Elementary Particles in a Finite World Geometry

Abstract

To account for the elementary particles a new physical geometry may be needed. A previous suggestion that this geometry may be finite is followed up by determining the representations of the orthogonal (Lorentzlike) group. Because of the existence of a new type of orthogonality-preserving transformation some of the representations are multiple-valued. A change of value is identified with a gauge transformation and electric charge is recognized as a certain number determining the many-valuedness of the representation. This charge number reverses sign under space inversion. The charged pions and sigma particles are correlated with some of the new representations.

The Lorentz group and its finite field analogs: Local isomorphism and approximation

“…6, then by Shapiro, 15 Ahmavaara, 1,2 Yahia, 17 Joos, 8 Beltrametti and Blasi, 3 and Nambu. In relation to theoretical alternatives in particle physics, Lorentz transformations over finite fields seem to have been first considered by Coish in Ref.…”

Lehmann—Källén-Theorem und Geometrie der Minkowski-Welt

“…COISH schl/igt eine sehr radikale Abiinderung der Geometrie unserer Raum-Zeit-Welt vor [13]: Die m5glichen Werte der Koordinaten der Punkte unserer Raum-Zeit-Welt sollen nicht den K5rper der reellen Zahlen bilden, sondern nur einen endlichen Zahlen-Ring. Durch geeignete Wahl dieses Zahleniinges kann man erreichen, dass diese Geometrie fª sehr kleine und sehr grosse Entfernungen entscheidend ver~indert wird, aber fª mittlere Entfernungen sehr genahert mit der pseudoeuklidischen Geometrie ª In einer solchen Theorie sind natª infinitesimale Translationen nicht mehr wohldefiniert, deshalb entf/illt Voraussetzung 1. des LEHMAr~r~--KXLL• Aber leider ist die Feldtheorie in einer solchen Geometrie noch nicht vollstiindig entwickelt.…”

On Minkowski space and finite geometry

“…Perhaps the first to consider an approximation between structures over the field of real numbers and structures over a very large finite field were astronomers Kustaanheimo [44,45] and Järnefelt [41]. Motivated by some alternative theories in particle physics, Lorentz transformations over finite fields were considered by many authors: see for example Coish [24], Shapiro [67], Ahmavaara [1][2][3][4], Yahya [72], Joos [42], Beltrametti and Blasi [12,13]. Recently, Foldes [29] derived an approximation result between Lorentz transformations over real numbers and Lorentz transformations over a finite field.…”