Ultimate Symmetry

According to the above Duality of Time postulate, the dynamic Universe is the succession of instantaneous discrete frames of space, that extend in the outward level of time that we normally encounter, but each frame is internally created in one chronological sequence within each inward level of the real flow of time. This is schematically demonstrated in Figure I.7, where space is conventionally shown in two dimensions, as the  plane, and we will mostly consider the  axis only, for simplicity.

In reality, however, we can conceive of at leastseven levels of time, whichcurlto make the four dimensions of space-time: , that are the three spatial and one temporal dimensions; since each spatial dimension is formed by two of the six inner levels, while the seventh is the outer time that we normally encounter.

Figure I.8: Representing the inner and outer levels of time together as genuinely-complex time-time space. In the real flow of time, one frame of space is re-created in internal chronological sequence, that then appears as one instance of the imaginary time , so the total complex time is: , as split-complex or hyperbolic numbers. We can also notice that the instantaneous velocity  in the normal time is always zero, while the real speed  in the inner time is always , and the apparent physical velocity  is the dynamic combination between them as can be calculated from equation 3.1.

As it will be explained further in section II.4.2 below, each spatial dimension is dynamically formed by the real flow of time, and whenever this flow is interrupted, a new dimension starts, which is achieved by multiplying with the imaginary unit that produces an abrupt rotation by , creating a new dimension that is perpendicular on the previous level, or hyperbolically orthogonal on it, to be more precise. This subtle property is what introduces discreteness, as a consequence of the duality nature of time, that is flowing either inwardly or outwardly, not both together. This is what makes space-time geometry genuinely complex and granular, otherwise if we consider all the dimensions to be coexisting together it will appear continuous and real, as we normally imagine , which may lead to space-time singularities at extreme conditions.

The concept of imaginary time is already being used widely in various mathematical formulations in quantum physics and cosmology, without any actual justification apart from the fact that it is a quite convenient mathematical trick that is useful in solving many problems. As Hawking states: It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed, but also effects we have not been able to measure yet nevertheless believe in for other reasons. .

Hawking, however, considers the imaginary time as something that is perpendicular to normal time that exists together with space, and that is how it is usually treated in physics and cosmology. According to the Duality of Time postulate, however, since space is now (dynamically re-created in) the real time, the normal time itself becomes genuinely imaginary, or latent.

Employing imaginary time is very useful because it provides a method for connecting quantum mechanics with statistical mechanics by using a Wick rotation, by . In this manner we can find a solution to dynamics problems in  dimensions, by transposing their descriptions in  dimensions, i.e. by trading one dimension of space for one dimension of time, which means substituting a mathematical problem in Minkowski space-time into a related problem in Euclidean space. Schrodinger equation and the heat equation are also related by Wick rotation. This method is also used in Feynman s path integral formulation, which was extended in 1966 by DeWitt into gauge invariant functional-integral. For this reason, there has been many attempts to describe quantum gravity in terms of Euclidean geometry, because in this way it is possible to avoid singularities which are unavoidable in General Relativity, since it is primarily constructed on curved space-time continuum that uses Riemannian manifolds, in which the geometry becomes ill-defined at the points of singularities.