An Introduction to The Lie–Santilli Isotopic Theory

Abstract

Communicated by B. BrosowskiLie's theory in its current formulation is linear, local and canonical. As such, it is not applicable to a growing number of non-linear, non-local and non-canonical systems which have recently emerged in particle physics, superconductivity, astrophysics and other fields. In this paper, which is written by a physicist for mathematicians, we review and develop a generalization of Lie's theory proposed by the Italian-American physicist R. M. Santilli back in 1978 when at the Department of Mathematics of Harvard University and today called Lie-Santilli isotheory. The latter theory is based on the so-called isotopies which are non-linear, non-local and non-canonical maps of any given linear, local and canonical theory capable of reconstructing linearity, locality and canonicity in certain generalized spaces and fields. The emerging Lie-Santilli isotheory is remarkable because it preserves the abstract axioms of Lie's theory while being applicable to non-linear, non-local and non-canonical systems. After reviewing the foundations of the Lie-Santilli isoalgebras and isogroups, and introducing seemingly novel advances in their interconnections, we show that the Lie-Santilli isotheory provides the invariance of all infinitely possible (well-behaved), non-linear, non-local and non-canonical deformations of conventional Euclidean, Minkowskian or Riemannian invariants. We also show that the non-linear, non-local and non-canonical symmetry transformations of deformed invariants are easily computable from the linear, local and canonical symmetry transforms of the original invariants and the given deformation. We then briefly indicate a number of applications of the isotheory in various fields. Numerous rather fundamental and intriguing, open mathematical and physical problems are indicated during the course of our analysis.

Representation of Galactic Dynamics Via Isoshifts Without Universe Expansion, Dark Matter and Dark Energy

“…112, thus experiencing a isoredshift in full agreement with measurements [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53]. [60].…”

“…This constitutes visual evidence, confirmed by systematic measurements [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52], on the lack of the expansion of the universe as well as on the lack of its acceleration.…”

“…for a general independent review of mathematical, physical and chemical studies; and Ref. [58] provides detailed evidence for the dismissal of interpretations of measurements [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53] in such a way to comply with the 20th century realization of special relativity.…”

“…In view of these contributions, the loss of energy by light to the inter-galactic medium is now referred to as the Zwicky-Santilli effect [ [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53].…”

“…Systematic experimental evidence has been provided by Santilli [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53] on the fact that the redness of the Sun at the horizon is indeed a redshift of the entire visible spectrum of light without any appreciable relative motion. Consequently, said redness is visual evidence of the lack of expansion of the universe (Figs.…”