**5. Final remarks**

We have formally prove that a class of gauge theories can be deformed into a first order gravity theory and, possibly, with an extra set of matter fields. For that we have employed the theory of fiber bundles. The relevance (and motivation) of the present work is that it can be applied to quantum gravity models which are based on gauge theories that can generate an emergent gravity theory. The main problem in quantizing gravity is that the principles of general relativity are incompatible with those of quantum field theory. In fact, a quantum field theory can only be formulated in an Euclidean spacetime. For example, a quantum field is, by definition, an object that carries uncertainty fluctuations and is parametrized through spacetime coordinates, *i.e.*, a set of well defined real parameters. Now, if a coframe field is a quantum field5, *e*ˆ(*x*), and from the fact that it defines a mapping from tangent coordinates *x<sup>a</sup>* to world coordinates *xμ*, then quantum fluctuations of *e*ˆ will induce spacetime to fluctuate as well, *x*ˆ*<sup>μ</sup>* = *e*ˆ *μ <sup>a</sup> xa*. Thus, a paradox is encountered because *x* must be a set of parameters instead of a fluctuating object.

On the other hand, if the starting gauge theory is constructed over an Euclidean manifold and it is renormalizable, then it can be an excellent candidate for a quantum gravity theory. All needed is that it emerges as a geometrodynamics at classical level. The class of theories that fits on this program are determined essentially by theorems 4.1 and 4.3.

A few practical examples are in order: In [15] a 4-dimensional *SU*(2) gauge theory generates a deformation of the 3-dimensional space. Time is left untouched by he mapping. In this example, the resulting theory contains the Einstein-Hilbert action for the extrinsic curvature and the solution of the equations of motion predicts not only curvature but also torsion. Another example can be found in [9], where a deformed 4-dimensional spacetime emerges from a de Sitter type gauge theory over an Euclidean spacetime. In this case, a dynamical mass scale is responsible for the separation between the gauge and gravity phases. In general,

<sup>5</sup> The hat indicates the quantum nature of the field.

**6** 

L. Marek-Crnjac

 *Slovenia* 

*Technical School Center of Maribor, Maribor* 

**Quantum Gravity in Cantorian Space-Time** 

Einstein's contribution to relativity was initially an intuitive approach based on a basic elimination of simultaneousity and a mathematical reformulation using the Lorenz transformation. In this respect Einstein just added some more physics to what Poincaré and Lorenz have done much earlier. However, it was Minkowski who introduced the geometrical ideas and the use of a four-dimensional space with time as the fourth dimension. Einstein took over Minkowski's idea and initiated what we may call the program of geometrizing physics, starting with gravity. Later on Einstein and Hilbert attempted the unification of electro-magnetism and gravity while Kaluza and Klein tried the same using an extra fifth dimension. This may have been the beginning of the higher dimensional space-time theories culminating in super strings, super gravity and the

In special relativity there is no absolute time. We have a space and each slice has its own time. Thus each point in the Minkowski's space is specified by four coordinates, three spatial and one temporal in a four-dimensional space-time rather than the 3+1 space plus

A fundamental role in this new geometry is played by the constancy of the velocity of light that cannot be exceeded without violating the causal structure as well documented experimental facts show. A change of things began by adding quantum mechanics to special

By replacing Euclidean geometry by curved Riemannian one, Einstein was the first to give gravity a geometrical interpretation as a curvature of space-time due to matter. Einstein never fixed the topology of his theory nor did he use or was aware of the existence of nonclassical geometry which was in any case in its infancy [1-5]. The possibly only encounter of Einstein with M. S. El Naschie's Cantorian like transfinite geometry was when K. Menger

Let us examine the basic concept of a line or more generally a curve. Classical geometry used in classical mechanics and general relativity the fact that a line is a one-dimensional object, while a point is zero-dimensional. Furthermore, it would seem at first sight that a line consists of infinite number of points and that it is simply the path drawn by a zero-

**1. Introduction** 

relativity.

Cantorian space-time theory [1].

time coordinates of the classical mechanics [1].

presented a paper in a conference held in his honour [1, 6-17].

**2. Transfinite sets and quantum mechanics** 

several emergent gravity theories fit to the results of this work, see for instance [1–8] where the Higgs mechanism is largely used to separate gauge and gravity phases.

We end this work by remarking that several issues are left for future investigation. Just to name a few: The role of matter fields living in the starting gauge theory; the generalization of the present results to include metric-affine gravities before the reduction of the coframe bundle; the role of the extra matter fields in the dark matter/energy problem; explicit computations in order to make reliable predictions that fit with actual data; and so on.
