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Journal of Advances in Applied Mathematics
JAAM > Volume 6, Number 2, April 2021

On the Mandelbrot Set for i^2 = ±1 and Imaginary Higgs Fields

Download PDF  (2618.3 KB)PP. 26-53,  Pub. Date:February 21, 2021
DOI: 10.22606/jaam.2021.62001

Author(s)
Jonathan Blackledge
Affiliation(s)
Stokes Professor, Science Foundation Ireland; Distinguished Professor, Centre for Advanced Studies, Warsaw University of Technology, Poland; Visiting Professor, Faculty of Arts, Science and Technology, Wrexham Glyndwr University of Wales, UK; Professor Extraordinaire, Faculty of Natural Sciences, University of Western Cape, South Africa; Honorary Professor, School of Electrical and Electronic Engineering, Technological University Dublin, Ireland; Honorary Professor, School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, South Africa.
Abstract
We consider the consequence of breaking with a fundamental result in complex analysis by letting i^2 = ±1 where i = √−1 is the basic unit of all imaginary numbers. An analysis of the Mandelbrot set for this case shows that a demarcation between a Fractal and a Euclidean object is possible based on i^2 = −1 and i^2 = +1, respectively. Further, we consider the transient behaviour associated with the two cases to produce a range of non-standard sets in which a Fractal geometric structure is transformed into a Euclidean object. In the case of the Mandelbrot set, the Euclidean object is a square whose properties are investigate. Coupled with the associated Julia sets and other complex plane mappings, this approach provides the potential to generate a wide range of new semi-fractal structures which are visually interesting and may be of artistic merit. In this context, we present a mathematical paradox which explores the idea that i^2 = ±1. This is based on coupling a well known result of the Riemann zeta function (i.e. (0) = −1/2) with the Grandi’s series, both being examples of Ramanujan sums. We then explore the significance of this result in regard to an interpretation of the fundamental field equations of Quantum Mechanics when a Higgs field is taken to be produced by an imaginary mass im such that (±im)^2 = +m^2. A set of new field equations are derived and studied. This includes an evaluation of the propagators (the free space Green’s functions) which exhibit decay characteristics over very short (sub-atomic) distances.
Keywords
non-standard Mandelbrot set, transient characteristics, imaginary mass, causal tachyons, Higgs fields
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