Arithmoí - A Study of Mathematical Cohesiveness in Physics and Metaphysics
Primordial Light Begets Phenomenon…
Spring 2024
Phenomenology and the Geometry of Light is the first book in the Arithmoí Physics series, studying phenomena as a direct consequence of mathematical light.
A unified physics first requires a geometric synthesis. Synthetic Geometry begins with the unification of quaternions and geometric algebra, showing these algebras are the same using Lie representation theory. This synthesis ultimately extends into higher-dimensional representation forms to explain the Standard Model in a cohesive geometric framework, including spacetime and its mathematical creation, and the phenomenon contained therein.
The compartmentalization of physics does not represent cohesive physics. Fundamental to cohesiveness is the proper geometric representation of spacetime and the phenomenon that plays out in it. A description of spacetime that only involves a metric to describe special relativity will not suffice. Gauge theories represent a type of field theory in which invariance is accommodated under local transformations. The term invariance in physics defines that an observable does not change locally under some transformation. Physics uses Lagrangian field theory to analyze the dynamics of a system of specific particles with a finite number of degrees of freedom. So, gauge theories entail that the Lagrangian maintains invariance under local transformations. In particular, gauge field theories the reader should be aware of are as follows:
The abelian gauge theory, with symmetry group U(1), represents the one-gauge field of the electromagnetic four-potential of quantum electrodynamics (QED) and has the photon as its gauge boson.
The non-abelian gauge theory of the Standard Model, with the symmetry group U(1)xSU(2)xSU(3), has twelve total gauge bosons, the photon, three weak bosons, and eight gluons.
Gravitational gauge theory, which differs from the symmetry group usage of the aforementioned gauge theories, and instead uses the Lanczos tensor to describe gravitation in general relativity.
From their particular gauge theories, the symmetry group generators then mathematically describe the forces of electromagnetism, the weak force, and the strong force, which act on vectors to describe the dynamics of their associated fields.
Multi-variable Lissajous Curves mapped to 3D
What is missing from these gauge theories is how the geometry of spacetime comes about, how we tend to perceive space as flat, how we can move in three spatial dimensions by way of two-dimensional rotations, why it is subject to relativity, why the vacuum of space teams with virtual particles coming in and out of existence, why quantization occurs in it, and how spin/angular momentum fits cohesively with it. Physics feels the apex of its science is to unify gravitation with quantum field theory, which has proved daunting due to the difficulty with renormalization with quantum gravity approaches using the graviton as its force carrier.
The difficulty with renormalization occurs with the coupling of mass dimension, where the process of power counting typically helps with divergences encountered with the force carriers. Gravity has a negative-dimensioned coupling due to the gravitational constant, which becomes problematic with renormalization at higher energies.
Achieving a grand unification in physics requires a mathematic and geometric unification first to be elucidated. Physics is a blend of mathematical approaches, including symmetric group representations, quaternions, geometric algebra, octonions, the real sphere, tensors, differential equations, Lagrangians, Hamiltonians, and probability, to name a few. Are all of these mathematics approaches relatable? The answer is no, there is physics founded on beautiful mathematics, and unfortunately, there is the contrary. Essential to know is that light is not just the symmetry group with the photon as its gauge boson, it is much more than this and represents the prerequisites for phenomenal geometry and physics. It is the apex generator that orchestrates the whole, and we will define how and why this is the case.
Empiricism and its experiments are bound to a subspace of the whole, and the physicist who scoffs at that which cannot physically be perceived is fatally flawed in their approach. They may not know it, but those who accept quantum spin and angular momentum already accept non-locality and the “zero-dimensional point.” Why? The intrinsic angular momentum of an electron is non-classical, given it is described as a point particle with a radius equal to zero. Another widely accepted non-local phenomenon is the quantum nature of the vacuum; accept this, and you accept that which cannot be understood by physical experiment.
The inflationary cosmologist Alan Guth once commented about the “universe as the ultimate free lunch,” which alludes to the Universe as being created from nothing. The assertion underlies a non-logical point of view held by many physicists and cosmologists, that being the undue emphasis on only what is seen/detected by the physical senses/apparatus. The law of the Universe is that which extends must also contract in its analog mathematical form by a degree that is inversely related to the said extension, and that which rotates by the dynamics of a specific geometry must rotate in its analog form in inverse relation to the said rotation. This is the net-zero principle, and implicit in the inverse relationship are that mathematical forms and their analogs form products equal to the identity and that net phase sums equal zero in the tangent space.
To accommodate the geometric requirements of the space capable of manifesting all the phenomena that physics is attempting to describe, the inherent mathematics of the space must include spherical/wave-like, hyperbolic, flat-light specific, and discrete geometries. Further, geometric unification must include the non-local geometry and the dynamics behind its interaction with the local geometry. Immediate consequences of the mathematical unification are geometric explanations for the matter/antimatter discrepancy, intrinsic spin, spacetime causality, and the rationale for invariance and conservation. More importantly, mathematical forms exist for which physical reality is predicated.