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European Journal of Applied Sciences – Vol. 12, No. 4

Publication Date: August 25, 2024

DOI:10.14738/aivp.124.17404.

Giannini, J. (2024). Fractal Representation of Composite Elementary Fermions with Positive and Negative Mass Components.

European Journal of Applied Sciences, Vol - 12(4). 293-303.

Services for Science and Education – United Kingdom

Fractal Representation of Composite Elementary Fermions with

Positive and Negative Mass Components

Judith Giannini

Independent Researcher/USA

ABSTRACT

The Fractal Rings and Composite Elementary Particles (FRACEP) Model provides

an alternate view to the elementary fermion picture. FRACEP represents a dual

universe where there are equal amounts of positive and negative matter (mass).

Its dual universe construction allows for composite particles of mixed (positive

and negative) mass with fractal-based components. The fractal dimension of these

composite particles is D = ~1.55, and is consistent with other estimates of D for the

elementary fermions and other fractal-based composite models. Initial efforts

developed an empirically-based two-parameter fit for the mass hierarchy formula,

relating the fractal dimension to the composite particle masses. The formula

predicts the cross-family (first-generation particles) masses, and the intra-family

masses (the three generations within each family). The mass predictions for the

structure-based composite particles, as well as, the mass hierarchy are consistent

with the Particle Data Group’s 2016/2020-update estimates for the elementary

fermions.

Keywords: Composite Elementary Particles, Fractals, Negative Mass, Negative Matter,

Preons, Mass Heirarchy, Dual Universe.

INTRODUCTION

Currently in particle physics, the elementary fermions are generally considered to have no

internal components. However, the spontaneous decay of most of them suggests the

possibility that they are composite in nature [1]. The Fractal Rings and Composite Elementary

Particles (FRACEP) Model, is an empirically-based effort to describe a possible construction

for the elementary fremions.

The idea of compositeness in the elementary fermions began taking shape, in the 1970s, with

a quantum-based preon theory describing possible particle substructure. Most of the early

models contained a basic set of two to four preons with an equal number of anti-preons in the

first-generation particles, lacked intrinsic mass, and required Higgs or some other mechanism

to provide the mass. The models often considered preons as having no internal structure [2-

3].

Harari [2] hypothesizes the higher generation fermions are excited states of the first- generation configurations. Shupe [3] explicitly notes that the constituent preons and anti- preons have masses that represent self-energy of the photon and gluon fields coupling to the

leptons and quarks, and that solitons have some of the desired properties for representing

fermionic preon-anti-preon pairs. Numerous models address the possibility of compositeness

in the particles [4-10]. Some predict particle masses considering simple preons as the

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European Journal of Applied Sciences (EJAS) Vol. 12, Issue 4, August-2024

substructure [11-12], while others like Salam [13] address composite preons or preons with a

fractal nature [14-16].

The general assumption is that all levels of the preons are positive mass, with simple, non- complex structures. The FRACEP model takes a broader view, considering them as coherent

composite quantities of matter with positive or negative mass that is associated with a Unified

Potential Field [17] defining the force they exert at any point in space.

Jammer indicates that the concept of matter and mass is an elusive one that’s meaning has

evolved over time [18]. From a classical point-of-view, mass is the amount of matter in a body,

and it is attractive with a force that is inversely proportional to the square of the separation

distance (expressed as the negative gradient of the potential). The general assumption is that

all matter and anti-matter needs to be positive [19], [20] (only one kind of polarity), but

Jammer notes (in Chapter 10) that Bondi pointed out that this assumption is purely empirical

indicating the idea of negative mass is not inconsistent with theory.

Quantum mechanics treats particles as having a dual nature with wave-particle duality.

Jammer notes (in Chapter 14) that it can be viewed as an analogous extension of the classical

picture in comparing the dynamical behavior of wave packets with that of Newtonian

particles. If one assigns “density” to the wave function (as a distribution over an infinite wave

front), integration over three dimensions would yield the mass of the particle’s wave packet.

The (assumed) inherent properties of the particles are assigned to be consistent with

empirical data, but Osmera [21] notes they are not generally related to any physical structure

or internal motions of the particles. This also generally assumes positive mass, though

negative energy states are not excluded, but Chen and Chou [22] studied the potential

usefulness of the negative mass concept in nonrelativistic quantum mechanics.

Quantum field theory is a generalization of quantum mechanics, viewing particles as a

disturbance in the field, and mediating particle exchanged between any two particles

produces a force that is recognized as associated with a particle’s mass in the classical picture

[23]. This also generally assumes positive mass because chiral symmetry considerations allow

the negative sign of the mass to be mathematically rotated away.

Dismissing the idea of the existence of negative mass is tempting because it is unsettling to

readily accept something you cannot see. However, negative mass has been the subject of

considerable theoretical study since the 1800s as discussed in Jammer. Among the more

recent works, Bondi [24] developed a non-singular solution to Einstein’s equations showing a

repulsive force between bodies with mass densities of opposite sign. Bonner [25] considered

mechanics in a universe with negative mass and the influence of the negative mass on the

Schwarzschild black hole solution. Hoyle, Burbidge and Narlikai [26] noted that allowing only

positive mass results in the standard hot Big Bang model, and they developed a scale- invariant form of the gravity equations that reduce to general relativity under the right

conditions, leading to the creation of equal numbers of positive-mass and negative-mass

particles (pairs) during creation events.

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Giannini, J. (2024). Fractal Representation of Composite Elementary Fermions with Positive and Negative Mass Components. European Journal of

Applied Sciences, Vol - 12(4). 293-303.

URL: http://dx.doi.org/10.14738/aivp.124.17404

Most recently, Chang [27] explored the force of negative matter and its potential relation to

Dark Matter, using single-metric field equations describing repulsive interactions, and he

studied the relevance of fractal geometry in composite preon characterization of the fermions

[28]. In another approach, Petite and his colleagues [29] developed a bimetric model for a

dual universe describing the interaction of positive and negative masses where the two types

of matter have different light speed (that is, both positive-energy photons and negative- energy photons). Further, they studied the concept of negative masses in standard relativistic

quantum mechanics (the Dirac equation), showing that negative energies are acceptable

provided the masses are simultaneously negative [30].

The FRACEP model has similarities and differences from the traditional preon-based models.

Like the traditional models, it has preon-type components, referred to here as “complex- constituents” (CCs), that make up the fermions. (Because it is not formalized as a quantum- based model, there are not, at this time, quantum numbers associated with the CCs). Unlike

traditional models, the CCs can have complex structures containing both positive and negative

masses because they are built from two fundamental particles with inherent mass resulting

from total symmetry at their pair-creation (Gp with positive mass, and Gn with negative mass)

[31]. (This differs from the standard picture that assumes there is no inherent mass in the

fundamental particles.)

The fractal structures of the FRACEP particles have two levels of complexity in their CCs. The

carrier CCs are the lowest level of complexity, and are the basis of all the other structures. The

radical CCs have a higher level of complexity, and contain carrier CCs and other radical CCs.

This paper describes the fractal structure of the composite fermions, and considers a possible

empirically-based two-parameter fit for its mass hierarchy formula.

THE FRACEP MODEL AND THE BASIC CARRIER CCS

The FRACEP model [31], with its Unified Potential [17], represents a dual universe containing

equal amounts of both positive and negative mass (or perhaps more properly positive and

negative matter associated with a force that is attractive or repulsive). This dual universe

concept allows for particles of purely positive mass, purely negative mass, and mixed-mass

having both positive and negative mass elements. The Bright Universe (BU) is the universe we

see. The Dark Universe (DU) is the universe we cannot see. We hypothesize it contributes to

the cosmological dark matter and energy that compose most of our universe. The basic

elements in this construction are the carrier CCs (Table 1).

Table 1: The basic set of carrier CCs. The p-suffix indicates positive mass; the n-suffix

indicates negative mass. The B designation indicates Bright Universe particles; the D

designation indicates Dark Universe particles.

Universe Type Momentum Carriers Charge Carriers Spin Carriers Mass Type

Bright Universe

MGXp QBp SBp All positive

MRXp

QBn SBn Mixed

Dark Universe

QDp SDp

MGXn QDn SDn All negative

MRXn