3.3.1 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. 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, in addition to the abundance of various fractal structures on all physical and biological levels in nature, as was previously described by Mandelbrot [(Mandelbrot et al., 1983)].

Accordingly, some theories were constructed based on fractal space-time, including Causal Dynamical Triangulation and Scale Relativity, which also share some fundamental characteristics with Loop Quantum Gravity, that is trying to quantize space-time itself. 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, 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, while other theories suggest that space-time dimensionality developed gradually from 2D at the Planck scale, to become 4D at large galactic scales. 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.

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 0D, 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 Chapter V of Volume II, 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 3D in our normal perception, and it is expected to be 4D 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 3 for a perfect three-dimensional vacuum, which corresponds to the state of super-energy (c, c), but it may condense down to 0 for void, which is absolute darkness that is the super-mass state (0, 0).