An Introduction to The Lie-Santilli Isotopic Theory

Abstract

Lie'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.

Isotopic Liftings of Clifford Algebras and Applications in Elementary Particle Mass Matrices

“…The isotopic lifting of such structures allows the physical theories to be described in a straightforward canonical, unitary and Lagrangian formalism [2][3][4][5][6][7][8][9][10][11][12][13], by maps from Lagrangian, linear and local theories to more general ones, involving a non-linear, nonlocal and non-Lagrangian character. These later are led to the former when formulated in an isospace, endowed with a new product in the context of the Clifford algebras, with respect to which the unit is now a fixed, but arbitrary, element ζ of the Clifford algebra.…”

“…The fields C and ζ C are shown to be isomorphic [2]. Note that given an operator A ∈ A, the isoproduct between isoscalars and such operator is given by a A = aζ ζ −1 A = aA.…”

“…It is well known that the Lie algebra so(p, q) spin(p, q) is isomorphic to the algebra (Λ 2 (R p,q ), [ , ])-where [ , ]) denotes the commutator-when Spin(n,0) Spin(0,n) and Spin + (n − 1,1) Spin + (1,n − 1), n > 4 [27,28]. Then besides the product given by (1), there is defined the isocommutator [2,3,10,11,13,17] [ , ] ζ defined by…”

The novel mathematics underlying hadronic mechanics for matter and antimatter

“…Tsagas and D. Sourlas; Refs. [14,15] for a study of Santilli isoalgebars and isogeometris by the Georgian mathematician J. Kadeisvili; Ref. [15] for studies on Santilli's isotopies by the Spanish mathematicians R. M. Falcon Ganfornina and Juan Nunez Valdes; Ref.…”

Comments on the Regular and Irregular IsoRepresentations of the Lie-Santilli IsoAlgebras

“…In a series of pioneering works [1][2][3][4][5][6][7][8][9][10][11], R. M. Santilli has constructed a new mathematics, today known as Santilli IsoMathematics, for the representation of extended, non-spherical and deformable particles under Hamiltonian as well as non-Hamiltonian interactions, which new mathematics has seen contributions by numerous important mathematicians (see, e.g. Rfs., [12][13][14][15][16][17][18][19][20][21]).…”