Discrete Mathematics and Physics on the Planck-scale Exempliied by Means of a Class of 'cellular Network Modells' and Their Dynamics

di erential algebra. Note e.g. that in our algebra an element like n123 is admissible (i.e. non-zero) if n1; n2 and n2; n3 are connected. n123 may however arise from a di erentiation process (i.e. from an insertion) like du(n13) with n1; n3 not(!) connected. This is exactly the situation discussed above: n13 2 I but du(n13) = 2 I (88) Dividing now by I 0 maps du(n13) onto zero whereas there may be little physical/geometric reason for n123 or a certain combination of such admissible elements being zero in our network. 3.32 Conclusion: Given a concrete physical network one has basically two choices. Either one makes it into a fulledged di erential algebra by imposing further relations which may however be unnatural from a physical point of view and very cumbersome for complicated networks. This was the strategy e.g. followed in [8]. Or one considers as the fundamental object and each admissible element in it being non-zero. As a consequence the corresponding algebraic/di erential structure on may be less smooth at rst glance but on the other side more natural (the 'graded Leibniz-rule' or d d = 0 may only hold up to a certain order). At the moment we refrain from making a general judgement whereas we would probably prefer the latter choice. 4 Intrinsic Dimension in Networks, Graphs and other Discrete Systems There exist a variety of concepts in modern mathematics which generalize the notion of 'dimension' one is accustomed to in e.g. di erential topology or linear algebra. In fact, 'topological dimension' is a notion which seems to be even closer to the underlying intuition (cf. e.g. [24]). Apart from the purely mathematical concept there is also a physical aspect of something like dimension which has e.g. pronounced e ects on the behavior of, say, many-body-systems, especially their microscopic dynamics and, most notably, their possible 'phase transitions'. But even in the case of e.g. lattice systems they are usually considered as embedded in an underlying continuous background space (typically euclidean) which supplies the concept of ordinary dimension so that the 'intrinsic dimension' of the discrete array itself does usually not openly enter the considerations. Anyway, it is worthwhile even in this relatively transparent situations to have a closer look on where attributes of something like dimension really come into the physical play. Properties of models of, say, statistical mechanics are almost solely 25 derived from the structure of the microscopic interactions of their constituents. This is more or less the only place where dimensional aspects enter the calculations. Naive reasoning might suggest that it is the number of nearest neighbors (in e.g. lattice systems) which re ects in an obvious way the dimension of the underlying space and in uences via that way the dynamics of the system. However, this surmise, as we will show in the following, does not re ect the crucial point which is considerably more subtle. This holds the more so for systems which cannot be considered as being embedded in a smooth regular background and hence do not get their dimension from the embedding space. A case in point is our primordial network in which Planck-scalephysics is assumed to take place. In our approach it is in fact exactly the other way round: Smooth space-time is assumed to emerge via a phase transition or a certain cooperative behavior and after some 'coarse graining' from this more fundamental structure. 4.1 Problem: Formulate an intrinsic notion of dimension for model theories without making recourse to the dimension of some embedding space. In a rst step we will show that graphs and networks as introduced in the preceding sections have a natural metric structure. We have already introduced a certain neighborhood structure in a graph with the help of the minimal number of consecutive bonds connecting two given nodes. In a connected graph any two nodes can be connected by a sequence of bonds. Without loss of generality one can restrict oneself to 'paths'. One can then de ne the length of a path (or sequence of bonds) by the number l of consecutive bonds making up the path. 4.2 Observation/De nition: Among the paths connecting two arbitrary nodes there exists at least one with minimal length which we denote by d(ni; nk). This d has the properties of a 'metric', i.e: d(ni; ni) = 0 (89) d(ni; nk) = d(nk; ni) (90) d(ni; nl) d(ni; nk) + d(nk; nl) (91) (The proof is more or less evident). 4.3 Corollary: With the help of the metric one gets a natural neighborhood structure around any given node, where Um(ni) comprises all the nodes with d(ni; nk) m, @Um(ni) the nodes with d(ni; nk) = m. With the help of the above neighborhood structure we can now develop the concept of an intrinsic dimension on graphs and networks. To this end one has at rst to 26 realize what property really matters physically (e.g. dynamically) independently of the model or embedding space. 4.4 Observation: The crucial and characteristic property of, say, a graph or network which may be associated with something like dimension is the number of 'new nodes' in Um+1 compared to Um for m su ciently large or m!1. The deeper meaning of this quantity is that it measures the kind of 'wiring' or 'connectivity' in the network and is therefore a 'topological invariant'. Remark: In the light of what we have learned in the preceding section it is tempting to relate the number of bonds branching o a node, i.e. the number of nearest neighbors or order of a node, to something like dimension. On the other side there exist quite a few di erent lattices with a variety of number of nearest neighbors in, say, twoor threedimensional euclidean space. What however really matters in physics is the embedding dimension of the lattice (e.g. with respect to phase transitions) and only to a much lesser extent the number of nearest neighbors. In contrast to the latter property dimension re ects the degree of connectivity and type of wiring in the network. In many cases one expects the number of nodes in Um to grow like some power D of m for increasing m. By the same token one expects the number of new nodes after an additional step to increase proportional to mD 1. With j j denoting number of nodes we hence have: jUm+1j jUmj = j@Um+1j = f(m) (92) with f(m) mD 1 (93) for m large. 4.5 De nition: The intrinsic dimension D of a regular (in nite) graph is given by D 1 := lim m!1(ln f(m)= lnm) or (94) D := lim m!1(ln jUmj= lnm) (95) That this de nition is reasonable can be seen by applying it to ordinary cases like regular translation invariant lattices. It is however not evident that such a de nition makes sense for arbitrary graphs, in other words, a (unique) limit point may not always exist. It would be tempting to characterize the conditions which entail that such a limit exists. We, however, plan to do this elsewhere. 4.6 Observation For regular lattices D coincides with the dimension of the euclidean 27 embedding space DE. Proof: It is instructive to draw a picture of the consecutive series of neighborhoods of a xed node for e.g. a 2-dimensional Bravais lattice. It is obvious and can also be proved that for m su ciently large the number of nodes in Um goes like a power of m with the exponent being the embedding dimension DE as the euclidean volume of Um grows with the same power. Remarks:i) For Um too small the number of nodes may deviate from an exact power law which in general becomes only correct for su ciently large m. ii) The number of nearest neighbors, on the other side, does not(!) in uence the exponent but rather enters in the prefactor. In other words, it in uences jUmj for m small but drops out asymptotically by taking the logarithm. For an ordinary Bravais lattice with NC the number of nodes in a unit cell one has asymptotically: jUmj NC mDE and hence: (96) D = lim m!1(ln(NC mDE)= lnm) = DE + lim m!1(NC= lnm) = DE (97) independently of NC . Before we proceed a remark should be in order concerning related ideas on a concept like dimension occurring in however completely di erent elds of modern physics: When we started to work out our own concept we scanned in vain the literature on e.g. graphs accessible to us and consulted various experts working in that eld. From this we got the impression that such ideas have not been pursued in that context. (It is however apparent that there exist conceptual relations to the geometry of 'fractals'.) Quite some time after we developed the above introduced concept we were kindly informed by Th. Filk that such a concept had been employed in a however quite di erent context by e.g. A.A. Migdal et al and by himself (see e.g. [25] and [26]). As a consequence one should say that, while a concept like this may perhaps not be widely known for discrete structures like ours, it does, on the other side, not seem to be entirely new. We hope to come back to possible relations between these various highly interesting approaches elsewhere (see our remarks before 4.6 Observation). Matters become much more interesting and subtle if one studies more general graphs than simple lattices. Note that there exists a general theorem showing that practically every graph can be embedded in IR3 and still quite a few in IR2 ('planar graphs'). The point is however that this embedding is in general not invariant with respect to the euclidean metric. But something like an apriori given euclidean length is unphysical for the models we are after anyhow. This result has the advantage that one can visualize many graphs already in, say, IR2 whereas their intrinsic dimension may be much larger. 28 An extreme example is a 'tree graph', i.e. a graph without 'loops'. In the following we study an in nite, regular tree graph with node degree 3, i.e. 3 bonds branching o each node. The absence of loops means that the 'connectivity' is extremely low which results in an exceptionally high 'dimension' as we will see. Starting from an arbitrary node we can construct the neighborhoods Um and count the number of nodes in Um or @Um. U1 contains 3 nodes which are linked with the reference node n0. There are 2 other bonds branching o each of these nodes. Hence in @U2 = U2nU1 we have 3 2 nodes and by induction: j@Um+1j = 3 2m (98) which implies D 1 := lim m!1(ln j@Um+1j= lnm) = lim m!1(m ln2= lnm + 3= lnm) =1 (99) Hence we have: 4.7 Observation(Tree): The intrinsic dimension of an in nite tree is 1 and the number of new nodes grows exponentially like some n(n 1)m with m (n being the node degree). Remark: D =1 is mainly a result of the absence of loops(!), in other words: there is exactly one path, connecting any two nodes. This is usually not so in other graphs, e.g. lattices, where the number of new nodes grows at a much slower pace (whereas the number of nearest neighbors can nevertheless be large). This is due to the existence of many loops s.t. many of the nodes which can be reached from, say, a node of @Um by one step are already contained in Um itself. We have seen that for, say, lattices the number of new nodes grows like some xed power of m while for, say, trees m occurs in the exponent. The borderline can be found as follows: 4.8 Observation: If form!1 the average number of new nodes per node contained in @Um, i.e: jUm+1j=jUmj 1 + " (100) for some " 0 then we have exponential growth, in other words, the intrinsic dimension is 1. Proof: If the above estimate holds for all m m0 we have by induction: jUmj jUm0 j (1 + ")m m0 (101) Potential applications of this concept of intrinsic dimension are manifold. Our main goal is it to develop a theory which explains how our classical space-time and 29 what we call the 'physical vacuum' has emerged from a more primordial and discretebackground via some sort of phase transition.In this context we can also ask in what sense space-time dimension 4 is exceptional,i.e. whether it is merely an accident or whether there is a cogent reason for it.As the plan of this paper was mainly to introduce and develop the necessary con-ceptual tools and to pave the ground, the bulk of the investigation in this particulardirection shall be presented elsewhere as it is part of a detailed analysis of the (statit-ical) dynamics on networks as introduced above, their possible phase transitions,selforganisation, emergence of patterns and the like.In the following we will only supply a speculative and heuristic argument in favorof space-dimension 3. We emphasized in section 2 that also the bond states, modellingthe strength of local interactions between neighboring nodes, are in our model theorydynamical variables. In extreme cases these couplings may completely die out and/orbecome 'locked in' between certain nodes, depending on the kind of model.It may now happen that in the course of evolution a local island of 'higher order'(or several of them) emerges via a spontaneous uctuation in a, on large scales,unordered and erratically uctuating network in which couplings between nodes areswitched on and o more or less randomly.One important e ect of the scenario we have in mind (among others) is that theremay occur now a pronounced near order in this island, accompanied by an increasein correlation length and an e ective screening of the dangerously large 'quantumuctuations' on Planck scale, while the global state outside remains more or lessstructureless. We assume that this will be e ected by a reduction of intrinsic dimen-sion within this island which may become substantially lower than outside, say, niteas compared to (nearly) in nity.If this 'nucleation center' is both su ciently large and its local state 'dynamicallyfavorable' in a sense to be speci ed (note that a concept like 'entropy' or somethinglike that would be of use here) it may start to unfold and trigger a global phasetransition.As a result of this phase transition a relatively smooth and stable submanifold ona certain coarse-grained scale (alluding to the language of synergetics we would liketo call it an 'order parameter manifold') may come into being which displays certainproperties we would attribute to space-time.Under these premises we could now ask what is the probability for such a spe-ci c and su ciently large spontaneous uctuation? As we are at the moment talkingabout heuristics and qualitative behavior we make the following thumb-rule-like as-sumptions:i) In the primordial network 'correlation lengths' are supposed to be extremely short(more or less nearest neighbor), i.e. the strengths of the couplings are uctuatingmore or less independently.ii) A large uctuation of the above type implies in our picture that a substantial30 fraction of the couplings in the island passes a certain threshold (cf. the models ofsection 2) i.e. become su ciently weak/dead and/or cooperative. The probabilityper individual bond for this to happen be p. Let L be the diameter of the nucle-ation center, const Ld the number of nodes or bonds in this island for some d. Theprobability for such a uctuation is then roughly (cf. i)):Wd = const p(Ld)(102)iii) We know from our experience with phase transitions that there are favorabledimensions, i.e. the nucleation centers may fade away if either they themselves aretoo small or the dimension of the system is too small. Apart from certain non-genericmodels d = 3 is typically the threshold dimension.iv) On the other side we can compare the relative probabilities for the occurrence ofsu ciently large spontaneous uctuations for various d's. 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