2.  Introduction:

  Relativity, and its classical predecessor, consider space and time to be continuous, and everywhere differentiable, whereas quantum mechanics is based on discrete quanta of energy and fields, albeit they still evolve in continuous background. Although both theories have already passed many rigorous tests, they inevitably produce enormous contradictions when applied together in the same domain. Most scholars believe that this conflict may only be resolved with a successful theory of quantum gravity (Calcagni, 2017).

In trying to resolve the discrepancy, some space-time theories, such as Causal Dynamical Triangulation (Ambjørn et al., 2005), Quantum Einstein Gravity (Lauscher and Reuter, 2005) and Scale Relativity (NOTALE, 1992), attempted to relax the condition of differentiability, in order to allow for fractal space-time, which was first introduced in 1983 (Ord, 1983). In addition to the abundance of all kinds of fractal structures in nature, this concept was also supported by many astronomical observations which show that the Universe exhibit a fractal aspect over a fairly wide range of scales (Joyce et al., 2005), and that large-scale structures are much better described by a scale-dependent fractal dimension (Hogg et al., 2005), but the theoretical implications of these observations are not very well understood, yet.

Nonetheless, the two most celebrated approaches to reconcile Relativity with Quantum Mechanics are Strings Theory and Loop Quantum Gravity (LQG). The first tries to develop an effective quantum field theory of gravity at low energies, by postulating strings instead of point particles, while LQG uses spin networks to obtain granular space that evolves with time. Therefore, while Strings Theory still depends on the background continuum, LQG tries to be background-independent by attempting to quantize space-time itself (Rovelli, 2011).

In this regard, the author believes that any successful theory of quantum gravity must not rely on either the continuum or discretuum structures of space-time. Rather, these two contrasting and mutually-exclusive views must be the product of such theory, and they must become complementary on the microscopic and macroscopic scales. The only contestant that may fulfill this criterion is “Oneness”, because on the multiplicity level things can only be either discrete or continuous; there is no other way. However, we need first to explain how the apparent physical multiplicity can proceed from this metaphysical oneness, and then exhibit various discrete and continuous impressions. The key to resolve this dilemma is in understanding the “inner levels of time” in which “space” and “matter” are perpetually being “re-created” and layered into the three spatial dimensions, which then kinetically evolve throughout the “outer level of time” that we encounter. This will be fully explained in sections 3 and 5 below.

Due to this “dynamic formation of dimensions”, in the inner levels of time, the Duality of Time Theory leads to granular and self-contained space-time with fractal and genuinely-complex structure, which are the key features needed to accommodate both quantum and relativistic phenomena. Many previous studies have already shown 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 they either do not justify the use of fractals, or they are forced to make new unjustified assertions, such as the relativity of scale, that may lead to fractal space-time. On the other hand, imaginary time had been successfully used in the early formulation of Special Relativity by Poincare (Poincaré, 1906), and even Minkowski (Einstein, 2010), but it was later replaced by the Minkowskian four-dimensional space-time, because there were no substantial reasons to treat time as imaginary. Nevertheless, this concept is still essential in current cosmology and quantum field theories, since it is employed by Feynman’s path integral formulation, and it is the only way to avoid singularities which are unavoidable in General Relativity.

In the Duality of Time Theory, since the dimensions of space and matter are being re-created in the inner (complete) levels of time, the final dimension becomes multi-fractal and equals to the dynamic ratio of “inner” to “outer” times. Additionally, and for the same reason, space-time becomes “genuinely complex”, since both its “real” and “imaginary” components have the same nature of time, which itself becomes as simple as the “recurrence”, or counting the number of geometrical nodes as they are re-created in one chronological sequence. Without postulating the inner levels of time, both the complex and fractal dimensions would not have any “genuine” meaning, unless both the numerator and denominator of the fraction, and both the real and imaginary parts of the complex number, are all of the same nature (of time).

In this manner, normal time is an imaginary and fractional dimension of the complete dimensions of space, which are the real levels of time. Because they are complete integers, the dimensions of space are mutually perpendicular, or spherically orthogonal, on each other, which is what makes (isotropic and homogeneous) Euclidean geometry that can be expressed with normal complex numbers , in which the modulus is given by . In contrast, because it is fractional or non-integer dimension, (normal, or the outer level of) time is hyperbolically orthogonal on the dimensions of space, and thus expressed by the hyperbolic split-complex numbers , in which the modulus is given by . This complex hyperbolic geometry is the fundamental reason behind relativity and Lorentz transformations, and it provides the required tools to express the curvature and topology of space-time, away from Riemannian manifolds, in which the geometry becomes ill-defined at the points of singularities.