Introduction to Causal Sets: an Alternate View of Spacetime Structure

Home , Causal sets

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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 theory 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 identification B. Why do we need quantum gravity? we accept the of the area of the black hole (actually 1/4 of it) as its entropy. In traditional thermodynamics the concept of entropy is best understood 1. The Einstein field equations in terms of discrete quantum states; not surprisingly, attempts 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 approaches to quantum gravity [13]. suggests the need for a quantum mechanical interpretation 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 interpretation 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 singularity were dominated by quantum mechanical influences scopic version is intimately related to macroscopic space- of on time structure, it seems a good working hypothesis that both and 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 requirements 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 structure 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 interaction. 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 divergence 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 0 1 2 3 constructed of a series of snapshots so rapidly paced that Spacetime consists of events xµ = (x , x , x , x ) = 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 2 µ ν cus on that fact that it is a continuous spectrum of in- ds = gµν dx dx 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 2 arbitrarily large frequency has arbitrarily large energy. cannot influence each other and in such cases ds 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 2 also an argument for discrete space. Nevertheless, a sim- gµν and gµν are conformally equivalent if gµν = Ω 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 equivalente spacetimes have the samee 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 spacetime 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 complete 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. poset really a partially ordered set, or 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 framework. 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 causal set theory to as . 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 quantum 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- scale is called coarse-graining. U(aµ,bν ) ∼ exp(iS/ℏ), (3) Some insight into this issue can be gained by looking at allX 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 throughout 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 ℏ the role played by exp(iS/ ). 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. πτ 4 V = ab , (4) Perhaps the most significant advance along the dy- 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 1 n n <τ > should be proportional to the number of links [29], = tk. the precise relationship between them is yet unknown. qn k kX=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,” sets have been performed [30]. In the thermodynamic 2nd. ed. Springer-Verlag (New York, 1999). limit, these studies exhibit phase transitions correspond- [3] D. Finkelstein, “The space-time code”, Phys. Rev. 184, ing to successively increasing numbers of layers of the 1261-71 (1969). lattice causing the poset to appear more and more con- [4] J. Myrheim, “Statistical Geometry,” CERN preprint TH- tinuous. In this analogy, the thermodynamic limit corre- 2538 (1978). 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 The Creation effects that are absent from the classical dynamics men- ,” pp. 167-179 in J. Leplin (editor), “ of Ideas in Physics: Studies for a Methodology of The- tioned previously can be found in Ref. [31]. ory Construction,” Proceedings of the Thirteenth Annual Symposium in Philosophy (1989). [7] Luca Bombelli, Joohan Lee, David Meyer, and Rafael D. VI. CONCLUDING REMARKS Sorkin, “Space-Time as a Causal Set,” Phys. Rev. Lett. 59, 521-24 (1987); Phys. Rev. Lett. 60, 656 (1988). In this paper we have tried to communicate the pri- [8] P. D. B. Collins, A. D. Martin, E. J. Squires, “Particle mary motivation and key ideas behind the causal set hy- Physics and Cosmology,” Wiley (New York, 1989), Ch. pothesis. Causal sets has emerged as an important ap- 11. proach to quantum gravity having been found to impact [9] William B. Rolnick, “The Fundamental Particles and 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]. 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