s,
of the progress that has been forged in recent years will where ℓP is called the Planck length and tP is called the
be discussed in sections III - VI. Planck time.
Size, however, is only one part of what makes a the-
ory “quantum.” Consider, once again, the atom. If we

1
dig deeper than just size and ask why quantum effects 2. Black holes
are important for atoms, the answer is that a relatively
small number of states are occupied (or excited). This A black hole is the final stage in the evolution of mas-
fact is more commonly stated in reverse as a correspon- sive stars. Black holes are formed when the nuclear en-
dence principle requiring that quantum mechanics merge ergy source at the core of a star is exhausted. Once the
with classical mechanics in the limit of large quantum nuclear fuel has run out, the star collapses. If the remain-
numbers, that is, a large number of occupied states. It ing mass of the star is sufficiently high, no known force
is this latter point that truly characterizes quantum be- can halt the collapse. General relativity predicts that the
havior. A quantum theory must therefore enumerate and stellar mass will collapse to a state of zero extent and in-
describe the states of a system in such a way that the finite density – a singularity. In this singular state there
known classical behavior emerges for large numbers of is no spatial extent, time has no meaning, and the ability
states. to extract any physical information is lost. This predic-
What, then, do we mean by “quantum gravity?” In tion may be a message which tells us that a quantum
this paper, my working definition is that theory of gravity is needed if we are to truly understand
the inner workings of black holes.
quantum gravity is a theory that describes While it may be obvious that processes deep within
the structure of spacetime and the effects of black holes must be treated in the framework of quan-
spacetime structure down to sub-Planckian tum gravity, it is less obvious that processes well away
scales for systems containing any number of from the singularity not only require quantum gravity,
occupied states. but may also provide important clues to the form a the-
ory of quantum gravity should take. In 1975 Hawking [10]
In the above definition, the “effects of spacetime struc-
showed that black holes radiate thermally with a black-
ture” include not only the phenomenon of gravitational
body spectrum. This finding, together with a previous
attraction, but also any implications that the spacetime
conjecture that the area of a black hole’s event horizon
dynamics will have for other interactions that take place
can be interpreted as its entropy [11], has shown that the
within this structure.
laws of black hole mechanics are identical to the laws of
thermodynamics. This equivalence only comes about if
we accept the identification of the area of the black hole
B. Why do we need quantum gravity?
(actually 1/4 of it) as its entropy. In traditional ther-
modynamics the concept of entropy is best understood
1. The Einstein field equations in terms of discrete quantum states; not surprisingly, at-
tempts to better understand the reasons for this area
The content of the Einstein field equations of general identification using classical gravity fail. It is widely ex-
relativity, pected that only a quantum mechanical approach will
produce a satisfactory explanation [12]. For this reason,
Gµν = κTµν , (2) black hole entropy is an important topic for most ap-
proaches to quantum gravity [13].
suggests the need for a quantum mechanical interpreta-
tion of gravity [8]. Here Gµν is the Einstein curvature
tensor representing the curvature of space-time, Tµν is 3. The early universe
the energy-momentum tensor representing the source of
gravitation, while κ is just a coupling constant between
the two. The energy-momentum content of spacetime One of the many triumphs of relativistic cosmology is
the explanation of the observed redshift of distant galax-
is already known to be a quantum operator from other
fundamental theories such as quantum electrodynamics ies as an expansion of the universe. However, the univer-
sal expansion extrapolates backward to an early universe
(QED). We have confidence in the reliability of this inter-
pretation because, despite the fact that QED may have that is infinitesimally small and infinitely dense – the big
flaws (discussed below), it has led to extremely accu- bang singularity. Here then, is another situation in which
general relativity predicts something it is not equipped to
rate agreement between theory and experiment [9]. Since
energy-momentum is a quantum operator whose macro- describe. It is fully expected that events near the singu-
larity were dominated by quantum mechanical influences
scopic version is intimately related to macroscopic space-
time structure, it seems a good working hypothesis that both of and on spacetime which necessarily affects the
subsequent evolution of the universe. Presently, cosmo-
its quantum mechanical version should correspond to a
structure of space and time appropriate in the quantum logical implications of the early universe are studied with
the techniques of quantum cosmology [14] which is the
mechanical regime.
quantum mechanics of classical cosmological models. It
has been pointed out that quantum cosmology cannot
be trusted except in very specialized cases [15]. While

2
we have good theories for doing quantum mechanics on do not meet the requirements of a quantum theory as dis-
background spacetimes, the early universe problem re- cussed in section II.A above. The breakdown of general
quires a theory for the quantum mechanics of spacetime relativity near the singularity of a black hole, or more ac-
itself – a theory of quantum gravity. curately, the prediction of a singularity inside of a black
hole, is just one of many examples. Given that general
relativity was formulated prior to quantum mechanics,
4. Unification the fact that it does not meet quantum mechanical re-
quirements is not surprising.
Throughout the history of physics, great strides have The dual role of the metric tensor makes formulating
been made through the unification of seemingly differ- a theory of quantum gravity very different from the for-
ent aspects of nature. One of the most prominent ex- mulations of the other interactions. In quantum gravity
amples is Maxwell’s unification of the laws of electricity we must determine the spacetime structure that acts as
and magnetism. Einstein’s theory of special relativity background to the classical structure of space and time
amounts to a unification of Maxwell’s electromagnetism that we have used to understand all other phenomena.
and Newton’s mechanics showing Newton’s laws to be Furthermore, this ultimate background to classical space-
merely a “low speed” approximation to a more accurate time structure must also be dynamic because it is this dy-
relativistic dynamics. Following relativity theory, quan- namics that will describe quantum gravity just as the dy-
tum mechanics was born. Soon thereafter, Dirac unified namics of classical spacetime describes general relativity.
quantum mechanics and special relativity giving rise to This latter point is the key reason for the incompatibility
modern quantum field theory. between general relativity and quantum mechanics. All
With the above successful unifications behind us, we of our successful experience is with quantum dynamics on
are left with the present situation of having several fun- a spacetime structure, but we have had very little success
damental forces known as the strong, weak, and electro- handling the quantum dynamics of spacetime structure.
magnetic interactions as well as gravitation. Given the This incompatibility challenges some of the most fun-
benefits that we have reaped from past unifications it damental concepts in physics. In field theory, we take as
seems natural that the search for deeper insight through the source of the field some distribution Tµν (of charge,
unification should continue. The recent success of the matter, energy, etc.). However, the concept of a spatial
electroweak theory has confirmed the value of this search. distribution of charge, for example, has no meaning apart
There are now some seemingly consistent models for the from the knowledge of the spatial structure. Without the
unification of the strong and electroweak theories. The rules for how to measure relative positions we cannot de-
very fact that these interactions can be mathematically fine the spatial distribution of anything. In quantum me-
unified in a manner consistent with macroscopic obser- chanics, Schrödinger’s equation describes the time evolu-
vations suggests that a truly physical unification exists. tion of the wave function Ψ(r, t). The concept of time
Gravity is the only fundamental force yet to have a evolution, however, requires an existing knowledge of
consistent quantum mechanical theory. It is widely be- temporal flow which comes from the spacetime structure.
lieved that until such a quantum mechanical description As a final example, consider the concept of interaction.
of gravity is attained, placing our understanding of grav- We generally think of interactions in a manner intimately
ity on the same level as that of the other interactions, connected with causality: an interaction precedes and
true unification of gravity with the other forces will not causes an effect. Causality, however, is a concept that
be possible [8]. can only be defined once the structure of spacetime is
known.

C. The incompatibility between general relativity

and quantum mechanics III. THE CAUSAL SET HYPOTHESIS

For all of the interactions except gravity, our present The above discussion implies the need for a spacetime
theoretical understanding of physics is such that systems structure that will underpin the classical spacetime struc-
interact and evolve within a background spacetime struc- ture of general relativity. The causal set hypothesis pro-
ture. This background structure serves to tell us how poses such a structure. Causal sets are based on two
to measure distances and times. In general relativity primary concepts: the discreteness of spacetime and the
it is the spacetime structure itself that we must deter- importance of the causal structure. Below I discuss these
mine. This spacetime structure then, acts both as the two founding concepts in more detail.
background structure for gravitational interactions and
as the dynamical phenomenon giving rise to this inter-
action. In general relativity the structure of spacetime
is determined by the Einstein field equations (2). These
field equations are, however, purely classical in that they

3
 A. Spacetime is discrete lieved that the perturbation series diverges. This diver-
gence is generally overlooked because QED is only a par-
The causal set hypothesis assumes that the structure tial theory and not a complete theory of elementary in-
of spacetime is discrete rather than the continuous struc- teractions (see Ch. 1 of Ref. 1). Therefore, we use QED
ture that physics currently employs. Discrete means that under the assumption that it is accurate for the phe-
lengths in three-dimensional space are built up out of a nomenon it was created to describe and that some aspect
finite number of elementary lengths ℓe which represents of a more fundamental theory will eventually solve its di-
the smallest allowable length in nature and the flow of vergence problem. The causal set idea proposes that the
time occurs in a series of individual “ticks” of duration aspect of more fundamental theory that will naturally
te which represents the shortest allowable time interval. solve this divergence problem in QED is a discrete struc-
The idea that something which appears continuous is ture of spacetime.
actually discrete is very common in physics and everyday
life. Any bulk piece of matter is made up of individual
atoms so tightly packed that the object appears contin- B. Causal structure contains geometric information
uous to the naked eye. Likewise, any motion picture is
constructed of a series of snapshots so rapidly paced that Spacetime consists of events xµ = (x0 , x1 , x2 , x3 ) =
the movie appears to flow continuously. (ct, x, y, z), that is, points in space at various times. At
Why hold a similar view of spacetime? There are some events physical processes take place. Processes that
many arguments for a discrete structure. The most fa- occur at one event can only be influenced by those oc-
miliar ones are related to electrodynamics. Here, I will curring at another event if it is possible for a photon
first try to motivate the idea of discreteness by consider- to reach the latter event from the earlier one. To cap-
ing the electromagnetic spectrum. The electromagnetic ture the essence of what one means by “causal struc-
spectrum gives us the range for the frequencies and wave- ture,” consider the example of the flat Minkowski space
lengths of electromagnetic radiation, or, photons. Of the of special relativity. In flat spacetime, two events are said
many interesting aspects of this spectrum, here let’s fo- to be causally connected and their spacetime separation
cus on that fact that it is a continuous spectrum of in- ds2 = gµν dxµ dxν is called timelike if it is positive (using
finite extent. Current theory predicts that the allowed a + − −− signature) and null if it is zero. Two events
frequencies of photons extend continuously from zero to are not causally connected if it is not possible for a pho-
infinity. The relation E = hν implies that a photon of ton from one event to arrive at the other; these events
arbitrarily large frequency has arbitrarily large energy. cannot influence each other and in such cases ds2 is neg-
The local conservation of energy suggests that such infi- ative and referred to as spacelike. When we speak of the
nite energy photons should not exist. If one adopts the causal structure of a spacetime, we mean the knowledge
(somewhat controversial) view that what cannot physi- of which events are causally connected to which other
cally exist should not be predicted by theory, there ought events. For a more general discussion of causal structure
to be a natural cutoff of the electromagnetic spectrum see reference [16].
corresponding to a maximum allowed frequency. Since It is now well established that the causal structure of
the frequency of a photon is the inverse of its period, a spacetime alone determines almost all of the informa-
discrete temporal structure provides a natural cutoff in tion needed to specify the metric [17, 18] and therefore
that the minimum time interval te implies a maximum the gravitational field tensor. The causal structure de-
frequency νmax = 1/te . termines the metric up to an overall multiplicative func-
Because of relativity any argument for discrete time is tion called a conformal factor. We say that two metrics
also an argument for discrete space. Nevertheless, a sim- gµν and geµν are conformally equivalent if geµν = Ω2 gµν ,
ilar argument for discrete space can be given in terms of where Ω is a smooth positive function. Since all confor-
wavelength. The electromagnetic spectrum, being con- mally equivalent spacetimes have the same causal struc-
tinuous, allows arbitrarily small wavelengths. The de- ture [19], the causal structure itself nearly specifies the
Broglie relation p = h/λ implies that a wavelength arbi- metric.
trarily close to zero corresponds to a photon of arbitrarily
large momentum. The local conservation of momentum
suggests that photons of infinite momentum should not C. Causal sets
exist. The minimum length implied by a discrete space-
time structure provides a natural cutoff for the wave- Lacking the conformal factor from the causal structure
length λmin = ℓe . essentially means that we lack the sense of scale which
In terms of QED, this problem can be seen in the fact allows for quantitative measures of lengths and volumes
that the infinite perturbation series requires the existence in spacetime. However, if spacetime is discrete, the vol-
of all the photons in the electromagnetic spectrum. In ume of a region can be determined by a procedure al-
this sense QED predicts the existence of these photons most as simple as counting the number of events within
of infinite energy-momentum. However, it is widely be- that region. Therefore, if nature endows us with discrete

4
spacetime and an arrow for time (the causal structure), A. Taketani stages
we have, in principle, enough information to build com-
plete spacetime metric tensors for general relativity. This In 1971 Mituo Taketani used Newtonian mechanics as
combination of discreteness and causal structure leads a prototype to illustrate his ideas on the development of
directly to the idea of a causal set as the fundamental physical theories [21]. According to Taketani, physical
structure of spacetime. theories are developed in three stages that he referred to
A causal set may be defined as a set of events for which as the phenomenological, substantialistic, and essential-
there is an order relation ≺ obeying four properties: istic stages.
The phenomenological stage is where the initial obser-
1. transitivity: if x ≺ y and y ≺ z then x ≺ z; vations occur that place the existence and knowledge of
2. non-circularity: if x ≺ y and y ≺ x then x = y; the new phenomenon (or “substance”) on firm standing.
In Ref. [21] this stage in the development of Newtonian
3. finitarity: the number of events lying between any mechanics is associated with the work of Tycho Brahe
two fixed events is finite; who observed the motions of the planets with unprece-
dented accuracy.
4. reflexivity: x ≺ x for any event in the causal set. Within the substantialistic stage, rules that describe
the new substance are discovered; that is, we come to rec-
The first two properties say that this ordered set is ognize “substantial structure” in the new phenomenon.
really a partially ordered set, or poset for short. Specifi- These rules would then play an important role in helping
cally, non-circularity amounts to the exclusion of closed to shape the final understanding. For Newtonian me-
timelike curves more commonly known as time machines chanics, Taketani associated this stage with the work of
[20]. The finitarity of the set insures that the set is dis- Johannes Kepler who provided the well known three laws
crete. The reflexivity requirement is present as a conve- of planetary motion.
nience to eliminate the ambiguity of how an event relates In the essentialistic stage, “the knowledge penetrates
to itself. In the present context of using a poset to repre- into the essence” of the new phenomenon. This is the
sent spacetime, reflexivity seems reasonable in that the final stage when the full theory of this new substance is
spacetime separation between an event and itself cannot known within appropriate limits of validity. Of course,
be negative requiring an event to be causally connected the work of Issac Newton himself represents this stage.
to itself. We can combine these statements to give the Even though Taketani used Newtonian mechanics,
following definition: there are many examples to which his ideas apply. Sakata
used Taketani’s philosophy to discuss the development of
A causal set is a locally finite, partially or-
quantum mechanics [22]. Similarly, the development of
dered set.
electromagnetic theory falls neatly into Taketani’s frame-
work. The phenomenological stage of electromagnetism
IV. THE DEVELOPMENT OF CAUSAL SET could be associated with the work of Benjamin Franklin
THEORY and William Gilbert. The substantialistic stage is nicely
represented by the work of Michael Faraday and Hans
Oerstead. The essentialistic stage is then represented by
With the conceptual foundation of causal sets clearly James Maxwell’s completion of his famous equations.
laid, let us now turn to the issue of developing a phys- Taketani realized that his three stages will not al-
ical and mathematical formalism which I loosely refer ways apply identically to the development of all physical
to as causal set theory. The development of causal set ideas. Since we currently know of no observed phenom-
theory is still far from complete. In fact, it is even less ena whose explanation clearly requires a complete theory
developed than some of the other approaches to quantum of quantum gravity, it is clear that the problem of quan-
gravity such as superstring theory and canonical quan- tum gravity is not based on experimental observations.
tization mentioned previously. For this reason, quan- As a result of this fact, the development of causal set
tum gravity by any approach is an excellent theatre for theory largely skips the phenomenological stage. There-
fine tuning ideas about how theory construction should fore, think of the development of causal sets in a two-step
proceed from founding observations, hypotheses, and as- process corresponding roughly to Taketani’s substantial-
sumptions. Hence, to better understand current thought istic and essentialistic stages. As a matter of terminol-
on how the development of causal set theory should pro- ogy, note that the substantialistic stage, in which phe-
ceed, I will first discuss some of the ideas on theory de- nomenology is described, plays the role of kinematics in
velopment in general that have influenced causal set re- Newtonian mechanics, while in the essentialistic stage the
search. full dynamics is developed. Consequentially, I will refer
to the two processes in the development of causal set
theory as “kinematics” and “dynamics.”

5
 B. Causal set kinematics are important because the founding ideas behind causal
sets in Sec. III suggests that a manifold (M, gµν ) emerges
The kinematic stage concerns gaining familiarity with from the causal set C if and only if an appropriately
and further developing the mathematics needed to de- coarse-grained version of C can be produced by a unit
scribe causal sets. This mathematics primarily falls un- density sprinkling of points into M [25]. This shows us
der the combinatorial mathematics of partial orders [23]. that an important problem in the development of causal
These techniques are not part of traditional physics train- set kinematics is to determine how to appropriately form
ing and have, therefore, not been widely used to analyze a coarse-graining of a causal set.
physical problems. Moreover, research in this branch of
mathematics has been performed largely by pure math-
ematicians; the problems they have chosen to tackle are C. Causal set dynamics
generally not those that are of most interest to physicists.
What we need from the kinematic stage are the math- The final stage in the development of causal set theory
ematical techniques for how to extract the geometrical is the stage in which we come to understand the full dy-
information from the causal order (i.e., working out the namics of causal sets. In this stage we devise a formalism
correspondence between order and geometry) and how to for how to obtain physical information from the behav-
do the counting of causal set elements that will allow us ior of the causal set and how this behavior governs our
to determine spacetime volumes. sense of space and time. Here we require something that
For an important, specific example of where causal set might be considered a quantum mechanical analog to the
kinematics is needed, consider the correspondence prin- Einstein field equations (2). Since our present framework
ciple between spacetime as a causal set and macroscopic for physical theories is based on a spacetime continuum,
spacetime. General relativity tells us that spacetime is a our experience is of limited use to us in this effort. De-
four-dimensional Lorentzian manifold. If causal sets com- spite this limitation, one commonly used approach stands
prise the true structure of spacetime they must produce out as the best candidate for a dynamical framework for
a four-dimensional Lorentzian manifold in macroscopic causal sets. This method is most commonly known as the
limits (such as a large number of causal set elements). path-integral formulation of quantum mechanics [26].
The mathematics of how we can see the manifold within This path-integral technique seems best suited to
the causal set is a kinematic issue that must be addressed. causal sets because at its core conception (a) it is a space-
On the natural length scale of the causal set one does time approach in that it deals directly (rather than indi-
not expect to see anything like a manifold. Trying to see rectly) with events; that is, we propagate a system from
a manifold on this scale is like trying to read this article one event configuration to another; and (b) it works on
under a magnification that resolves the individual dots a discrete spacetime structure. As currently practiced,
of ink making up the letters. To discern the structure of the path-integral approach determines the propagator
these dots we look upon them at a significantly different U (aµ , bν ) by taking all paths between the events aµ and
scale than the size scale of the dots. Similarly, we need bν in a discretized time and summing over these paths
a mathematical change-of-scale in order to extract the using an amplitude function exp(iS/ℏ):
manifold structure from the causal set. This change-of- X
scale is called coarse-graining. U (aµ , bν ) ∼ exp(iS/ℏ), (3)
Some insight into this issue can be gained by looking at all paths
the reverse problem of forming a causal set from a given
metric manifold (M, gµν ). This is achieved by randomly where S is the action for a given path. Continuous space-
sprinkling points into M . The order relation of this set time enters in at two places. In a continuum there are an
of points is then determined by the light-cone structure infinite number of paths between two events, “all paths”
of gµν . Since we need to ensure that every region of the are generated by integrating over all intermediate points
spacetime is appropriately sampled, that is, that highly between the two events; this is the “integral” part of
curved regions are represented equally well as nearly flat path-integration. Since each of these paths were dis-
regions are, the sprinkling is carried out via a Poisson cretized into a finite number of points N , the second
process [24] such that the number of points N sprinkled place where continuous spacetime is recovered is to take
into any region of volume V is directly proportional to the limit N → ∞.
V . Using a two-dimensional Minkowski spacetime, Fig. In a discrete setting the number of paths and the num-
1 provides a picture of such a sprinkling at unit density ber of points along the paths are truly finite. Hence, the
ρ = N/V = 1. final limit as well as the integration to generate all paths
Since the causal sets generated by random sprinklings are not performed. Since causal sets would not require
are only expected to be coarse-grained versions of the integration, calling this the “path-integral formulation”
fundamental causal set, their length and time scales are seems inappropriate. This method essentially says that
not expected to be the fundamental length and time of the properties of a system in a given event configuration
nature. Nevertheless, these studies of random sprinklings depend on a sum over all the possible paths through-
out the history of this system. Therefore, the alternative

6
name for this technique, the sum-over-histories approach, A. Volume
is better suited for causal sets. The word “histories” is
particularly appropriate because, as mentioned earlier, As discussed in Sec. III, the causal set hypothesis is
we take the arrow of time to be fundamental. partially founded on the fact that the causal structure
There are several key issues that must be resolved be- of spacetime contains all of the geometric information
fore a sum-over-histories formulation of causal sets can needed to specify the metric tensor up to a conformal
be completed. One such issue involves the need to iden- factor which prevents us from determining volumes. One
tify an amplitude function for causal sets analogous to example which shows how, in principle, volumes might
the role played by exp(iS/ℏ). Secondly, the required for- be extracted from the causal set has been discussed by
mulation must do more than just propagate the system Bombelli [28]. Gerard ’t Hooft has shown [5] that, in
because the entire dynamics must come from this for- Minkowski space, the volume V of spacetime bounded
malism. The procedure outlined above is presently inad- by two causally connected events a and b is given by
equate for these purposes; a modified, or better, general-
ized sum-over-histories method must be developed. 4
πτab
Perhaps the most significant advance along the dy- V = , (4)
24
namical front is the recent development by Rideout and
Sorkin of a general classical dynamics for causal sets [27]. where τab is the proper time between events a and b. We
In this model, causal sets are grown sequentially, one can apply this expression to causal sets by relating τab
element at a time, under the governance of reasonable to the number of links in the longest path between a and
physical requirements for causality and discrete general b. The volume V is then identified with the number of
covariance. When a new element is introduced, in go- elements in this region of spacetime.
ing from an n-element causal set to an (n + 1)-element Spacetime is dynamic, however. The above procedure
causal set, it is associated with a classical probability qn of counting the number of links between two events is
of being unrelated to any existing element according to subject to (perhaps large) statistical fluctuations. There-
fore, while it is believed that the expected proper time
n  
X
1 n < τ > should be proportional to the number of links [29],
= tk . the precise relationship between them is yet unknown.
qn k
k=0 Attempts to numerically determine this relationship via
computer simulations remain inconclusive [28].
The primary restriction is that the tk ≥ 0; hence, there
is a lot of freedom with which different models can be
explored. This framework has the potential to teach
B. Coarse-graining
us much about the needed mathematical formalism for
causal sets, the effects of certain physical conditions, and
the classical limit of the eventual quantum dynamics for While the labeling of Fig. 2 clearly suggests that it is
causal sets. a two-dimensional example of a causal set, note that our
physical sense of dimensionality (given us by relativity)
is intimately related to the manifold concept. Although
V. ILLUSTRATIVE EXAMPLES the causal set in Fig. 2 looks suspiciously regular it may
not be immediately obvious whether or not this set can
be embedded into any physically viable two-dimensional
To illustrate some of the points discussed above I
spacetime.
present the 72 element causal set shown in Fig. 2. The
As an example of one form of coarse-graining, we can
black dots represent the elements of the causal set. The
look at this causal set on a time scale twice as long as its
graphical form in which this causal set is shown is known
natural scale. Figure 3a is a coarse-grained version of the
as a spacetime-Hasse diagram. The term “Hasse dia-
causal set in Fig. 2 for which only even time steps are
gram” is borrowed from the mathematical literature on
shown and Fig. 3b is a subset of Fig. 2 showing events
posets [23]. Figure 2 is also a spacetime diagram in
that only occur at odd time steps. In both cases we
the usual sense. The solid lines in the figure are causal
find causal sets that clearly can be embedded into two-
links, i.e., lines are only drawn between events that are
dimensional Minkowski space. In a realistic situation this
causally related; however, for clarity only those relations
fact would suggest that the fundamental causal set just
that are not implied by transitivity (the links) are ex-
might represent a physically discrete spacetime.
plicitly shown. The causal set shown has 15 time steps
as enumerated along the right side of the figure. Hence,
the first time step at the bottom shows 7 “simultaneous”
C. Dynamics
events.
As stated above, a sum-over-histories dynamical law
for causal sets requires the identification of an amplitude

7
function. As an example, one could start by consider- branch of quantum gravity research. Those with further
ing an amplitude modeled after the familiar amplitude interest can find more detailed discussion of causal sets
of the continuum path-integral formulation in Eq. (3), in Refs. [35, 36].
i.e., exp(iβR). Here, β plays the role of 1/ℏ and R plays
the role of the action S. In quantum field theory the oscil-
latory nature of this amplitude causes problems that are VII. ACKNOWLEDGMENTS
sometimes bypassed by performing a continuation from
real to imaginary time (often referred to as a Wick rota- The author would like to thank Drs. R. D. Sorkin,
tion). Similarly, it is convenient here to consider the case J. P. Krisch, N. L. Sharma, and E. Behringer for help-
β → iβ giving an amplitude ful suggestions. Support of the Michigan Space Grant
Consortium is gratefully acknowledged.
A = exp(−βR), (5)

where we can take R, for example, to be the total number

of links in the causal set.
This model is interesting because the amplitude A,
which acts as a weight in the sum-over-causal sets, has
the form of a Boltzmann factor exp(−E/kT ). The math- [1] Abhay Ashtekar, “Lectures on Non-perturbative Canon-
ematical structure of the causal set dynamics then be- ical Quantum Gravity,” World Scientific (New Jersey,
comes very similar to that of statistical mechanics. Stud- 1991).
ies of the statistical mechanics of certain partially ordered [2] M. Kaku, “Introduction to Superstrings and M-Theory,”
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ing to successively increasing numbers of layers of the 1261-71 (1969).
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sponds to one macroscopic limit of causal set theory in [5] G. ’t Hooft, “Quantum gravity: a fundamental prob-
which the number of causal set elements goes to infinity. lem and some radical ideas,”, pp. 323-45, in “Recent
Such results, therefore, are somewhat suggestive that an Developments in Gravitation” (Proceedings of the 1978
appropriate choice of amplitude might indeed lead to the Cargese Summer Institute) edited by M. Levy and S.
expected kind of continuum limit. Deser (Plenum, 1979).
Another, more detailed, example of a quantum dynam- [6] R. D. Sorkin, unpublished (1979). See, for example, “A
ics for causal sets that exhibits the kind of interference Specimen of Theory Construction from Quantum Grav-
ity,” pp. 167-179 in J. Leplin (editor), “The Creation
effects that are absent from the classical dynamics men-
of Ideas in Physics: Studies for a Methodology of The-
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Symposium in Philosophy (1989).
[7] Luca Bombelli, Joohan Lee, David Meyer, and Rafael D.
VI. CONCLUDING REMARKS
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pothesis. Causal sets has emerged as an important ap- 11.
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other approaches such as the spin network formalism [32]. Their Interactions,” Addison-Wesley (Reading, 1994), p.
Adding to the importance of the causal set approach 72.
is the fact that it led Sorkin to predict a non-zero cos- [10] S. W. Hawking, “Particle Creation by Black Holes,”
mological constant nearly a decade ago [25]. In light of Comm. in Math. Phys. 43, 199-220 (1975).
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Chicago Press, (Chicago, 1994) Ch. 7.
Before a final causal set theory can be constructed,
[13] For discussions on how various approaches address this
much work remains. Studies of random sprinklings in
problem see the following papers. A. Ashtekar, J. Baez,
both flat and curved spacetimes, the mathematics of par-
A. Corichi, and K. Krasnov, “Quantum Geometry and
tial orders, and the behavior of fields that sit on a discrete Black Hole Entropy,” Phys. Rev. Lett. 80, 904-907
substructure are just a few areas of needed investigation. (1998); G. Horowitz, “The Origin of Black Hole Entropy
Enough progress on causal sets has been made, however, in String Theory,” talk given at Pacific Conference on
to establish the causal set hypothesis as a very promising

8
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[16] R. M. Wald, “General Relativity,” University of Chicago VIII. FIGURE CAPTIONS
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Figure 2. A spacetime-Hasse diagram of a two-
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[19] See reference 16, Appendix D. the right-hand-side of the figure.
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stein’s Outrageous Legacy,” Norton (New York, 1994), by the odd time steps only. (b) The subset formed by
Ch. 14. the even time steps only. Both coarse-grainings are more
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