Toward a "fundamental theorem of quantal measure theory"
aa r X i v : . [ h e p - t h ] A p r Toward a “fundamental theorem of quantal measure theory” ⋆ Rafael D. SorkinPerimeter Institute, 31 Caroline Street North, Waterloo ON, N2L 2Y5 CanadaandDepartment of Physics, Syracuse University, Syracuse, NY 13244-1130, U.S.A.address for email: [email protected]
Abstract
We address the extension problem for quantal measures of path-integraltype, concentrating on two cases: sequential growth of causal sets, and aparticle moving on the finite lattice Z n . In both cases the dynamics canbe coded into a vector-valued measure µ on Ω, the space of all histories.Initially µ is defined only on special subsets of Ω called cylinder-events,and one would like to extend it to a larger family of subsets (events)in analogy to the way this is done in the classical theory of stochasticprocesses. Since quantally µ is generally not of bounded variation, a newmethod is required. We propose a method that defines the measure of anevent by means of a sequence of simpler events which in a suitable senseconverges to the event whose measure one is seeking to define. To this end,we introduce canonical sequences approximating certain events, and wepropose a measure-based criterion for the convergence of such sequences.Applying the method, we encounter a simple event whose measure is zeroclassically but non-zero quantally. Keywords and phrases : quantal measure theory, path integral, measuretheory, descriptive set theory.
1. Introduction
In order to define area, even for something as simple as a disk of unit radius, one needsto invoke an extension theorem. In a systematic development [1] of plane-measure, one ⋆ To appear in a special issue of the journal,
Mathematical Structures in Computer Science edited by Cris Calude and Barry Cooper (Cambridge University Press).1egins by defining the measure µ of an arbitrary rectangle, and one then seeks to extendthe set-function µ unambiguously to subsets of the plane that can be made from rectanglesvia countable processes of union and complementation (these sets comprising the σ -algebragenerated by the rectangles). The unit disk is such a subset, and (if we take it to be open)it obviously can be built up as the disjoint union of a countable family of rectangles. Butthis can be done in an infinite number of different ways, and one needs to know that thenet area of the rectangles is always the same, no matter which decomposition one choosesand no matter in which order one chooses to perform the resulting sum. The theorem thatguarantees this consistency is known as the Kolmogorov-Carath´eodory extension theorem,but it might also be called the “fundamental theorem of classical measure theory”. Not onlyis it used to construct Lebesgue measure, but it plays a central role in defining stochasticprocesses like the Wiener process, a mathematical model of Brownian motion that alsodescribes the Wick rotated path-integral for a non-relativistic free particle on the line.In this sort of application, one is dealing with a probability-measure on a space ofpaths or more generally “histories”, and the possible values of µ are therefore positive realnumbers between 0 and 1. When one seeks to define a genuine path integral in real timehowever (as opposed to Wick-rotated, imaginary time), one encounters complex amplitudesthat can be arbitrarily large and of any phase. Once again, there are specially simple sets ofpaths, analogs of the rectangles called “cylinder sets”, from which the more general sets ofinterest can be built up, but the sums that arise in this case no longer converge absolutely.In technical terms the complex measure one is trying to extend is not of bounded variation,and the available extension theorems cannot be used [2].The problems that one faces vary, depending on context. There are “ultraviolet”problems springing from the infinite divisibility of the paths or “histories” one is trying tosum over, and there are “infrared” problems that arise in connection with histories thatare unbounded in time. By limiting ourselves to spatio-temporally discrete processes wenullify the former problems, and that will be the context of the rest of this paper, wherewe will encounter only discrete histories like those that occur in a random walk. It willthus be only issues of infinite time that will occupy us.The concrete instances we will consider will be of one of two types, which we cancharacterize by the kind of “sample space” or “history space”, Ω, on which one builds.The first instance arises in the context of quantum gravity and more specifically within the2ausal set programme. There the discreteness reflects the finiteness of Planck’s constant,and the underlying physical process is a kind of “birth” or “accretion” process by means ofwhich the causal set is built up or “grows”. The corresponding sample-space of “completed”causal sets consists of all the countable, past-finite partial orders P ; and one is seeking todefine a certain type of vector-valued measure µ on it. (The dynamics determines µ onlyup to a unitary transformation. The object of direct physical interest is not µ itself buta certain scalar-valued set-function belonging to the class of strongly positive decoherencefunctionals or quantal measures on Ω. However, any such a functional can be represented[3] as a measure on Ω which is valued in some Hilbert space H .) In the second type ofexample, the elements of Ω will be discrete-time trajectories moving in a lattice that willbe either the integers modulo n ( Z n ) or just the integers as such ( Z ). These examplescorrespond to a widely studied class of processes known as “quantal random walks”, butfor us they will be important primarily as simplified analogs of causal set growth processes.In that role, they are particularly illuminating because their sample spaces are essentiallythe same ones that “descriptive set theory” investigates.How certain are we, though, that the quantal measures in these all instances really needto be defined in a new way? With the lattices, the dynamical laws in question are those ofthe evolution generated by a unitary operator or “transfer matrix”. In their path-integralformulation, such unitary laws inevitably lead to measures of unbounded variation [3], andthe theorems of the Kolmogorov-Carath´eodory type are thus guaranteed to fail. In themore important, causal set case however, there remains some doubt, especially given theanticipated breakdown of unitary evolution in that case. The only fully developed dynamicsone has for causal sets is that of the classical sequential growth (CSG) models, which inthemselves are not quantal in nature. For them, the usual extension theorems do sufficebecause one is dealing with a classical probability measure [4]. But if one complexifies theparameters of a CSG model, one obtains straightforwardly a family of quantal measures(decoherence functionals) which are in general neither unitary nor of bounded variation[3]. Although none of these complexified CSG dynamics is likely to exhibit quite the typeof interference required by quantum gravity, the fact that the measures that arise are notof bounded variation suggests that this might turn out to be a general feature of quantalcausal sets, just as it is a general feature of quantal path integrals in other contexts.Nevertheless it’s worth keeping in mind the possibility that the physically appropriatequantal measures for causal set dynamics will turn out to be σ -additive in the traditional3ense. Were that to happen, quantum gravity would have revealed itself to be moretractable mathematically than the nominally much “simpler” non-relativistic free particle!The problems addressed in the present paper would then be pseudo-problems, as far asquantum gravity went.In our current state of ignorance, however, it seems prudent not to count on so muchgood fortune. And besides, one might still like to have a well defined path-integral forsystems like the free particle, without having to embed them in a full-blown theory ofquantum gravity. What, then, can one do when bounded variation fails? As urged inreference [3], such a failure need not be the end of the story, because in a concrete physicalsituation, the space of histories has more structure than what is available in an arbitrarymeasure space. Indeed, physicists routinely work with infinite sums and integrals thatconverge only conditionally. Typically one introduces a “cutoff” or integrating factor ina manner mandated by physical considerations, in effect doing the sums or integrals ina particular order so that their convergence need not be absolute. In the case of theplanar disk, for example, instead of expressing it as a disorganized sum of an infinitenumber of rectangles, one might think to employ a definite sequence of approximations,each consisting only of rectangles bigger than a certain size ǫ . The area would then begiven by the ǫ → A ⊆ Ω of histories or paths, a “canonical” sequence of approx-imations A n to A in terms of cylinder sets, and this sequence should be as near to uniqueas feasible. Then, given such a sequence, we will try to decide what further convergenceproperties it ought to have in order that we can form a limit µ ( A ) of the individual µ ( A n )and consistently attribute this limit to A as its quantal measure.In what follows, we take only a few steps in the direction indicated, pointing out alongthe way various pitfalls that one needs to avoid. Hopefully this can at least illustrate thekind of approach one might take to the extension problem for quantal measures. Sections 6and 7 are the heart of the paper. In section 6, we will define canonical approximations for alimited class of events , as (sufficiently regular) subsets A ⊆ Ω are normally designated. Wewill then introduce, in section 7, a convergence criterion for an approximating sequence A n , and we will prove that the resulting extension of µ is additive for disjoint unions4f open sets. Limited as our approximation scheme will be, it will at least embrace thetype of event A which is most important for the sake of causal sets, namely the covariantstem-event . Among all the sets of histories to which one might wish to assign a measure,the only indispensable ones are these. Without their aid it would be nearly impossible toproduce a generally covariant dynamical scheme in any useful sense [5] [4].An appendix lists some of the symbols used in the body of the paper. Contents
1. Introduction2. Sample-spaces and amplitudes for causal sets and the 2-site hoppercausal sets n -site hopper3. Some events whose measures one would like to define4. Ω as a compact metric spaceopen and closed setsthe tree of truncated histories5. Set-theoretic limits of events6. Canonical approximations for certain events7. Evenly convergent sequences of eventsExamples8. Epilogue: does physics need actual infinity?Appendix. Some symbols used, in approximate order of appearance
2. Sample-spaces and amplitudes for causal sets and the 2-site hopper causal sets
A causal set (or causet ) [6] [7] [8] [9] [10] in its most general conception can be any locallyfinite partial order or poset , but in the context of the dynamics of sequential growth andquantal cosmology no element of the causet will possess more than a finite number ofancestors. For present purposes we may thus define a causet as a past-finite countableposet , i.e. a countable (possibly finite) set of elements endowed with a transitive, acyclicorder-relation, ≺ , which I will also take to be irreflexive. These concepts are exposed ingreater detail in [11], where the notion of sequential growth is also explained. Here I willjust summarize the main definitions and introduce the notation we will use.A sequential growth process proceeds as a succession of “births” of new elements, andin this sense is never ending. If however, one idealizes it has having “run to completion”,5t will have produced a completed causet as defined above: a countable set of elements,each having a finite number of predecessors or ancestors but a possibly infinite number ofdescendants. The set of all such causets constitutes the natural sample-space Ω for thisprocess. Actually, one must distinguish here two distinct sample spaces, which one may callΩ gauge and Ω physical . The latter, which in some sense is the true sample-space, consists of unlabeled causets, or equivalently isomorphism equivalence classes of causets. The former,which I’ll normally denote simply as Ω, then consists of the naturally labeled causets, anatural labeling being a numbering, 0 , , , . . . of the elements which is compatible withthe defining order ≺ : if x ≺ y then y carries a bigger label than x . Here again, one ofcourse really intends isomorphism equivalence classes of labeled causets (or if you like theelements could be taken to be the integers themselves in this case).The labels record the order of the respective births, and what is most important forus here is that this order is supposed to be fictitious in the same sense as is a choiceof coordinate system for a continuous spacetime is fictitious. The physically meaningfulor covariant events will thus correspond to subsets of Ω physical , whereas the measure µ defining the growth process is in the first instance defined on Ω gauge . But even a verysimple subset of Ω physical , even a singleton, will equate to a much less accessible subsetof Ω gauge , namely the subset obtained by taking every possible natural labeling of everymember of the original subset. Thus arises the need for an extension of µ that will assignwell defined measures to such “covariant” subsets of Ω gauge . Unlike for the example of thehopper to be discussed next, this is not just a matter of convenience if one wants to be ina position to ask truly label-independent questions about the causet.Henceforth in this paper all causets will be labeled unless otherwise specified. (Inreference [4] the true or “covariant” sample space was denoted simply by Ω, while itslabeled counterpart was e Ω. Here however, it seems simpler to use Ω for the latter, since itis the space we will usually be dealing with.)In reference [4], the measures defining the CSG dynamical models were defined rigor-ously by extending a probability measure given originally on the space Z of cylinder events (or cylinder sets), where a cylinder event cyl( c ) ∈ Z is by definition the set of all completedcausets containing a given, naturally labeled, finite causet c . A finite causet will also becalled a stem and on occasion a “truncated history”. In conjunction with these definitions,let us also define Ω( n ), the space of all naturally labeled causets of n elements, and Z ( n ) or Z n , the space of cylinder events of the form cyl( c ) for c ∈ Ω( n ). The cylinder sets comprise6hat is called a “semiring” of sets in the sense that given any two cylinder sets, Z and Z , their intersection, Z Z ≡ Z ∩ Z , is also a cylinder event, and their difference Z \ Z is the disjoint union of a finite number of cylinder events. (In fact the cylinder eventsform an especially simple kind of semiring, because any two of them are either disjoint ornested.)To rehearse the definition of the CSG models in general would take us too far afield, butthe special case of “complex percolation” is simple enough to be given here in illustrationof the general scheme. The vector measure µ is determined in this case by a single complexparameter p , and it takes its values in a one-dimensional Hilbert space that we may identifywith C , so that µ ( A ) is itself just a complex number. Now let c ∈ Ω( n ) be a labeled causet of n elements and let Z = cyl( c ) be the corresponding cylinder set. Then µ ( Z ) = p L (1 − p ) I ,where L = L ( c ) is the number of links in c and I = I ( c ) is the number of incomparabilities.Here an incomparability is simply a pair of unrelated elements, and a link is a causalrelation, x ≺ y , which is “nearest neighbor” in the sense that there exists no intervening z for which x ≺ z ≺ y .Observe now that the collection of naturally labeled finite causets, i.e. the space S n Ω( n ), has itself the structure of a poset in a natural way. Indeed this poset is actuallya tree T , because its elements are labeled. (The corresponding structure formed by theunlabeled stems is a more interesting poset called poscau in reference [11].) Clearly, aparticular realization of the growth process, or equivalently the resulting completed causetin Ω, can be conceived of as an upward path through this tree. An analogous conceptionwill be possible for the two site hopper, and in this guise seems to be exploited heavily indescriptive set theory [12][13]. (See figures 1 and 2.)7 igure 1. The first 4 levels of the tree T of naturally labeled causetsFinally, let us define an event algebra to be a family of subsets of the sample space Ωclosed under the operations of intersection and complementation. An event algebra is thusa Boolean algebra or “ring of sets”. To the extent it can be achieved, one normally wantsthe domain of µ to be such an algebra, because for example, if the events “ A happens”and “ B happens” are of interest, then so also is the event “either A or B happens”. Thecylinder sets Z do not themselves form an algebra, but the family S of finite unions ofcylinder sets does. It is in fact R ( Z ) the Boolean algebra generated by Z . In all casesof interest µ will automatically extend uniquely from Z to S , yielding a finitely-additivemeasure thereon. The space S thus constitutes a minimum domain of definition for thevector-measure µ . The question then will be how far µ can be extended beyond S intothe σ -algebra generated thereby, the hope being that the enlarged domain A will itselfbe an event algebra, and that it will contain enough events so that, at a minimum, thephysically most important questions will become well posed. (Some noteworthy instancesof covariant questions/events will be discussed in the next section.) n -site hopper By “2-site hopper” I mean the formalization of a particle residing on a 2-site lattice andat each of a discrete succession of moments either staying where it is or jumping to theother site [14]. For definiteness, I will assume that the moments are labeled by the naturalnumbers, the sites by Z , and that at moment 0 the hopper begins at site 0. The definitionsof sample space, cylinder event, etc. are closely analogous to those given above for causets,8nd references to them should be understandable without their formal definitions, whichI will postpone until after the transition amplitudes have been specified. The full courseof the motion, idealized as having run to completion, will be called a path or “history”.Notice that, modulo the small ambiguity in how a real number can be expressed as a“binary decimal”, each such path can be identified uniquely with a point in the unitinterval [0 , ⊆ R .Aside from a simpler sample space than in the causet case, the hopper offers us inaddition a fuller illustration of the problems of defining the vector-measure correspondingto a path integral. Unlike the former case, where the correct choice of quantal ampli-tudes is only conjectural, there exists for the hopper a choice that can be interpreted as astraightforward discretization of the Schr¨odinger dynamics of a non-relativistic free particlemoving on a circle (cf. [15]).These amplitudes can be understood more easily if one sets them up, not just for twosites, but for the more general case of the circular lattice Z n (“ n -site hopper”). Perhapsthey will look most familiar if presented as the unitary evolution operator or “transfermatrix” analogous to the propagator that solves the Schr¨odinger equation in the continuouscase. To that end, let x ∈ Z n be the location of the particle at some moment t , let x ′ beits location at the next moment t ′ = t + 1, and write for brevity exp(2 πiz ) ≡ z . Theamplitude to go from x to x ′ in a single step is then1 √ n ( x − x ′ ) /n for n odd and 1 √ n ( x − x ′ ) / n for n even. For example, for n = 6 and with q = / , the (un-normalized) amplitudesto hop by 0, 1, 2 or 3 sites are respectively q = 1, q = q , q , and q = − i . For the 2-and 3-site hoppers, the above amplitudes are particularly simple, yielding for n = 3 thetransfer matrix 1 √ ω ωω ωω ω ( ω = 1 / )and for n = 2 the transfer matrix 1 √ (cid:18) ii (cid:19) . (1)9rom these expressions and the definition of the decoherence functional it is not hardto carry out the construction of the equivalent vector measure along the lines of [3]. Inthe simplest case of two sites, which will be our main example herein, µ is valued in atwo-dimensional Hilbert space C and, with a convenient choice of basis vectors, can beexpressed as follows. Let (0 x x x . . . x m ) be a truncated path and let Z ⊆ Ω be thecorresponding cylinder event. Then µ ( Z ) ≡ | Z i will be the two-component complex vector v α where (no summation implied) ⋆ v α = ( U − m ) αx m U x m x m − · · · U x x U x x U x , (2) U being the unitary matrix of equation (1). Notice incidentally that U j is periodic withperiod 8 and is very easy to compute explicitly, since U = − U = (cid:18) ii (cid:19) is alsovery simple.Finally the formal definitions for the 2-site hopper. A truncated history, the coun-terpart of a finite causal set, is for the hopper an initial segment of a path, for example(0 , , , , n when the initial 0 is omitted will be Ω( n ), andthe corresponding cylinder events will be the elements of Z n . The semiring Z will be theunion of the Z n . For example, cyl(0 , , , , ∈ Z is the set of all completed paths of theform (0 , , , , , x , x , · · · ). Exactly as above, S will be the Boolean algebra generatedby Z . One can check straightforwardly that µ , as defined by (2), extends uniquely andconsistently to each S n and therefore to S as a whole. Again, the truncated histories canbe construed as the nodes of a tree T , the “branches” or “edges” being given by extensionof path. (See Figure 2.) For example, there will be an edge from (0 , , ,
0) to (0 , , , , ⋆ Another notation for v α could be h α | x x x . . . x m i igure 2. The first 4 levels of the tree T for the 2-site hopperIn what follows, it will sometimes be enlightening to consider hopper-paths on theinfinite lattice Z . In that case the paths will be restricted to move no more than onesite per step (“random walk”), in order that the resulting tree T continue to have a finitenumber of branches emanating from each node.For an extensive discussion of the quantal 2-site hopper see [14]. For more generalsorts of quantal random walks see [16].
3. Some events whose measures one would like to define
The event algebra S generated by the cylinder events supplies enough events to allow oneto ask any question † about the process under consideration, as long as it doesn’t refer tohappenings arbitrarily far into the future. But often one does not want to be bound bythis limitation, especially since in the causet case, the “time” referred to contains a largeelement of gauge, as explained above.To illustrate how “infinite-time” events enter the story, let us dwell on a few examples,beginning with the n -site hopper. Perhaps the simplest and most familiar example of thiskind is the event R of return , which occurs if and when the particle returns to its startingpoint at some later time. This event, in other words, is the set of all paths (0 x x . . . ) forwhich one of the x i = 0. Plainly R is not in S , because the return, although it must occurat a finite time if it occurs at all, can take place arbitrarily late. For a classical hopper on † The words “event” and “question” are in a certain sense synonyms. To an event A ⊆ Ωcorresponds the question “Does A happen?”. Note in this connection that (except in theclassical case) it would lead to confusion if one read “ A happens” as “the path is an elementof A ”, cf. [17]. 11 finite lattice, one knows that µ ( R ), the measure of the return-event (which classically isits probability), is unity, but to express this fact directly, we need R to be in the domain of µ . Of course, we could circumvent any direct reference to R by introducing the finite-timeevent R n that the particle returns on or before the n th step. Instead of asserting that µ ( R ) = 1, we could then say “The sequence µ ( R n ) converges to 1 as n → ∞ ”. Plainly, thefirst formulation is simpler and less cumbersome to work with. Notice in this case that notonly at the level of the measures, but even at the level of the events themselves, R is thelimit of the R n in a natural sense, since the latter are nested and “increase monotonicallyto R ”. That is, one has R ⊆ R ⊆ R · · · , with R itself being the union of the R n , orlogically speaking their “disjunction”. Were µ a classical measure, this would guaranteeconvergence of the µ ( R n ) and consistent extension of the domain of µ to include the event R ; in the quantal case it guarantees nothing.A similar event to “return”, but one which is related even less directly to any cylinderevent, is the event R ∞ that the particle visits x = 0 infinitely often. This event also hasa well-defined probability of unity in the classical case. Since, however, it cannot come tofruition at any finite time, it cannot — unlike the event R of simple return — be expressedas a union of cylinder sets or other members of S . Instead it is a countable intersectionof events, each of which is a countable union of events in S . For example, let E ( j, k ) for j < k be the event that x k = 0. Then R ∞ = T j S k E ( j, k ). (In words: for each moment j there is a later moment k at which the particle visits the origin. ♭ ) To give meaning to µ ( R ∞ ) by prolonging the initially defined measure with domain S one would thus have tothink in terms of a limit of limits.As a third example (restricted this time to one of the lattices, Z or Z n with n > x = 3 but never reaches x = 5. Intermediatebetween the two previous examples in its remoteness from S , this event is naturally ex-pressed as the set-theoretic difference of two limits of finite-time events, the first being,naturally the event F that the particle reaches x = 3 and the second G that it reaches x = 5. Just as with the return event R , the event F \ G is, in a well defined sense to which ♭ One can often arrive at such combinations by beginning with a formal statement ofwhat it means for the event to happen. In this case one might first write down whatit means for R ∞ not to occur: ( ∃ n )( ∀ n > n )( x n = 0), and then negate it to obtain( ∀ n )( ∃ n > n )( x n = 0). The nested combination of unions and intersections is basicallyjust a translation of this second statement into set-theoretic language.12e will return below, a limit of events in S , but it is not simply the union or intersectionof a monotonically increasing or decreasing sequence.It is useful at this point to introduce some further notation to help in discussingthe types of events we have just met. Let X be any collection of subsets of Ω closedunder pairwise union and intersection. Then W X will be the family of events of the form S ∞ n =1 X n , where X n ∈ X and X ⊆ X ⊆ X · · · . It is easy to see that W X is alsoclosed under union and intersection, also that it would not change if we dropped themonotonicity condition, X ⊆ X ⊆ X · · · . In words, the members of W X are the unionsof monotonically increasing events in X . For the intersections of monotonically decreasingevents in X , I will write dually V X . And for the Boolean algebra generated by X , I willwrite, as above, R X or R ( X ) . Our first example, “return”, is then an element of W S ,our second of V W S , and our third of R W S , while for S itself, we have S = R ( Z ).Turning now to events for causal sets, we will encounter some types very similar tothose just discussed. Foremost in importance are the unlabeled stem-events mentionedearlier. Given two causets c and c ′ , of which the first is finite, we say that c ′ admits c as a stem (or “partial ⋆ stem”) if c ′ contains a downward-closed subset that is isomorphic to c .In the context of sequential growth, this can also be expressed by saying that it might havehappened that elements of c were all born before any of the remaining elements of c ′ . Astem thus generalizes the notion of “initial segment”. The stem-event ‘ stem( c )’ is then theset of all c ′ ∈ Ω which admit c as a stem. The stem c that enters this definition is taken tobe unlabeled, because our purpose is to produce a label-independent or “covariant” event.It is evident that stem( c ) is indeed covariant in this sense, since the condition that definesit does not refer to the labeling of c ′ .The importance of the stem events physically is that essentially any covariant questionthat we care to ask about the causet can in principle be phrased in terms of stem-events.The precise result proven in [4] is that every covariant event is equal, up to a set of measurezero, to a member of the σ -algebra generated by the stem-events. One can also prove thatany covariant event which is open in the topology of section 4 below is a countable unionof stem events, a purely topological result that holds independently of any assumptionabout the measure µ . Ideally then, the domain of µ would embrace the whole σ -algebra ⋆ One can also define “full stems” [11], but there is no special reason to consider themhere. 13enerated by the stem-events. At a minimum, one would hope that it would embrace thestem-events themselves.Now the event ‘ stem( c )’ does not belong to the domain S on which µ is initiallydefined, because it is not a finite-time event when referred to “label-time”. If it were, thenthere would exist some integer N such that if the growing causet c ′ admitted c as a stem,it would already admit it as soon as the first N elements had been born. But in fact thereis nothing in principle to stop the stem in question appearing at an arbitrarily late stageof the growth process. Evidently, the situation is like that of the hopper event “return”.Based on this analogy, one would expect the stem events to be found in W S , and so theyare, as follows directly from the fact that any stem-event is a union of cylinder sets:stem( c ) = [ n cyl( e b ) ∈ Z : e b admits c as a stem o . (3)The problem of extending the vector-measure µ from S to W S is thus the most basic onefor causal sets.Starting from the stem-events, one can build up other covariant events, whose occur-rence or non-occurrence is of interest for cosmology. The simplest of these is the eventthat the causet is “originary”, meaning that all its elements descend from a unique min-imal element or “origin”. To say that a completed causet is originary is simply to saythat it contains no second minimal element, for which it is necessary and sufficient that itfail to admit the 2-element antichain as a stem. (An antichain is a set of elements whichare mutually unrelated or “spacelike” to one another.) Thus the event ‘originary’ is thecomplement of the event ‘ stem( a )’, where a is the antichain of two elements. As such itbelongs to V S , since as we have seen, the stem events all belong to W S , and union turnsinto intersection under complementation.If an originary causet represents a certain kind of “big bang” then a causal set contain-ing what is called in the combinatorics literature a post describes a “cosmic bounce”. (Apost is an element of a poset which is spacelike to no element.) In its degree of remotenessfrom the elementary cylinder events, the post event is comparable to the event of “infinitereturn” in the case of a random walk, the similarity being even closer if we compare thepost event to the complement of the infinite return event. In fact both events belong to W V S , although this is less easy to demonstrate for the post-event than it is in the caseof return. To see why it is nevertheless true, imagine watching a succession of births ofcauset elements, x , x , x . . . and waiting for a post to be born. If the birth in question14s that of element x n then x n must have every previous element as an ancestor: x j ≺ x n for all j < n . This renders x n momentarily a “candidate for becoming a post”, but it doesnot guarantee that x n will remain a viable candidate forever. In order for that to occur,every subsequent element, x n +1 , x n +2 , · · · , must arise as a descendant of x n , i.e. x j ≻ x n for all j > n . By thinking of the post event P in this manner, namely as the set of allsequences of births satisfying the condition that a candidate post appear at some stage n and then not lose its viability at any later stage m > n , one can deduce that P belongs to W V S . The most “covariant” (albeit not the most direct) construction along these linesproceeds by first expressing P in terms of stem-events; this will also illustrate the thesisthat all covariant questions of interest can be expressed in terms of stem-events.Proceeding in this manner, note first that if x is a post then its exclusive past T = { y : y ≺ x } is not only a stem, but what has been called a “turtle” [18], meaning in thepresent context a stem that wholly precedes its complement: ( ∀ x ∈ T )( ∀ y / ∈ T )( x ≺ y ). Some thought reveals that a causet contains a turtle of n elements iff every stem ofcardinality n + 1 has a unique maximal element. Introducing the term principal for sucha stem, together with the terms n -stem [resp. n -turtle] for a stem [turtle] of n elements,we can say succinctly that a causet contains an n -turtle iff every ( n + 1) -stem is principal .Furthermore it’s easy to demonstrate that x is a post iff both its exclusive and inclusivepasts are turtles (the exclusive past being { y = x : y ≺ x } and the inclusive past being { y : y (cid:22) x } ). Therefore, in a labeled causet, element x n is a post iff every stem of either n + 1 or n + 2 elements is principal. Let P n be the event that this happens, then the postevent itself is P = ∪ ∞ n =1 P n .Now let us examine the event P n more closely. It fails to happen iff some ( n + 1)-stemor ( n + 2)-stem fails to be principal. Let S n , S n , . . . , S nK n be an enumeration of all suchstems (there being only a finite number of n -stems, for any n ), and let Q nj = stem( S nj ) bethe corresponding stem-events. We then obtain P n in the “manifestly covariant” form P n =Ω \ ( ∪ j Q nj ) = Ω \ ( ∪ j stem( S nj )). P is thus a countable union of finite Boolean combinationsof stem events: P = ∞ [ n =0 (Ω \ ( K n [ j =1 stem( S nj ))) . (4)If one knew how to take stem events as primitive, P would thus be a rather simple typeof event, inasmuch as the inner union only ranges over a finite number of events. But15iven that the extant dynamical schemes all begin with labeled causets, we will still needto trace everything back to the cylinder events Z .First, though, a simple example might be in order, say for n = 1. (The event P isjust the originary event, which one might not even want to count as a post.) The event P requires that all 2- and 3-stems be principal. The only 2-stem that can occur is thusthe 2-chain ( a ≺ b ), while the admissible 3-stems are the 3-chain ( a ≺ b ≺ c ) and the“Λ-order” ( a ≺ c, b ≺ c ). The stems that must be excluded — those denoted above by S nj — are correspondingly the 2- and 3-stems which are not principal: the 2-antichain, the3-antichain, the “L-order” ( a ≺ b, c ), and the “V-order” ( a ≺ b, a ≺ c ). (See figure 3.) Figure 3.
The non-principal 2- and 3-stems.To complete the demonstration that P ∈ W V S , let us return exclusively to labeledcausets, observing first that in view of equation (4), it suffices to show that the complementof a finite union of stem-events belongs to V S . (Strictly speaking, given how we havedefined the operation W , we also need to convert the outer union in (4) into an increasing countable union of events in V S . That this is possible follows readily from the relation (6)of Section 5, which informs us that, when A n and B n are both decreasing sequences of sets,then the union of their limits coincides with the limit of the decreasing sequence A n ∪ B n ,in consequence of which V S is closed under finite union, and we can replace a countableunion ∪ n F n of events F n ∈ V S with the increasing union ∪ n F ′ n , where F ′ n = ∪ m ≤ n F n .)To that end, recall that any stem-event A is an increasing union of events in S . Itscomplement, Ω \ A , is therefore a decreasing intersection of complements of events in S ,each of which is itself in S since the latter, being a Boolean algebra, is closed undercomplementation. Hence, the complement of a stem-event belongs to V S , and the sameholds for the complement of a finite union of stem-events, such as occurs in (4).16or reasons that will become clear shortly, it is natural to designate the elements of S as clopen , meaning “both closed and open” in the sense of point-set topology. † Theevents in W S will then be open , those of V S will be closed , and we will have expressedour post-event P ⊆ Ω as an increasing limit of closed subsets of Ω. Continuing in this vein,more elaborate combinations of the clopen events can be formed, including for example theevent that infinitely many posts occur. But the physical relevance of such combinationsseems to shrink rapidly as their complexity grows. Indeed, one might feel that, questionsof convenience aside, no event more complicated than a finite Boolean combination ofstem-events can claim to be indispensable. One might even go farther and call into doubtthe status of complementation (negation), leaving unquestioned only those events formedas finite unions and intersections of stem-events. Ω as a compact metric space open and closed sets Whenever the idea of convergence plays a role, one can expect, almost by definition,that topology will make an appearance. In the present situation, we are talking aboutconvergence to a given event A ⊆ Ω of a sequence of approximating events A n , in the firstinstance events formed as finite unions of cylinder events and thus belonging to the event-algebra S . In setting up such a sequence of approximations, we would like, as explainedearlier, to regard those cylinder events that specify a greater portion of the history asmore “fine grained” than those that specify a lesser portion. This leads very naturally to adefinition of distance between histories that makes Ω into a compact metric space [4] [12].As implemented for causal set growth processes, the definition runs as follows. Foreach pair of completed labeled causets a , b ∈ Ω, we set: d ( a, b ) = 1 / n , (5)where n is the largest integer for which elements a a · · · a n produce the same poset (withthe same labeling) as elements b b · · · b n . It is easy to verify that this yields a metric onΩ; indeed d satisfies a condition stronger than the triangle inequality: for any three causets a , b and c , we have d ( a, c ) = max( d ( a, b ) , d ( b, c )). This “ultrametric” property derives from † In the context of abstract measure theory, the term “elementary sets” was used in ref-erence [1] to refer to events analogous to those of S .17he tree structure of the space Z of cylinder sets (or equivalently truncated histories), asdescribed earlier. The maximum distance between two causets is 1 / ♭ Moreover the cylinder sets, being the balls of some radius, are bothopen and closed: clopen. It follows by definition that Ω has a basis of clopen sets and thatevery open set is a countable union of cylinder sets (there being only a countable number ofcylinder sets because the finite causets are only countable in number). Since each elementof the event algebra S is itself a (finite) union of cylinder sets, we can conclude that theopen sets are precisely the members of W S , the closed sets, their complements, being thenthe members of V S . The events that belong to both these families are the clopen events,and they clearly include all of S , because a finite union of open [respectively closed] setsis also open [closed]. Let us prove the converse, that every clopen event belongs to S . Lemma (4.1) W S ∩ V S = S Proof
We are asked to prove that S comprises precisely the clopen subsets of Ω. Sincewe already know that every A ∈ S is clopen, it suffices to verify that any clopen A , i.e.any A ∈ W S ∩ V S , also belongs to S . Let A ∈ W S . By definition A is a union ofcylinder sets: A = Z ∪ Z ∪ Z . . . . Now this sequence either terminates at a finite stage orit does not. If it does terminate then A is a finite union of cylinder sets, whence a memberof S , and we are done. If it does not terminate, then we can find a sequence of points x j ∈ A which escapes from every Z j ; and because Ω is compact, we can suppose that thissequence converges to some x ∈ Ω. This x cannot lie in any given Z k because it is a limitof points x j which eventually belong to the closed set Ω \ Z k ; consequently x / ∈ A = ∪ k Z k .We have thus constructed a sequence of points of A which converge to a point outside A ,meaning that A is not closed. It thus cannot be clopen. Or if it is clopen, then we areback to the terminating sequence and the conclusion that A ∈ S .Turning to the 2-site hopper, we need to make only one change to what has beenwritten above for causets. The histories are now sequences of digits, 0 or 1, beginningwith 0, and the integer n that occurs in the definition (5) is now the largest index such ♭ Proven explicitly in the next sub-section. 18hat the two subsequences (0 a a · · · a n ) and (0 b b · · · b n ) coincide. The rest is all thesame. The history space Ω is still a compact metric space, the cylinder sets are clopen andgenerate the topology, etc. the tree of truncated histories We have already seen in section 2, that a point of Ω, that is to say a history, can beconstrued as a path γ through the tree T , each node of which is a “truncated history”,meaning, as the case may be, either a finite causet or a finite sequence of binary digits. ⋆ Flowing from this correspondence between histories and paths through T is a different wayto characterize certain types of events, including the open sets W S and more generallythe events in R W S .Consider first a cylinder set Z = cyl( h ), where h ∈ Ω( n ) is a truncated history.Which paths γ correspond to this cylinder set? By definition they are just the pathswhose corresponding histories reproduce h when truncated at the n th stage, that is, theyare precisely the paths that pass through the node in T that represents h , which I willdenote either as h itself or as node( h ) in order to emphasize that h is being treated asa node in T . Because T is a tree, such a path necessarily follows one of the branchesemanating from node( h ); it then remains forever in the “upward subset” of T consisting ofall descendants (in T ) of h . In this way, every open event A ⊆ Ω can be represented by anupward-closed subset α ⊆ T , and vice versa, given such a subset the paths that enter (andconsequently remain in) α comprise an open event A ⊆ Ω. More generally, every subset α ⊆ T gives rise to an event S ( α ) by the same rule: Definition S ( α ) = { γ : γ is eventually in α } Here, γ is a point of Ω, represented as a path γ = ( h , h , h · · · ) through T , and thestatement that this path is eventually in α means that ( ∃ n )( ∀ n > n )( h n ∈ α ). Evidently, S ( · ) commutes with the Boolean operations: S ( αβ ) = S ( α ) S ( β ), S ( α \ β ) = S ( α ) \ S ( β ), S ( α + β ) = S ( α ) + S ( β ), etc. (where α + β := ( α ∪ β ) \ ( αβ ) is the Boolean operation of“addition modulo 2”). ⋆ The paths under consideration in what follows will usually begin at the “root” of T (corresponding to the cylinder-set Ω), but sometimes they will originate at some othernode of T . The two cases are actually interchangeable because any path not originatingat the root has a unique extension back to it, T being a tree.19s a further aid to intuition, one can conceive of certain types of events in terms of“properties” acquired or lost in the course of the process under consideration. Formally,this corresponds closely to the characterization by sets of nodes in T , but it carries perhapsa more “evolutionary” feeling. For example consider the event of return analyzed earlier.One can cook up a “property” which the particle possesses when, and only when, it hasreturned to the origin. By definition, this property of “having returned” is hereditary inthe sense that, once acquired, it can never be lost. Topologically, the set of all paths γ which acquire a hereditary property yields an open subset of Ω, as is easily corroborated ifone thinks through the definitions. Dually, a property that once lost can never be regained,but that every path begins with, corresponds to a closed set (causet example: being anoriginary). And a property that can be acquired but never regained if lost yields an eventof the form A \ B , where both A and B are open. (Hopper example: visiting x = 1 but not x = 2 . Notice that this third type of property includes both of the previous two as specialcases.) In terms of sets of nodes like the sets α discussed above, the first type of propertyis an upward-closed subset of T , the second is a downward-closed subset, and the third isa convex subset, defined as a subset of T that contains, together with nodes h and h ,every node that lies on some path from h to h . In order-theoretic language for the poset T , this just says that α includes the order-interval between any two of its elements. † InSections 5 and 6, the events of the form S ( α ) for some convex α will be among those forwhich we will able to produce a canonical representation as a limit of clopen events.As an application of some of these ideas, let us prove the assertion made earlier thatΩ is topologically compact. By a standard criterion for compactness, it suffices to provethat any covering of Ω by cylinder sets has a finite sub-covering, so consider an arbitrarycollection of cylinder sets Z ∈ Z that covers Ω. In relation to T , such a covering is acollection of nodes which no path γ can avoid forever. Now the (incomplete) paths that doavoid these nodes fill out a subtree ♭ T ′ of T , with the property that no path γ can remainwithin T ′ forever. But it is well-known that such a tree can have only a finite number ofnodes, assuming that no node has an infinite branching number. (This has been calledthe “infinity lemma” of graph theory.) The maximal elements of T ′ thus furnish a finite † The order-interval delimited by elements x and y of some poset is { z : x ≺ z ≺ y } . ♭ a downward-closed subset of T . 20ollection of nodes that every path must encounter. In their guise as cylinder-sets thesenodes constitute then a finite subcover of Ω.
5. Set-theoretic limits of events
The most elementary kind of limit that one can imagine for a sequence of events partakesof neither metric nor topology nor measure; it is purely set-theoretic. We have alreadyseen how increasing sequences of clopen sets yield the open sets W S , while decreasingsequences of clopen sets yield the closed sets V S . In both cases the operative concept oflimit emerges more or less automatically. Going beyond these two types of approximation,we can recognize a more general concept of which W and V are special cases. Let X j be asequence of subsets of Ω, and deem it to be convergent when we have for any point x of Ωthat eventually x ∈ X j or eventually x / ∈ X j . In such a case, we will write X = lim X j ,where of course X consists of those x that realize the first alternative of being eventuallyin X j . The set of all events obtainable in this way as limits of events A j ∈ S , I will denoteas Lim S . Notice that ‘lim’ commutes with the Boolean operations:lim( A n ∪ B n ) = (lim A n ) ∪ (lim B n ) , etc. (6)In trying to extend our vector-measure µ beyond the clopen events, one might hopethat one could at least get as far as Lim S . Were µ an ordinary measure, this would betrue, because lim A j would be sandwiched between the measurable sets lim sup A j = ∩ j ∪ k>j A k and lim inf A j = ∪ j ∩ k>j A k , both of which are equal to lim A j when the latterexists. This would ensure that lim A j was measurable and that lim µ ( A j ) = µ (lim A j ) .But with quantal measures this argument is not available, and it turns out that convergencecan fail already for certain decreasing sequences of clopen events whose measures diverge[14] to infinity. On the other hand, convergence succeeds for many other sequences, andone might hope that the failures were confined to physically uninteresting questions.Of course, the failure of convergence in even some cases is likely to contaminate othercases, making it dubious that µ ( A ) can be defined without some further limitation on thesequence A j beyond the mere requirement that lim A j = A . In the next two sectionswe will investigate some restrictions of this sort. For now, let’s notice that for open sets A there exists a very naturally defined canonical sequence of events A n ∈ S n convergingto A . Namely, we can take for A n the union of all the cylinder sets from Z n that are21ontained within A . This yields a “best approximation to A at stage n ” in the sense that A n couldn’t be enlarged without the sequence losing its increasing nature.Dually, one immediately obtains a canonical choice of sequence for any closed event B (just apply complementation to the sequence of clopen events approximating Ω \ B ),but having thus two different classes of canonical sequences introduces an ambiguity forevents that are both open and closed. Fortunately, the ambiguity in this case does noharm because a clopen event necessarily belongs to S , according to Lemma 4.1. Both theincreasing and decreasing canonical sequences thus terminate at a finite stage: they differonly transiently.Leaving aside questions of convergence and uniqueness, one might ask how manyevents the above limit process can access, even in the best case. That is, how many of theinteresting questions even belong to Lim S at all? With reference to the causal set case,recall first of all that one encounters all the stem-events without ever leaving the open sets W S . Remembering also that Lim S is closed under the Boolean operations, we can thussay on the positive side that every finite logical combination of stem-events is availablewithin Lim S (as also the entire event-algebra R W S of course). On the negative sidehowever, we can notice that events like the post-event and (for the particle case) the eventof infinite return fall outside of Lim S , as a consequence of the following lemma. Lemma (5.1) Let A ⊆ Ω. If both A and Ω \ A are dense subsets of Ω then A / ∈ Lim S . Proof
In the following A ⊥ B will mean that A and B are disjoint. Suppose, for contra-diction, that A = lim A n with A n ∈ S , and write A n for its complement Ω \ A n , also takingnote of the fact that A n , like A n itself, is clopen. We will find inductively a subsequence A n , A n , A n , · · · of the A n and a matched sequence of clopen sets B ⊇ B ⊇ B · · · suchthat B j is alternately included in and disjoint from A n j .step 1. To start with, put n = 1 and B = A n . We have B ⊆ A .step 2. Next observe that since B is open and Ω \ A is dense, there exists x ∈ B ∩ (Ω \ A ).Then since x / ∈ A = lim n A n , there exists by hypothesis some n > n such that x / ∈ A n ,i.e. x ∈ A n . Put B = A n ∩ B , which is again clopen since both A n and B areclopen. We have B ⊆ B with B ⊥ A n .step 3. For step 3, we proceed exactly as in step 2 with the roles of A and Ω \ A interchanged.Namely we observe that since B is open and A is dense, there exists x ∈ B ∩ A . Thensince x ∈ A = lim n A n , there exists by hypothesis some n > n such that x ∈ A n .22ut B = A n ∩ B , which is again clopen since both A n and B are clopen. We have B ⊆ B ⊆ B with B ⊆ A n .Now proceed inductively to produce B ⊆ A n , B ⊥ A n , etc. Finally put B = lim n B n = ∞ T n =1 B n and note that B is non-empty since the B n are all compact. (Indeed, every eventin S is compact, being a closed subset of the compact space Ω.) Pick any x ∈ B . Forodd j we have x ∈ B j ⊆ A n j ⇒ x ∈ A n j . For even j we have x ∈ B j ⊥ A n j ⇒ x / ∈ A n j .Thus the A n vacillate between including and excluding x , contradicting our assumptionthat lim A n exists.The lemma applies to the post-event because, no matter how far the growth process hasproceeded, the growing causet “still has a free choice” whether to end up with a post orwithout one (and exactly the same thing can be said for the event of infinite return). Butthis freedom means precisely that both the post-event and its complement are dense in Ω.Lemma 5.1 shows that Lim S is far from containing every event of potential interest,but one might wonder exactly how far. One answer comes from Exercise (22.17) of reference[12], according to which Lim S equals what is called ∆ , defined to be the intersection of W V S and V W S . This places Lim S at a very low level of the so called “Borel hierarchy”,which continues on for ℵ steps beyond ∆ before it exhausts the Borel subsets of Ω. Inthis sense the limiting process ‘lim’ does not take us very far beyond the clopen events. Onthe other hand, we have also seen that by applying ‘lim’ more than once, one can reach,for example, the post-event. How many events can one reach in this manner? Whencombined with other results in [12], (22.17) therein also answers this question by implyingthat (transfinite but still countable) iteration of the ‘lim’ operation suffices to produce anyBorel set. In this sense the lim operation is quite far reaching, given that an event ofinterest but not falling within the Borel domain would be hard to conceive of.
6. Canonical approximations for certain events
For an event A ⊆ Ω which is open with respect to the topology defined in Section 4, thatis for A ∈ W S , we have already discovered one canonical sequence A n of approximationsto A . The cylinder sets Z n “at stage n ” provide a kind of “mesh” in Ω whose finenessincreases with n , and our canonical choice of approximating event at stage n was A n = [ { Z ∈ Z n : Z ⊆ A } , (7)23he biggest member of S n = RZ n which can fit inside A . As we have seen, the A n convergeto A in the sense defined in section 5, but of course there exist many other sequences B n ∈ S n which also converge to A in this sense, and when the vector-measure µ is notof bounded variation, there is no guarantee that the corresponding sequences µ ( A n ) and µ ( B n ), if they converge at all, will converge to the same limit. In general they doubtlesswill not if B n is chosen with sufficient malice. In the face of such ambiguity, one mightstill hope to find some reasonably inclusive event-algebra A ⊇ S and for each event A ∈ A a canonical approximating sequence of events A n ∈ S n with lim A n = A and such that µ ( A n ) was a convergent sequence in Hilbert space. The vector lim n µ ( A n ) could then beadopted as the definition of µ ( A ).One snag that this perspective encounters is apparent already for the case where weare approximating open sets A and B , and our canonical approximations A n and B n arethe ones given by (7). From lim A n = A and lim B n = B it does indeed follow, as wehave already noted, that lim( A n ∩ B n ) = A ∩ B and lim( A n ∪ B n ) = A ∪ B . Forthe case of intersection it even follows that the events ( A n ∩ B n ) provide the canonicalapproximations to the open event A ∩ B , but the analogous conclusion fails for the caseof union because the canonical approximation ( A ∪ B ) n will in general be larger than( A n ∪ B n ), since some cylinder set Z ∈ Z n can, by “straddling the boundary” between A and B , be included in A ∪ B without being included in either A or B . One would thusobtain different approximating sequences for A ∪ B , depending on whether one regarded itas an open set in its own right or as the result of uniting A with B . In the next section wewill begin to see what it would take to render this kind of ambiguity harmless. For nowhowever, let’s ignore that issue and consider simply the question of finding unambiguousapproximating sequences for as many members of R W S (= R V S ) as possible.To that end, let’s return to the tree T of truncated histories and the method ofrepresenting certain events by subsets α ⊆ T . Although we didn’t make it explicit earlier,it is clear that a sequence of events A n ∈ S n is equivalent to a set of nodes α ⊆ T . Indeed,each A n is a union of cylinder sets Z ∈ Z n , and each such cylinder set corresponds to anode in T n , the n th level of T . This associates to each A n a set of nodes at level n , andamalgamating the nodes of all levels into a single collection yields α . Conversely, given α ⊆ T we obtain A n as the union of the cylinder sets that correspond to the α -nodesat level n . Since the correspondences between cylinder sets Z ∈ Z n , nodes in T n , andtruncated histories γ ∈ Ω( n ) are so close, I will often identify all three with one another,24peaking for example of a cylinder set Z as a node in T . When this is done, we can expressthe correspondence between node-sets α and approximating sequences ( A n ) in a simpleformula by writing A n = S ( α ∩ Z n ).Now let α be any set of nodes and let the A n be the corresponding sequence of events.Recall that we defined S ( α ) as the event that γ is eventually in α : S ( α ) = { γ : ( ∃ N )( ∀ n > N )( γ n ∈ α ) } . Dually one can also define e S ( α ) as the event that γ is repeatedly in α : e S ( α ) = { γ : ( ∀ N )( ∃ n > N )( γ n ∈ α ) } . It follows, simply by tracing through the definitions, that S ( α ) = lim inf A n , e S ( α ) = lim sup A n . (8)Since lim A n exists if and only if lim inf A n = lim sup A n (in which case their commonvalue equals lim A n ), we learn that the events of the form lim A n are precisely thosefor which S ( α ) = e S ( α ) , which in turn are precisely those such that no path γ can leaveand re-enter α more than a finite number of times. Evidently this property generalizesthe concept of convexity which we met with earlier. Notice incidentally that equations(8) imply that the forms S ( α ) and e S ( α ) don’t reach beyond W V S and V W S , knownin descriptive set theory as Σ and Π , respectively. Roughly, they reach as far as eventswhose complexity is that of the post event. Very optimistically, one might hope to gobeyond this and find for any Borel set A ⊆ Ω, some sort of canonical presentation interms of clopen events, but in this section we will not venture outside of R W S , the finiteBoolean combinations of opens. Since R W S ⊆ Lim S , all such events can be expressedas S ( α ) for some subset α ⊆ T .What we are asking for is a sort of “normal form” for events E in R W S . As a firststep in that direction, let us prove that every such event can be expressed as a disjointunion of events, each of which has the form, open \ open, or equivalently, open ∩ closed. Lemma (6.1) Let E ∈ R W S be a finite logical combination of open events. Thenthere exists a decreasing sequence of open events E ⊇ E ⊇ E · · · ⊇ E K such that E = E + E + E · · · + E K = E \ E ⊔ E \ E ⊔ · · · , where ‘ ⊔ ’ denotes disjoint union.25oreover the E j are formed from the original events using only the operations of unionand intersection. Proof
In this proof, as in the statement of the lemma, we use the operation of Booleanaddition, A + B = ( A ∪ B ) \ ( A ∩ B ) , (9)and we write the intersection of two sets as their product. Any Boolean combination ofsets is then a polynomial in these sets, and since products of open sets are open, anyBoolean combination of open events can be expressed simply as a Boolean sum of openevents. Given these facts, a proof by induction is not hard to devise, but it seems clearerjust to illustrate the pattern involved with the cases of K = 2 ,
3. For two events wehave A + B = ( A + B + AB ) + AB = A ∪ B + AB . For three we have A + B + C = A + ( B + C ) = A + ( B ∪ C + BC ) = ( A + B ∪ C ) + BC = ( A ∪ B ∪ C + A ( B ∪ C )) + BC = A ∪ B ∪ C + ( A ( B ∪ C ) + BC ) = A ∪ B ∪ C + A ( B ∪ C ) ∪ BC + A ( B ∪ C ) BC = A ∪ B ∪ C +( AB ∪ AC ∪ BC ) + ABC . The “inclusion-exclusion” pattern that is evident here emergeswith particular clarity when one interprets Boolean addition as addition of characteristicfunctions modulo 2. The final equation in the statement of the lemma then follows directlyfrom the fact that the E j are decreasing. One can also restate the essence of the proof in asimple formula: P Kj =1 A j = P Kj =1 B j , where B j = { x : x belongs to at least j of the A k } ,this being clearly a union of intersections of the A k .Given any set E expressed as in the lemma, we get immediately the approximations E n = E n + E n + · · · + E Kn , where E jn is our canonical n th approximation to the open set E j , and thence the corresponding sets of nodes α n = α n + α n + · · · + α Kn together withtheir union α = ∪ n α n . However, this construction is only a first step toward uniqueness,because the resulting α still depends on the original choice of the E j , which are not givento us uniquely by the lemma.In working toward a unique approximating sequence, let us concentrate on the sim-plest case of an event E = A \ B which is the difference of only two open sets B ⊆ A (corresponding to K = 2 in the lemma). Can we render A and B unique in this case? It’snot difficult to demonstrate that if we gather together all pairs A ⊇ B such that E = A \ B ,then the union of all the sets A and the union of all the sets B yields another such pair.Evidently this “biggest pair” is unique and uniquely determined by the original event E .This in turn yields [by (6)] a canonical sequence of approximations E n to E of the form, E n = A n \ B n , where A n ∈ S n and B n ∈ S n are the canonical n th approximations to A B . In terms of the equivalent node-sets α n these approximations are given by α n \ β n ,whose union over n I’ll designate simply by α , following our earlier notation.Although the node-set α that we have found is canonical and concisely defined, onemight wish for a more constructive route to it, or at least a characterization of it in termsof more easily verifiable necessary and sufficient conditions. The remainder of the presentsection will develop a prescription of this sort. In fact, I am not certain that the secondprescription will be strictly equivalent to the first. If it is, that is all to the good since wewill then not be forced to choose between the two. If it isn’t, that doesn’t really matter,since the second prescription stands on its own and, being more concrete, is likely to bemore useful in practice. Lemma (6.2) If α and β are upward-closed subsets of T with α ⊇ β then α \ β is convex. Proof
We are to show that no path between two nodes x and y in α \ β can contain nodesoutside of α \ β . Equivalently, no path which has left α \ β can ever re-enter it. But since α isupward-closed no path from x ∈ α can leave α , therefore it can leave α \ β only by entering β ; it then must remain in β (which is also upward-closed) forever, and consequently cannever re-enter α \ β .Now let E = A \ B as above and let α and β be the corresponding node-sets. Since A and B are open, both α and β are upward-closed subsets of T . We also know that A = S ( α ), B = S ( β ) and A \ B = S ( α \ β ). The lemma then teaches us that A \ B = S ( b α ),with b α a convex subset of T . The converse is true as well: Lemma (6.3) If b α ⊆ T is convex then S ( b α ) = A \ B for some open sets A and B . Proof
Recalling that we have identified points of Ω with infinite paths γ through T , let A be the set of all paths that enter b α , and let B be the subset of these that subsequentlyleave b α . By definition S ( b α ) = A \ B , but both A and B are open because the property of“having entered b α ” and the property of “having left b α ” are both hereditary.Henceforth, we will just deal with the convex subset b α , renaming it to plain α for simplicity.That is, we will be concerned with a fixed event E of the form (open \ open) and with aconvex set of nodes α ⊆ T such that ⋆ E = S ( α ).Let us say that α ⊆ T is prolific if it lacks maximal elements. A second requirementthat adds itself very naturally to convexity is the condition that α be prolific in this sense. ⋆ In view of (8) we would also want in general to require that S ( α ) = e S ( α ), but this holdsautomatically when α is convex. 27iven convexity, this is equivalent to saying that every node x ∈ α originates a path thatremains forever within α . In the opposite case, α will contain “sterile” nodes from whichall paths eventually leave α for good. It is clear that removing these sterile nodes will notalter E , nor will it spoil the convexity of α . We can therefore always arrange that α beboth convex and prolific. The “pruning” of the “sterile” nodes in order to render α prolificalso appears as a very natural operation when it is expressed in terms of cylinder sets Z .It simply removes from α those Z which are disjoint from E .We have now arranged for α to be convex and prolific, but this does not yet makeit unique, since for example we could remove all the nodes up to any fixed finite level n without altering S ( α ). If we did so, however, we might create a situation where, forexample, some cylinder set Z was wholly included in E without Z itself (regarded as anode in T ) belonging to α . To remedy this kind of lacuna, we can adjoin to α every node Z such that every path originating at Z eventually enters α . It is again easy to see thatadjoining these nodes will not interfere with α being convex and prolific.In this last step we have, in a manner of speaking, completed α toward the past, butin fact there is cause to carry this process of “past-completion” farther by adjoining stillother nodes to α . These additional nodes are perhaps not such obvious candidates as theprevious ones, but throwing them in as well (which I think corresponds to enlarging theopen set A ) will provide us with the uniqueness we are seeking. Definition
Let x ∈ T and α ⊆ T . Then x ≺ α means that x precedes some node in α :( ∃ y ∈ α )( x ≺ y ) . Remark
In terms of cylinder sets, Z ≺ Z ⇐⇒ Z ⊇ Z . Definition
The exclusive past of α is the set of nodes strictly below α : { x / ∈ α : x ≺ α } .Using this definition, let us say that α is past-complete if its exclusive past P is prolific,which in turn says that any node in P originates a path that repeatedly visits P . I claimwe can render α past-complete be adjoining to it all nodes below α that fail to satisfy thislast condition, and furthermore that the resulting set of nodes α ′ will yield the same event E as α and will be convex and prolific if α itself was. Lemma (6.4) Let α ⊆ T be any set of nodes and let α ′ be its “past-completion” as justdescribed. Then α ′ is past-complete. Moreover S ( α ′ ) = S ( α ) and e S ( α ′ ) = e S ( α ) . Proof
That S ( α ′ ) ⊇ S ( α ) is obvious. To prove that they are equal it suffices to showthat no path can eventually remain within α ′ without also remaining eventually within α . Suppose the contrary, and let γ ∈ S ( α ′ ) be a path which is repeatedly outside α . By28assing to a tail of γ we can suppose that it is always within α ′ . Let x ∈ γ be a nodewhich is not in α and let y ∈ γ be a later node of the same type. Since y ∈ α ′ \ α it is bydefinition in P , the exclusive past of α . Hence x originates a path (namely γ ) which visits P at y ; and since there are an infinite number of nodes like y , γ visits P repeatedly. Butthis contradicts the criterion for having included x in α ′ in the first place.The proof that e S ( α ′ ) = e S ( α ) is similar. It suffices to show that every path in e S ( α ′ ) visits α repeatedly. Suppose the contrary, and let γ ∈ e S ( α ′ ) be a path which is eventually outside α . By passing to a tail we can suppose that γ is always outside of α . Let x ∈ γ be anode which is in α ′ and let y , y , · · · be a sequence of later nodes of γ which are also in α ′ . Since the y j belong to α ′ \ α they are by definition in P , the exclusive past of α . Hence x originates a path which returns repeatedly to P , contradicting the criterion for havingincluded x in α ′ in the first place.To complete the proof, we need to show that α ′ is past-complete. To that end let x bein the exclusive past of α ′ . Since any y in α ′ is either in α or in its exclusive past, andsince α ′ ⊇ α , x is also in the exclusive past of α . Consequently, since x was not put into α ′ , it originates a path γ that repeatedly visits the exclusive past of α . But by definition,no node in such a path would have been put into α ′ either, whence γ repeatedly visits theexclusive past of α ′ , as was to be shown.We also wish to prove that past-completion preserves the attributes of being prolificand convex. The first is easy because past-completion only adds nodes which are belowsome element of α , and this can introduce no new maximal element.For the second, we need to demonstrate † that if α is convex, and if x ≺ y are nodes in α ′ then the order-interval I delimited by x and y is also within α ′ . When x ∈ α the proof issimple, since y is either within α itself or precedes some element z which is. In either casethe I is included in some second interval I ′ (possibly the same as I ) with endpoints in α .Then I ′ ⊆ α because α is convex, whence also I ⊆ I ′ ⊆ α ⊆ α ′ , as desired. The remainingpossibility is that x ∈ α ′ \ α , in which case it seems more convenient to deal with pathsrather than intervals. From the definition of convexity, proving that α ′ is convex amountsto showing that no path originating from x can leave α ′ and then re-enter it. First, observe † The demonstration that follows seems somehow longer than it ought to be. Intuitivelyit suffices to observe first that α ′ is built up from α by successive adjunction of maximalelements of its exclusive past, and second that adjoining such an element cannot spoilconvexity. 29hat since every node of α ′ precedes some node of α , no node of α ′ \ α can follow a nodeof α ; for if it did, it would also lie within the convex set α . In consequence, any path thatexits α permanently exits α ′ as well. Now let γ be any path originating from x ∈ α ′ \ α ,and as before write P for the exclusive past of α . By the definition of α ′ , every path from x must eventually leave P . If it does so by leaving the past of α , { x ∈ T : x ≺ α } , thenit certainly can never re-enter α ′ . If it does so by entering α , then it can exit α ′ only byexiting α , in which case it can never re-enter α or (as we just observed) α ′ .So far, we have established the existence, for our event E , of a node-set α which isconvex, prolific, and past-complete. Let us complete the story by proving that α is alsounique. To that end, let α and β be two prolific, convex and past-complete node-sets suchthat S ( α ) = S ( β ). Does it follow that α = β ? In demonstrating that the answer is “yes”,I will use the ad hoc notation P ( α ) for the exclusive past of α as defined earlier, with P ( α ) = α ∪ P ( α ) = { x ∈ T : ( ∃ y ∈ α )( x (cid:22) y ) } being the inclusive past .First, let us establish that α and β have equal inclusive pasts: P α = P β . In factif x ∈ P α then x originates a path γ that visits α . Since α is prolific this path can bearranged to visit α repeatedly, and since α is convex, such a path can never leave α . Hence γ ∈ S ( α ), implying in particular that γ visits β , whence x ∈ P β . The converse followssymmetrically.Now suppose for contradiction that there exists x ∈ α \ β . Such an x belongs bydefinition to P α , hence to P β , hence to P β , which in turn is prolific by definition of past-completeness. Thus x originates a path γ that repeatedly visits P β . I claim that γ mustleave α at some stage. (Otherwise γ ∈ S ( α ) ⇒ γ ∈ S ( β ) ⇒ γ eventually in β , whence γ could never again visit P β .) And since α is convex, γ must remain outside of α onceit has left. On the other hand, γ must continue to visit P β , which in turn is a subset of P β = P α . But if γ really visited some y ∈ P α , then by definition we could divert it at y to some other γ ′ that would re-enter α , something that we just proved to be impossible.This completes the proof of: Theorem (6.5) Every event E of the form E = A \ B with A and B open can be expressedas E = S ( α ) = e S ( α ) for a unique set of nodes α which is convex, prolific and past-complete.The theorem furnishes a canonical set α of nodes corresponding to E , and as explainedearlier, one obtains immediately from such an α a canonical sequence of approximants E n to E such that E = lim n A n . We have thus reached our immediate goal.30he canonical approximating sequences of the theorem provide a good reference pointfor further developments, and we have learned how to arrive at them step by step, startingfrom the open sets A and B . Nevertheless it seems unlikely that we can limit ourselves tothese particular approximants in general. Rather, as remarked already at the beginningof this section, one will in general have to deal with many different sequences convergingto the same event, unless perhaps it is possible to devise canonical sequences which areclosed under the Boolean operations.We already encountered an ambiguity of this nature when we noticed that our original,increasing canonical approximants (7) for open events (call them “C1”) are not fully com-patible with the Boolean operation of complementation. Specifically for a clopen event E , these C1 approximants depend on whether one derives them directly from E or bycomplementing the corresponding approximants for the open event Ω \ E . We run into afurther, but related conflict if we now compare the C1 approximants with those of theabove theorem (call them “C2”). For an open event E the C1 approximant E n is nothingbut the biggest member of S n included within E . But if we view E as the difference E = A \ B , with A being E itself and B = 0 being the empty event, the theorem providesa different set of approximants E n . In general, the two disagree, as one can appreciate ifone notices that the pair ( E
0) is not the “biggest one” yielding E .Consider for example the 2-site hopper event that the particle does not remain foreverat its starting site 0, but that the first time it hops to site 1 it immediately returns to 0.This event is a union of cylinder sets corresponding to truncated trajectories of the shape(0 , , , · · · , , , ∗ ) where the star ‘ ∗ ’ represents any finite sequence of zeros and ones.For this event, the node-set α of type C1 consists of precisely the truncated trajectoriesjust indicated. But that set of nodes is not past-complete. Its completion, the type C2node-set α , contains in addition the truncated trajectories (0 , , , · · · , α differs from α by an infinite number of nodes in this case. (Figure 4 illustrates thisphenomenon.) 31 igure 4. Illustrating past-completion in the tree shown. The nodes circledin blue “complete” those in solid red. Let the node-set be α beforecompletion and α after completion. Evidentally S ( α ) = S ( α ) but only α is upward-closed.We thus have to reckon with overlapping but in general incompatible prescriptionsfor different types of events. If one prescription were to be adopted exclusively, it shouldprobably be C2, which covers more events than C1 does. (Incidentally, C2 resolves theaforementioned ambiguity in the C1 prescription in favor of treating clopen events asclosed, not open.) On behalf of C1 one might make the counter-argument that monotonicconvergence of the approximants is to be preferred, but this does not seem so compellingin the context of a quantal measure, which itself is not a monotonic set-function. Betterthan either choice, however, would be not having to choose at all because the alternativeapproximating sequences would all lead to the same extension of our initial quantal mea-sure. The main thing for now is that we’ve discovered at least one canonical choice ofclopen events E n converging to any event of the form E = A \ B with A and B open.In the face of these various ambiguities it seems well to emphasize that none of themaffect, in the causet case, the stem-events themselves, essentially because the latter arenot only open but dense in Ω, or more physically because any growing causet that has notyet produced a given stem always retains a choice whether or not to do so. It follows that32ot only does the C1 prescription coincide with the C2 prescription for stem-events (itsexclusive pasts being already prolific), but also the “biggest pair” prescription with whichwe began provably agrees with the C1 prescription. The same ought to apply to finiteunions and intersections of stem-events, and similar comments could be made about theevent of “return” in the hopper case.Let me conclude this section by sketching very briefly how one might try to carryour successful “canonization” of E = open \ open over to the general case where E = E + E · · · + E K , the E j being open and nested. Just as earlier we found a “biggestpair”, A ⊇ B , by forming unions of the individual events A and B , one can do the samething here with the E j to obtain (at least in principle) a canonical set of “biggest” openand nested events E j such that E = E + E · · · + E K . Associating to each such E j itscanonical approximants E jn (in the C1 sense, say) then yields for E itself the approximants E n = E n + E n · · · + E Kn such that lim E n = E . In principle this achieves our goal, butit remains once again at a rather abstract level.As before, we can attempt a more constructive development by working with thenode-set α ⊆ T corresponding to our approximating sequence E n , or perhaps with somesimilar node-set whose uniqueness can be established directly, and for which we can provethat E = S ( α ) = e S ( α ) . But how would our construction of α go in this more general case,and what would generalize the conditions that α be convex, prolific and past-complete? Itis clear from Lemmas (6.2) and (6.3) that convexity is now too restrictive. In its place, onewould probably put the more general requirement that no path γ could enter and leave α more than K times. Correspondingly one might then expect that α would decomposeinto subsets α j that were convex in the strict sense. One might also try to arrange foreach α j to be prolific and past-complete, hoping that this would again confer uniquenesson the whole collection. If all this worked out, one would have constructed a canonicalapproximating sequence for any Boolean combination of open events, in particular for anyBoolean combination of stem events.A next step beyond R W S , if one could take it, would be to devise canonical approx-imations for larger families of events, starting with the collections W V S and V W S inwhich the post-event and its complement are to be found. An event A in either of thesecollections is accessible from S as a limit of limits, but such a double limiting processcan only make the potential ambiguities worse. For example, the event R ∞ of repeatedreturn is in V W S . It could be expressed as lim A n , where A n = “returns at least n B n , where B n = “returns at least onceafter t = n ”. Both A n and B n give decreasing sequences of open events and both convergeto R ∞ , but which sequence, if either, should be favored as canonical? Perhaps in certaincases, one could arrive at a canonical presentation by generalizing further our treatmentabove in terms of sets of nodes α ⊆ T , but beyond this, it’s not easy to guess how onemight proceed. To devise canonical approximations for events of still greater complexitywould seem to demand a fresh approach.Finally, it might bear repeating here that uniqueness in and of itself does not guar-antee compatibility with the Boolean operations. And I believe in fact that none of theprescriptions that this section has considered are compatible with the full set of such con-nectives, albeit some are compatible, for example, with complement or disjoint union (cf.figure 5). If there did exist a compatible prescription — or even a prescription compatible“modulo initial transients”, which is just as good — that would weigh very heavily in itsfavor. Figure 5.
Two sets of nodes shown in red and blue. Both sets are convex,prolific and past-complete, but their union is not convex.34 . Evenly convergent sequences of events
We are given a vector-valued measure µ : S → H defined initially on the finite unions ofcylinder-events, and we wish to enlarge this initial domain S so that it can embrace eventslike the stem-events and some of the other events we have been using as illustrations. Inattempting such an extension or “prolongation” of µ , it is natural to think in terms ofapproximations, or more formally of limits. Let A ⊆ Ω be some event A outside the initialdomain. In order to define | A i ≡ µ ( A ) as a limit, one would aim to identify a sequence A n ∈ S n of “best approximations to A ” and one would then hope that the correspondingmeasures | A n i ∈ H would also converge. If they did, then one would take their limit in H to be the measure of A : | A i = lim n → ∞ | A n i . Notice here that in attempting to define | A i = µ ( A ) this way, we have relied on twoindependent notions of convergence, first the purely set-theoretic convergence of A n to A in the sense of section 5 above, and second the topological convergence of the measures | A n i to | A i in Hilbert space (say in the norm topology or perhaps the weak topology).One might question whether the first notion is really needed, given that the extensiontheorems of ordinary measure theory do without it, relying solely on the measure µ itself.Would it be possible to proceed similarly here? Unfortunately, this looks dubious, eventhough the vector | A n i carries a certain amount of information about the event A n (a verylimited amount since, owing to quantal interference, very different events can share thesame vector-measure.)In the ordinary setting, where µ is real and positive, it defines a distance on the spaceof initially measurable sets modulo sets of measure zero, such that two events A, B ∈ S areclose when µ ( A + B ) is small. Extension of the measure then corresponds to completionof the metric space thereby defined [1]. Quantally, however, an event of small or zeromeasure is not negligible in the same way as it is classically, because of interference. Thus,if we tried to use the norm of | A + B i as a distance, it wouldn’t even obey the triangleinequality. (Example: three disjoint events A , B , C as in the 3-slit experiment of [19] [20]; | A + B i = | B + C i = 0 but | A + C i 6 = 0.) Similarly trying to quotient the event-algebra bythe events of measure zero would yield nonsense; it can even happen that all of Ω is coveredby events of measure zero [17]. To establish an association between a vector v = lim | A n i and a definite event A in Ω, one thus seems to need an independent notion of convergencelike that introduced above in section 5 and developed in section 6.35ccepting this apparent necessity, let us investigate how a limiting procedure mightgo in the important case of an open event E ∈ W S . In so doing, let us employ for E thecanonical approximants E n ∈ S n “of type C1”, these being the simplest to work with andprobably the first to suggest themselves for most people: E n = [ { Z ∈ Z n : Z ⊆ E } . (10) = (7)As we know, there is no guarantee in general that the corresponding vectors | E n i willconverge, but when they do, we’d like to regard E as “measurable” and to associate to itthe measure | E i = lim n | E n i . Below I will illustrate this procedure with the two-site andthree-site hoppers, but first let us consider whether or not our criterion of convergence isadequate as it stands or whether it needs to be strengthened.Recall in this connection that we had defined a second sequence of approximantsfor E “of type C2”, related to the first ones by past-completion of the correspondingnode-sets α . Would these approximants have led to the same set of measurable openevents and to the same measures for them? A second question concerns compatibility withthe Boolean operations, some form of which is needed if the extended measure is to beadditive on disjoint events. Consider for example the disjoint union G = A + B of twoopen events A and B , and let G n , A n and B n be the corresponding C1 approximants. If | A n i + | B n i = | G n i held automatically it would follow immediately that | A i + | B i = | G i ,as desired. But plainly this is not automatic because G = A ∪ B can include cylinder sets Z that are not included separately in either A or B (see figure 6). We’d like the contributionfrom such Z to go away in the limit n → ∞ , and we’d also like any mismatch between ourC1 and C2 sequences to go away. These two desiderata turn out to be closely related. Figure 6.
A cylinder set Z contributing to the difference (11).36et us examine the difference, | G n i − | A n + B n i = | G n i − | A n i − | B n i , (11)more closely. In light of equation (10), this difference is just X {| Z i : ( Z ∈ A + B )( Z / ∈ A )( Z / ∈ B ) } , but the Z here are not arbitrary cylinder sets. Rather, I claim that any Z which contributesto the above sum is special in that its overlaps with A and B are clopen (which implies that— within Z — the discrepancy disappears entirely in a later approximation). By the nextlemma, something similar holds for the difference between the C1 and C2 approximantsto any individual open set A . Definition
The cylinder set Z straddles the event A if it meets both A and its complementand if Z ∩ A is clopen. In symbols: 0 = ZA = Z and ZA ∈ S . Remark Z straddles A iff it straddles the complement Ω \ A To see why the Z contributing to equation (11) are “straddlers” in this sense, it sufficesto observe first that ZA ≡ Z ∩ A is open because both A and Z are open, and second that ZB is therefore closed, having the form closed \ open: ZB = Z \ ZA . By symmetry both ZA and ZB are consequently both open and both closed. Lemma
Let A be an open event and let α be the corresponding node-set of type C1. Ifpast-completion adds Z ∈ Z to α then Z straddles A , and conversely. Proof
Since A is open, α is upward-closed, while for any α at all, it’s true that β = { x ∈ T : x ≺ α } is downward closed. Hence the difference β \ α , the exclusive past of α , isalso downward closed; i.e. it is a subtree of T . Now suppose without loss of generality that Z ∈ β \ α . By definition, past-completion will adjoin Z to α iff no path originating at Z can remain within the subtree β \ α . (It cannot leave β \ α and later return to it in this casebecause subtrees are convex.) But this means that the portion of β \ α above Z is actuallyfinite (by the infinity lemma). Consequently, for n sufficiently big, every descendant of Z is either in α or below no node of α . Translated into the language of subsets of Ω, this saysthat at a sufficient degree of refinement n , every cylinder set Z ′ ∈ Z n and within Z is eitherfully included within A (the former alternative) or disjoint from A (the latter alternative).Now suppose that past-completing α does adjoin Z to it. Then ZA is the union of the Z ′ belonging to the first family and is therefore clopen, straddling A . Conversely, if ZA isclopen, then it is a union of cylinder sets Z ′ ∈ Z n for some n .37n view of these results, we can to some extent deal with the issues raised above byprovisionally adding to our criterion of convergence the requirement that any “straddling”cylinder sets contribute negligibly in measure as n → ∞ . Definition
Let A n ∈ S n be a sequence of clopen events. Call this sequence evenlyconvergent with respect to µ if the following hold:(i) A = lim A n for some A ⊆ Ω(ii) | A i = lim | A n i for some | A i ∈ H (iii) ( ∀ ε > ∃ N )( ∀ n > N ) P || | Z i || < ε , where the sum ranges over all Z ∈ Z n thatstraddle A .In view of the previous lemma, condition (iii) implies immediately that the C1 and C2approximants for any open event E yield equivalent results. As desired, it also gives riseto additivity on disjoint open events, as we will see in the next theorem. In the statementof the theorem, the canonical approximants may for definiteness taken to be those “oftype C1”. As we have just seen, exchanging any of them for type C2 would have no effect.(Notice also in the statement of the theorem that the Boolean sum A + B coincides withthe union A ∪ B when A and B are disjoint: AB = 0.) Theorem
Let A and B be disjoint open events and let A n and B n be their canonicalapproximating sequences, with G n being the canonical approximating sequence for G = A + B . If the first two sequences are evenly convergent then the third is also, and themeasures add: | A i + | B i = | G i . Proof
It will be convenient in the following to work with the canonical sequences given by(10), since for them the approximants to A and B will be disjoint. In the above definitionof being evenly convergent , we need to establish conditions (i)–(iii) with A replaced by G .To begin with, condition (i), viz. G = lim G n , is true by construction.Turning to condition (ii), we must verify that | G i = lim | G n i with | G i = | A i + | B i .We’ve already learned in connection with equation (11) that | G n i − | A n i − | B n i is thesum of the measures | Z i of all those cylinder sets Z ∈ Z n that straddle both A and B .But this sum can be made arbitrarily small by choosing n big enough, since by hypothesisthe sequence A n is itself evenly convergent . (In more detail: || | G n i − | A n i − | B n i || = || P {| Z i : Z straddles both A and B } || ≤ P {|| | Z i || : Z straddles both A and B } || ≤ P {|| | Z i || : Z straddles A } || → n → ∞ .) Therefore lim | G n i = lim( | A n i + | B n i ) =lim | A n i + lim | B n i = | A i + | B i , as required.38inally, we need to verify that the G n themselves fulfill the third condition for beingevenly convergent . To that end we will demonstrate that any cylinder event Z ∈ Z n that straddles G also straddles either A or B . The total norm of the straddlers of G willthus be bounded by the sum of the bounds for A and B , both of which go to zero as n goes to ∞ ; and therewith the proof will be complete. Suppose then that Z straddles G = A + B = A ⊔ B , where the symbol ‘ ⊔ ’ denotes the union of disjoint sets. We havethen ZG = Z ( A ⊔ B ) = ZA ⊔ ZB . By definition Z meets A + B , so suppose it meets A : ZA = 0. Now ZA is obviously open since both Z and A are open. It is also closed, beingthe difference of the clopen set ZG = ZA ⊔ ZB and the open set ZB . Hence Z straddles A if it meets A at all, and in general it will straddle either A or B , as announced.The theorem takes a first step toward arranging additivity of the extended measure,but of course disjoint open events is a special case. More generally, one would like tohave similar theorems covering, say, arbitrary events in R W S (not just open events)and arbitrary Boolean operations (not just disjoint union). For example, it’s easy toestablish for any open event E that | E i is defined iff | Ω \ E i is defined, and that then | Ω \ E i + | E i = | Ω i . To what extent such results can be obtained in general remains to beinvestigated. Examples
Our friend, the return-event R , can serve to illustrate some of the definitions we havemade. Let us start with the 2-site hopper, in which case R ′ = Ω \ R , the event of “non-return”, consists of the single history, (0 , , , · · · ) . As we know, R itself is topologicallyopen, and correspondingly Ω \ R is closed, as one can see directly from the fact that it is thelimit of a decreasing sequence of clopen events of the form R ′ n = cyl(0 , , , · · · , R ′ . In this case no cylinder event in Z n straddles R ′ n since it itself is a cylinder event. To check that | R ′ i is well defined, then, we haveonly to check that the sequence | R ′ n i converges. In fact, it converges trivially to 0, since || | R ′ n i || = (1 / n/ . Thus, | Ω \ R i = 0 and non-return is precluded for the two-site hopper[14]. Taking complements shows then that | R i is defined and has the value | R i = | Ω i .In the context of the three-site hopper, the events of return and non-return becomemuch more interesting. Classically, non-return “almost surely” does not occur in a finitelattice; its measure vanishes. Moreover, this conclusion obtains independently of whatinitial conditions one cares to assume. What we will find quantally? More generally, whatwill we find if, instead of asking whether the particle visits site 0, we ask whether it visits39ite 1 or 2? By symmetry, these questions become equivalent if we generalize our initialcondition to admit different starting sites. Let us therefore consider (still more generally)an initial condition, in which each possible initial location contributes its own complexamplitude ψ ( j ) , j = 0 , , ∈ Z . The measure of a cylinder-set of trajectories can thenbe derived from the 3-site analog of equation (2), generalized to allow for an arbitraryinitial position x , and with an additional factor of the initial amplitude ψ ( x ) thrownin: v y = ( U − n ) yx n U x n x n − · · · U x x U x x ψ ( x ) . For the event of non-return, one must sum this expression over all trajectories x j suchthat x j = 0 for all j >
0. Evidently the resulting vector of components v y is then givenby a matrix product, of the form ( U − n ) V n ψ , where the matrix V is nothing but thematrix U with its first row set to zero. We can also set the first column of V to zero ifwe re-express v as ( U − n ) V n − ψ , wherein ψ is just U ψ with its first entry set to zero.This way, V becomes effectively a 2 × ω = 1 / ) U = 1 √ ω ωω ωω ω whence also V = 1 √ ω ω For these matrices, powers of U and V can both be evaluated in essentially the samemanner, by writing U or V as a linear combination of orthogonal projectors. Taking U asexemplar, we obtain by adding and subtracting a multiple of the identity matrix to U : U = λ (1 − P ) + σP , where P = 13 and 1 − P = 13 − − − − − − , λ = (1 − ω ) / √ σ = (1 + 2 ω ) / √
3. In the same way, defining Q = 1 / (cid:18) (cid:19) (or more correctly as the 3 × V in the form, V = λ (1 − Q ) + ρQ , where Q = 12 and 1 − Q = 12 − − , with ρ = − ω / √
3. It follows immediately that U n = λ n (1 − P ) + σ n P and V n − = λ n − (1 − Q ) + ρ n − Q . Noticing now that | ρ | = 1 / √ <
1, while | λ | = | σ | = 1, wesee that in the limit n → ∞ we can drop the second term in V , without affecting | R ′ i ,i.e. without affecting whether the sequence of approximations | R ′ n i converges or what itconverges to. And noticing further that P (1 − Q ) = 0, we see that we can also dropthat term in the product U − n V n − , leaving the simple asymptotic form, U − n V n − ∼ λ − n (1 − P ) λ n − (1 − Q ) = (1 /λ )(1 − P )(1 − Q ) . We thus obtain, modulo an exponentiallysmall correction, | R ′ n i = 1 λ (1 − P )(1 − Q ) ψ . This formula leads to a somewhat odd conclusion. With our original initial conditionthat the particle begins at 0, the components of ψ are just the last two entries of the firstcolumn of U , namely ( ω/ √ , , − Q . Hencethe event of non-return is again precluded: | R ′ i = 0; and once again | R i = | Ω i = (1 , , , , | R ′ i does not vanish! In particular, if theparticle starts at site 2 instead of site 0, then the event that it fails to visit site 0 hasthe non-zero vector-measure | R ′ i = (1 / , − / , − / h R ′ | R ′ i , or 1 / | R i = | Ω i − | R ′ i = ( − / , / , /
6) .Our analysis of the 3-site case used tacitly the fact that the events R and R ′ are freeof straddling cylinder-sets, for the same reason that stem-events are. Convenient thoughthis is, it means that our example fails to illustrate condition (iii) in our definition of anevenly convergent sequence. It would be good to work out an example where (iii) doescome into play, since doing so could indicate whether that condition is a reasonable one to41ave added, or whether on the contrary it tends to rule out events that one would want toinclude.It would also be good to work out some physically interesting instances of our approx-imation procedure in the causal set case. One might begin, for example, with the event“originary” for the relatively simple dynamics of complex percolation.
8. Epilogue: does physics need actual infinity?
Does the description of nature require actual infinities? Or is a truly finitary physicspossible, in which infinite sets would figure only as potentialities?Inasmuch as the theories to which we have grown accustomed employ real numbersheavily, they thereby presuppose an actual infinity of cardinality ℵ , as emphasized in [22].In itself, however, this seems more a matter of convenience than of principle, since onecould imagine making do with rational numbers of a very fine but finite precision thatcould be made still finer as the need arose — in other words a potential infinity. ⋆ The other prominent continuum in present-day physics is of course spacetime. Non-relativistically, one could again imagine circumventing the actual infinities that continuousspace and time seem to imply, but when it comes to relativistic field theories, the newrequirement of locality appears to force strict continuity on us. Perhaps one could get bywith only ℵ points, say points with rational coordinates, but even that would still be anactual infinity.Quantum gravity raises all these questions anew, of course. String theory and loopquantum gravity both presuppose background continua, at least in their current formu-lations. Causal dynamical triangulations and the “asymptotic safety” approach retainlocality and presuppose the same type of continuum as classical gravity, albeit not asbackground.With causal sets, the situation seems more fluid. On one hand, they transcend locality,but on the other hand they still maintain covariance in the sense of label-invariance, andthat brings with it an “infrared” infinity, as discussed earlier. An important new feature,however, is that now the infinity is in some sense pure gauge: we need it only becausewe have introduced both an auxiliary time parameter and a space of “completed causets” ⋆ In writing ‘ ℵ ’, I have adopted the continuum hypothesis, ℵ = 2 ℵ , for . . . notationalreasons. 42n order to give a precise meaning to the concept of sequential growth. Could it bethat a manifestly covariant formulation of growth dynamics could dispense with this “lastremaining infinity”? Limited to measure theoretic tools inherited from the classical theoryof stochastic processes, we apparently lack the technical means to ask the question properly.As things stand, we can acknowledge at a minimum that being able to refer to completedcausets is very convenient even if it ultimately turns out not to be physically necessary.(One can also comment here that the cardinality of a completed causet, though not finite,is reduced to that of the integers. On the other hand, the associated sample-space Ω stillhas the cardinality of the continuum.)Based on this evidence, one could perhaps agree that physics is tending toward morefinitary conceptions, even if it hasn’t genuinely reached them yet. In particular, even ifcausal sets are implicitly free of actual infinities, the available mathematical tools don’t letus express this fact clearly. Might some of the tools that we seem to lack arise naturallyin the course of attempts, like those above, to extract well-defined generalized measuresfrom quantal path-integrals and path-sums?43 ppendix. Some symbols used, in approximate order of appearance Ω (the sample-space or space of histories), Ω physical , Ω gauge , Ω( n )0 ⊆ Ω (the empty subset)cyl( c ) (the cylinder event corresponding to the truncated history c ) Z (the semiring of cylinder events), Z n S = RZ (the Boolean algebra generated by Z = the finite unions of cylinder sets) S n T (the tree of truncated histories), T n z ≡ exp 2 πiz W S , V S , W V S S ( α ), e S ( α )lim, Lim, lim inf, lim sup ≺ P , P A ⊔ B | Z i = µ ( Z )I would like to thank Sumati Surya for numerous corrections and/or suggestions for im-proving the clarity of the manuscript. Research at Perimeter Institute for TheoreticalPhysics is supported in part by the Government of Canada through NSERC and by theProvince of Ontario through MRI. References [1] A.N. Kolmogorov and S.V. Fomin,
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