http://www.msel-naschie.com/pdf/cantorian-theory.pdf

1. Introduction

Einstein’s Relativity Theory was a revolutionary departure from our habitual classical picture of a mechanical pro-

cesses taking place on a passive spacetime stage uninfluenced by it and vice versa[1]. It may be likened to the dif-

ference between the classical theatre of Racine and a modern play by Luigi Pirandello.

While Einstein’s relativity-theory has shown us that, in the large, spacetime geometry is curved and quite different

from our passive, flat and smooth classical space and time, modern results in high energy physics are forcing us to

reconsider the properties not only of flatness but also of the smoothness of the geometry of spacetime at the quantum

scale[2,3]. This is exactly what we are proposing in this paper. We will present a substantially novel approach to

quantum gravity and particle physics based on the idea that spacetime is basically a large infinite-dimensional but

hierarchical, disconnected and thus non-differentiable Cantor Set[4,5].

1.1. Clash of symmetries

To meet the special requirements of particle physics as well as gravity many excellent unification spacetime the-

ories have been proposed with varying degrees of success[6,7]. A crucial stumbling block of a consistent revision

of Euclidean spacetime topology and geometry is the potential and frequently detrimental clash between classical

spacetime symmetry in which our experiments are inevitably conducted and the internal Gauge symmetry required

by particle physics. Such a clash of symmetries leads to what is known in quantum field theory as anomalies[8,9].

It turns out that there are two strategies to completely eliminate these anomalies. The first possibility is to formulate

our geometry and topology without any direct reference to the concept of points. A well-known example for this type

of theory is Connes’ non-commutative geometry which is a natural extension of von Neumann’s continuous geometry[10,11]. In fact, von Neumann used to joke about his own proposal by calling it point-less geometry.

The second possibility is to avoid having any points at all in the theory. This possibility was made use of in String

theory and one of the main reasons for the phenomenal success and popularity of String-Theory lies in this fact[3].

Without excluding points the anomaly cancellation procedure of Green and Schwarz leading to a ten dimensional

super symmetric spacetime theory, replacing the old 26-dimensional Bosonic String theory would not have been possible [3].

1.2. Fractal spacetime and Cantor sets

In our work which began about two decades ago, we started exploring the possibility of a geometry which in a

sense reconciles the irreconcilable namely having points which are no points in the ordinary sense[4,12]. In other

words we could have our cake and eat it by using geometry with points which upon close examination reveal them-

selves not as a point but as a cluster of points. Every point in this cluster, when re-examined, reveals itself again as

another cluster of points and so on ad infinitum. Not surprisingly this type of geometry is well-known since a long

time to mathematicians and seems to have been discovered first by the German mathematician Georg Cantor, the

inventor of Set-Theory [13,14]. The famous Triadic Cantor Set is probably the simplest and definitely the best known

example of such geometry (see Fig. 1). Cantor sets are at the heart of modern mathematics and a particular form of non-metric spaces and geometry, known in modern parlance as fractals. However, Cantor Sets have never been used explicitly to model spacetime in physics. Material scientists, mechanical engineers, chemists, and biologists, all apart of the hard-core non-linear dynamicists apply Cantor Sets across the fields [15]. By contrast it seems indeed that our group is the first to take the idea of a Cantorian fractal spacetime seriously and give it a viable mathematical formulation with the help of which specific and precise computations can be made [16–57]. It is this approach that we wish to introduce in this paper.

1.3. Basic assumptions

Starting from the basic assumption that spacetime is essentially a very large Cantor Set we are going to extract

from this simple theory a great deal of information about particle physics, gravity and the combination of the two,

namely quantum gravity[21].

We recall that gravity is a property of spacetime in the large. By contrast, particle physics is a property of space-

time at the very small, or more accurately on the quantum level of observation. This disparity in size which corre-

sponds to disparity in the energy scale is one, if not the main reasons behind the unyielding resistance against all

attempts to reconcile the two fundamental theories. However, in Cantorian fractal spacetime where unlike the smooth

classical case there is no a-priori given natural scale it is very easy to make the very large meet the very small exactly

as in P-Adic number theory[22]. In this sense we can speak of a coincidentia oppositorium, to use a Hegelian termi-

nology. In theoretical physics this is what is called by Edward Witten T-duality in connection with M-theory[23].In other words, a Cantorian fractal spacetime has an inbuilt T-duality or a P-Adic property as well as being free of anom-

alies and scale invariant all for the simple reason of not having any ordinary points in it[24].

We will start in the next section by discussing in more detail how to construct the said Cantorian spacetime man-

ifold from scratch as well as deriving its dimensionality and finally looks at the dynamics induced by it. ]]>