A manifestly covariant framework for causal set dynamics

We propose a manifestly covariant framework for causal set dynamics. The framework is based on a structure, dubbed covtree, which is a partial order on certain sets of finite, unlabeled causal sets. We show that every infinite path in covtree corresponds to at least one infinite, unlabeled causal set. We show that transition probabilities for a classical random walk on covtree induce a classical measure on the $\sigma$-algebra generated by the stem sets.


Introduction
General Relativity (GR) has the property of general covariance. As discussed in section 7 of [1], this gauge invariance of GR has two facets. The first is that physical statements -or, equivalently, properties or predicates or events -in GR must be diffeomorphism invariant. The second is that the equations of motion of GR are diffeomorphism invariant so that metrics in a diffeomorphism equivalence class must either all satisfy or all not satisfy the equations of motion. As stated in [1], "[In GR] the second facet of covariance flows directly from the first as a consistency condition, because it would be senseless to identify two metrics one of which was allowed by the equations of motion and the other of which was forbidden; and conversely, the kinematical identification must be made if one wishes the dynamics to be deterministic. Thus, the first or "ontological" facet of general covariance tends 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 the development of GR and one might expect that grappling with the corresponding issues within quantum gravity will be important to its development too [2]. The requirement that the diffeomorphism invariance of GR must emerge from quantum gravity in the large scale approximation is indeed used to formulate guiding precepts for each approach to quantum gravity, though the form that these precepts take varies from approach to approach.
In the case of the causal set approach to the problem of quantum gravity, any deeper precept of general covariance cannot be literally diffeomorphism invariance because diffeomorphism is a continuum concept and causal set theory is discrete.
Causal set theory postulates that the fundamental structure of spacetime is atomic at the Planck scale and takes the form of a causal set or locally finite partial order. 1 The elements of the causal set are the atoms of spacetime and continuum spacetime is an approximation to the causal set at large scales. The order relation of the underlying causal set reveals itself as the causal structure of the approximating continuum spacetime and the number of causal set elements manifests itself as the spacetime volume of the approximating continuum spacetime. For a recent review of causal set theory see [3].
The structure of continuum spacetime, then, emerges from Order and Number and this central conjecture of causal set theory has an immediate consequence: the physical content of a causal set is independent of what mathematical objects the causal set elements are and is also independent of any additional labels those causal set elements might carry: only the order relation of the elements and the number of elements has physical meaning. This "mathematical-identity-and-label independence" is a good candidate for a condition of general covariance in causal set theory, at least as far as the first facet, mentioned above, goes 2 .
This form of general covariance is a consequence of a more general guiding heuristic, Occam's Razor, applied in the particular case of causal set quantum gravity and grounded in earlier seminal work, theorems in Lorentzian geometry by Penrose, Kronheimer, Hawking and Malament [4,5,6]. These theorems show that 1 At least, this is the kinematics of the theory. In the full quantum theory this statement will be revised to take account of quantum interference between causal sets in the sum over histories. 2 Whether this condition alone is enough to give rise to diffeomorphism invariance is subsumed in the question of whether causal sets can give rise to a continuum approximation at all. the 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 including all globally hyperbolic spacetimes. This is strong evidence that order and number in the discrete substructure are together sufficient to encode approximate Lorentzian geometrical information at large scales. From this then arises the principle that the mathematical identity of the spacetime atoms is not physical.
The second facet of general covariance in GR mentioned above, in which all diffeomorphic manifold-metric pairs either do or do not satisfy the Einstein equations, does not have such a direct analogue in causal set theory as it currently stands. The dynamical models developed thus far are stochastic and there is no analogue of "equations of motion" that a given causal set can either satisfy or not in a binary distinction. Nevertheless, a specific proposal for a condition of "discrete general covariance" was made in the context of a particular paradigm for causal set dynamics, namely classical random models of causal set growth, and this condition was used to construct an interesting family of classical stochastic dynamical models for causal sets, the Classical Sequential Growth (CSG) models [7]. Each CSG model is a stochastic process of growth of the causal set spacetime in which new elements are born in sequence, forming relations with the previous elements in the sequence 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 contains unphysical, gauge information. As described further in section 2.3, the definition of a CSG model is given in terms of this sequence, as is the discrete general covariance condition 3 . This condition is well-justified within the context of the framework of sequential growth but the framework depends on and refers to the unphysical sequential label of "stage". The question arises: is it possible to define physically interesting causal set growth dynamics that only ever refer to the physical degrees of freedom, to the physical partial order with no reference to any other labels? This paper describes work motivated by this question and provides a positive first step 3 The discrete general covariance (DGC) condition could be considered as the "dynamical" facet within causal set theory. The DGC condition imposes that, given a pair of order-isomorphic causal sets with cardinality n, the probabilities of growing each by stage n are equal. This is somewhat akin to the "dynamical" facet in GR where all diffeomorphic manifold-metric pairs have the same action and therefore the same weight in the path integral.
4 in that direction.

Preliminaries
In this subsection we list some of the terminology and assumptions used in this paper. A more complete glossary of causal set terminology is given in [8]. All the infinite causal sets we consider in this paper are countable and past-finite (see below).
Let (C, ≺) be a causal set. We use the irreflexive convention in which x ≺ x. The word 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 a descendant of x.
The past of x ∈ C is the subcauset past(x) := {y ∈ C|y ≺ x}. This is the non-inclusive past: x ∈ past(x). The future of x is defined similarly.
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 is a direct ancestor of y or y is a direct descendant of x. A link is also called a covering 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 also links of C.
The element x of C is in level n if the longest chain of which x is the maximal element has cardinality n. Level 1 of C comprises the minimal elements of It is useful to represent a causet as a graph in a Hasse diagram in which elements 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 by transitivity. All the pictures of causal sets in this paper are Hasse diagrams.
An isomorphism, f , between two causets, C, D, is a bijection f : we write C ∼ = D.

Labeled causets and n-orders
For concreteness, we fix a collection of ground sets for the causal sets we will work with in this paper, following notation and terminology adapted from [9,10].
We defineΩ to be the set of partial orders on the ground set N satisfying i ≺ j =⇒ i < j. We call an element ofΩ an infinite labeled causet. By this definition, every infinite labeled causet is past finite since element j can be above at most j other elements. The converse is also true: any infinite, past finite causet admits a natural labeling by the natural numbers [11].
We denote labeled causets (finite or infinite) and their stems with a tilde. Figure   1 gives some examples of stems of a labeled causets. By our definitions, not all stems of a labeled causet are themselves labeled causets because the ground set of the stem may 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 atoms of spacetime have no structure. It is of no physical relevance what mathematical objects the elements of a causal set are. One way to express this is to say we are ultimately interested only in isomorphism equivalence classes of causal sets.
Isomorphism is an equivalence relation on eachΩ(n), and onΩ. We define unlabeled causets, or orders for short, to be isomorphism classes of labeled causets. An unlabeled causet of cardinality n, or n-order for short, is an An infinite unlabeled causet, or infinite order for short, is an isomorphism class, C = [C] = {D ∈Ω |D ∼ =C}, whereC ∈Ω is some representative of C. We define Ω(n) to be the set of n-orders, and Ω(N) := n∈N + Ω(n) is the set of finite orders. 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 stem in order C if there exists a representative of S which is a stem in a representative of 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 T is a stem in order U then S is a stem in U : "a stem in a stem is a stem" . So there 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 = C and the two orders have the same stems. If infinite orders C and D have the same stems

Dynamics
Guided by the insight that path integral quantum theory is a form of generalised measure theory [12], and by the heuristic of becoming [1,13], a major breakthough in the development of a dynamics for causal sets was the construction of the Classical Sequential Growth (CSG) models by Rideout and Sorkin [7]. i . This is a one-to-one correspondence and so the histories in the model can be though of, equivalently, as elements ofΩ or as infinite paths in labeled poscau.
An event in such a stochastic process is a measureable subset ofΩ. For example, corresponding to each node,C n of cardinality n, in labeled poscau is the cylinder set [9], The measure of each cylinder set is given byμ(cyl(C n )) = P(C n ), where P(C n ) is the probability that the random walk reachesC n . This measure can be uniquely extended to a measure on the σ-algebraR generated by the collection of cylinder sets, by standard results in stochastic processes and measure theory [14]. Now, not all events inR are physical because they are not all covariant. For example, the cylinder set cyl(C n ) is the event "the causet at the end of stage n − 1 of the process isC 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 labeled causets isomorphic toC 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 all covariant measureable events [15,9].

Classical Sequential Growth models
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 more interesting class, the Classical Sequential Growth (CSG) models. The transition probabilities for CSG models were derived by RS by imposing on the random walk two conditions: Bell Causality (BC) and Discrete General Covariance (DGC) [7].
DGC is the condition that the probability of arriving at any node of labeled poscau depends only on the isomorphism class of the node. For example, the probabilities to arrive at the three nodes in figure 4 which are in the isomorphism class of the "L" 3-order ( q q q q) are equal in a CSG model. BC is analogous to the local causality condition that enters in the derivation of the Bell inequalities in Bell's no-localhidden-variables theorem. At stage n, consider two possible transitions from a parent causetC either to childÃ or to childB. Suppose there is an element, k, of C, which is not in the ancestor set of the newborn element n, neither inÃ nor inB.
Such an element k is called a spectator of both transitions. Now consider transitions at stage n − 1,C →Ã andC →B , which are formed from the previous ones by deleting the spectator k fromC,Ã andB and consistently relabeling the remaining causal sets so their base sets are integer intervals. Bell Causality is the condition 10 RS showed that these two conditions of BC and DGC imply that a CSG model is specified by a sequence of non-negative real numbers, {t 0 , t 1 , t 2 , . . . }, which determine the transition probability for each possible transitionC n →C n+1 in the following way. The newly born element n chooses a subset Y from amongst all the subsets ofC 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 the transition amplitude in a CSG model for the transitionC n →C n+1 : where is the cardinality of the ancestor set of the newborn element n, m is the number of maximal elements of the ancestor set of n and The covariant events in CSG models were fully characterised and given a physical interpretation in [15,9]. Here we give a brief summary of those results which were 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 Let S denote the collection of all stems sets, and let R(S) denote the stem algebra, 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 all the 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 causets which is of measure zero in any CSG model because the set of all rogues is of measure zero. In other words, in CSG models covariant events are, for all practical purposes, stem events. This is important. It means that every physical statement in 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 causet universe.

A covariant framework
There exist several successful gauge theories, including GR, that are defined mathematically in terms of their gauge dependent degrees of freedom but in which physical statements can be made, purged of any unphysical gauge dependence introduced along the way. CSG models make sense physically in this way. Although the definition of a CSG model is given in terms of an unphysical sequence of birth events, the model provides an exhaustive set of physically comprehensible, covariant measureable events from which we make physical predictions. Were we only seeking a class of interesting classical growth models to explore, we might be content with CSG models as we have them. But for quantum gravity in the causal set approach, the task in hand is to find a quantum dynamics for causal sets, from which GR must then emerge as a large scale approximation. We seek quantum growth models.
One possible route to a Quantum Causet Dynamics would be to try to generalise what was done for CSG to the quantum case, finding appropriate analogues of the DGC and BC conditions on a decoherence functional or double path integral for a growing causal set [16]. In this paper, we take a slightly different path by asking whether there is an explicitly label-independent framework for classical causet growth, an alternative to labeled poscau, which might suggest novel possibilities for quantal generalisations. We frame the question as: is it possible to construct a physically well-motivated measure on the stem algebra R(S) directly, in a manifestly label-independent way that does not rely on any gauge dependent notion and which respects the heuristic of growth and becoming?
There already exists a structure in the literature, poscau [7], which at first sight might seem to furnish such a framework. Poscau is a partial order on finite orders, only have a single stem of each finite cardinality. 6 Rideout and Sorkin originally used poscau to introduce CSG models. Thinking in this way, however, suggests the solution: the walk should be on a tree 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 to the 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 of stages, each of which is a transition from a node in one level of covtree to a node in the level above, just as the CSG models are defined on labeled poscau. We will make this precise in the rest of this section.

Certificates
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 3 in figure 6. We use Λ to denote the collection of sets of n-orders, for all n, which have certificates: Note also that if Γ n has a certificate then it has infinitely many certificates and that if Γ n has a certificate then it has a finite certificate.
13 Figure 6: Ω(3) and two of its subsets, Γ 3 and Γ 3 , are shown. C, D and E are certificates of Γ 3 . F is not a certificate of Γ 3 because F contains the 3-antichain (circled in the figure) as a 3-stem. Γ 3 has no certificates because every order which contains the 3-chain and the 3-antichain also contains the "L" 3-order as illustrated by G.
Definition. Given some Γ n ∈ Λ, we order its finite certificates as follows: let C 1 , C 2 be finite certificates of Γ n , then C 1 C 2 if C 1 is a stem in C 2 . A minimal certificate of Γ n is minimal in this order.
If Γ n ∈ Λ has more than one minimal certificate, these minimal certificates need not have the same cardinality 7 as each other. Also, an n-order in Γ n may be embedded in a minimal certificate of Γ n in more than one way. 8 Examples are shown in figure 7. Proof. Consider a labeled representativeC of C. For each A i , i = 1, 2, . . . k, take a subset ofC that is a stem inC, isomorphic to A i . Take the unionŨ of all those subsets.Ũ is a stem inC.Ũ is isomorphic to a labeled representative of a finite order, U , which is a stem in C and has cardinality |U | ≤ kn. U is also a certificate of 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 certificate of Γ n and |C| = n. If Γ n is not a singleton then any minimal certificate must have cardinality greater than n.
We will also need the concept of a labeled certificate.

Definition.
A labeled causetC is a labeled certificate of Γ n ∈ Λ ifC is a representative of a certificate of Γ n . A labeled causetC is a labeled minimal certificate of Γ n ∈ Λ ifC is a representative of a minimal certificate of Γ n .
An example is shown in figure 8.

Construction of covtree
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. 7 Recall that we define the cardinality, |C|, of an n-order C as |C| := n. 8 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 representativeC of C, there are k different subcausets ofC which are stems and which are isomorphic to a representative of X.  Definition. For any n and any set, Γ n , of n-orders, the map O − takes Γ n to the set of (n − 1)-stems of elements of Γ n : 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 that maximal element to form (a representative of) an (n − 1)-order. The set O − (Γ n ) is the set of all (n − 1)-orders which can be formed in this way.
Lemma 3.2. Let C be an n-order and 0 < k ≤ n. The set of k-stems of C is Proof. Consider, X, a k-stem in C. There exist labeled representativesX of X and  We are now ready to define covtree. Recall that Λ is the collection of sets of n-orders, for all n, which have certificates.

Causal set dynamics on covtree
Covtree allows us to realise the idea described previously of defining a dynamics on a tree in which the nodes in level n are sets of n-orders and each node corresponds to 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-going random walk on covtree, starting at the root. In preparation for exploring the relationship between paths in covtree and infinite orders -the histories in a causal set cosmological model -we generalise the notion of a certificate of a node to the certificate of a path: Definition. An infinite order is a certificate of a path P in covtree if it is a certificate of every node in P. A labeled certificate of a path P is a representative of a certificate of P.
One relationship between infinite orders -elements of Ω -and paths in covtree is straightforward to state and understand: Proof. Let Γ n be the set of n-stems of C, for each n > 0. C is a certificate of each Γ n . Each Γ n is a node in covtree and corollary 3.3 shows that these nodes form a path in covtree down to the root.  The map from infinite orders to paths in covtree implied in the lemma above is not one-to-one because a rogue order is not specified by its stems: if C and C are equivalent rogues then they are both certificates of the same path in covtree.
This means that our stochastic process cannot, in principle, distinguish between equivalent rogues.
It is not immediately apparent whether or not every infinite path in covtree has an infinite order as a certificate but in fact it is true and we have: Let P be an infinite path in covtree starting at the root. There exists 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 then the desired order. We begin with some lemmas.
. . } be a path in covtree and let Γ n ∈ P not be a singleton. 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 of Γ N . By corollary 3.3, D is a certificate of every node below Γ N so D is a certificate of Γ n . Now, at least one minimal certificate of Γ n occurs as a stem in D. Choose one, call it C, let m := |C| and consider Γ m ∈ P. C is an m-stem in D. Γ m is the set 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 of Γ N and a choice of a stem in it which is a minimal certificate of Γ n .
Lemma 4.4. Let P = {Γ 1 , Γ 2 , ...} be a path in covtree and let Γ n ∈ P. There is a node 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 property is 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 a certificate of Γ n . (ii)C m k is a subcauset, a stem, inC m k+1 ; (iii)C m k+1 is a labeled certificate of Γ m k and also therefore a labeled certificate of all nodes below Γ m k ; (iv) C m k+1 , the isomorphism class ofC m k+1 , is an element of Γ m k+1 .
Step 1: 1.0) Pick some nonzero natural number m 0 to start and consider Γ m 0 ∈ P.
1.1) By lemma 4.4 there exists an m 1 such that m 1 > m 0 and such that Γ m 1 contains a certificate of Γ m 0 as an element. Call that certificate C m 1 . Its cardinality is |C m 1 | = m 1 .
1.2) PickC m 1 , a labeled causet which is a representative of C m 1 .

1.3) Go to step 2.
Step k > 1: k.1) By lemma 4.4 there exists an m k such that m k > m k−1 and such that Γ m k contains a certificate of Γ m k−1 as an element. Call that certificate C m k . Its cardi- k.2) Consider C m k−1 and its labeled representativeC m k−1 from the previous step.  This corollary is theorem 4.2.

Measures on R(S)
We propose random walks upwards on covtree as dynamical models in which an order grows and in which arriving at a node Γ n corresponds to the occurrence of First we introduce the covtree measure space. The certificate set, cert(Γ n ), of a node, Γ n , in covtree is the subset The node certificate sets are the covtree "cylinder sets". Let Σ denote the set of node certificate sets cert(Γ n ) for all nodes in covtree, together with the empty set. A random walk on covtree, defined by the transition probabilities for each link 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 path from the root to Γ n . The tree structure of covtree means that Σ is a semi-ring and that a measure µ on Σ generated by a set of Markovian transition probabilities on covtree is countably-additive 9 . Hence we can apply the Fundamental Theorem of Measure Theory [14] which says that the measure µ extends to R(Σ), the σ-algebra generated by Σ.
We are now in a position to prove that: Proof. First we note that as defined, these two σ-algebras are defined over different sample spaces: an element of Σ is a set of infinite orders and an element of S is a set of infinite labeled causets. However, both the covariant algebra R and the stem algebra R(S) can be thought of, in an obvious way, as σ-algebras on the sample space Ω of infinite orders, since their elements are covariant. This is the sense in which the claim is to be interpreted.
We will show that any stem set -thought of as a set of infinite orders -can be constructed by finite set operations on the certificate sets and vice versa, and the result follows.
Consider an n-order B. Let Γ i n be the nodes in covtree such that B ∈ Γ i n , where i labels the nodes. Suppose C ∈ cert(Γ i n ) for some i. Then B is a stem in C and hence C ∈ stem(B). Suppose C / ∈ cert(Γ i n ) for all i. Then B is not a stem in C and hence C / ∈ stem(B). It follows that stem(B) = i cert(Γ i n ).
Suppose C ∈ cert(Γ n ). Then A 1 , ..., A k are stems in C, and B 1 , ..., B l are not stems stem(B j ). Suppose C / ∈ cert(Γ n ). Then either Hence every walk on covtree induces a unique measure on R(S), and every measure on R(S) induces a unique walk on covtree: the transition probability in the 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 between walks on labeled poscau and measures onR so we will call a measure on R(S) a poscau measure if it is a restriction to R(S) of some measureμ onR. A CSG measure on R(S) is a poscau measure such thatμ is induced by a CSG walk.
It follows from lemma 4.8 that every poscau measure on R(S) is a covtree measure on R(S). In fact, it is also true that every covtree measure is a poscau measure: Lemma 4.9. For every measure µ on R(S) there exists an extensionμ toR.
Proof. First note that there is a metric onΩ with respect to which (Ω,R) is a Polish space [9]. Since every Polish space is a Lusin space [17], (Ω,R) is a Lusin space. Note also that R(S) is a separable sub-σ-algebra ofR since there exists a countable collection of subsets ofΩ which generates R(S), namely S (or Σ). The result follows from the theorem that if (Y, B) is a Lusin space, then every measure defined on a separable sub-σ-algebra of B can be extended to B [18].

Discussion
At the beginning of section 3 we posed the question: "Is it possible to construct a physically well-motivated measure on the stem algebra R(S) directly, in a manifestly label-independent way that does not rely on any gauge dependent notion?" The key phrase here is physically well-motivated. We have shown that we can generate a mathematically well-defined measure on the stem events R(S) via a growth process conceived as a random walk up covtree. There is no reason to expect, however, that a generic such walk will be physically interesting: the class of walks is too vast to be interesting. We need physically motivated conditions to restrict the models to a sub-class worth studying. This is what was done by Rideout and Sorkin in the context of walks up labeled poscau by imposing the conditions of discrete general covariance (DGC) and Bell causality (BC) [7]. These conditions restrict the class of walks on labeled poscau to the CSG models.
The relationship between the "labeled" conditions of DGC and BC and any conditions on covtree walks is not understood. Note that lemma 4.9 means that although every covtree walk is apparently completely covariant in its setup, for every walk on labeled poscau -whether it satisfies Discrete General Covariance or not -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 walk and the manifest "covariance" of a covtree walk. We can frame the sort of progress we'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 such that the covtree measure is a poscau measure from a walk on labeled poscau that satisfies DGC only, measures which Brightwell and Luczak call "orderinvariant"? [11,19,20]. (v) What form might a quantum random walk on covtree take and might it be possible to formulate a quantum relativistic causality condition for it, even while 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 when considering what a condition of relativistic causality might look like in a theory in which the spacetime causal order itself is dynamical and stochastic/quantal and in 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 these questions.
stitute and the Raman Research Institute for hospitality while this work was being completed. SZ is partially supported by the Kenneth Lindsay Scholarship Trust.
A Table of   Recall that a random walk on covtree, defined by the transition probabilities for 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 path from the root to Γ n (and µ(∅) = 0).
Next we will show that countable additivity of µ is trivially satisfied as no certificate set is a countable disjoint union of certificate sets 11 . Consider some Γ m in covtree and suppose for contradiction that cert(Γ m ) = i∈N cert(Γ i n i ). Consider the following suborder in covtree, {Γ n ∈ Λ|Γ n Γ m and Γ n Γ i n 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 König's lemma, T m contains an infinite upward-going path starting at Γ m [23]. It follows that there is an infinite path P in covtree such that Γ m ∈ P and Γ i n i / ∈ P for all i ∈ N. Therefore there exists a certificate C of P and hence of Γ m such that C / ∈ cert(Γ i n i ) for all i ∈ N, which is a contradiction.