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On the"renormalization"transformations induced by cycles of expansion and contraction in causal set cosmology

We study the ``renormalization group action'' induced by cycles of cosmic expansion and contraction, within the context of a family of stochastic dynamical laws for causal sets derived earlier. We find a line of fixed points corresponding to the dynamics of transitive percolation, and we prove that there exist no other fixed points and no cycles of length two or more. We also identify an extensive ``basin of attraction'' of the fixed points but find that it does not exhaust the full parameter space. Nevertheless, we conjecture that every trajectory is drawn toward the fixed point set in a suitably weakened sense.


Introduction
There is good reason to believe that the appropriate dynamical framework for "quantum gravity" will be -just as the ordinary quantum dynamical framework already is -a generalization of classical probability theory. If in addition, we take the deep structure of spacetime to be that of a causal set [1], then the required dynamics will be expressible as a generalized probability measure on the space of all causal sets, the generalization being in the sense of [2] and [3] or something similar.
In the search for such a dynamics, principles like general covariance and relativistic causality can offer much needed guidance. Indeed, in the limit in which quantal probabilities reduce to classical ones, it has proved possible to derive [4] from versions of these two principles a unique family of dynamical laws (Markov processes) parameterized by a sequence of "coupling constants" where t 0 ≡ 1 and t n ≥ 0 for n = 1, 2, 3, · · ·. Although this family of stochastic processes is tremendously special compared to the most general relevant Markov process, it still contains a countable infinity of coupling constants, and the question arises how further to narrow down the field of possibilities. Any answer to this question can be expected to be of use in the attempt to find a quantal generalization of the dynamical scheme in question; it might also, with luck, lead us to a choice of the t n which would define a causal set version of classical general relativity, arising as the classical limit of quantum causet dynamics. (We will often use the contraction "causet" in place of the longer "causal set".) In the absence of any further general principle 1 that might be brought to bear in narrowing down the possibilities (1), one can consider an "evolutionary" approach to this problem, that is, one can try to let the causet "choose its own dynamics". Such a possibility was discussed in [5], where it was shown that each cosmic cycle of expansion and contraction for the causet has the effect of "renormalizing" the parameters t n in a definite manner. 2 The renormalization reduces, in fact, to the repeated action of a single transformation M, given by equation (6) below. By understanding better the action of M, one might hope to identify a distinguished sub-family of the dynamical laws parameterized by (1). For example, if M had a single fixed point (t 0 n ) and if every other set of parameters (t n ) were carried toward (t 0 n ) by the action of M, then one might designate (t 0 n ), or more generally some neighborhood thereof, as distinguished since it would be the dynamics toward which the universe would naturally evolve. 3 We will see below that the actual situation is more complicated than this, but with many points of similarity. Instead of a single fixed point, there is a one-parameter family of them, and not all sequences (t n ) are attracted to the fixed point set, at least in the most obvious sense. We will also find that the transformation M has no "limit cycles", but we do not know whether it might have "attractors" of a more complicated sort. Thus, we will make a start on determining the nature of the "RG flow" associated with M, but we will not characterize it fully. Nor will we obtain much information about the flow in the neighborhood of the fixed point set itself. Some such results for specific trajectories can be found in [6] and [5], however. Indeed, the very suggestive nature of some of those results for helping to explain the most salient features of the early universe furnished a good part of the motivation for the present investigation.
2 Formulation of the problem and summary of our main results Let us now pose more precisely the problem we intend to address. In the course of the growth/evolution of a causal set C, a post occurs when an element ω is born such that every other element of C is (or will be) either an ancestor or a descendant of ω. That is, a post is an element ω of C for which where past(ω) := {x ∈ C|x ≺ ω} and future(ω) := {x ∈ C|ω ≺ x} (Our basic definitions and notation are as in [4], in particular ≺ is irreflexive.) Cosmologically speaking, this may be interpreted as a kind of collapse of the universe to zero diameter followed by a subsequent reexpansion. It is a simple consequence of the form taken by the dynamical law derived in [4] that the portion {ω} ∪ (future ω) of C comprising the post and its descendants is governed by an effective dynamics of the same nature as that valid for C as a whole, but with different values of the t n .
To see what these new parameters are, recall from [4] that any given transition probability leading from an n-element causet to one of n + 1 elements takes the form where Here the transition is occasioned by the birth of some new element x and ̟ = | past(x)| is the number of all its ancestors, while m = | maximal (past(x))| is the number of its immediate ancestors or "parents". Now, by definition, no element x born to the future of a post can have vanishing m (or ̟). In view of (3), it follows simply from this that the parameter t 0 becomes irrelevant subsequent to a post. 4 By the same token, we see from (2) that only the ratios of the remaining t's, (t 1 , t 2 , t 3 , · · ·), have meaning in such a region of C, not their overall normalization. Moreover, if ̟ and m are the effective sizes of past(x) and of maximal (past(x)) for some element x born after the post ω, then their true values are plainly ̟ + p and m, respectively, where p = | past ω| is the number of ancestors of the post. The (un-normalized) probability of this birth is therefore given by the effective quantity λ(̟, m), where Given these facts, it is an easy matter to derive the effective coupling constants for the afterpost dynamics, and we claim that t (p) n , the effective or "renormalized" value of t n in the after post region, is given by or equivalently by p applications of the transformation M to the fundamental coupling constants, where p = | past ω| and M takes (t 1 , t 2 , t 3 , · · ·) to ( t 1 , t 2 , t 3 , · · ·) via t n = t n + t n+1 (n = 1, 2, 3, · · ·) .
Perhaps the simplest way to demonstrate these claims, if we accept that a relationship of the form (3) holds good for the effective dynamics 5 , is just to observe that, as a trivial special case of that equation, t n = λ(n, n) from which we find immediately, for the special case p = 1, Our claim that the t (p) n are given by p applications of M is then a consequence of the manifest fact that adding p > 1 elements to past(ω) all at once is no different from adding them one by one. A logically impeccable proof is almost as easy, but requires a short calculation, given in the Appendix, to establish (40). In the light of the latter, one sees immediately from (4) that the effective coupling constants are indeed those of (5), and these in turn are readily seen to be given by M p (t n ), for example by noting that M can be expressed as 1 + a, where a is a shift operator acting on the t n and comparing (4) with the binomial expansion of (1 + a) p .
The "renormalization (semi)group" we will study in the sequel is that generated by the transformation M. For consistency with ω being a post, at least one of the parameters t 1 , t 2 , t 3 · · · t p+1 must be nonzero. Henceforth we will assume for simplicity that t 1 > 0. If this is not true initially, it will become so after a sufficient number of applications of the operator M.
For completeness, we record here the correctly normalized transition probability T for the birth of an element x to the future of our post ω. For this purpose, let ̟ = | past x| − p be the effective number of ancestors of x (excluding the p elements preceding ω), let m be its number of parents, and let n be the effective size of the pre-existing causet (the size of C\(past ω) before x is born). Then T can be written as λ(̟, m) λ(n, 0) − λ(0, 0) where λ(̟, m) is the same function of the t k as λ(̟, m) is of the t k : and where, consequently, the denominator can be written more properly as n k=1 n k t k (eliminating the apparent reference to t 0 ).
In the next section, we will prove those properties of the "renormalization map" M : t n → t n that we have been able to establish. We assume throughout that (t n ) = (t 1 , t 2 , t 3 , · · ·) is a sequence of nonnegative real numbers, with t 1 > 0, and we let M act by (6), it being understood that the space T on which it acts is actually the set of equivalence classes of sequences (t n ), where (t n ) and (t ′ n ) are equivalent iff t n = λt ′ n for all n and for some fixed λ > 0. Our principal results are then as follows.
(i) The fixed points of M are given by the sequences (t n ) such that for some t ≥ 0 [6]. They thus form a 1-parameter set, whose parameter t is related to the parameter p of originary percolation [4] by p = t/(t + 1). In (8) the t = 0 case is to be interpreted by taking the limit t ց 0, which is equivalent to putting t n = δ n1 , a dynamics which produces originary causets C that are always trees [5,4]. Note that the limit t → ∞ also makes sense, and corresponds to originary percolation with p = 1 (cf. (3) and (7)), a dynamics which always produces the same causet: the "purely one dimensional" poset, or chain.
(ii) Aside from its fixed points, M possess no other cycles.
(iii) Suppose that (t n ), though not necessarily of the form (8), is such that t 1/n n has a limit in [0, ∞) as n → ∞. Then, under repeated action of M, the sequence (t n ) converges pointwise to (8) with t = lim n → ∞ t 1/n n . (Of course, this convergence can only be modulo the overall scale ambiguity in (t n ). One way to lift this ambiguity is to deal with the ratios t n+1 /t n , and what we prove below is that lim where ( t (p) n ) is the result of acting p times with M on (t n ). Notice that this asserts more than simply pointwise convergence to t n = δ n1 in the case t = 0. Notice also that we have not included the case t = ∞ in the result just stated.) For the case where t 1/n n has no n → ∞ limit, we have no general result, although one can show using the generating functions defined below that if ( t (p) n ) does converge pointwise to some fixed point (8), then the latter must be the one with parameter t = lim sup t 1/n n . In general, however, one can not expect pointwise convergence to any sequence (see the counterexample in Section 7 below.) As a matrix, our transformation M is just Given the simplicity of this matrix (it is already in "Jordan normal form" and in fact is just the identity plus a shift operator), one might think it an easy matter to understand the "flow" its powers define. However one meets with two complicating circumstances: the infinite dimensionality and the fact that M only appears to be linear, because the space T on which it acts is really a projective space (rather than a vector space), since its points are given by the ratios, t 1 : For these reasons, we have chosen to study the transformation (6) directly, without attempting to utilize any of the general results (such as the spectral theorem, for example) which one might have tried to bring to bear on the problem.

Functional representation
We are working on sequences t n (n = 1, 2, 3, · · ·) defined up to an overall multiplicative constant, and such that t n ≥ 0, t 1 = 0 .
One way to remove the overall multiplicative constant freedom is to set t 1 = 1, however it will be more convenient to keep the freedom in the following. The renormalization scheme on these sequences is given by It is rather clear that after renormalization, the new sequence satisfies the same conditions (9) as the initial sequence. Thus it is possible to iterate the renormalization, and the issue is then to decide what kind of limiting behavior arises after a large number of iterations.
It is sometimes useful to represent these sequences as formal power series defined up to a multiplicative constant: The sequence can be recovered from the power series by differentiation: In this new representation, the renormalization (10) is mapped to a functional relation between power series which can be derived easily as Iterating this functional relation is still rather difficult, however a change of variable induces a new renormalization map on g(y) which is simply given by Note that since the change of variable is analytic, g(y) is still a formal power series with new coefficients which depend on the original ones as There is no obvious sufficient condition on the g p which will ensure the positivity of the t n , so that this condition can not be checked easily in the y representation. However, it is easy to see that it implies that g p is bounded below by some strictly positive number (because g p ≥ t 1 > 0). The renormalization equation (12) written in terms of the coefficients g n takes a particularly simple form which can evidently be iterated p times to get

Fixed points
Fixed points are sequences (t n ) which do not change under renormalization. Since our sequences are only defined up to a multiplicative constant, this means that t n = ct n or equivalently t n + t n+1 = ct n the solution of which is t n+1 = (c − 1)t n where we must have c ≥ 1 in consequence of the positivity of the t j . For c > 1, this implies t n = t n−1 t 1 , where we have put c − 1 = t; this can be written most compactly if we use the scale freedom to set t 1 itself to be t, in which case we get simply For c = 1, we just obtain the c → 1 limit of these relationships, namely The equivalent relationships in terms of the generating functions introduced in the last section are as follows. Fixed points are power series which do not change under renormalization. Since these power series are only defined up to a multiplicative constant, this means that t a real (positive) constant. Putting Eq. (11) in (14) then gives This corresponds for the sequence (t n ) to a geometric series of ratio t, and gives a power series g(y) ∝ y 1 − cy of the same type after change of variable, where we have put t + 1 = c.

Cycles
Cycles are such that, after a finite number of renormalizations p, one gets back to the initial sequence. Taking into account the multiplicative constant freedom, this gives the equation for all n ≥ 1, where the constant, which is necessarily positive since both t n and t (p) n are, was written as a power c p . In light of (5), the equations (15) can be rewritten as a recursion relation for the sequence t n as For the sequence to be positive, this implies in particular c ≥ 1. A sequence given by such a linear recursion relation with constant coefficients independent of the index n can always be rewritten as a linear combination of p geometric progressions satisfying (16). The corresponding polynomial characteristic equation for the ratios can be solved easily as q j = −1 + ce 2iπj/p , and a cycle must therefore have the general form where the last term (in parentheses) is only there if p is even α 0 and α p/2 are real constants, the other α j are complex, and the combination was chosen such that t n be real.
Note that α p/2 (−c − 1) n = (−) n α p/2 (c + 1) n will never be purely positive for n large. Similarly, Re(α j q n j ) can never be purely positive for large n. Indeed, if the phase of q j is called θ j , it is well known that for θ j /2π irrational, cos(nθ j ) is dense in [−1, 1], whereas if it is rational, it is periodic with values of the form cos(2kπ/r), 0 ≤ k < r, which is positive for k = 0 and negative for k = [r/2], the integer part of r/2.
As a linear combination of geometric progressions, the behavior, and the sign, of t n for large n is dominated by the geometric progression with non zero coefficient α j and ratio q j with largest modulus. For t n to be positive for large n, the geometric progression (c − 1) n must therefore dominate there. As it happens, the modulus of q j grows with j from j = 0 to j = [p/2], which implies that for 0 < j ≤ p/2, all the coefficients α j must be 0. This gives for the sequence t n the necessary form which is percolation with t = c − 1 ≥ 0, and therefore a fixed point.
Thus, we have proved that the only cycles are the fixed points.

Flows
The issue is now to determine the behavior of a given initial sequence after a large number of renormalizations. Among other possibilities, one may expect either convergence to a fixed point, or some kind of oscillatory behavior. The convergence studied here will simply be pointwise convergence of sequences. In the absence of cycles, and since fixed points are geometric progressions (percolation), a reasonable hypothesis is that an initial sequence will converge to a fixed point described by t = lim(t 1/n n ), assuming this limit exists. In the following, this hypothesis will be validated in two qualitatively different cases: t > 0 finite, and its limiting case, t = 0. If t 1/n n does not have a limit (or converges to ∞), the issue becomes more complex, as will be discussed in the last part of this section and subsequently.

t finite and non-zero
In this subsection, the assumption will be that lim n→∞ t 1/n n = t = 0.
As a consequence, one can write where ε n is an auxiliary sequence which goes to 0 as n → ∞. Another useful auxiliary sequence is which also goes to 0 as n → ∞.
Taking into account the scaling freedom of the sequences, the pointwise convergence of the sequence to a fixed point in the limit of a large number of renormalizations is expressed as for n a fixed integer number. This is equivalent to showing that for any given n ≥ 1, First, it will be useful to prove the following Lemma.
Lemma If t n ≥ 0 is a sequence satisfying (17) for some particular t > 0 and if i 1 , i 2 , i 3 , · · · is a sequence of positive integers such that i p = o(p/ ln p), then ip i=0 We will need the lemma only in the special case i p = √ p.
The proof will be obtained by showing first that the sequence t n can be replaced by another sequence u n with no zeros so that the sequence u 1/n n be bounded from below by a strictly positive number. From (17), it is clear that an integer i 0 can be fixed such that t i > 0 for all i ≥ i 0 . Then introducing the auxiliary sequence u n such that because the left hand side is a polynomial in p of degree i 0 while the right hand side is a polynomial of degree i 0 + 1. Thus (with 'A ∼ B' meaning as usual that A/B → 1), ip i=0 and the sequence t n has been replaced by a strictly positive sequence that fulfills (20) iff the original sequence does. So in the following, it will be simply assumed that t n was strictly positive to start with.
Let 0 < T − ≤ T + be the lower and upper bounds of the sequence t 1/n n . Then, ip i=0 The numerator can be bounded from above by comparing it with a geometric series. Thus, call . So, by taking p large enough, it can be ensured that The numerator on the right hand side can be bounded using that where we have put i p = k for short. From this it follows that or taking logarithms, which implies in turn that the right hand side of (21) converges to zero for p → ∞ and concludes the proof of the Lemma.
Main proof The desired limiting behavior, equation (19), can be rearranged using Eqs. (5) and (18) as The goal is to prove that the latter expression goes to 0 when p → ∞. To do so, we will split the sum in the numerator into four pieces corresponding to the ranges and where we leave the auxiliary sequences n p and m p free for the moment, subject only to the conditions, n p , m p = o(p) = o(r p ) , n p , m p → ∞ as p → ∞ .
We will also assume, without loss of generality, that p is large enough so that 0 < √ p < r p − n p , whence we will have 0 < √ p < r p − n p < r p + m p < p .
Observe here that r p has been chosen to make the expression vanish when i = r p . Now, on performing our split in (22), we obtain where (with ' b i=a ' interpreted to imply a ≤ i < b in the first three cases) we can write In these deductions, we used the Lemma in the final step of (26), and we used it also (in the rather trivial special case, t n →[t(1 − E √ p )] n ) to extend the sums in the denominators for S 2 and S 4 to the full range, 0 ≤ i ≤ p. In (28), by narrowing the sum in its denominator, we converted the second expression into a weighted mean of | i p−i+1 − t| with the positive weights, p i t n+i , thereby obtaining the third expression, which is o(1) thanks to (24). Obviously, the only thing left to prove is that S 2 and S 4 go to zero as p → ∞. For that, it will be sufficient to show that their bounds A p and B p go to zero.
For future reference, we quote here the Stirling formula, n! ∼ √ 2πn n e n and its direct consequence (valid for m, n ≫ 1) log (m + n)! m! n! = m log m + n m + n log m + n n In order to bound A p and B p , one can observe that the general term of the sums appearing in them grows for 0 ≤ i ≤ r p and decreases for r p ≤ i ≤ p. This is because r p is effectively defined such that the terms with i = r p and i = r p − 1 be equal. Then the sums can be bounded by their largest term times the number of terms in the sum (which is itself smaller than p). From there, the following bounds for A p and B p are easily deduced: where the simplified notations were introduced. This expression can be estimated with the aid of the asymptotic formula (30). Substituting and expanding in powers of small quantities like m/p produces, after some tedium, which we will need only in the simplified form, The only question now is whether (consistently with (24)) we can choose the auxiliary sequence m (representing either m p or −n p + 1) so that the right hand side of (31) diverges to −∞, i.e. whether we can choose m = m(p) so that the first term in (31) dominates the others for large p. But this is not difficult. For example, meets all our requirements. With this choice, both A p and B p converge to zero as p → ∞, and thus S 2 and S 4 , which concludes the proof that the sequence t for with this choice u i+1 /u i ≤ √ E→0 for p → ∞, and the conditions (33) are also satisfied. Than, by taking p large enough, we can insure so that (omitting the subscripts on m p and E √ p for brevity) Invoking the Stirling formula (30) once again then yields, after some simplification, where the final step used that ln(E) ≤ 2 ln(m/p) because of (34). This proves that ln(A p ) → −∞ since m ≥ √ p → +∞ and m/p = o(1) → 0 as p → ∞. In consequence, A p → 0, which entails the desired convergence t=∞ In this case, the coefficients t n grow faster than any geometric progression, and one might think, consistent with the t → ∞ limit of (19), that t (p) n would tend to +∞ after a large number of renormalizations. Unfortunately this is not necessarily so, but it seems clear, at least, that it would follow under the stronger hypothesis that not only the (t n ) 1/n , but the ratios ρ n = t n+1 /t n converged to +∞ with n. In such a case, the renormalized dynamics would be trivial in the sense that it would produce only chains, as one can see from (2) and (3) above (because the unique λ(̟, m) with ̟ = n would swamp all others in the limit p → ∞).

Multiple limits
When the sequence t 1/n n does not converge, it is difficult to conclude anything in general. However, if, for a given integer q, the q subsequences t k(q+1)+i all converge as k → ∞, for 0 ≤ i ≤ q: then the p-times renormalized sequence t (p) n tends as p → ∞ toward the fixed point corresponding to t = sup(T i ). This is because, as we are going to show, after q renormalizations one has so that the previous result applies. We can restrict ourselves to the case when t = 0, because otherwise, since T i ≥ 0 automatically from the positivity of t n , we would have T i = 0 for all 0 ≤ i ≤ q and therefore t 1/n n itself would converge to 0, a case treated in the previous subsection.
To show (36), we define the auxiliary sequence The latter inequality is because at least one of the terms t n+j in the sum is equal to u n and the binomial coefficient q q/2 is the largest of the lot. The inequalities (38) show in particular that Now, to study the convergence of the sequence u 1/n n , we split it into q + 1 subsequences with indices k(q + 1) + i, 0 ≤ i ≤ q, which cover the whole sequence. Thus, reordering the elements of the supremum in (37), we have where H(p) is an Heaviside-like function defined by Then from the ansatz (35), and therefore the subsequence u k(q+1)+i converges: Since each of the subsequences converges to the same limit t and since they cover the whole sequence u n , we deduce lim and hence the result (36) from eq. (39).
In more general cases, it seems to be impossible to decide the question of convergence with our present methods. However, since the sequences we have not studied will have converging subsequences in stretches of diverging size, it seems that, as the number p of renormalizations goes to infinity, longer and longer initial stretches of the renormalized t n will, for almost all p, look like percolation for some t (which will in general vary with p). However, this same idea also gives a way to construct sequences which will not flow to any single fixed point, but alternate between various ones, and we describe such a counterexample in the next section.
If, for any k, we were to send p k to infinity, then obviously lim n → ∞ (t n ) 1/n would be a k and our main result above would assure us that f (n) n → ∞ −→ a k ; hence we can certainly find p k great enough that |f (p k ) − a k | < ǫ. Moreover, we can subsequently alter the values of t n for n > p k + 2 without affecting f (p k ), since the latter clearly depends only on t (p k ) 1 and t (p k ) 2 , and these in turn depend only on t n for n ≤ p k + 1 and n ≤ p k + 2, respectively. Hence, we can always select the p k to verify our claim. Figure 1 illustrates this technique with a closely related example in which t oscillates between (near to) 1/2 and (near to) 2, the specific choice of (t n ) in that case being: t n = n if n is a power of 2, and t n = 0 otherwise, i.e. (t n ) = (1, 2, 0, 4, 0, 0, 0, 8, 0, 0, 0, · · ·).  under renormalization Closer inspection of this example, and in particular of Figure 1, reveals that the t (p) n are not behaving completely chaotically. Rather, the precipitous jumps are narrowly localized in "time", while between them, the ratios ρ n = t n+1 /t n vary only gradually. Together with the evidence from other examples, this suggests that the "moments" when the renormalized t n deviate from percolation-like values are few and far between. In some, yet to be specified sense, then, (t n ) would be spending most of the time near the "percolation submanifold", with but brief excursions to other regions. In order to render this idea somewhat more definite, let us cast it in the form of a conjecture. For brevity, let us say that a real number r is "within ǫ of ∞" when r > 1/ǫ. Then we conjecture that: (∀ǫ > 0) (∀m) (the fraction of n < N for which the m ratios are all within ǫ of each other (or all within ǫ of ∞) tends to unity as N → ∞).

Further reflections
The main theorem proved above guarantees pointwise convergence to percolation dynamics when lim n → ∞ (t n ) 1/n exists. However the topology of pointwise convergence is rather coarse, and using it, we could not even claim, for example, that the "basin of attraction" of the fixed point set was extensive enough to include any open neighborhood of the latter (because no open set in this topology can control more than a finite number of terms of the sequence (t n )). On the contrary, the set of sequences not converging to the fixed-point set would be dense in the space T of all sequences. Thus this topology does not seem to provide a useful language for discussing the global features of our "RG flow".
From a physical point of view, pointwise convergence provides information with limited temporal validity. For example, pointwise convergence to t n = 0 means that, following a cosmic cycle comprising a large number of causet elements, the ensuing expansion will be tree-like for a long, but in general finite period. To guarantee permanent tree-like behavior, one would need something like uniform convergence of the t n , or perhaps more appropriately, uniform convergence of their ratios, t n+1 /t n . An interesting question, therefore, is whether, by suitably strengthening its hypotheses, one could prove an analog of our main theorem for the topology of uniform convergence. Another natural extension of the present work, possibly of greater urgency, would be to explore, not just which trajectories approach the fixed point set, but also the manner in which they approach it; for this could help answer the question of how general is the phenomenon discovered in [6] according to which the "cosmic big number" associated with the so-called "flatness problem" could be explained by the hypothesis that the universe has undergone several previous cycles of expansion and collapse.
Finally, let us remark that the question of which topology (or topologies) is most suited to discuss the "renormalization group flow" with which we have been concerned in this paper is inseparable from the wider question of which sequences (t n ) represent dynamical laws that are genuinely conceivable from the physical point of view. One can imagine, for example, uncontrollably divergent sequences, and if T is really the set of all sequences, then one might expect its dimensionality as a projective space to exceed ℵ 0 , the dimension of a separable Hilbert space. In that case, the normal tools of functional analysis would seem to be unavailing.
But does it really make sense that t n could grow arbitrarily rapidly (or, for that matter, be a number whose precision required for its expression arbitrarily many significant digits)? Our mathematical framework allows this, but only -it would seem -as an artifact of the procedure by which one introduces the dynamical laws in a "non-material" manner, as if "from outside the universe". It would be more satisfactory if the laws could somehow be understood as embodied in the structure of material universe -in this case in the structure of the causal set. But then, the number of possible laws should itself be limited at any stage of the growth of the causal set, meaning that the number of possibilities for t 0 , · · · , t n (the parameters that determine the dynamics at stage n) would be bounded by something like the number of possibilities for a causet of n elements, a number which grows only as 2 O(n 2 ) . It might be, therefore, that the space in which our renormalization transformation M acts is in reality not just of countable dimension but actually of countable cardinality. Important as this would be for the deeper understanding of the questions studied in this paper, it would be premature at present to speculate on how such a limitation on dynamical laws might work out in detail someday. Here, we wished only to raise the possibility, and in doing so to call to mind the idea that kinematics must ultimately fuse with (or absorb) dynamics as part of the further progress of fundamental physical theory.
R.D.S would like to thank the Aspen Center for Physics, where parts of this paper were written.