7.  Fractal Space-Time and Quantum Phenomena:

Soon after the discovery of fractals, fractal structures of space-time were suggested in 1983 (Ord, 1983), as an attempt to find a geometric analogue for relativistic quantum mechanics, in accordance with Feynman’s path integral formulation, where the typical infinite number of paths of quantum-mechanical particles are characterized as being non-differentiable and fractal (Abbott and Wise, 1981). This theoretical concept suggests that the structure of space-time itself has a fractal dimension, and not only the large-scale distribution of galaxies as confirmed by some observations (Joyce et al., 2005; Hogg et al., 2005), in addition to the abundance of various fractal structures on all physical and biological levels in nature, as was previously described by Mandelbrot (Mandelbrot, 1983).

Accordingly, some theories were constructed based on fractal space-time, including Causal Dynamical Triangulation (Ambjørn et al., 2005) and Scale Relativity (NOTALE, 1992), which also share some fundamental characteristics with Loop quantum gravity, that is trying to quantize space-time itself (Rovelli, 2011). Actually, there are many studies that have successfully demonstrated how the principles of quantum mechanics can be derived from the fractal structure of space-time (Notale and Célérier, 2007; JUMARIE, 2001; Cresson, 2003; Adda and Cresson, 2005; Jumarie, 2007), but there is yet no complete understanding of how the dimensionality of space-time evolved to the current Universe. Some multiverse and eternal-inflation theories exhibit fractality at scales larger than the observable Universe (Linde, 1987), while other theories suggest that space-time dimensionality developed gradually from 2D at the Planck scale, to become 4D at large galactic scales (Ambjørn et al., 2005; Lauscher and Reuter, 2005; NOTALE, 1992). Fractality also arises in the non-commutative geometry approach to quantum gravity, which tries to understand how fractal space couples with gravitation, and suggests that time is an emergent property (Connes and Rovelli, 1994).

In this regard, based on the concept of re-creation according to the Duality of Time Theory, the Universe is constantly being re-created from one geometrical point, that is , from which all the current dimensions of space and matter are re-created in the inner levels of time before they evolve in the outer time. Therefore, the total dimension of the Universe becomes naturally multi-fractal and equals to the dynamic ratio of “inner” to “outer” times, because spatial dimensions alone, as an empty homogeneous space, are complete integers, while fractality arises when this super-fluid vacuum, as described in section 5.1 above, starts oscillating in the outer time, which causes all types of vortices that we denote as elementary particles. So we can see how this notion, of space-time having fractal dimensions, would not have any “genuine” meaning unless both the numerator and denominator of the fraction are both of the same nature of time, and this can only be fulfilled by interpreting the complete dimensions of space as inner levels of time.

In the absolute sense, the ratio of inner to outer times is the same as the speed of light, which only needs to be “normalized” in order to express the fractality of space-time; to become time-time. For example, if the re-creation process, that is occurring in the inner levels of time, is not interrupted in the outer time, i.e. when the outer time is zero, this corresponds to absolute vacuum, that is an isotropic and homogeneous Euclidean space, with complete integer dimensions, that is  in our normal perception, and it is expected to be  on large cosmological scales. So the speed of light, in the time-time frame, is a unit-less constant that is equal to the number of dimensions that are ideally  for a perfect three-dimensional vacuum, which corresponds to the state of super-energy , as described in section 5.1, but it may condense down to  for void, which is absolute darkness that is the super-mass state .

The standard value of the speed of light in vacuum is now considered a universal physical constant, and its exact value is  meters per second. Since 1983, the length of the Meter has been defined from this constant, as well as the international standard for time. However, this experimentally measured value corresponds to the speed of light in actual vacuum that is in fact not exactly empty. The true speed that should be considered as the invariant Speed of Creation is the speed of light in absolute “void” rather than “vacuum”, which still has some energy that may interact with the photons, and delay them, but void is real “nothing”. Of course, even high vacuum is very hard to achieve in labs, so void is absolutely impossible.

Because we naturally distinguish between space and time, this speed must be measured in terms of meters per second, and it should be therefore exactly equal to . The difference between this theoretical value and the standard measured value is what accounts for the quantum foam, in contrast to the absolute void that cannot be excited. Of course all this depends also on the actual definition of the meter, and also the second, which may appear to be conventional, but in fact they are based on the same ancient Sumerian tradition, included in their sexagesimal system which seems to be fundamentally related to the structure of space-time (Haj Yousef (2017), Ch. VII).

Therefore, the actual physical dimensions of the (local) Universe are less than three, and they change according to the medium, and they are expected to be more than three in extra-galactic space, to accommodate negative mass and super-symmetry. For example the fractional dimensions of the actual vacuum is simply , and the fractional dimensions of water would be , and so on for all transparent mediums according to their relative refraction index. Opaque materials could be also treated in the same manner according to their refraction index, but for other light wave-lengths that they may transfer. Dimensionality is a relative and dynamic property, so the Universe is ultimately described by multi-fractal dimensions that change according to the medium, or the inner dimensions (of space), and also wavelength, that is the outer dimension (or time).