II.2.6 Yang-Mills Theory

Yang-Mills fields are gauge fields used in modern physics to describe physical fields that play the role of carriers in fundamental interaction. In 1954, C.N. Yang and R.L. Mills suggested that the space of intrinsic degrees of freedom of elementary particles depends on the points of space-time, and the intrinsic spaces corresponding to different points are not canonically isomorphic. In geometrical terms, this suggestion means that the space of intrinsic degrees of freedom is a vector bundle over space-time that does not have a canonical trivialization, and physical fields are described by cross-sections of this bundle. To describe the differential evolution equation of a field one has to define a connection in the bundle, that is, a trivialization of the bundle along the curves in the base. Such a connection with a fixed holonomy group describes a physical field, usually called a Yang-Mills field. The equations for a free Yang-Mills field can be deduced from a variational principle. They are a natural non-linear generalization of Maxwell s equations.

The most important Quantum Field Theories describing elementary particles are gauge theories formulated in terms of a principal fibre bundle over the Minkowski space-time with particular choices of the structure group. Given an arbitrary compact gauge group, does a non-trivial quantum Yang-Mills theory with a finite mass gap exist?

This problem is listed as one of the Millennium Prize Problems in mathematics given by the Clay Institute, which is to prove that for any compact simple gauge group , a non-trivial quantum Yang-Mills theory exists on  and has a mass gap . Existence includes establishing strong axiomatic properties, such as those cited in Streater & Wightman (1964), Osterwalder & Schrader (1973) and Osterwalder & Schrader (1975).

As it was accurately anticipated by Jaffe and Witten in the Official Problem Description [17], that establishing the existence of Yang-Mills theory and mass gap requires the introduction of fundamental new ideas both in physics and in mathematics. Hence, the innovative concept of the Duality of Time and perpetual creation of space provide a dynamic granular structure of space-time which clearly indicates the existence of a minimum physical mass as a result of the coupling between at least two geometrical points, or the individual monads that are the temporal images of the only one Single Monad that is actually described by real existence at the real flow of time. Since the massless monads are abruptly bursting from vacuum to the speed of light, and because they cannot coexist together in the inner level of time, when two monads couples to form a physical particle their minimum average velocity on the normal outward level of time is necessarily smaller than the speed of light, because the speed of each one of them must be either zero or , which gives the particle a minimum positive energy or rest mass, as we shall describe with more detail in section III.3.3.

The problem in most current theories is that they consider space-time as a background continuum, where particles propagate in continuous existence, which means that they can acquire any infinitesimal energy. With the new concept of the Duality of Time and perpetual creation of space, there is no motion in the common sense that the object or particle gradually leaves its place to occupy new infinitesimally adjacent places, but it is perpetually re-created in those new places. Therefore, there is no instantaneous transmutation but an average change between the discrete points of space, which also takes a discrete duration of time. These discrete instances of space-time are created dynamically to form the aether which then may condense into matter particles at some higher excited states. This dynamic aether does not contradict the null result of Michelson-Morley experiment which proved that light travels with the same speed regardless whether it was moving in the direction of the movement of the Earth or perpendicular to it. On the contrary, this constancy and invariance of the speed of light can now be explained as the speed of the dynamic creation of space, and it is the only real speed in nature, while the limited velocities of objects and matter particles are a result of averaging over the inner levels of time.

As it will be clarified by the end of Chapter IV, using the normal-complex and hyperbolic-complex numbers, it can be shown that Hilbert spaces, of any dimension, can be dynamically constructed in the inner levels of time by the perpetual coupling of genuinely-complex time-time instances, which form the nested dimensions of the Euclidean space, and what it may contain of matter particles that evolve in the outer (split-complex or hyperbolic) time, with their different magnetic, electric, (weak and strong) nuclear and gravitational properties, depending on the number of degrees of freedom, or dimensions.

According to the Duality of Time Theory, all physical dimensions, particles and objects are formed by the coupling between at least two metaphysical or abstract geometrical points of this time-time space, which introduces the various physical properties, such as charge and mass, that become naturally quantized and proportional to the number of geometrical points constituting each particle, starting from one individual point for massless bosons. Therefore, the reason why some bosons appear to have heavy masses is simply because they are confined in lower dimensions;  and , while the photon is a  spatial wave that evolves in our  space, and the anticipated graviton is a  wave that evolves in the outer time that may only be observed in the encompassing  space, which is expected to include negative masses and negative energy, as we shall discuss further in Chapter III.

The fundamental reason behind this coupling is that, in the real flow of the inner level(s) of time, only one geometrical point is created every instance, which then makes the vacuum that is a flat homogeneous and isotropic, or continuous, space, without the outer time that we encounter. Depending on the number of degrees of freedom, this flat flow of time, in the inner levels, creates the different bosons, which behave as waves, or energy moments, as far as they remain undisturbed in the outer time. Henceforth, the individual moments of our normal time cause this perpetual flow to be interrupted and flow in a new dimension orthogonal to the space that was being created. This interruption of the real flow of time that was creating the various bosonic states in the continuous space, causes their collapse into specific discrete states that then behave as particles because they have become coupled into one point in the higher dimension. Therefore, from the various possible combinations between the real flow of the inner time and the orthogonal, or imaginary, flow of the outer time, all the states of matter are created.

For example, a photon is a hollow spherical wave of space, which is one instance of two-dimensional inner time moments flowing simultaneously in three dimensions, thus described by normal complex numbers, as far as it remains unobserved or absorbed by other particles in the outer time. When at least two such instances are entangled together to become one moment in the outer time, a solid three-dimensional fermion is created, which then acquires mass because its individual points cannot propagate together at their original speed of light at which they were propagating before they became entangled. This is the origin of inertial mass as it will be clarified further in section III.3.3 in Chapter III.

Therefore, tt is clear from this brief description that there is a minimum mass above the ground state of vacuum. Although this mass gap is strictly positive, it can be arbitrarily small because it is proportional to the ratio between the outer and inner flow of time moments. As we explained above, the state of vacuum corresponds to zero outer time, and the various bosons are the ground states of the different dimensions of this empty space that is dynamically created in the inner levels of time. In our normal three-dimensional space, the possible levels of time are: , ,  and , so the bosons correspond to , while for fermions , and for ant-fermions .