A manifestly covariant framework for causal set dynamics
Fay Dowker, Nazireen Imambaccus, Amelia Owens, Rafael Sorkin, Stav Zalel
AA manifestly covariant framework for causal setdynamics
Fay Dowker , Nazireen Imambaccus , Amelia Owens , Rafael Sorkin ,and Stav Zalel Blackett Laboratory, Imperial College London, SW7 2AZ, U.K. Perimeter Institute, 31 Caroline Street North, Waterloo ON, N2L 2Y5,Canada.March 27, 2020
Abstract
We propose a manifestly covariant framework for causal set dynamics. Theframework is based on a structure, dubbed covtree , which is a partial order oncertain sets of finite, unlabeled causal sets. We show that every infinite path incovtree corresponds to at least one infinite, unlabeled causal set. We show thattransition probabilities for a classical random walk on covtree induce a classicalmeasure on the σ -algebra generated by the stem sets. Contents n -orders . . . . . . . . . . . . . . . . . . . . . . 62.3 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 a r X i v : . [ g r- q c ] M a r .4 Classical Sequential Growth models . . . . . . . . . . . . . . . . . . . 10 R ( S ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 B.1 Appendix to section 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . 28B.2 Appendix to section 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 29B.3 Appendix to section 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . 29
General Relativity (GR) has the property of general covariance. As discussed insection 7 of [1], this gauge invariance of GR has two facets. The first is that physicalstatements – or, equivalently, properties or predicates or events – in GR must bediffeomorphism invariant. The second is that the equations of motion of GR arediffeomorphism invariant so that metrics in a diffeomorphism equivalence class musteither all satisfy or all not satisfy the equations of motion. As stated in [1], “[InGR] the second facet of covariance flows directly from the first as a consistencycondition, because it would be senseless to identify two metrics one of which wasallowed by the equations of motion and the other of which was forbidden; andconversely, the kinematical identification must be made if one wishes the dynamicsto be deterministic. Thus, the first or “ontological” facet of general covariancetends to coalesce with its second or “dynamical” facet.”There is a widespread expectation that GR will turn out to be an approximation,at large scales and in certain circumstances, to a deeper theory of quantum gravity.A. Einstein’s struggles over the understanding of general covariance were central to he development of GR and one might expect that grappling with the correspondingissues within quantum gravity will be important to its development too [2]. Therequirement that the diffeomorphism invariance of GR must emerge from quantumgravity in the large scale approximation is indeed used to formulate guiding preceptsfor each approach to quantum gravity, though the form that these precepts takevaries from approach to approach.In the case of the causal set approach to the problem of quantum gravity, anydeeper precept of general covariance cannot be literally diffeomorphism invariancebecause diffeomorphism is a continuum concept and causal set theory is discrete.Causal set theory postulates that the fundamental structure of spacetime is atomicat the Planck scale and takes the form of a causal set or locally finite partialorder. The elements of the causal set are the atoms of spacetime and continuumspacetime is an approximation to the causal set at large scales. The order relation ofthe underlying causal set reveals itself as the causal structure of the approximatingcontinuum spacetime and the number of causal set elements manifests itself as thespacetime volume of the approximating continuum spacetime. For a recent reviewof causal set theory see [3].The structure of continuum spacetime, then, emerges from Order and Numberand this central conjecture of causal set theory has an immediate consequence:the physical content of a causal set is independent of what mathematical objectsthe causal set elements are and is also independent of any additional labels thosecausal set elements might carry: only the order relation of the elements and thenumber of elements has physical meaning. This “mathematical-identity-and-labelindependence” is a good candidate for a condition of general covariance in causalset theory, at least as far as the first facet, mentioned above, goes .This form of general covariance is a consequence of a more general guidingheuristic, Occam’s Razor, applied in the particular case of causal set quantumgravity and grounded in earlier seminal work, theorems in Lorentzian geometry byPenrose, Kronheimer, Hawking and Malament [4, 5, 6]. These theorems show that At least, this is the kinematics of the theory. In the full quantum theory this statement will berevised to take account of quantum interference between causal sets in the sum over histories. Whether this condition alone is enough to give rise to diffeomorphism invariance is subsumed in thequestion of whether causal sets can give rise to a continuum approximation at all. he spacetime causal order plus spacetime volume are sufficient, in the continuum,to provide the full geometry of a Lorentzian spacetime for a very large class includingall globally hyperbolic spacetimes. This is strong evidence that order and number inthe discrete substructure are together sufficient to encode approximate Lorentziangeometrical information at large scales. From this then arises the principle that themathematical identity of the spacetime atoms is not physical.The second facet of general covariance in GR mentioned above, in which alldiffeomorphic manifold-metric pairs either do or do not satisfy the Einstein equa-tions, does not have such a direct analogue in causal set theory as it currentlystands. The dynamical models developed thus far are stochastic and there is noanalogue of “equations of motion” that a given causal set can either satisfy or notin a binary distinction. Nevertheless, a specific proposal for a condition of “discretegeneral covariance” was made in the context of a particular paradigm for causal setdynamics, namely classical random models of causal set growth, and this conditionwas used to construct an interesting family of classical stochastic dynamical mod-els for causal sets, the Classical Sequential Growth (CSG) models [7]. Each CSGmodel is a stochastic process of growth of the causal set spacetime in which newelements are born in sequence, forming relations with the previous elements in thesequence at random with a probability distribution given by the particular model.The sequence of the births is a total order on the spacetime atoms and containsunphysical, gauge information. As described further in section 2.3, the definition ofa CSG model is given in terms of this sequence, as is the discrete general covariancecondition . This condition is well-justified within the context of the framework ofsequential growth but the framework depends on and refers to the unphysical se-quential label of “stage”. The question arises: is it possible to define physicallyinteresting causal set growth dynamics that only ever refer to the physical degreesof freedom, to the physical partial order with no reference to any other labels? Thispaper describes work motivated by this question and provides a positive first step The discrete general covariance (DGC) condition could be considered as the “dynamical” facetwithin causal set theory. The DGC condition imposes that, given a pair of order-isomorphic causal setswith cardinality n , the probabilities of growing each by stage n are equal. This is somewhat akin tothe “dynamical” facet in GR where all diffeomorphic manifold-metric pairs have the same action andtherefore the same weight in the path integral. n that direction. In this subsection we list some of the terminology and assumptions used in thispaper. A more complete glossary of causal set terminology is given in [8]. Allthe infinite causal sets we consider in this paper are countable and past-finite (seebelow).Let ( C, ≺ ) be a causal set. We use the irreflexive convention in which x (cid:54)≺ x . Theword causet is short for causal set.If x ≺ y we say x is below y , y is above x , x is an ancestor of y or y is adescendant of x .The past of x ∈ C is the subcauset past ( x ) := { y ∈ C | y ≺ x } . This is thenon-inclusive past: x (cid:54)∈ past ( x ). The future of x is defined similarly. C is past-finite if | past ( x ) | < ∞ , ∀ x ∈ C .A stem in C is a finite subcauset S of C such that if x ∈ S and y ≺ x then y ∈ S .An n - stem is a stem with cardinality n .A relation x ≺ y is called a link if there is no element in the order between x and y . In that case we say x is directly below y , y is directly above x , x isa direct ancestor of y or y is a direct descendant of x . A link is also called acovering relation and we can also say y covers x .An antichain is a causet whose elements are unrelated to each other.A chain is a causet whose elements are all related.A path in C is a subcauset of C which is a chain all of whose links are alsolinks of C .The element x of C is in level n if the longest chain of which x is the maximalelement has cardinality n . Level 1 of C comprises the minimal elements of , level 2 comprises the minimal elements of what remains of C after theelements in level 1 are deleted, etc .It is useful to represent a causet as a graph in a Hasse diagram in whichelements are represented by nodes, and there is an upward-going edge from x to y if and only if x ≺ y is a link. The other relations are implied bytransitivity. All the pictures of causal sets in this paper are Hasse diagrams.An isomorphism , f , between two causets, C, D , is a bijection f : C → D such that f ( x ) ≺ D f ( y ) ⇐⇒ x ≺ C y, ∀ x, y ∈ C . If C and D are isomorphic,we write C ∼ = D . n -orders For concreteness, we fix a collection of ground sets for the causal sets we willwork with in this paper, following notation and terminology adapted from [9, 10].Consider, for each n >
0, the interval of natural numbers, [ n −
1] := { , , ..., n − } of cardinality n . Let ˜Ω( n ) denote the set of partial orders, ≺ , on the ground set[ n −
1] satisfying i ≺ j = ⇒ i < j . We call an element of ˜Ω( n ) a finite labeledcauset . We define ˜Ω( N ) := (cid:83) n ∈ N + ˜Ω( n ), the set of all finite labeled causets.We define ˜Ω to be the set of partial orders on the ground set N satisfying i ≺ j = ⇒ i < j . We call an element of ˜Ω an infinite labeled causet . By thisdefinition, every infinite labeled causet is past finite since element j can be aboveat most j other elements. The converse is also true: any infinite, past finite causetadmits a natural labeling by the natural numbers [11].We denote labeled causets (finite or infinite) and their stems with a tilde. Figure1 gives some examples of stems of a labeled causets. By our definitions, not all stemsof a labeled causet are themselves labeled causets because the ground set of the stemmay not itself be an interval, as shown in figure 1.As described in the introduction, it is a tenet of causal set theory that the atomsof spacetime have no structure. It is of no physical relevance what mathematical More generally, for a causal set C of cardinality n we call a bijection f : [ n − → C a naturallabeling if f ( i ) ≺ f ( j ) = ⇒ i < j . Using the interval [ n −
1] itself as the ground set for C , together withthe condition that i ≺ j = ⇒ i < j , makes the identity map into a natural labeling. C, ˜ D and ˜ F are labeled causets. ˜ D and ˜ E are stems in ˜ C . ˜ F is not a stem in˜ C because it is not a subcauset of ˜ C . ˜ E is not a labeled causet by our definition becauseits ground set is not an interval of integers. objects the elements of a causal set are. One way to express this is to say we areultimately interested only in isomorphism equivalence classes of causal sets.Isomorphism is an equivalence relation on each ˜Ω( n ), and on ˜Ω. We define unlabeled causets , or orders for short, to be isomorphism classes of labeledcausets. An unlabeled causet of cardinality n , or n - order for short, is anisomorphism class, C = [ ˜ C ] = { ˜ D ∈ ˜Ω( n ) | ˜ D ∼ = ˜ C } , where ˜ C ∈ ˜Ω( n ) is somerepresentative of C .An infinite unlabeled causet , or infinite order for short, is an isomorphismclass, C = [ ˜ C ] = { ˜ D ∈ ˜Ω | ˜ D ∼ = ˜ C } , where ˜ C ∈ ˜Ω is some representative of C . Wedefine Ω( n ) to be the set of n -orders, and Ω( N ) := (cid:83) n ∈ N + Ω( n ) is the set of finiteorders. Ω is defined to be the set of infinite orders.We generalise the concept of stem to orders. We say a finite order, S , is a stemin order C if there exists a representative of S which is a stem in a representativeof C and in this case we say, variously, S is a stem in C , or S occurs as a stem in C or C contains S as a stem. We say a finite order, S , is a stem in labeled causet˜ C if the order S is a stem in the order [ ˜ C ]. So, the meaning of stem depends onthe context.Examples are shown in figure 2. Note that if order S is a stem in order T and T is a stem in order U then S is a stem in U : “a stem in a stem is a stem” . Sothere is an “order-by-inclusion-as-stem” on the set of all finite orders.Finally, we introduce a concept that will be important later. An infinite order C ∈ Ω is a rogue [9] if there exists an infinite order D such that D (cid:54) = C and thetwo orders have the same stems. If infinite orders C and D have the same stems C, ˜ S and ˜ L are representatives of orders C, S and L , respec-tively. ˜ S is a 3-stem in ˜ C . ˜ L is not a subcauset of ˜ C so it is not a stem in ˜ C . S and L are 3-stems in ˜ C and in C . we write C ∼ R D . If C ∼ R D and C (cid:54) = D , we say that C and D are equivalentrogues. An example of a pair of equivalent rogues is given in figure 3. Figure 3: C is a countable union of 2-chains and D is the union of C with a singleunrelated element. C and D have the same stems – any union of finitely many 2-chainsand a finite, unrelated antichain – so C and D are equivalent rogues. Guided by the insight that path integral quantum theory is a form of generalisedmeasure theory [12], and by the heuristic of becoming [1, 13], a major breakthough inthe development of a dynamics for causal sets was the construction of the ClassicalSequential Growth (CSG) models by Rideout and Sorkin [7]. A CSG model is astochastic process consisting of the sequential coming into being, or birth , of newcauset elements and the formation of relations between each newly born elementand a randomly chosen subset of the elements born previously in the sequence. Theprocess can be represented as an upward-going random walk on a partially orderedtree called labeled poscau (short for the poset of labeled causal sets):
Definition.
Labeled poscau is the partial order ( ˜Ω( N ) , ≺ ) , where ˜ S ≺ ˜ R if andonly if ˜ S is a stem in ˜ R . We use the symbol ≺ to denote the relation for several different partial orders in this work. Themeaning of ≺ in each case is to be inferred from the context. abeled poscau is a tree formed of countably many levels, the first three ofwhich are shown in figure 4. A growth model based on labeled poscau is a randomwalk formed of a sequence of stages. At stage n , a labeled causet, ˜ C n , of cardinality n transitions to a labeled causet, ˜ C n +1 , of cardinality n + 1 such that ˜ C n is a stemin ˜ C n +1 . The transition can be thought of as the birth of the new element n , of˜ C n +1 , which comes into being above a randomly chosen subset of ˜ C n , the ancestorset of the newborn. The labeled causet ˜ C n +1 is one of the ‘child’ causets thatare directly above the ‘parent’ ˜ C n in labeled poscau. Any assignment of transitionprobabilities, satisfying the Markov sum rule, to all the links in labeled poscau givesa well-defined stochastic process. Each infinite path, ˜ C ≺ ˜ C ≺ ... ≺ ˜ C k ≺ . . . , inlabeled poscau beginning at the root is identified with the infinite labeled causet (cid:83) i ∈ N + ˜ C i . This is a one-to-one correspondence and so the histories in the model canbe though of, equivalently, as elements of ˜Ω or as infinite paths in labeled poscau.An event in such a stochastic process is a measureable subset of ˜Ω. For example,corresponding to each node, ˜ C n of cardinality n , in labeled poscau is the cylinderset [9], cyl ( ˜ C n ) := { ˜ D ∈ ˜Ω | ˜ D | [ n − = ˜ C n } . (1)The measure of each cylinder set is given by ˜ µ ( cyl ( ˜ C n )) = P ( ˜ C n ), where P ( ˜ C n ) isthe probability that the random walk reaches ˜ C n . This measure can be uniquelyextended to a measure on the σ -algebra ˜ R generated by the collection of cylindersets, by standard results in stochastic processes and measure theory [14].Now, not all events in ˜ R are physical because they are not all covariant. Forexample, the cylinder set cyl ( ˜ C n ) is the event “the causet at the end of stage n − C n ” which refers to the unphysical, gauge information of the stage.An event, E , is covariant if whenever a labeled causet, ˜ C , is in E then all labeledcausets isomorphic to ˜ C are also in E . E can then be identified, in an obvious way,with a set of orders. We define the sub- σ -algebra, R ⊂ ˜ R , as the algebra of allcovariant measureable events [15, 9]. The collection of random walks up labeled poscau is vast, so Rideout and Sorkin(RS) imposed physically motivated conditions to restrict the models to a moreinteresting class, the Classical Sequential Growth (CSG) models. The transitionprobabilities for CSG models were derived by RS by imposing on the random walktwo conditions: Bell Causality (BC) and Discrete General Covariance (DGC) [7].DGC is the condition that the probability of arriving at any node of labeled poscaudepends only on the isomorphism class of the node. For example, the probabilitiesto arrive at the three nodes in figure 4 which are in the isomorphism class of the“ L ” 3-order ( (cid:113)(cid:113)(cid:113) (cid:113) ) are equal in a CSG model. BC is analogous to the local causalitycondition that enters in the derivation of the Bell inequalities in Bell’s no-local-hidden-variables theorem. At stage n , consider two possible transitions from aparent causet ˜ C either to child ˜ A or to child ˜ B . Suppose there is an element, k , of˜ C , which is not in the ancestor set of the newborn element n , neither in ˜ A nor in ˜ B .Such an element k is called a spectator of both transitions. Now consider transitionsat stage n −
1, ˜ C (cid:48) → ˜ A (cid:48) and ˜ C (cid:48) → ˜ B (cid:48) , which are formed from the previous ones bydeleting the spectator k from ˜ C , ˜ A and ˜ B and consistently relabeling the remainingcausal sets so their base sets are integer intervals. Bell Causality is the condition P ( ˜ C → ˜ A ) P ( ˜ C → ˜ B ) = P ( ˜ C (cid:48) → ˜ A (cid:48) ) P ( ˜ C (cid:48) → ˜ B (cid:48) ) . (2) S showed that these two conditions of BC and DGC imply that a CSG modelis specified by a sequence of non-negative real numbers, { t , t , t , . . . } , which de-termine the transition probability for each possible transition ˜ C n → ˜ C n +1 in thefollowing way. The newly born element n chooses a subset Y from amongst all thesubsets of ˜ C n with relative probability t | Y | and n is put above all elements of Y and the transitive closure taken. For completeness, we give the explicit form of thetransition amplitude in a CSG model for the transition ˜ C n → ˜ C n +1 : P ( ˜ C n → ˜ C n +1 ) = λ ( (cid:36), m ) λ ( n, , (3)where (cid:36) is the cardinality of the ancestor set of the newborn element n , m is thenumber of maximal elements of the ancestor set of n and λ ( k, p ) := k − p (cid:88) i =0 (cid:18) k − pi (cid:19) t p + i . (4)The covariant events in CSG models were fully characterised and given a phys-ical interpretation in [15, 9]. Here we give a brief summary of those results whichwere based on the concept of stem set. Given an n -order C n , the stem set stem ( C n )is the event “ C n is a stem in the growing order” and is given mathematically by stem ( C n ) := { ˜ D ∈ ˜Ω | C n is a stem in ˜ D } . (5)Let S denote the collection of all stems sets, and let R ( S ) denote the stemalgebra , the σ -algebra generated by S . We call an element of R ( S ) a stem event .Stem events are covariant, and R ( S ) is a sub- σ -algebra of R . This inclusion is strict,mathematically, but in a well-defined, physical sense the stem algebra exhausts allthe covariant events. Indeed, for every covariant event, E , one can find a stem event F such that the symmetric difference between E and F is a set of rogue causetswhich is of measure zero in any CSG model because the set of all rogues is ofmeasure zero. In other words, in CSG models covariant events are, for all practicalpurposes, stem events. This is important. It means that every physical statementin a CSG model for which the dynamics provides a probability is a (countable)logical combination of statements about which finite orders are stems in the causetuniverse. A covariant framework
There exist several successful gauge theories, including GR, that are defined mathe-matically in terms of their gauge dependent degrees of freedom but in which physicalstatements can be made, purged of any unphysical gauge dependence introducedalong the way. CSG models make sense physically in this way. Although the defini-tion of a CSG model is given in terms of an unphysical sequence of birth events, themodel provides an exhaustive set of physically comprehensible, covariant measure-able events from which we make physical predictions. Were we only seeking a classof interesting classical growth models to explore, we might be content with CSGmodels as we have them. But for quantum gravity in the causal set approach, thetask in hand is to find a quantum dynamics for causal sets, from which GR mustthen emerge as a large scale approximation. We seek quantum growth models.One possible route to a Quantum Causet Dynamics would be to try to generalisewhat was done for CSG to the quantum case, finding appropriate analogues of theDGC and BC conditions on a decoherence functional or double path integral fora growing causal set [16]. In this paper, we take a slightly different path by ask-ing whether there is an explicitly label-independent framework for classical causetgrowth, an alternative to labeled poscau, which might suggest novel possibilitiesfor quantal generalisations. We frame the question as: is it possible to constructa physically well-motivated measure on the stem algebra R ( S ) directly , in a mani-festly label-independent way that does not rely on any gauge dependent notion andwhich respects the heuristic of growth and becoming?There already exists a structure in the literature, poscau [7], which at first sightmight seem to furnish such a framework. Poscau is a partial order on finite orders,(Ω( N ) , ≺ ), where A ≺ B if and only if A is a stem in B . Figure 5 shows a Hassediagram of the first three levels of poscau. If one identifies each node A in poscauwith its stem set, stem ( A ), one might be tempted to try to define a dynamics asa random walk up poscau such that arriving at the node A corresponds to theoccurrence of the covariant event stem ( A ). This does not work because in such adynamics only one stem set at each level can occur and the growing order wouldonly have a single stem of each finite cardinality. Rideout and Sorkin originally used poscau to introduce CSG models.
Thinking in this way, however, suggests the solution: the walk should be on atree formed of countably many levels in which the nodes in level n are not single n -orders but sets of n -orders. Each set of n -orders in level n will correspond tothe covariant event “the n -stems of the growing order are the elements of this set.”We will call this tree covtree (short for covariant tree ) and in the classical case,the dynamics will take the form of a stochastic process consisting of a sequence ofstages, each of which is a transition from a node in one level of covtree to a nodein the level above, just as the CSG models are defined on labeled poscau. We willmake this precise in the rest of this section. Let Γ n be a (non-empty) set of n -orders, i.e. a subset of Ω( n ). Definition.
An order C is a certificate of Γ n if Γ n is the set of all n -stems in C . Note that a given Γ n ⊆ Ω( n ) may have no certificate: see the example Γ (cid:48) infigure 6. We use Λ to denote the collection of sets of n -orders, for all n , which havecertificates: Λ := (cid:91) n ∈ N + { Γ n ⊆ Ω( n ) |∃ a certificate for Γ n } . (6)Note also that if Γ n has a certificate then it has infinitely many certificates andthat if Γ n has a certificate then it has a finite certificate. and Γ (cid:48) , are shown. C , D and E are certificatesof Γ . F is not a certificate of Γ because F contains the 3-antichain (circled in the figure)as a 3-stem. Γ (cid:48) has no certificates because every order which contains the 3-chain andthe 3-antichain also contains the “L” 3-order as illustrated by G .14 efinition. Given some Γ n ∈ Λ , we order its finite certificates as follows: let C , C be finite certificates of Γ n , then C (cid:22) C if C is a stem in C . A minimalcertificate of Γ n is minimal in this order. If Γ n ∈ Λ has more than one minimal certificate, these minimal certificatesneed not have the same cardinality as each other. Also, an n -order in Γ n maybe embedded in a minimal certificate of Γ n in more than one way. Examples areshown in figure 7.
Lemma 3.1.
Let Γ n = { A , A , ..., A k } be a set of n -orders. If C is a minimalcertificate of Γ n then n ≤ | C | ≤ kn . | C | = n if and only if Γ n is a singleton set( k = 1 ).Proof. Consider a labeled representative ˜ C of C . For each A i , i = 1 , , . . . k , takea subset of ˜ C that is a stem in ˜ C , isomorphic to A i . Take the union ˜ U of all thosesubsets. ˜ U is a stem in ˜ C . ˜ U is isomorphic to a labeled representative of a finiteorder, U , which is a stem in C and has cardinality | U | ≤ kn . U is also a certificateof Γ n and since C is a minimal certificate, C = U and so | C | ≤ kn .If Γ n is a singleton set then its single element is the unique minimal certificateof Γ n and | C | = n . If Γ n is not a singleton then any minimal certificate must havecardinality greater than n .We will also need the concept of a labeled certificate. Definition.
A labeled causet ˜ C is a labeled certificate of Γ n ∈ Λ if ˜ C is arepresentative of a certificate of Γ n . A labeled causet ˜ C is a labeled minimalcertificate of Γ n ∈ Λ if ˜ C is a representative of a minimal certificate of Γ n . An example is shown in figure 8.
Given any Γ n ∈ Λ, we will be interested in the set of all k -stems of elements of Γ n for k < n . The following definition will be useful. Recall that we define the cardinality, | C | , of an n -order C as | C | := n . This is shorthand for a more precise statement. Let the certificate of Γ n be C and let the n -order be X ∈ Γ n . We say that X can be embedded in C in k ways if, for any labeled representative ˜ C of C , thereare k different subcausets of ˜ C which are stems and which are isomorphic to a representative of X . C , D and E areminimal certificates of Ω(3). C is a stem in C , C is a stem in C , and similarly for D and E . | C | = | D | = 7 and | E | = 8. The dotted outlines on E show that the “L” orderis embedded in E in more than one way.Figure 8: Two labeled minimal certificates of Ω(3).16 efinition. For any n and any set, Γ n , of n -orders, the map O − takes Γ n to theset of ( n − -stems of elements of Γ n : O − (Γ n ) := { B ∈ Ω( n − | ∃ A ∈ Γ n s . t . B is a stem in A } . (7)One way to think about the operation of O − on Γ n is to take an n -order in Γ n ,choose a maximal element of (a representative of) that n -order, and delete thatmaximal element to form (a representative of) an ( n − O − (Γ n )is the set of all ( n − Lemma 3.2.
Let C be an n -order and < k ≤ n . The set of k -stems of C is O − n − k ( { C } ) .Proof. Consider, X , a k -stem in C . There exist labeled representatives ˜ X of X and˜ C of C such that ˜ X is a stem in ˜ C . The ground set of ˜ C is the interval [ n − n − k )-step process of deleting the elements n − n − k in turn from˜ C results in ˜ X . This shows that X is in O − n − k ( { C } ). And conversely, deleting amaximal element from a representative of C n − k times results in a representativeof a k -stem of C . Corollary 3.3.
Let Γ n be a set of n -orders. If C is a certificate of Γ n , then C isalso a certificate of O − k (Γ n ) for any k , ≤ k < n . The converse is not true: if C is a certificate of O − (Γ n ), then C may or maynot be a certificate of Γ n . In fact, Γ n may have no certificates at all. Examples areshown in figure 9.We are now ready to define covtree. Recall that Λ is the collection of sets of n -orders, for all n , which have certificates. Definition.
Covtree is the partial order (Λ , ≺ ) , where Γ n ≺ Γ m if and only if n < m and O − m − n (Γ m ) = Γ n . Covtree is a tree formed of levels labeled by { , , , . . . } . The nodes in level n are sets of n -orders. A set of n -orders, Γ n , is a node in level n of covtree if and onlyif Γ n has a certificate. (This is the motivation for the term certificate: a certificateof Γ n certifies that Γ n is a node in covtree.) The partial order on covtree is definedby putting Γ n directly above O − (Γ n ), for every node Γ n , and taking the transitiveclosure. a) Γ has no certificates. C is a certificate of O − (Γ ).(b) D is a certificate of Γ (cid:48) , and therefore a certificate of O − (Γ (cid:48) ). E is acertificate of O − (Γ (cid:48) ) and is not a certificate of Γ (cid:48) . Figure 9: Illustration of the O − operation.18 he singleton set containing the one-element order is the root of covtree. Thefirst three levels of covtree are shown in figure 10. There are 22 nodes in level 3 ofcovtree out of a possible 2 − O − sequentially to the node. In particular,every singleton set { C } where C is an n -order is a node in covtree because C isits certificate and the path in covtree down from { C } to the root is formed of thenodes O − k ( { C } ), k = 0 , , , . . . , n −
1. In the upward direction, generating thenodes directly above a given Γ n in covtree is a difficult problem. Covtree allows us to realise the idea described previously of defining a dynamics ona tree in which the nodes in level n are sets of n -orders and each node correspondsto the covariant event “the n -stems of the growing order is this set of n -orders.”Consider a classical dynamical model for a growing causal set as an upward-goingrandom walk on covtree, starting at the root. In preparation for exploring therelationship between paths in covtree and infinite orders – the histories in a causalset cosmological model – we generalise the notion of a certificate of a node to thecertificate of a path: Definition.
An infinite order is a certificate of a path P in covtree if it is a cer-tificate of every node in P . A labeled certificate of a path P is a representativeof a certificate of P . One relationship between infinite orders – elements of Ω – and paths in covtreeis straightforward to state and understand:
Lemma 4.1.
Let C be an infinite order. The nodes of covtree of which C is acertificate form a path in covtree starting at the root.Proof. Let Γ n be the set of n -stems of C , for each n > C is a certificate of eachΓ n . Each Γ n is a node in covtree and corollary 3.3 shows that these nodes form apath in covtree down to the root. a) The structure of the first three levels of covtree.(b) Nodes in level three which are directly above the node { (cid:113)(cid:113) , (cid:113) (cid:113) } . Figure 10: The first three levels of covtree.20 he map from infinite orders to paths in covtree implied in the lemma aboveis not one-to-one because a rogue order is not specified by its stems: if C and C (cid:48) are equivalent rogues then they are both certificates of the same path in covtree.This means that our stochastic process cannot, in principle, distinguish betweenequivalent rogues.It is not immediately apparent whether or not every infinite path in covtree hasan infinite order as a certificate but in fact it is true and we have: Theorem 4.2.
Let P be an infinite path in covtree starting at the root. Thereexists an infinite order C which is a certificate of P . To prove theorem 4.2 we will demonstrate an algorithm to generate a labeled certificate of any path P . The isomorphism class of this labeled certificate is thenthe desired order. We begin with some lemmas. Lemma 4.3.
Let P = { Γ , Γ , Γ . . . } be a path in covtree and let Γ n ∈ P not be asingleton. Then there exists a node in P above Γ n that contains a certificate of Γ n as an element. In other words, there exists an m > n and an m -order C such that C is a certificate of Γ n and C ∈ Γ m ∈ P .Proof. By lemma 3.1, the cardinality of any minimal certificate, C , of Γ n satisfies n < | C | ≤ N where N := n | Γ n | . Consider Γ N ∈ P and let D be a finite certificate ofΓ N . By corollary 3.3, D is a certificate of every node below Γ N so D is a certificateof Γ n . Now, at least one minimal certificate of Γ n occurs as a stem in D . Chooseone, call it C , let m := | C | and consider Γ m ∈ P . C is an m -stem in D . Γ m is theset of all m -stems of D and so C is an element of Γ m .Note the choices made in the proof above: a choice of a particular certificate ofΓ N and a choice of a stem in it which is a minimal certificate of Γ n . Lemma 4.4.
Let P = { Γ , Γ , ... } be a path in covtree and let Γ n ∈ P . There is anode in P above Γ n which has a certificate of Γ n as an element.Proof. In the case that Γ n is not a singleton, a node with the required propertyis Γ m as defined in the proof of lemma 4.3. In the case that Γ n is a singleton,then a node with the required property is Γ n +1 because every element of Γ n +1 is acertificate of Γ n . emma 4.5. Let P = { Γ , Γ , ... } be an infinite path in covtree. There exists aninfinite subsequence m < m < m < . . . of the natural numbers, and a set oflabeled causal sets { ˜ C m , ˜ C m , ˜ C m . . . } such that, for all k ,(i) | ˜ C m k | = m k ;(ii) ˜ C m k is a subcauset, a stem, in ˜ C m k +1 ;(iii) ˜ C m k +1 is a labeled certificate of Γ m k and also therefore a labeled certificate ofall nodes below Γ m k ;(iv) C m k +1 , the isomorphism class of ˜ C m k +1 , is an element of Γ m k +1 .Proof. Consider an infinite path P = { Γ , Γ , ... } in covtree. The required sequenceof causal sets { ˜ C m , ˜ C m , ˜ C m . . . } is constructed by the following inductive algo-rithm. Step 1: m to start and consider Γ m ∈ P .1.1) By lemma 4.4 there exists an m such that m > m and such that Γ m con-tains a certificate of Γ m as an element. Call that certificate C m . Its cardinalityis | C m | = m .1.2) Pick ˜ C m , a labeled causet which is a representative of C m .1.3) Go to step 2. Step k > m k such that m k > m k − and such that Γ m k contains a certificate of Γ m k − as an element. Call that certificate C m k . Its cardi-nality is | C m k | = m k .k.2) Consider C m k − and its labeled representative ˜ C m k − from the previous step. C m k − is an element of Γ m k − . Because C m k is a certificate of Γ m k − , C m k − is an m k − -stem of C m k . Pick a representative ˜ C m k of C m k such that ˜ C m k − from theprevious step is a sub-causet of ˜ C m k .k.3) Go to step k + 1.The subsequence m < m < m < . . . of the natural numbers, and the set oflabeled causal sets { ˜ C m , ˜ C m , ˜ C m . . . } have the required properties by construc-tion. Lemma 4.6.
An infinite path in covtree has a labeled certificate. roof. The union of the nested sequence of labeled causets { ˜ C m , ˜ C m , ˜ C m . . . } ofthe previous lemma is a labeled certificate of the path. Corollary 4.7.
An infinite path in covtree has a certificate.
This corollary is theorem 4.2. R ( S ) We propose random walks upwards on covtree as dynamical models in which anorder grows and in which arriving at a node Γ n corresponds to the occurrence ofthe event “the set of n -stems of the order is Γ n .” A question that arises is: whatis the relationship between dynamical models on covtree and dynamical modelson labeled poscau? Do the kinematical structures of covtree and labeled poscaugive rise to different classes of causet growth models? We will show that the setof measures induced on R ( S ) by walks on labeled poscau and the set of measuresinduced on R ( S ) by walks on covtree are equal, and equal to the set of all measureson R ( S ).First we introduce the covtree measure space. The certificate set, cert (Γ n ), ofa node, Γ n , in covtree is the subset cert (Γ n ) := { C ∈ Ω | C is a certificate of Γ n } . (8)The node certificate sets are the covtree “cylinder sets”. Let Σ denote the setof node certificate sets cert (Γ n ) for all nodes in covtree, together with the emptyset. A random walk on covtree, defined by the transition probabilities for eachlink in covtree satisfying the Markov sum rule, gives a measure µ on Σ, where µ ( cert (Γ n )) is the product of the transition probabilities on the links of the pathfrom the root to Γ n . The tree structure of covtree means that Σ is a semi-ring andthat a measure µ on Σ generated by a set of Markovian transition probabilities oncovtree is countably-additive . Hence we can apply the Fundamental Theorem ofMeasure Theory [14] which says that the measure µ extends to R (Σ), the σ -algebragenerated by Σ.We are now in a position to prove that: This is standard measure theory for stochastic processes. For completeness, we present proofs inappendix B.3. emma 4.8. R ( S ) = R (Σ) .Proof. First we note that as defined, these two σ -algebras are defined over differentsample spaces: an element of Σ is a set of infinite orders and an element of S is aset of infinite labeled causets. However, both the covariant algebra R and the stemalgebra R ( S ) can be thought of, in an obvious way, as σ -algebras on the samplespace Ω of infinite orders, since their elements are covariant. This is the sense inwhich the claim is to be interpreted.We will show that any stem set – thought of as a set of infinite orders – can beconstructed by finite set operations on the certificate sets and vice versa, and theresult follows.Consider an n -order B . Let Γ in be the nodes in covtree such that B ∈ Γ in , where i labels the nodes. Suppose C ∈ cert (Γ in ) for some i . Then B is a stem in C andhence C ∈ stem ( B ). Suppose C / ∈ cert (Γ in ) for all i . Then B is not a stem in C and hence C / ∈ stem ( B ). It follows that stem ( B ) = (cid:83) i cert (Γ in ).Consider some node Γ n = { A , ..., A k } in covtree. Let Ω( n ) \ Γ n = { B , ..., B l } .Suppose C ∈ cert (Γ n ). Then A , ..., A k are stems in C , and B , ..., B l are not stemsin C . Hence C ∈ k (cid:84) i =1 stem ( A i ) \ l (cid:83) j =1 stem ( B j ). Suppose C / ∈ cert (Γ n ). Then either( i ) there exists some A i ∈ Γ n which is not a stem in C = ⇒ C / ∈ k (cid:84) i =1 stem ( A i ), or( ii ) there exists some B j ∈ Ω( n ) \ Γ n which is a stem in C = ⇒ C ∈ l (cid:83) j =1 stem ( B j ).It follows that, cert (Γ n ) = k (cid:84) i =1 stem ( A i ) \ l (cid:83) j =1 stem ( B j ).Hence every walk on covtree induces a unique measure on R ( S ), and everymeasure on R ( S ) induces a unique walk on covtree: the transition probability inthe covtree walk from node Γ n to node Γ n +1 directly above it is the measure of cert (Γ n +1 ) divided by the measure of cert (Γ n ). Therefore, let us call a measure on R ( S ) a covtree measure .By a similar argument to the above, there is is a 1-1 correspondence betweenwalks on labeled poscau and measures on ˜ R so we will call a measure on R ( S ) a poscau measure if it is a restriction to R ( S ) of some measure ˜ µ on ˜ R . A CSGmeasure on R ( S ) is a poscau measure such that ˜ µ is induced by a CSG walk.It follows from lemma 4.8 that every poscau measure on R ( S ) is a covtree easure on R ( S ). In fact, it is also true that every covtree measure is a poscaumeasure: Lemma 4.9.
For every measure µ on R ( S ) there exists an extension ˜ µ to ˜ R .Proof. First note that there is a metric on ˜Ω with respect to which ( ˜Ω , ˜ R ) is aPolish space [9]. Since every Polish space is a Lusin space [17], ( ˜Ω , ˜ R ) is a Lusinspace. Note also that R ( S ) is a separable sub- σ -algebra of ˜ R since there exists acountable collection of subsets of ˜Ω which generates R ( S ), namely S (or Σ). Theresult follows from the theorem that if ( Y, B ) is a Lusin space, then every measuredefined on a separable sub- σ -algebra of B can be extended to B [18]. At the beginning of section 3 we posed the question: “Is it possible to construct aphysically well-motivated measure on the stem algebra R ( S ) directly , in a manifestlylabel-independent way that does not rely on any gauge dependent notion?” The keyphrase here is physically well-motivated . We have shown that we can generate amathematically well-defined measure on the stem events R ( S ) via a growth processconceived as a random walk up covtree. There is no reason to expect, however, thata generic such walk will be physically interesting: the class of walks is too vast tobe interesting. We need physically motivated conditions to restrict the models toa sub-class worth studying. This is what was done by Rideout and Sorkin in thecontext of walks up labeled poscau by imposing the conditions of discrete generalcovariance (DGC) and Bell causality (BC) [7]. These conditions restrict the classof walks on labeled poscau to the CSG models.The relationship between the “labeled” conditions of DGC and BC and anyconditions on covtree walks is not understood. Note that lemma 4.9 means thatalthough every covtree walk is apparently completely covariant in its setup, forevery walk on labeled poscau – whether it satisfies Discrete General Covariance ornot – there exists a covtree walk that produces the same measure on R ( S ). So,there is no easy relationship between the DGC condition on a labeled poscau walkand the manifest “covariance” of a covtree walk. We can frame the sort of progresswe’d like to make from here as a set of interrelated questions. i) Is there a condition on the transition amplitudes of a walk up covtree suchthat the covtree measure is a poscau measure from a walk on labeled poscauthat satisfies DGC only, measures which Brightwell and Luczak call “order-invariant”? [11, 19, 20].(ii) Is there a condition on the transition amplitudes of a walk up covtree suchthat the covtree measure equals a CSG measure?(iii) Is there a condition on a random walk up covtree which expresses the physicalcondition of relativistic causality? How is this related to the condition of BellCausality satisfied by CSG models as walks on labeled poscau? Is this newcondition enough to reduce the class to a physically interesting one or areother conditions needed and what are they?(iv) What is the role of the rogues, if any, in understanding the physics of covtreewalks? Could the condition that the set of rogues has measure zero – as itdoes for any CSG model – be considered as a physical condition in itself andwhat conditions on the transition amplitudes for the walk would imply thiscondition?(v) What form might a quantum random walk on covtree take and might it bepossible to formulate a quantum relativistic causality condition for it, evenwhile the labeled BC condition has thus far resisted a quantal generalisation?Here we start to grapple with the kinds of knotty questions that crop up whenconsidering what a condition of relativistic causality might look like in a theoryin which the spacetime causal order itself is dynamical and stochastic/quantal andin which labels/coordinates are banned, even as a prop to kick away at the end.Here, in covtree, at least we now have a concrete arena in which to investigate thesequestions. Acknowledgments:
We thank Jeremy Butterfield for useful discussions. Thisresearch was supported in part by Perimeter Institute for Theoretical Physics. Re-search at Perimeter Institute is supported by the Government of Canada throughIndustry Canada and by the Province of Ontario through the Ministry of Eco-nomic Development and Innovation. FD is supported in part by STFC grantST/P000762/1 and APEX grant APX/R1/180098. SZ thanks the Perimeter In- titute and the Raman Research Institute for hospitality while this work was beingcompleted. SZ is partially supported by the Kenneth Lindsay Scholarship Trust. A Table of sets defined in the text ˜Ω( n ) The set of labeled causets of cardinality n ˜Ω( N ) The set of finite labeled causets˜Ω The set of infinite labeled causetsΩ( n ) The set of n -ordersΩ( N ) The set of finite ordersΩ The set of infinite orders cyl ( ˜ A ) cyl ( ˜ A ) = { ˜ C ∈ ˜Ω | ˜ A is a stem in ˜ C } ˜ R The σ -algebra generated by the cylinder sets R The covariant sub- σ -algebra of ˜ R stem ( A ) stem ( A ) = { ˜ C ∈ ˜Ω | A is a stem in ˜ C }S The set of stem sets R ( S ) The sub- σ -algebra of ˜ R generated by S Γ n A subset of Ω( n ) cert (Γ n ) cert (Γ n ) = { C ∈ Ω | C is a certificate of Γ n } Λ Λ = (cid:83) n ∈ N + { Γ n ⊆ Ω( n ) |∃ a certificate for Γ n }P An infinite path from the root in covtreeΣ The set of certificate sets together with the empty set R (Σ) The sub- σ -algebra of ˜ R generated by Σ Proofs
B.1 Appendix to section 3.2
Figure 11: The level 3 nodes which are directly above the level 2 doublet are showntogether with their respective certificates.Figure 12: The sets shown in the figure have no certificates and therefore are not nodes.For every set shown, if an order contains all the elements of that set as stems then it alsocontains the L as a stem. 28 .2 Appendix to section 4
Alternative proof to theorem 4.2.
Recall that ∼ R denotes the rogue equivalence re-lation, and let p : Ω → Ω / ∼ R be the associated canonical quotient map. Let [ A ] R denote an element of Ω / ∼ R , where A ∈ Ω is a representative of [ A ] R .Define the following metric on Ω / ∼ R : d ([ A ] R , [ B ] R ) = n , where n is thehighest integer such that the set of n -stems of A is the set of n -stems of B . Onecan show that (Ω / ∼ R , d ) is a complete metric space.Let [ cert (Γ n )] R ⊂ Ω / ∼ R denote the image of the certificate set cert (Γ n ) ⊂ Ωunder the quotient map p . Then one can show that [ cert (Γ n )] R is both open andclosed in (Ω / ∼ R , d ).The diameter of [ cert (Γ n )] R , d ([ cert (Γ n )] R ), is defined to be the maximumdistance between any two elements in [ cert (Γ n )] R and is equal to n . Hence d ([ cert (Γ n )] R ) → n → ∞ .Given a path in covtree, P = { Γ , Γ , ... } , then the following is a nested sequence:[ cert (Γ )] R ⊃ [ cert (Γ )] R ⊃ . . . .Now, Cantor’s Lemma states that a metric space ( X, d ) is complete if and onlyif, for every nested sequence { F n } n ≥ of nonempty closed subsets of X , that is, (a) F ⊇ F ⊇ . . . and (b) d ( F n ) → n → ∞ , the intersection (cid:84) ∞ n =1 F n containsone and only one point [21]. B.3 Appendix to section 4.1
Recall that Σ is the collection of certificate sets with the empty set.
Lemma B.1. Σ is a semi-ring.Proof. A family F of subsets of a set M is a semi-ring if ( i ) ∅ ∈ F , ( ii ) A ∩ B ∈ F for all A, B ∈ F , and ( iii ) for every pair of sets
A, B ∈ F with A ⊂ B , the set B \ A is the union of finitely many disjoint sets in F [14].Let Γ m and Γ n be nodes in covtree, m > n . Suppose Γ n ≺ Γ m . Then cert (Γ m ) ⊂ cert (Γ n ) = ⇒ cert (Γ m ) ∩ cert (Γ n ) = cert (Γ m ) ∈ Σ. Suppose Γ n (cid:54)≺ Γ m . Then cert (Γ m ) ∩ cert (Γ n ) = ∅ ∈ Σ. [ cert (Γ n )] R are exactly the open balls under the metric topology. et cert (Γ m ) ⊂ cert (Γ n ). Then cert (Γ n ) \ cert (Γ m ) is the set of all certificates ofΓ n which are not certificates of Γ m . Let Γ im , i = 1 , , . . . k , denote the nodes in level m such that Γ im (cid:31) Γ n and Γ im (cid:54) = Γ m . Then cert (Γ n ) \ cert (Γ m ) = (cid:70) i cert (Γ im ).Recall that a random walk on covtree, defined by the transition probabilitiesfor each link in covtree satisfying the Markov sum rule, gives a measure µ on Σ: µ ( cert (Γ n )) = the product of the transition probabilities on the links of the pathfrom the root to Γ n (and µ ( ∅ ) = 0). Lemma B.2.
The measure µ on Σ is countably-additive.Proof. We defined µ : Σ → [0 ,
1] by µ ( cert (Γ m )) = P (Γ m ) where P (Γ m ) is theprobability of a random walk to pass through Γ m . Also µ ( ∅ ) = 0.Suppose cert (Γ n ) = (cid:70) ki =1 cert (Γ in i ), where i labels the individual nodes. Then µ ( (cid:70) ki =1 cert (Γ in i )) = µ ( cert (Γ n )) = P (Γ n ) = (cid:80) i P (Γ in i ). Hence, finite additivity issatisfied.Next we will show that countable additivity of µ is trivially satisfied as nocertificate set is a countable disjoint union of certificate sets . Consider some Γ m in covtree and suppose for contradiction that cert (Γ m ) = (cid:70) i ∈ N cert (Γ in i ). Considerthe following suborder in covtree, { Γ n ∈ Λ | Γ n (cid:23) Γ m and Γ n (cid:54)(cid:31) Γ in i ∀ i } , and let T m be the transitive reduction of it.We note that ( i ) T m is infinite, ( ii ) every node in T m has finite valency, and( iii ) T m is a connected tree.Then by K¨onig’s lemma, T m contains an infinite upward-going path starting atΓ m [23]. It follows that there is an infinite path P in covtree such that Γ m ∈ P andΓ in i / ∈ P for all i ∈ N . Therefore there exists a certificate C of P and hence of Γ m such that C / ∈ cert (Γ in i ) for all i ∈ N , which is a contradiction. References [1] Rafael D. Sorkin. Relativity theory does not imply that the future alreadyexists: a counterexample. In Vesselin Petkov, editor,
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