Studies on Santilli’s Isonumber Theory
Arun S. Muktibodh
Mohota College of Science, Nagpur, India
To cite this article:
Arun S. Muktibodh. Studies on Santilli’s Isonumber Theory. American Journal of Modern Physics. Special Issue: Issue II: Foundations of Hadronic Mechanics. Vol. 5, No. 2-1, 2016, pp. 17-36. doi: 10.11648/j.ajmp.2016050201.12
Received: August 10, 2015; Accepted: August 11, 2015; Published: May 18, 2016
Abstract: Beginning with studies in the 1980s at the Department of Mathematics of Harvard University, the Italian-American scientist R. M. Santilli discovered new realizations of the abstract axioms of numeric fields with characteristic zero, based on an axiom-preserving generalization of conventional associative product and consequential positive-definite generalization of the multiplicative unit, today known as Santilli isonumbers , and the resulting novel numeric fields are known as Santilli isofields. By remembering that 20th century mathematics was formulated on numeric fields, their generalization into isofields stimulated a corresponding generalization of all of 20th century mathematics and its application to mechanics, today known as Santilli isomatheatics and isomechanics, respectively, which is used for the representation of extended-deformable particles moving within physical media under Hamiltonian as well as contact non-Hamiltoian interactions. Additionally, Santilli discovered a second realization of the abstract axioms of a numeric field, this time with arbitrary (non-singular) negative definite generalized unit and related multiplication, today known as Santilli isodual isonumber  that have stimulated a second covering of 20th century mathematics and mechanics known as Santilli isodual isomathematics and isodual isomechanics. The latter methods are used for the classical as well as operator form of antimatter in full democracy with the study of matter. In this paper, we present a comprehensive study of Santilli's epoch making discoveries of isonumbers and their isoduals along with their application to isomechanics and its isodual for matter and antimatter, respectively.
Keywords: Isonumber, Isodual Number, Isodual-Isonumber, Genonumber
As it is well known, modern mathematics has a strong foundation on number theory, algebraic structures such as groups, rings, algebra, vector spaces and related methods have found vast applications in all quantitative sciences. More general structures like groupoids, semigroups, monoids, quasigroups and loops were also being studied in 20th century, although their applications in quantitative sciences are under development. The detailed consolidated account of these generalized structures is found in Survey of Binary Systems by R.H.Bruck .
While the scientific discoveries and mathematical knowledge were moving hand in hand, towards the end of 20th century there were few mathematically unexplained physical phenomena in Quantum Physics and Quantum Chemistry. These new physical situations could not be faithfully described by the existing mathematical structures and called for more generalized mathematical structures.
It was Enrico Fermi,  beginning of chapter VI, p.111 said "..... there are some doubts as to whether the usual concepts of geometry hold for such small region of space." His inspiring doubts on the exact validity of quantum mechanics for the nuclear structure led to the genesis of the whole new kind of generalized mathematics, called isomathematics and generalized mechanics, called as Hadronic mechanics.
In fact, the prevailing Newtonian and Einsteinian ‘Dynamical systems’ called as ’Exterior Dynamical systems’ which are characterized as ‘local’, ‘linear’ ‘Lagrangian’ and ‘Hamiltonian’ could not accommodate these obscure situations. Thus it was the pressing demand of time to formulate new mathematical theory which could deal with the obscure phenomena and develop a new physical theory. This stupendous task was taken up by the Italian-American theoretical physicist Ruggero Maria Santilli, President of Institute for Basic Research, Palm Harbor, Florida, USA and did the pioneering work by defining axiom-preserving, nonlinear, nonlocal and noncanonical isotopies of conventional mathematical structures, including units, fields, vector spaces, transformation theory, algebras, groups, geometries, Hilbert spaces etc. while at Department of Mathematics of Harvard University in the early 80’s. Prof. Santilli has rightly said;
"There can not be really new physical theories without
really new mathematics, and there can not be really
new mathematics without new numbers".
The founders of analytic mechanics, such as Lagrange, Hamilton  and others classified dynamical systems in to two kinds. First one is the ‘Exterior Dynamical system’ and the second one is the more complex but generalized ‘Interior Dynamical system’.
However, over a period of time the the above distinction was abandoned preventing the identification of limitations of the prevailing mathematical and physical theories. One can easily notice that Lie’s Theory is exactly applicable to the exterior dynamical systems. It was Prof. Santilli who at the Department of Mathematics of Harvard University, for the first time, drew the attention of the scientific community towards the crucial distinction between exterior and interior dynamical systems and presented insufficiencies of prevailing mathematical and physical theories by submitting the so-called axiom-preserving, nonlinear, nonlocal, and noncanonical isotopies of Lie’s theory  under the name Lie Isotopic theory. Further generalization as Lie-admissible theory [6,7] was also achieved by him.
Exterior Dynamical Systems: In this system Point-like particles are moving in a homogeneous and isotropic vacuum with local-differential and potential-canonical equations of motion. These are linear, local, Newtonian Lagrangian and Hamiltonian. Conventional Mathematical structures such as Algebras, Geometries, Analytical Mechanics, Lie Theory can faithfully represent these systems.
Interior Dynamical Systems: In this system we consider extended non-spherical deformable particles moving within non-homogeneous anisotropic physical medium. These are non-linear, non-local, non-Newtonian, non-Lagrangian and non-Hamiltonian. The mathematical structures needed to describe these systems are most general possible which are axiom preserving; non-linear and non-local formulations of current mathematical structures.
During a talk at the conference Differential Geometric Methods in Mathematical Physics held in Clausthal, Germany, in 1980, Ruggero Maria Santilli submitted new numbers based on certain axiom preserving generalization of the multiplication, today known as isotopic numbers or isonumbers in short. This generalization induced the so-called isotopies of the conventional multiplication with consequential generalization of the multiplicative unit, where the Greek word "isotopy" from the Greek word " implied the meaning "same topology" [8,9]. Subsequently, Ruggero Maria Santilli submitted a new conjugation, under the name isoduality which yields an additional class of numbers, today known as isodual isonumbers .
The discovery of isonumbers was made with the specific need of quantitative representation of the transition from Exterior Dynamical Systems to Interior Dynamical System.
It should be quite clear that there can not be new numbers without new fields. This led Santilli to define ’Isofield’ which is the first new algebraic structure defined by him. This concept of ‘Isofield’ further led to a plethora of new isoalgebraic structures and a whole new ‘Isomathematics’ which is a step further in Modern Mathematics. Subsequently, ‘Isomathematics’ has grown in to a huge tree with various branches like ‘Isofunctional Analysis’, ‘Isocalculus’, ‘Isoalgebra’, isocryptography etc.
Prof. Santilli attracted great attention from academic community at Chinese Academy of Sciences during a workshop in China on August 23, 1997. Since then Prof Santilli and his associates in various countries around the world have produced numerous papers, monographs, conference proceedings which cover approximately 10,000 pages of research work.
Today Number theory has advanced as an important branch of axiomatized mathematics with highly sophisticated applications in the Modern world of computer science and information technology. After some advances in 19th century due to Gauss , Abel , Hamilton , Cayley , Galois  and others, major important advances were made during 20th century which included axiomatic formulation, the algebraic number theory .
The classification of all normed algebras with identity, over reals, in view of the previous studies by Hurwitz, Albert , and N.Jacobson  may be expressed in the following important Theorem.
Theorem 1.1. All possible normed algebras with multiplicative unit over the field of real numbers are given by algebras of dimension 1 (real numbers), 2 (complex numbers), 4 (quaternions), and 8 (octonians).
Starting with the brief background of the origin of ’isounit’ and isofield, we present the theory of isonumbers, pseudoisonumbers, "hidden numbers" and their isoduals. Genonumbers, pseudogenonumbers and their isoduals are also of fundamental importance. We will study the isotopies and isodualities of the notions of numbers, fields and normed algebras with unit ref.. In short, in this paper we are going to study the properties of isonumbers and their isoduals .
In his study Santiili has taken into account the four normed algebras over reals as given in the above theorem. The isotopic lifting of these algebras give rise to isotopies of normed algebras with multiplicative unit of dimension 1,2,4 and 8 which includes realization of ’isoreal numbers’, ’isocomplex numbers’, ‘isoquaternions’ and ’isooctonions’. Isodualities of these structures give isodual isonumbers.
The mathematical non-triviality of these structures is evident due to lack of unitary equivalence of isotopic and genotopic theories to conventional ones, non-applicability of trigonometry and some other aspects. On the other hand, the physical non-triviality of these structures emerges from the fact that this theory of isonumbers is at the foundations of Li-isotopic theory used successfully to study nonlinear, nonlocal, and nonhamiltonian dynamical systems. The more general Lie-admissible theory emerges from the more general genonumbers.
In a nutshell, the theory of isonumbers is at the foundation of current studies of nonlinear-nonlocal-nonhamiltonian systems in nuclear, particle and statistical physics, superconductivity and other fields.
1.1. Origin of Isonumbers
The first and foremost algebraic structure defined by Santilli is ‘isofield’. Elements of an isofield are called as ‘isonumbers’. The conversion of unit to the isounit is of paramount importance for further development of ‘Isomathematics’.
The reader should be aware that there are various definitions of "fields" in the mathematical literature , ,  and  with stronger or weaker conditions depending on the given situation. Often "fields" are assumed to be associative under the multiplication.
Definition 1.1 Given a "field" , here defined as a ring with with elements , sum , multiplication , which is commutative and associative under the operation of conventional addition and (generally nonassociative but) alternative under the operation of conventional multiplication and respective units and , "Santilli’s isofields" are rings of elements where are elements of and is a positive- definite matrix generally outside equipped with the same sum of with related additive unit and a new multiplication , under which is the new left and right unit of in which case satisfies all axioms of the original field.
is called the isoelement. In the above definitions we have assumed "fields" to be alternative, i.e.
Thus, "isofields" as per above definition are not in general isoassociative, i.e. they generally violate the isoassociative law of the multiplication, i.e.
but rather satisfy isoalternative laws.
The specific need to generalize the definition of "number" to ‘real numbers’, complex numbers, ‘quaternions’ and ‘octonians’ suggested the above definition. The resulting new numbers are ‘isoreal numbers’, isocomplex numbers, ‘isoquaternions’ and ‘isooctonians’ respectively, where ‘isooctonians’ are alternative but not associative.
The ‘isofields’ are given by elements characterized by one-to-one and invertible maps of the original element equipped with two operations , the conventional addition of and a new multiplication called "isomultiplication" with corresponding conventional additive unit and a generalized multiplicative unit , called "multiplicative isounit" under which all the axioms of the original field are preserved.
Santilli has shown that the transition from exterior dynamical system to interior dynamical system can be effectively represented via the isotopy of conventional multiplication of numbers and from its simple possible associative form in to the isotopic multiplication, or isomultiplication for short, as introduced in .
The lifting of the product of conventional numbers in to the form
denoted by , where is a fixed invertible quantity for all possible called isotopic element.
This isomultiplication then lifts the conventional unit defined by to the multiplicative isounit defined by
Under the condition that preserves all the axioms of the lifting is an isotopy, i.e. the conventional unit and the iso unit (as well as the conventional product and its isotopic form ) have the same basic axioms and coincide at the abstract level by conception. The isounit is so chosen that it follows the axioms of the unit namely; boundedness, smoothness, nowhere degeneracy, hermiticity and positive-definiteness. This ensures that the lifting is an isotopy and conventional unit and the isounit coincide at the abstract level of conception.
Thus, the isonumbers are the generalization of the conventional numbers characterized by the isounit and the isoproduct as defined above.
The liftings , and can be used jointly or individually.
It is important to note that unlike isotopy of multiplication , the lifting of the addition implies general loss of left and right distributive laws. Hence the study of such a lifting is the question of independent mathematical investigation.
The first generalization was introduced by Prof. Santilli when he generalized the real, complex and quaternion numbers ,  based on the lifting of the unit into isounit as defined above. Resulting numbers are called isorealnumbers, isocomplex numbers and isoquaternion numbers.
In fact, this lifting leads to a variety of algebraic structures which are often used in physics. The following flowchart is self explanatory.
Isonumbers Isofields Isospaces Isotransformations Isoalgebras Isogroups Isosymmetries Isorepresentations Isogeometries etc.
The isounit is generally assumed to be outside the original field with all the possible compatible conditions imposed on it. For rudiments of isomathematics reader can refer to [1, 6, 7, 25].
The lifting of unit to isounit may be represented as,. where is time, is the position vector, is the momentum vector, is the wave function and are the corresponding partial differentials. The positive definiteness of the isounit is assured by, where is called the isotopic element, a positive definite quantity. The isonumbers are generated as, , .
Isofields are of two types, isofield of first kind; wherein the isounit does not belong to the original field, and isofield of second kind; wherein the isounit belongs to the original field. The elements of the isofield are called as isonumbers. This leads to number of new terms and parallel developments of conventional mathematics.
2. Isounits and Their Isoduals
As stated earlier, the isonumbers and their product can first be introduced as the generalization of conventional numbers by equations (1) and (2) as above.
called the isodual isounit following lifting of iso multiplication defined in (1) into the isodual multiplication called isoduality as
The isodual isonumbers were first conceived as characterized by isodual multiplication (4) with respect to the multiplicative isodual isounit .
• In the exterior dynamical system ordinary particles moving in the vacuum are characterized by conventional numbers.
• In the interior dynamical system ordinary particles moving in the physical medium are characterized by isonumbers.
• In the exterior dynamical system ordinary antiparticles moving in vacuum are characterized by isodual numbers.
In the interior dynamical system the antiparticles moving in the physical medium are characterized by isodual isonumbers.
Interpretation of customary characterization of antiparticles via negative-energy solutions of Dirac’s equations behave in an un-physical way when interpreted with respect to the same numbers and unit of particles, forcing various hypothetical assumptions and postulates, where as, reinterpretation of antiparticles with same negative energy solutions when interpreted as belonging to the field of isodual numbers behave in a fully physical way ref . This treatment of antiparticles with isodual numbers also leads to intriguing geometrical implications which predict another universe, called as isodual universe, interconnected to our universe via isoduality and identified by the isodualities of Riemannian geometry and their isoduals refs.[31,24,32]. Thus, the isodual theory emerged from the identification of negative units in the antiparticle component of the conventional Dirac equation and the reconstruction of the theory with respect to this new negative unit. Hence isoduality provides a mere reinterpretation of Dirac’s original notion of antiparticle leaving all numerical predictions electro-weak interactions essentially unchanged.
In view of the definition of an isofield , we can say that an isofield is an additive abelian group equipped with a new unit (called isounit) and isomultiplication defined appropriately so that the resulting structure becomes a field.
If the original field is alternative then the isofield also satisfies weaker isoalternative laws as follows. and
We mention two important propositions by Santilli.
Proposition 2.1. The necessary and sufficient condition for the lifting (where the multiplication is lifted but elements not the elements) to be an isotopy (that is for to verify all axioms of the original field F) is that is a non-null element of the original field .
Proposition 2.2. The lifting (where both the multiplication and the elements are lifted)
constitutes an isotopy even when the multiplicative isounit is not an element of the original field.
The above proposition guarantees the physically fundamental capability of generating Plank’s unit of quantum mechanics into an integro-differential operator for quantitative treatment of nonlocal interactions .
As the first application of the isotopies of numbers Santilli considers the set , the set of all purely imaginary numbers. This set is not closed (). On the other hand, the same set represented as with constitutes an isofield. i.e. it verifies all the axioms of a field including closure under isomultiplication because and .
This illustrates an important fact that, even when a given set does not constitute a field, there may exist an isotopy under which it verifies the axioms of a field.
As stated earlier the lifting of to does not necessarily produce an isotopy of a given field. This lifting does not preserve the distributivity in the resulting set as stated in the following proposition 2.3.
Proposition 2.3 The lifting where
where is an element of the original field and is an arbitrary invertible quantity, is not an isotopy for all nontrivial values of the quantity , because it preserves all the axioms of proposition 2.1 except the distributive law.
Based on the failure of distributivity Santilli defines "pseudoisofields" as follows.
Definition 2.1 Let be an isofield as defined above. Then the "pseudoisofields" are given by the images of under all possible liftings of the addition , with additive isounit in which case the elements are called the "pseudoisonumbers".
Images of field, isofield and pseudoisofield under the change of sign of the isounit is called the Isotopic conjugation or isoduality ref. [28, 29, 30].
Definition 2.2 Let be a field as per definition 1.1. Then the isodual field is constituted by the elements called "isodual numbers"
defined with respect to the "isodual multiplication" and related "isodual unit"
Definition 2.3 Let be an isofield as per definition 1.1. Then the isodual isofield is constituted by the elements called "isodual isonumbers"
where is the conventional conjugation of (e.g. complex conjugation) defined in terms of the " isodual isomultiplication"
Definition 2.4 Let be a pseudofield as per definition 2.1. Then the "isodual pseudofield" is given by the image of the original isofield under isodualities (6) and (7) plus the additional isoduality
and its elements are called "isodual pseudonumbers.
2.1. Classes of Isofields
Kadeisvilli  classified isounits into five primary classes according to their usefulness.
· CLASS I: Isounits:- These are the isounits when they are sufficiently smooth, bounded, nowhere singular, Hermitian and positive-definite. This class is of primary use in physics for characterization of ordinary particles moving in interior physical conditions. This class represents the isotopy of the conventional unit.
· CLASS II: Isodual Isounits:- They are same as isounits except that they are negative-definite. Isodual isounits are used in physics to characterize antiparticles via reinterpretation of the negative energy solutions of Dirac’s equation [31,36]. They represent isodual isotopy according to isodual conjugation.
· CLASS III: Singular Isounits:- These occur when isounits are considered as a divergent limit, . These are used to represent gravitational collapse into a singularity and other limit conditions ref.[37,23].
· CLASS IV: Indefinite Isounits :- This class represents isounits which are sufficiently smooth, bounded, nowhere singular, Hermitian and can smoothly interconnect positive definite with negative definite values. These are particularly used in mathematics.
· CLASS V: General Isounits, when they are solely Hermitian:- This is the most general class which includes preceding ones and permits a large variety of additional realizations including those in terms of discrete structures, discontinous functions, distributions etc.
Isofields can be classified according to the isounits as defined above. They are;
2. Isodual isofields.
3. Singular isofields.
4. Indefinite isofields.
5. General isofields.
The following four fundamental numbers are generated depending upon the isofield we consider;
1. (a) Ordinary numbers: real numbers , complex numbers , quaternions and octonians which are used in the characterization of particles in vacuum.
(b) Isonumbers: isoreal numbers , isocomplex numbers , isoquaternions and isooctonians which are used for the characterization of particles within the physical media.
(c) Isodual numbers: isodual real numbers , isodual complex numbers , isodual quaternions and isodual octonians which are used in the characterization of antiparticles in vacuum.
(d) Isodual isonumbers: isodual isoreal numbers,isodual isocomplex numbers ,isodual isoquaternions and isodual isooctonians which are used for the characterization of particles within the physical media.
2. Genofield is the generalization of isofield with the selection of an ordering of the multiplication to the left or to the right and applied for the more general Lie-admissible branch of hadronic mechanics.
3. Pseudofields, and
4. Pseudogenofields are the further generalization based on lifting of addition which relaxes at least one axiom of conventional fields, and which do have applications in other fields.
5. Hyper numbers can be constructed from hyperstructures ref..
Let be a metric (or pseudo metric) n-dimensional space with local coordinates and (Hermitean) metric over the field of reals . Then the isospace first introduced in  is characterized by;
Also the isodual isospace  is given by;
Note that isospaces coincide with spaces at the abstract level of conception. Spaces have the most general known curvature and integral character owing to the arbitrariness in the isotopic element . The isometries have the most general possible, nonlinear, nonlocal, noncanonical dependence in all variables,
The isospaces which are most important for physical and mathematical applications are isoeuclidean spaces , isominkowski spaces and isoriemanian spaces . These are the foundations of the representation of nonlinear, nonlocal, and noncanonical interior systems in nonrelativistic and gravitational interior problems [31, 23].
Also, pseudoisospaces can be defined as the images of the original space characterized by further lifting . Subsequently, isodual pseudoisospaces are also defined.
The concept of isoalgebra was fundamental in the correct description of interior dynamical systems. As conventional numbers constitute normed algebras with unit, isoalgebras were defined to represent isonumbers ref. [21, 8, 39]. An isovector space with elements and isomultiplication over an isofield with elements and isomultiplication with multiplicative isounit is called (associative or nonassociative) isoalgebra when it satisfies right and left scaler and distributive laws;
for all the elements and
Note that the isoalgebra may contain the matrices where as the iso field can contain ordinary numbers.
The isoalgebra is an isodivision algebra if the equation always admits a solution in , for nonzero .
Isonorm can be defined in the following manner;
Let be an "isobasis" of over the isofield . Then the generic element can be written as , with and . The isonorm of in the isobasis considered, is then given by;
The isoalgebra is said to be isoassociative if;
The isoalgebra is said to be Lie-isotopic when the isoproduct satisfies Lie-algebra axioms (anticommutativity and Jacobi laws) in the following form;
It is said to be lie-admissible when the antisymmetric bracket product is;
and is Lie-isotopic as in the realization;
We shall be mainly interested in the isoassociative isonormed algebras with isounit which can be extended to isoalternative algebras in order to include isooctonians.
Extension of and under the pseudofield implies loss of distributive laws and hence do not remain algebras in the real sense, however, we call them pseudoisoalgebras ref..
2.5. Isoreal Numbers and Their Isoduals
2.5.1. Real Numbers
Real numbers constitute a one-dimensional normed associative and commutative algebra ref..
Real numbers are realized ref. as a one-dimensional real Euclidean space which represents a straight line with origin at , local coordinates , metric , additive unit and multiplicative unit . Another characterization of real numbers is defined by the isomorphism of the reals into the commutative one-dimensional multiplicative group of dilations defined by;
The basis is given by
with the norm defined by
2.5.2. Isodual Real Numbers
Isodual Real numbers constitute a one-dimensional isodual associative and commutative normed algebra which is anti-isomorphic to ref..
Isodual real numbers are the conventional numbers defined with respect to the isodual unit . The isodual conjugation of real numbers is then written as
Note that, such a sign inversion occurs when the isodual real numbers are projected in the field of conventional real numbers. As a result, all the numerical values change sign under isoduality.
The one-dimensional real isodual Euclidean space is a straight line, with conventional additive unit , and isodual multiplicative unit . The represents the Euclidean space . Also, the isodual dilations are defined by
This establishes an isomorphism between and the isodual group of dilations (the conventional group reformulated according to the multiplicative unit ). Santilli points out that and are antiisomorphic and the same property holds for and . Also, the isodual dilations coincide with dilations as defined above. Santilli further says that "this could be the a reason for the lack of detection of isodual numbers until then." ref.s [38, 27, 28].
In the isodual case, the isodual basis is given by
with isodual norm
satisfying the axioms
2.5.3. Isoreal Numbers
Isoreal numbers constitute a one-dimensional, isonormed isoassociative and isocommutative isoalgebra ref..
Isoreal numbers are the numbers of an isofield of Class I, with isomultiplication defined by and isounit , generally outside the original field . These can be represented as the isoeuclidean spaces with , over the isotopes of conventional one-dimensional Euclidean spaces .
Some of the important remarks are as follows.
· The conventional Euclidean space and its isotopic covering are locally isomorphic due to the joint liftings and .
· is not a Riemannian space because of the intrinsic dependence of the isometric on the derivatives as well as the fact that the basic unit is not the conventional quantity .
· However, is a simple, yet bona-fide isoriemannian space , because where the local dependence is generally nonlinear, nonlocal and noncanonical in all variables.
In fact, the one-dimensional isospace represents a one-dimensional generalization of conventional straight line, called as isoline. This is because of its intrinsically nonlinear, nonlocal and noncanonical metric with multiplicative isounit . Then can be realized via isodilations on as;
which is isodual dilation and represents one-dimensional isogroup of isodilations same as the group realized with respect to isounit .
Again, the isobasis is given by
with isonorm defined as;
which is the conventional norm only rescaled to the new unit . We then also have
2.5.4. Isodual Isoreal Numbers
The isodual isoreal numbers are the realization of the one-dimensional isodual, isonormed, isoassociative and isocommutative isoalgebra ref..
These are the isodual numbers
in the isodual isofield .These correspond to the isoeuclidean space of Class II of dimension one with isodual isodilations
coinciding with dilations (24). This also characterizes an isomorphism isodual isoreal numbers with the one-dimensional isodual isogroup . The underlying isomorphism
implies the .
The isodual isobasis is defined by
The isodual isonorm
verifies the axioms
2.6. Isocomplex Numbers and Their Isoduals
2.6.1. Complex Numbers
Complex numbers constitute a two-dimensional, normed associative and commutative algebra ref..
Complex numbers where and are real numbers and is an imaginary unit, are represented in a Gauss plane which is a realization of two-dimensional Euclidean space satisfying
whose group of isometries is one dimensional Lie Group , the invariance of the circle. Hence, complex numbers can be represented via fundamental representation of as follows.
A one-to-one correspondence between complex numbers and points in the Gauss plane can be obtained by following dilative rotations
which preserve all the properties of a field.
Representation of a complex number via matrices has the following form
which are well known as the identity and fundamental representation of .
Norm can also be defined as
Also, the identification of basis in terms of matrices is and .
2.6.2. Isodual Complex Numbers
Isodual complex numbers constitute a two-dimensional isodual, normed, associative and commutative algebra anti-isomorphic to ref..
Isodual complex numbers are given by
where is the complex conjugation. Thus, given a complex number , its isodual is given by
Considering the group of isometries, the one-dimensional isodual Lie group i.e. the image of under the lifting of the two-dimensional isodual Euclidean space
with basic invariant
isodual complex numbers can be characterized by the isorepresentation of .
Now, the image of the conventional plane under isoduality is the isodual Gauss plane. Also, a one-to-one correspondence between the points and complex numbers can be defined by isodual dilative rotations as
following the multiplication rules
which preserve all the properties of a field.
Isodual transformations form an isodual group antiisomorphic to . Even the one-to-one correspondence between complex numbers and Gauss plane continues under isoduality.
Matrix representation of isodual complex numbers can be defined as
with the isodual unit and isodual representations of respectively.
The isodual norm can be defined as which may be written as
and verifies the axioms
The isodual basis in terms of matrices is given by
2.6.3. Isocomplex Numbers
Isocomplex numbers constitute a two-dimensional, isonormed, isoassociative and isocommutative isoalgebras over the isoreals ref..
In this case we consider the isofield of isocomplex numbers
withe generic element . Here we need the two-dimensional isoeuclidean space of class I, . The most important realization used in the physical literature has the diagonalized and positive-definite isotopic element and isounit
with basic isoseparation
The group of isometries of this space is the Lie group , the group constructed with respect to the multiplicative isounit which provides the invariance of all possible ellipses with semiaxes as the infinitely possible deformation of the circle . Thus, isocomplex numbers are characterizable via fundamental representation of .
Isocomplex numbers can also be characterized to be the set of points on the isogauss plane on .
In fact, a one-to one correspondence between isocomplex numbers and the points on the isogauss plane can be defined via following isodilative isorotations
characterized by the isomultiplication defined as
Isocomplex numbers also admit following two-by-two matrix representation.
which characterize the isounit and the fundamental (adjoint)representation of respectively.
The set of matrices (63) is closed under addition and isomultiplication. Also, each element possesses the isoinverse
where is the ordinary inverse. As a result, is an isofield with the local isomorphism . We note that the one-to-one correspondence between complex numbers and Gauss plane is preserved under isotopy. It is important know that the realization of complex numbers as matrices is not unique.
The isonorm is defined as
which readily verifies the axiom
The isobasis is given by
2.6.4. Isodual Isocomplex Numbers
The isodual isocomplex numbers constitute a two-dimensional, isodual, isonormed, isoassociative and isocommutative isoalgebras over the isodual isoreals ref..
Now the isodual isocomplex numbers are defined as
with generic element Here we need a two-dimensional isodual isoeuclidean space with the realization
with basic isoseparation
whose group of isosymmetries is the isodual isoorthogonal group
The isodual isogauss plane is defined as the set of points on which characterize the isocomplex numbers .
The correspondence between the isodual isocomplex numbers and the isodual gauss plane can be made one-to-one by the isodual isodilative isorotations
having rule for multiplication as
Isodual isoguass planes characterizes isodual isofield. Also the isodual isotransformations forms an isodual isogroup .
Isodual isocomplex numbers also admit the following two-by-two matrix representation.
This satisfies isomultiplication rule (74) characterizing the isodual isounit and fundamental representation of
The set of matrices representing isodual complex numbers , is closed under addition and isomultiplication. Each element possesses the isodual isoinverse
As a result we get a local isomorphism .
Now, the isodual isonorm can be defined as
The isodual isobasis is given by
2.7. Isoquaternions and Their Isoduals
Quaternions constitute a normed, associative, non-commutative algebra of dimension 4 over reals ref..
Quaternions admit a realization in the complex Hermitean plane with separation
with basic (unimodular) invariant . Hence quaternions have a fundamental representation by Pauli’s matrices.
Quaternions can be realized as the pairs of complex numbers, , and with multiplication . Hermitean dilative rotation on which leaves invariant is given by
where the dilation is represented by These transformations form a group . This group is associative but noncommutative resulting into a one-to-one correspondence with quaternions.
Quaternions can be represented via matrices over the field of complex numbers as
The matrix admits the representation
where are the Pauli’s matrices
with fundamental relations
where is the tensor of rank three. The norm of the quaternion can be defined as
The basis is defined by
2.7.2. Isodual Quaternions
Isodual quaternions constitute an isodual four-dimensional, normed associative and noncommutative algebra over the isodual reals which is anti-isomorphic to ref..
Isodual quaternions can be represented via the isodual Hermitean Euclidean space
Isodual complex numbers can also be realized via pairs of isodual complex numbers as .
Also, the isodual Hermitean dilative rotation on leaving invariant is given by
where the dilation is represented by the value .
These transformations form an associative but noncommutative isodual group which is in one-to-one correspondence with isodual quaternions .
As a result there is a matrix representation of isodual complex numbers over the field of isodual complex numbers as
under the condition
We can represent as
where i’s are the Pauli’s matrices. Note that Pauli’s matrices change sign under isoduality although their product with isodual numbers is isoselfdual.
Isodual norm is then defined as
The isodual basis is defined as
Isoquaternions constitute a four-dimensional, isonormed, isoassociative, non-isocommutative isoalgebra over the isoreals , ref..
Isoquatrnions can be represented using two-dimensional, complex Hermitean isoeuclidean space of class I, on the isofield with real separation given by
with basic isotopic element and isounit
The (unimodular) invariance group of this space is the Lie-isotopic group . Isoquaternions can also be characterized by fundamental representation of algebra. A Hermitean isodilative isorotation on is given by
where the dilation is represented by the value . Representation of isoquaternions into two-by-two matrices on is characterized by the isorepresentations of the Lie-isotopic algebra ref. [40,41,42]. These can be expressed in terms of the basic isounit
and fundamental representation of as
Note that the matrices above satisfy the properties of isotopic image
and hence are closed under commutators, which is a necessary condition for the existence of an isotopy. This results into a Lie-isotopic algebra
Isoquaternions can be represented in the form
Note that the set is a four dimensional vector space over the isoreals which is closed under the operation of conventional addition and isomultiplication and hence, is an isofield. Thus, .
The isonorm of the isoquaternions is defined as follows
and may be written as
The isobasis is defined as
2.7.4. Isodual Isoquaternions
The isodual isoquaternions constitute a four-dimensional, isodual, isonormed, isoassociative, non-isocommutative isoalgebra over the isodual isoreals ref. .
The isodual isoquaternions by a two-dimensional isodual complex Hermitean isoeuclidean space of class II over the isodual isocomplex field as
having basic isodual isotopic element and isodual isounit
having invariance as the isodual Lie-isotopic group . An isodual Hermitean isodilative isorotation on is given by
where dilation is represented by .
Isodual Isoquaternions can also be realized as the isodual isorepresentation of and can be written as
Note that the set of all the matrices is an isofield and hence .
The isodual isonorm is defined as
The isodual isobasis is defined as
2.8. Isooctonians and Their Isoduals
Octonians constitute and eight-dimensional normed, non-associative and non-commutative , alternative algebra over the field of reals .ref.[20, 21].
Octonians can be realized as two-dimensional quaternions with multiplication rules
The antiautomorphic conjugation of an octonian is defined as
The norm of an octonian is defined as
with the basic axioms
It is important to note that Octonions do not constitute a realization of the abstract axioms of a numeric field and, therefore, they do not constitute numbers as conventionally known in mathematics due to the non-associative character of their multiplication (see ref. ).
2.8.2. Isodual Octonians
The isodual octonians constitute an eight-dimensional isodual, normed, non-associative, and non-commutative algebra over the isodual real numbers ref. .
Isodual octonians are defined as
over the isodual reals . The isodual multiplication of isodual octonians is defined by
The isodual antiautomorphic conjugation of an octonian is defined as
The isodual norm of an octonian is defined as
with the basic axioms
2.8.4. Isodual Isooctonians
Isodual isooctonians form an eight-dimensional isodual, isonormed, non-isoassociative, non-isocommutative, but isoalternative isoalgebra over the isodual isofield , ref. .
Isodual isooctonians can be defined as the pair of isoquaternions over the isodual isoreals with the multiplication rule
The isodual isoantiautomorphism is defined as
The isodual isonorm is defined as
which readily verifies
Again it is important to note that Isodual isooctonians do not constitute a realization of the abstract axioms of a numeric field and, therefore, they do not constitute numbers as conventionally known in mathematics due to the non-associative character of their multiplication (see Ref. ).
3. Grand Unification of Numeric Fields
Isotopic generalization has brought about a grand unification of the conventional numbers into one single, abstract notion of isonumber. It is important to note that the unification of all numbers was conjectured by Prof. Santilli in numerous publications through out his research for many years. Finally it was proved by Kadeisville, Kamiya and Santilli ref.. The following theorem is the main result in this regard.
Theorem 3.1. Let be the fields of real numbers, complex numbers and quaternions, respectively, the isodual fields, the isofields, and the isodual isofields as defined in the preceding section. Then all these fields can be constructed with the same methods for the construction of from , under the relaxation of the condition of positive-definiteness of the isounit, thus achieving a unification of all the fields, isofields and their isoduals into the single, abstract isofield of Class III, denoted by R.
3.1. Hidden Numbers of Dimension 3, 5, 6, 7
Based on the historical problem ‘The four and eight square problem and division algebras’ ref., Prof. Santilli conjectured the possibility of ‘Hidden numbers’ of dimension 3, 5, 6 and 7’. The numbers studied by Santilli, namely, reals, complex, quaternions and octonians are the solution of the following problem.
where all the a’s, b’s and c’s are elements of a field with conventional operations and . It is well known that the only possible solutions of the problem are of dimension 1, 2, 4 and 8. These facts are in corporated in the theorem 1.1, restated here
Theorem 3.2 All possible normed algebras with multiplicative unit over the field of real numbers are given by algebras of dimension 1 (real numbers), 2 (complex numbers), 4 (quaternions), and 8 (octonians).
The question posed by Santilli: Is ‘Does the classification according to above theorem persist under isotopies, pseudoisotopies and their isodualities ?’ or ‘Is it incomplete ?’ First, we investigate this problem for isotopies of the multiplication. The above problem, equation (135) is reformulated under the isotopies of the multiplication as follows.
The isotopic lifting of the multiplication
transforms the problem (135) in to
where all the a’s, b’s and c’s are elements of an isofield in which is an element of the original field, can be simplified to the conventional operations as
Comparing the original problem and its isotopic conversion as formulated above, we observe that the reformulation of the problem is same as the original problem and hence the isotopic lifting and isoduality of the field does not change the solution of the problem. As the result we get the following theorem.
Theorem 3.3. All possible isonormed isoalgebras with multiplicative isounit over the field of the isoreals are the isoalgebras of dimension 1 (isoreals), 2 (isocomplex), 4 (isoquaternions), and 8 (isooctonians) and the classification persists under isoduality.
Further, lifting of addition gives the third formulation which is pseudoisotopic type
under which (137), (138) can be written over the pseudoisofield as
This can be written in the conventional operations as
The solution to (144) of dimension other than 1,2,4,8 under the pseudoisofield was envisaged by prof.Santilli as a conjecture under the loss of the needed axioms of a field, such as distributive laws.
It was found that the solution do exist, but under the loss of number of axioms of the original field, in addition to the loss of distributivity. We consider a representative example of "Hidden numbers" of dimension 3 as follows
Note that also the condition on is true, that the elements in the r.h.s can be written as the combinations of the elements on the l.h.s as
Hence we can rewrite the problem as
which on simplification gives a quadratic equation in as
Thus the solution exists, but is not an integer. This implies the loss of closure under isoaddition for the case of integers. However, the closure can be regained if the original field is enlarged to include all real numbers. The issue whether such solutions do indeed form a pseudoisofield is open for the mathematicians.
As algebras of dimensions higher than 8 are not alternative , also, as this property persists under isotopies and pseudoisotopies, leads to the fact that formulations (137 )and (142) are restricted to dimensions .
Prof. Santilli ref. identified following open problems with regards to the notion of isofields.
· Investigative study of "number with singular unit", i.e. isofields of class IV which are at the foundations of the isotopic studies of gravitational collapse.
· The study of isofields of characteristic , to see whether new fields and therefore new Lie-algebras are permitted by isotopies.
Author of this article has defined ‘Iso-Galois fields’ ref. which are basically finite isofields essentially of nonzero characteristic. As predicted by Santilli these isofields have important applications in Cryptography, Genetics, Fractal geometry etc.
· The study of the integro-differential topology characterized by isofields with local differential structure and integral isounits.
3.2. Genonumbers and Their Isoduals
We have seen that the two degrees of freedom due to isotopic lifting of addition and multiplication give rise to isofields and pseudoisofields respectively. These fields are at the foundation of the Lie-isotopic theory [8, 9, 45].
Also, there exists a third degree of freedom caused by the ordering of the above operations which leads to further generalization of a field which is at the foundation of Lie-admissible algebras [8, 9, 18].
Given a field of ordinary numbers with generic elements , with addition and multiplication , we can define the following.
Genoaddition: Addition of to from the left, denoted by and addition of to from the right denoted by are called genoadditions.
Genomultiplication: Multiplication of times from the left denoted by , and multiplication times from the right denoted by are called genomultiplications.
It is worthwhile to note that ordering of multiplication is fully compatible with its basic axioms, such as commutativity for real and complex numbers, associativity for quaternions, and alternativity for the octonions. In the case of real and complex numbers we will have
The identity of multiplication from left and right can be different and hence two genomultiplications can very well be different i.e.
where R and S are fixed isotopic elements, called the genotopic elements. These are sufficiently smooth, bounded and nowhere singular (not necessarily Hermitean) outside the original field.
The left and right generalized genounits can be defined in the following manner
Note that all the axioms and properties of the original field are preserved under the mentioned left or right multiplication and multiplicative units under the appropriate ordering for all the dimensions 1,2,4,8. This procedure leads to new fields called as genofield denoted by (right genofield) or (left genofield) or . Also, isodual genofields are defined by the antiautomorphic conjugations
denoted by .
Note that isofields are the particular case of genofields where the genotopic elements coincide. i.e.
R-S mutation of the Lie product: is defined as
which is Lie-admissible via the attached antisymmetric product
which is Lie-isotopic.
The lifting is called an isotopy. The lifting is called a genotopy, ref. [8, 1].
The Lie-isotopic algebras are defined by one single isotopy of the enveloping associative algebra and related unit
For the consistent formulation of Lie-isotopic algebras they must be defined over an isofield with isounit .
Note that for the conventional multiplication there is no ordering as . The above ordering can be defined for isomultiplication wherein we can have different isounits.
The Lie-admissible algebras can be generated by two different isotopies of the original associative algebra using left and right isounits with corresponding isotopies as
which must be defined over the genofields with isounits . Here, the isounits related with the left and right isomultiplication are disjoint and can indeed be Hermitean and real-valued, which admit Kadeisville classification into classes I. II, III, IV and V.
However, in physics the isounits (left and right) used have a real physical significance when they are inter-related by a Hermitean conjugation as
This representation of the genounits (and hence genofields) provides approximation of irreversibility ref..
It is important to note that conventional addition admits no meaningful ordering as . However, the ordering exists for the isoaddition as with . But there is loss of distributive law for the resulting genofield under genoadditions .
All the above discussion leads to a broadest generalization of the existing theory of numbers through
1. pseudogenofields defined via genotopies of all aspects of conventional fields and
2. isodual pseudogenofields defined via isoduality of pseudogenofields.
This new generalization of the conventional numbers leads to the following categorization of numbers:
· Conventional numbers of dimension 1,2,4,8 and their isoduals;
· Isonumbers of the same dimension and their isoduals;
· Genonumbers of the same dimensions and their isoduals;
· Pseudoisonumbers of the same dimension and their isoduals;
· Pseudogenonumbers of the same dimension and their isoduals;
· "Hidden pseudoisonumbers" of dimension 3,4, 5,7 and their osoduals;
· "Hidden pseudogenonumbers" of dimension 3,4,5,7 and their isoduals.
Note that each of these can be defined for the fields of characteristic 0 or for .
In addition to above generalization, we can have an ordered set of values for the multiplicative unit such as defined as applicable or to the right or left. This possibility leads to the new numbers called as hyper-Santillian numbers. These include hyper-real, hyper complex, hyper-quaternion numbers which have vast applications in biological sciences.
In the further generalization, the multiplicative unit can very well have non-zero negative values. This leads to a new class of numbers called iso-dual Santillian numbers. This further leads to a new kinds of conventional iso-dual numbers called as iso-topic isodual numbers, geno-topic iso-dual numbers and hyper-structural isodual. These numbers have applications for antimatter.
The above generalization of the conventional numbers gives us, in all, eleven classes of new numbers namely, the iso-topic numbers, genotopic to the right and left, right and left hyper-structural numbers, iso-dual conventional numbers, iso-dual iso-topic numbers, iso-dual geno-topic to the right and left numbers and hyper-structural iso-dual to the right and left numbers. Each class is applicable to the real, complex and quaternion numbers where each of the applications have infinite number of possible units.
4. Applications and Advances
Quantum mechanics was sufficient to deal with ’Exterior Dynamical systems’ which are liner, local, lagrangian and hamiltonian. The main purpose of formulating the new generalized mathematics was to deal with the insufficiencies in the modern mathematics to describe ’Interior Dynamical systems’ which are intrinsically non-linear, non-local, non-hamiltonian and non-lagrangian. The axiom-preserving generalization of quantum mechanics which can also deal with non-linear, non-local non-hamiltonian and non-lagrangian systems is called the Hadronic mechanics. The mechanics; built specifically to deal with ’hadrons’ (strongly interacting particles) ref. . Prof. Santilli, in 1978 when at Harvard University, proposed ’Hadronic mechanics’ under the support from U. S. Department of Energy, which was subsequently studied by number of mathematicians, theoreticians and experimentalists. Hadronic mechanics is directly universal; that is, capable of representing all possible nonlinear, nonlocal, nonhamiltonian, continuous or discrete, inhomogeneous and anisotropic systems (universality), directly in the frame of the experimenter (direct universality). In particular the hadronic mechanics has shown that quantum mechanics is completely inapplicable to the synthesis of neutron , as mass of the neutron is greater than the sum of the masses of proton and electron (called "mass defect") of which it is made. In this case quantum equations are completely inconsistent. Hadronic mechanics has achieved numerically exact results in the cases in which quantum mechanics results are not valid. For further details of isonumber theory we recommend refs. [47,1,48,46,49].
As far as mathematics is concerned, one of the major applications of isonumber theory is in Cryptography, ref. . Cryptograms can be lifted to iso-cryptograms which render highest security for a given crypto-system. Isonumbers, hypernumbers and their pseudo-formulations can be used effectively for the tightest security via new disciplines, isocryptology, genocryptology, hypercryptology, pseudocryptology etc. More complex cryptograms can be achieved using pseudocryptograms in which we have the additional hidden selection of addition and multiplication to the left and those to the right whose results are generally different among themselves. Yet more complex pseudocryptograms can be achieved in which the result of each individual operations of addition and multiplication is given by a set of numbers . Santillian iso-crypto systems have maximum security due to a large variety of isounits which can be changed automatically and continuously, achieving maximum possible security needed for the modern age banking and other systems related with information technology.
Reformulations of conventional numbers to the most generalized isonumbers and subsequently to genonumbers and hypernumbers led to a vast variety of parallel developments in the conventional mathematics including hyperstructures  and its various branches such as ’iso-functional analysis’ ref , iso-calculus ref , iso-cryptography  etc.
Iso-Galois fields , Iso-permutation groups [54,53] have been defined by this author, which can play an important role in cryptography and other branches of mathematics where finite fields are used. Investigations are underway.
Isomathematics can also explain complex biological structures and hence has applications in Fractal geometry. Further applications in Neuroscience and Genetics can provide new insight in these disciplines.