Along this path whenever possible, dimensionless physical constants and fundamental physical constants will be applied. Dimensional analysis and Buckingham π theorem will be employed. The concepts of nondimensionalization and natural units will be further analyzed until this new system of measurement begins to make sense to others.
Since the advent of dimensional analysis, renormalization, and cutoff regularization, the concept of infinity has been less of a problem for a few scholars. But, it is no secret that each works around the problems of infinity. After awhile, it is easy to stop looking at each result to ask, “What does this tell us about the very nature of infinity? What does it tell us about the four Planck base units?” The deeper inherent assumptions pulls us back into the homogeneity and isotropy of the universe; we would say that it is a result of deeper continuity, symmetry and harmonics that defines infinite qualitatively and the finite quantitatively. It has an active and constant role with all things dimensionless. It part of the definition of light and as such is intimate with space-time and mass-charge. All four values are a reflection of the speed of light.
The Buckingham π theorem is part of our understanding of homogeneity and isotropy; it’s the heart of dimensional analysis. The Buckingham π theorem takes π to the next level along with the pure numbers, i, e, and φ. It is a thrust in the intellectual direction that ratios and equations are real and the things of space and time are derivative.
Using dimensional analysis, a natural length scale emerges by quantizing gravitation with the curvature of space. It may also be true that another way to calculate it is within the entropy for black holes. A formula which is beyond our current comprehension is S = A / (4 L)2 where A is the area of the event horizon.
So, yes, we will continue our studies of those pure numbers, i, e, and φ and de Moivre numbers, as part of our work on the theory of group characters, and the discrete Fourier transform seeking to justify our belief that all these expressions of mathematics are inherent and active at the Planck scale.
The primary focus of this page will become this article, Regularization, Renormalization, and Dimensional Analysis (PDF): Dimensional Regularization…, Fredrick Olness & Randall Scalise Department of Physics, Southern Methodist University, Dallas, TX 75275-0175, (August 22, 2017)