Studies of the Regular and Irregular Isorepresentations of the Lie-Santilli Isotheory

As it is well known, the Lie theory is solely applicable to dynamical systems consisting of point-like particles moving in vacuum under linear and Hamiltonian interactions (systems known as exterior dynamical systems). One of the authors (R.M. Santilli) has proposed an axiom-preserving broadening of the Lie theory, known as the Lie-Santilli isotheory, that is applicable to dynamical; systems of extended, non-spherical and deformable particles moving within a physical medium under Hamiltonian as well as non-linear and non-Hamiltonian interactions (broader systems known as interior dynamical systems). In this paper, we study apparently for the first time regular and irregular isorepresentations of Lie-Santilli isoalgebras occurring when the structure quantities are constants or functions, respectively. A number of applications to particle and nuclear physics are indicated. It should be indicated that this paper is specifically devoted to the study of isorepresentations under the assumption of a knowledge of the Lie-Santilli isotheory, as well as of the isotopies of the various branches of 20th century applied mathematics, collectively known as isomathematics, which is crucial for the consistent formulation and elaboration of isotheories.


Introduction
Despite historical achievements in atomic physics [1], nuclear physics [2] and other fields, a number of authoritative scientists expressed doubts on the final character of quantum mechanics for the representation of all possible conditions existing in the universe.
For instance, Einstein, Podolsky and Rosen [3] expressed doubts on the "completeness" of quantum mechanics. Fermi [4] states in regard to the structure of strongly interacting particles "..... there are some doubts as to whether the usual concepts of geometry hold for such small region of space." Blatt and Weisskopf [2] have indicated then possibility that the deviations of the predictions of quantum mechanics from experimental data on nuclear magnetic moments are due to deformations of the charge distributions of protons and neutrons when under strong nuclear interactions.
One of us (Santilli) has conducted systematic studies on the above historical legacies over a number of decades [5][6][7][8] and papers quoted therein suggesting that the indicated limitations originate from the mathematics underlying quantum mechanics, rather than on its basic axioms.
More particularly, the limitations originated from the localdifferential character of 20th century mathematics because such a calculus is solely applicable to dynamical systems of point-like particles moving in vacuum under Hamiltonian interactions, which systems are known as exterior dynamical systems.
The resulting mathematics has been established as being valid for mutual distances of particles much bigger than their wavepackets and/or charge distributions, as it is the case for the atomic structure, particles in accelerators, the structural crystallography, and numerous other systems.
However, the same point-like approximation results to be insufficient for extended, non-spherical and deformable particles moving within a physical medium under potential-Hamiltonian as well as contact non-Hamiltonian interactions (Broader systems known as interior dynamical systems) as it is the case for the structure of hadrons, nuclei and stars.
The above assumptions immediately allowed an explicit representation of protons and neutrons as being extended, nonspherical and deformable charge distributions via realizations of the isotopic element of the type: where: 2 , = 1, 2,3 k n k represent the semi-axes of the particle considered (assumed to be a spheroid ellipsoid for simplicity) normalized to the values n K =1 for the vacuum; 2 4 n represents the density of the particle considered also normalized to the value n 4 =1 for the vacuum; and Γ represents non-linear, non-local and non-Hamiltonian interactions due to mutual penetrations of particles (interactions known as variationally non-selfadjoint) [5].
Subsequently, Santilli realized that an isomathematics solely based on isoproduct (1) would have other shortcomings because the emerging time evolution would be non-unitary, with ensuing lack of preservation of Hermiticity (and observability) over time, violation of causality, etc. [8].
In order to resolve this impasse, memoir [6] of 1993, Santilli introduced the isotopies ˆ( , , ) F n I × of conventional numeric fields F(n,×,I) characterized by isoproduct (1), generalized multiplicative unit called isounit,1 and generalized numbers, called isoreal, isocomplex and isoquaternionic isonumbers, and related isoaxiom: Despite the reformulation of all preceding works from their original formulation on conventional fields to new formulations on isofields, the emerging mathematics was still unable to achieve the crucial invariance possessed by 20th century theories, namely, the prediction of the same numerical values under the same conditions at different times.
Following a number of attempts, Santilli achieved the resolution of the latter impasse via the following isotopies of the ordinary differential calculus, today known as Santilli isodifferential,ˆˆˆ= with corresponding isoderivative [6],ˆ( It essentially emerged that the isotopic generalization of the product, Eq. (1), and its intrinsic representation of extended particles, Eq. (3), were incompatible with the local-differential character of the Newton-Leibnitz differential calculus that, consequently, to be lifted into a form eqs (6) and (7) admitting extended particles in their very structure.
Following the achievement of mathematical maturity in memoir [7], monographs [8] presented isotopic images of various branches of 20th century mathematics and their application to the isotopies of quantum mechanics known as hadronic mechanics, while monographs [9] presented applications and experimental, verifications in quantitative sciences.
It should be noted that this paper is devoted to one single application of isomathematics, the initiation of isorepresentations of the Lie-Santilli isoalgebras. Therefore, a technical understanding of this paper requires a knowledge of the Lie-Santilli isotheory at large, as well as of isomathematics since the latter is essential for the consistent formulation and elaboration of isotheories.
Besides the above rudimentary outline of the most salient aspects of isomathematics, and an equally rudimentary outline of the Lie-Santilli isotheory in the next section, the authors regret being unable to review the various branches of isomathematics to prevent a prohibitive as well as non-necessary length of the paper, since the foundations of isomathematics are available in the original proposals [5][6][7][8][9] as well as in numerous mathematical treatments [10][11][12][13][14][15].
The reader not aware of this rather large literature should be alerted that the mixing of mathematics and isomathematics generally leads to insidious inconsistencies. Fir instance, the formulation of the Lie-Santilli isotheory over conventional numeric fields, even though seemingly acceptable on mathematical grounds, carries exactly the same-inconsistencies of the formulation of the conventional Lie theory over isofields, both formulations implying the violation of causality in physical applications.
Hence, the correct formulation and elaboration of the Lie-Santilli isotheory requires the isotopies of the totality of mathematical notions used for the Lie theory with no exception known to the authors.

Rudiments of the Lie-Santilli Isotheory
Santilli's primary motivation for the isotopes of 20th century mathematics was its application to the isotopies of the various branches of Lie's theory that were first presented in ref. [5] on conventional spaces over conventional field and elaborated via the conventional differential calculus. Said isotopic theory was then reformulated [7][8][9] on isospaces over isofield and elaborated via the isofunctional analysis and the isodifferential calculus, in which form it is nowadays called the Lie-Santilli isotheory.
The main elements of the Lie-santilli isotheory can be outlined as follows. Let E=E(L) be the universal enveloping associative algebra of an N-dimensional Lie algebra L with ordered (Hermitean) generators X k , k=1,2,…,N, and attached antisymmetric algebra isomorphic to the Lie algebra, [E(L)] − ≈L over a field F (of characteristic zero), and let the infinite-dimensional basis I, X k , X i × X j , i≤j,… of E(L) be characterized by the Poincaré-Birkhoff-Witt theorem. We then have the following.
The first application of the above theorem, also formulated [4] and then reexamined by various authors, is a rigorous characterization of the isoexponentiation, i.e.,î where we continue to use the notation of Paper I according to which quantities with a "hat" are formulated on isospaces over isofields and those without are formulated on conventional spaces over conventional fields.
The non-triviality of the Lie-Santilli isotheory is illustrated by the emergence of the nonlinear, nonlocal and noncanonical or nonunitary isotopic element T directly in the exponent, thus ensuring the desired generalization.
As it is well known, Lie algebras are the antisymmetric algebras L≈[ξ(L)] − attached to the universal enveloping algebras ξ(L). This main characteristic is preserved although enlarged under isotopies as expressed by the following.

Theorem 2 [5b] (Lie-Santilli Second theorem)
The antisymmetric isoalgebras L attached to the isoenveloping algebras ˆ( ) E L verify the isocommutation rules:ˆˆˆˆ[ where the C's, called the "structure isofunctions" of L , generally have an explicit dependence on local variables, and are restricted by the conditions (Lie-Santilli Third Theorem) It was stated in the original proposal [4] that all isoalgebras L are isomorphic to the original algebra L for all positive-definite isotopic elements. In other words, the isotopies cannot characterize any new Lie algebra because all possible Lie algebras are known from Cartan classification. Therefore, Lie-Santilli isoalgebras merely provide new nonlinear, nonlocal and noncanonical or nonunitary realizations of existing Lie algebras.
Under certain integrability and smoothness conditions hereon assumed, Lie algebras L can be "exponentiated" to their corresponding Lie transformation groups G and, vice-versa, Lie transformation groups G admit corresponding Lie algebras L when computed in the neighborhood of the unit I .
These basic properties are preserved under isotopies although broadened to the most general possible, axiom-preserving nonlinear, nonlocal and noncanonical transformation groups according to the following.

Theorem 3 [5b] (Lie-Santilli isogroups)
The isogroup characterized by finite (integrated) form Ĝ of isocommutation rules (1.12) on an isospace ˆ( , ) S x F over an isofield F with common isounit ˆ= 1 / > 0 I T is a group mapping each element with the following isomodular action to the right.
1) The map 2)Î is the left and right unit:ˆˆ= 3) the isomodular action is isoassociative, i.e., 4) in correspondence with every element there is the inverse element 5) the following composition laws are verified:ˆ( with corresponding isomodular action to the left, and general expression:ˆ( Another important property is that conventional group composition laws admit a consistent isotopic lifting, resulting in the following. is the isogroup of all ordered pairs:

I I and inverse
The studies of the isotopies of the remaining aspects of the structure of Lie groups is then consequential. It is hoped the reader can see from the above elements that the entire conventional Lie theory does indeed admit a consistent and nontrivial lifting into the covering Lie-Santilli formulation.
Among a considerable number of mathematical papers on the Lie-Santilli isotheory listed in the Comprehensive Bibliography of Volume [9], we quote in particular the readable review by Kadeisvili [15], an excellent presentation of the all fundamental isotopology by Falcon Ganfornina and Nunez Valdes [11], and the unification of all simple Lie algebras of a given dimension (excluding exceptional algebras) into one single Santilli isotope of the same dimension by Tsagas [16].

Theorem 5 [5b]
The structure functions L of a Lie-isotopic algebra L verify the conditions:= 22) and the property (when commuting with the generators)ˆˆˆˆˆ= 0(23) p q p q p q ij pk jk pi ki pj Isorepresentations of Lie-Santilli isoalgebras is classified into; 1. Regular isorepresentations which occur due to C's of the rules(10) are constants; and,ˆˆ− 2. Irregular Isorepresentation occurring when the C's of the rules (10) are functions of the local variables (an occurrence solely possible for the Lie-Santill isotheory).
Following the construction of the above isotopies, Santilli applied the resulting isotheory to the step-by-step construction of the isotopes of all primary aspects of the Lorentz and Poincaré symmetries for the above identified regular case [17][18][19][20][21][22][23][24] and representative independent contributions [25,26]. A recent general outline of the applications of the Lie-Santilli isotheory to various branches of physics and chemistry is available from memoir [27]. The general rule for mapping Lie algebras into regular Lie-Santilli isoalgebras were identified for the first time by Santilli in [7] and then studied systematically in monographs [8]. They can be written as follows; †=

General Rules for Construction of Regular Isorepresentations General construction
This non-unitary transformation is applied to the entire mathematics of Lie's theory leading to Santilli's isomathematics. We get the following important fundamental transformations; † etc.

Invariance
The extremely important Invariance of the isorepresentation over time requires non-unitary transforms identically rewritten on Hilbert-Myung-Santilli isospaces over isofields.
to eqs (25)-(31) above, causes the lack of invariance, with consequential activation of the catastrophic inconsistency theorems reviewed in ref. [8] such as the change of basic isounit; †ˆˆ= This identical reformulation gives; under which we have the invariance of the isounit and isoproduct as; †ˆˆ= etc.
Namely, the numerical value of the isounit and isotopic element are invariant.ˆ' This is crucial since the isounit or the isotopic element represent non-Hamiltonian interactions.
Santilli constructed the following example of regular iso-Pauli matrices: where ∆=detQ=g 11 g 22 >0 . These representations verify the isotopic rules , and consequently the following isocommutator rules and generalized isoigenvalues for 1 2 ( ) = f ∆ ∆ and, This confirms the 'regular' character of the generalization considered here. The isonormalized isobasis is then given by a trivial extension of the conventional basis In fact, regular iso-Pauli matrices (41) admit the conventional eigenvalues 1 2 and 3 4 for ∆=1 which can be verified by putting It is important to emphasize the condition of isounitarity, i.e. †= UU I I ≠ for which 2 . This degree of freedom has major fundamental implications presented in ref. [20] as well as for the spin component of the first known representation of nuclear magnetic moments presented in the paper [29].

O S on a Santilli Isofield of Characteristic Zero
The regular isounitary isoirreducible isorepresentations of the Lie-Santilli isoalgebra 3 SO on a Santilli isofield of characteristic zero were first studied by Santilli in [17,30]. In this case we have the trivial isorepresentation characterized by U=Diag(n 1 ,n 2 ,n 3 ) with n k >0 . There exist only two compact Lie-isoalgebras: while the other remaining six algebras are noncompact.
The Lie-isotopic commutation rules for the compact algebras are: For the case of isotope we consider the compact isotopic rotation about the third axis. Then the group element: with underlying transformations: which leave invariant the hyperboloids: where the θ's are the Euler's angles; the skew Hermitian generators are given by: leading to universal enveloping algebra ξ with conventional associative product of matrices and unit: ξ: J i J j = associative product (54) The attached Lie algebra is characterized by the familiar commutation rules: while the second order Casimir invariant is given by: Now the isotopic rotation can be readily computed to give: This is the most general possible isotopic lifting of rotations, which includes as particular cases all other forms under consideration. Following illustration are the particular cases.