Quantum gravity, Clifford algebras, fuzzy set theory and the fundamental constants of nature

Abstract

In a recent paper entitled “Quantum gravity from descriptive set theory”, published in Chaos, Solitons & Fractals, we considered following the P-adic quantum theory, the possibility of abandoning the Archimedean axiom and introducing a fundamental physical limitation on the smallest length in quantum spacetime. Proceeding that way we arrived at the conclusion that maximising the Hawking–Bekenstein informational content of spacetime makes the existence of a transfinite geometry for physical “spacetime” plausible or even inevitable. Subsequently we introduced a mathematical description of a transfinite, non-Archimedean geometry using descriptive set theory where a similar conclusion regarding the transfiniteness of quantum spacetime may be drawn from the existence of the Unruh temperature. In particular we introduced a straight forward logarithmic gauge transformation linking, as far as we are aware for the first time, classical gravity with the electroweak via a version of informational entropy. That way we found using ε (∞) and complexity theory that α ̄ G =(2) α ̄ ew −1 =1.7×10 38 where α ̄ G is the dimensionless Newton gravity constant and α ̄ ew =128 is the fine structure constant at the electroweak unification scale. The present work is concerned with more or less the same category of fundamental questions pertinent to quantum gravity. However we switch our mathematical apparatus to a combination of Clifford algebras and set theory. In doing that, the central and vital role of the work of D. Finkelstein becomes much more tangible and clearer than in most of our previous works.