Mathematical Physics

Mathematical Physics A overall study to suggest set theory as nature’s logical description

Authors Arno P.L.M. Gorgels Shevkinaz Bulut

Introduction This study proposes to introduce Set Theory, based upon the sets of natural and real numbers, as the most appropriate mathematical tool to describe nature. It is a qualified suggestion meant to give an alternative logic possibility in the universal search for a unified theory. The authors believe that the axioms of Zermelo-Fraenkel[1] (1907-1930), forming the standard form of axiomatic Set Theory (including Cantor's theory of infinities[2] (1877)), allow for the most accurate description of the universe; natural laws and nature's constants should be derivable from it. Set Theory allows real, virtual and imaginary (concrete and abstract) elements of nature to be arranged individually as well as in groups. It assumes the existence of any elements to appear within a mathematical environment that is called "the set of those elements", in certain cases sets with (actual) infinite numbers in which case Cantors mathematics of infinities applies. Unlike the fully symmetrical Lie-Algebras (around 1870, theory of continued symmetry, [3]) and the KacMoody-Algebras, [4] and [5], Set Theory (when applied to physics) is asymmetrical with regards to the structure of its basic field (basic set). This basic set is assumed to be existing as to be the quantized vacuum that has its origin in an zero information-structure with an infinitely extended inner structure. Dynamic expansion of this zero-point is postulated to result in a full-blown vacuum in accordance with the development of numbers as proposed by Giuseppe Peano up to a point where actual infinity is reached and the vacuum system becomes instable. This vacuum following Set Theory of natural numbers allows to describe the early cosmic phase of space inflation (assumed t=0-4 sec., inflation starting at an initial speed of c²), in a way which eventually results in a stable residual basic field (the vacuum with gravitational characteristics), described by the set of natural numbers. At instability beyond its cardinality the splitting off of photons at residual speed of c (c = the speed of light) is suggested to be seen. The photons as EM-waves are suggested to be caused due to instability at the vacuum field's borders whilst exceeding actual infinity of the vacuum-field. This realm over and above the set of natural numbers is known as Cantor’s Dilemma because Cantor was unable to figure out the relation between the cardinalities of the set of natural numbers and the set of rational numbers. This extended field is mathematically described by the set of rational numbers (in fact: minus the set of natural numbers). The set of natural numbers is known to correspond to the gravitational field (Isaac Newton (1642-1727), Wheeler (1911-2008)). Applying methods of the theory of large numbers in combination with Peter Plichta’s theory of prime numbers1 and its consequences in Fourier series and corresponding waves this process can be proven to result in Maxwell Equations, i.e. the occurrence of electric-magnetic waves, i.e. photons. Two photons split under certain circumstances in one

1

Peter Plichta Das Primzahlkreuz, Bd.1, Im Labyrinth des Endlichen: BD I

1

electron and one positron which eventually leads – under further complex considerations that aren’t discussed within this course – to the forming of atoms and molecules. Vacuum quanta are undetectable. They are proposed as virtual directionless one-dimensional-volume-quanta of the format Xn = (X.sin(ωt + nφ) + j.X.cos(ωt + nφ)*(X.sin(ωt + nφ) - j.X.cos(ωt + nφ)) with n=1,2,3 … actual ∞

(1)

X = non-directional length in m; ω = frequency (random); φ = angular phase shift Thus, these virtual elements, numbered n = 1,2,3… are lined up like a sphere surrounding a fixed starting point according to an increasing n finally forming a structured but continuous vacuum as a vibration-background in which all natural events happen. The structure resembles a classical spin network (as proposed by Roger Penrose) but seems to make non-commuting processes possible as measured by quantum mechanics. In line with the established expression superstrings the numbered volume-quanta can be called: superquanta or spatial quanta. Spatial quanta factually correspond to three-dimensional strings. Their surface can mathematically be made visible in the Lorentz equations of Albert Einstein’s Special Relativity as follows: x² + y² + z² + j².c².t² = F

(2)

for which we can write: x² + y² + z² + j².c².t² = F.(sin²(ωt + nφ)-j²cos²(ωt + nφ))

(3)

whereas sin²(ωt + nφ) - j²cos²(ωt + nφ) = sin²(ωt + nφ) + cos²(ωt + nφ) = 1. F in m² is the basic invariable introduced by Max Born7 (1920). (It should be noted that the expression of the right member in (3) can (without F) be written as: (sin(ωt + nφ)+jcos(ωt + nφ))*(sin(ωt + nφ)-jcos(ωt + nφ)) or ej(ωt + nφ) *e-j(ωt + nφ), i.e. the product of two undamped vibrations. Each volume-point in vacuum appears to possess this feature.) Through formula (3) Special Relativity (SRT), volume-quanta (Quantum Mechanics) and strings (onedimensional vibrations in multidimensional space) can thus be seen to be interrelated at vacuum level. Each point of the vacuum is defined accodingly. The resulting well-ordering is considered to be the cause of gravity. This leads to a quantum loop theory with well-ordered virtual elements. The field asymmetry mentioned above should explain the phenomenon of dark matter (the extra gravitational force that is being observed in addition to the gravitational force caused by visible matter, which pulls stars to the centre of the galaxy they belong to). At material level, however, complete symmetry prevails. This complies with Set Theory and is also in line with current exclusively symmetrical theories. This study gives a number of examples of observations that can only be explained through the basic asymmetry mentioned here-above. Those observations include, at the macroscopic level, the concept of dark matter, as well as the concept of a galaxy being a closed space-time unit in a steady state universe with intergalactic space equal to 0. At the microscopic level, they include the allocation of positrons within the atom nucleus. Related to this is the redefinition of protons, neutrons and quarks as being measured results of subatomic structures that, according to Set Theory, can be compositions of positrons surrounded by newly proposed particles, called eons and peons, that carry -1/3 e and +2/3 e electrical charge, as well as electrons orbiting in subatomic paths in respectively 4*c and 2*c space-environments, for which c is the speed of light. The article thus provides arguments to accept Set Theory as the most fundamental mathematical description of nature. The related physical theory is proposed as Superquantum Theory (SQT). For a complete picture, please refer to the diagram “Generalized structure of mathematical physics”.

2

1) Methods of investigation. In 1959, Louise Volders demonstrated that spiral galaxy M33 does not spin as expected according to Keplerian dynamics, a result which was extended to many other spiral galaxies during the seventies. This extra mass is proposed by astronomers to be dark matter within the galactic halo. The phenomenon of Dark Matter first observed by Fritz Zwicky[6] in 1933, in addition to the almost classical problem of relating the theories of gravity (the General Relativity Theory[7] of Albert Einstein, 1912) and quantum mechanics (Heisenberg, 1929), seems to urge a new approach in theoretical physics. To do this, the field of gravity is, as a first step, to be understood as the basic set (a field of numbers with an originating point (midpoint of the galaxy) and, as such, asymmetrical) which unfolds into several power sets that form the natural forces. 2) Matter composed of atoms

Only two centuries ago scientists believed the atom to be the smallest unit of matter. A short time later it was found that the atom is composed of a nucleus surrounded by orbiting electrons. The present experiments with colliding hadrons (LHC) are an attempt to find further subatomic particles. Quantum physics assumes subatomic particles to be elementary. The Standard Model describes this world of elementary particles. Atoms are made up of a dense nucleus containing protons and neutrons. The nucleus is surrounded by one or more electrons. Electrons are lightweight, negatively charged particles. The nucleus is made up of positively charged particles called protons and neutral neutrons. Protons and neutrons are believed to be made up of even smaller

3

particles called quarks. In this paper, a slightly different composition is proposed, taking two new elementary particles called eons and peons into account. 3) The standard model and gravity

The standard model (SM) has mainly been developed in the years 1961-1973. It has extensively been investigated and tested (Glashow 1961, [8]; Weinberg/Salam 1967, [9]), and thus appears to perfectly describe the building blocks of the atomic world and the interactions of at least three of the four known natural forces. However, there are a number of open questions that it can not answer. There is, first of all, the question of a quantum related to gravity: so far gravity is excluded from the SM – the reason being that gravitational field elements called gravitons have never been observed. A hypothetical particle, a graviton called the Higgs Boson, has been proposed by Peter Higgs (1964, [10]) to cause that gravitational interaction that gives mass to particles. Secondly, we mentioned above the problem of Dark Matter: this substance which is necessary to explain fast star movements at the edge of galaxies can neither be explained using the particles of the Standard Model nor by the General Relativity Theory. The present understanding of nature describes four natural forces: gravitation, electromagnetism, the weak interaction and the strong interaction. Each of these forces is mediated by a fundamental particle (=quantum) known as a force carrying particle. Three of the four forces are unified through the Standard Model of particle physics. SM describes the universe in terms of matter and force. If the Higgs Boson experimentally could be found SM would gain considerably in terms of reliability and acceptance. However, SQT does not predict a Higgs Boson.

4

SQT argues that the genuine theoretical basis of nature is given by the mathematics of sets. The universal and simple structure of set theory that, at the same time, allows for the most complicated constructions, is understood as being able to formalize almost all mathematical concepts, [11]. Set theory is therefore generally taken to be the backbone structure of mathematics[12]. SQT proposes that it may also constitute the mathematical description of nature. Like the string theory, SQT assumes virtual particles for which no experiment can be designed to allow direct observation. Moreover those virtual particles are associated solutions of partial differential equations, i.e. the solutions contain a real and an imaginary portion of the form given above in (1). They are grouped as in the infinite set of numbers that leads to Cantor's set theory of infinities, also called Cantor's Continuum. In this paper the authors propose to use the mathematical tools of Set Theory to find solutions to some of the current problems of particle physics. 4) Power set of the gravitational field

5

The set theory was conceived in 1874 by the mathematician Georg Cantor[13]. He especially investigated sets with an infinite number of elements. Cantor's perception was that several number infinities do exist; that was called Cantor's Continuum. He defines actual infinity and uses the power set axiom to build a large numbers of infinities. Aczel (2000) suggests applying Set Theory to theoretical physics. We are convinced that Aczel is correct. This article proposes ways to test or falsify his hypothesis. SQT suggests connecting the four known natural forces by arranging them within the power sets of the basic set of real numbers, already identified as the gravitational field. The following graphic shows how. 5) The correspondence between power set and natural forces Assuming gravity as the basic field of the set of natural fields, it can be considered as a basic set with just one element: gravity. The first power set of this basic set contains, according to set theory, 2 elements that easily should be identifiable as the electric and magnetic field. These three measurable fields (gravity, electric field and magnetic field) can be arranged as elements of a further basic set, being the set of measurable fields. Besides the function of its elements as causing the natural forces of gravity (1), electricity (2) and magnetism (3), they allow for the existence of a further power set, that exists of 2³ elements, schematically made visible as follows:

6

6) Power Set and Quantum Chromodynamics

Gluons are commonly described as combinations of three "colour charges". SQT suggests that the three gluon colours seen in experiments (Weinberg, 1967) should therefore be identified as the equivalent of G, E, and M.

7

7) The atom nucleus

Cantor's continuum allows the calculation of the internal structure of elementary particles. It covers, in principle, the three-dimensional space x, y, z of Galileo, the four-dimensional space time x, y, z, t of Albert Einstein, and the extended six- and more dimensional spaces of Burkhard Heim and his successors. Cantor understood that the continuum is unlimited, however it should be realized that the corresponding physical continuum must be limited because its components and elements may physically be neither infinitely small nor infinitely large even though this would mathematically be thinkable. Cantor failed to resolve the dilemma called after him concerning the question of the cardinalities of the set of natural numbers as compared to the set of irrational numbers. This problem is resolved herein by introducing the observer and the magnitude time that leads to the observation of photons. It has to do with mathematics of the sort of 10000000=10000001=1000002=1000000g as long as g