http://www.santillifoundation.org/docs/RMS.pdf
CHAPTER 2:
HADRONIC MATHEMATICS
2.1. FOREWORD
2.2. DISCOVERY OF NEW NUMBERS 2.2A. Discovery of isonumbers (1983) 2.2B. Discovery of genonumbers (1993) 2.2C. Discovery of hypernumbers (1994) 2.2D. Discovery of isodual numbers (1993)
2.3. DISCOVERY OF ISO, GENO, HYPERDIFFERENTIAL CALCULI, FUNCTIONAL ANALYSIS AND THEIR ISODUALS (1996)
2.4. DISCOVERY OF ISO, GENO, HYPER, SPACES AND THEIR ISODUALS (1983).
2.5. DISCOVERY OF ISO, GENO, HYPERSYMPLECTIC GEOMETRIES AND THEIR ISODUALS (1996)
2.6. ISOTOPIC UNIFICATION OF MINKOWSKIAN, RIEMANNIAN AND FINSLERIAN GEOMETRIES (1998)
2.7. LIEISOTOPIC COVERING OF LIE'S THEORY AND ITS ISODUAL (1978)
2.8. LIEADMISSIBLE COVERING OF THE LIEISOTOPIC THEORY AND ITS ISODUAL (1967)
2.9. INTEGRABILITY CONDITIONS FOR THE EXISTENCE OF A LAGRANGIAN 2.9A. Integrability conditions in Newtonian mechanics (1978). 2.9B. Integrability conditions in field theory (1975)
CHAPTER 2:
SANTILLI'S DISCOVERIES IN MATHEMATICS
2.1. FOREWORD
Santilli has repeatedly stated that: The origin of protracted controversies or unsolved problems in physics, chemistry, biology, and other sciences, is generally due to the use of mathematics basically insufficient for the quantitative treatment of the problem at hand, with consequential need to develop new appropriate mathematics.
Most of the insufficiencies of the 20th century theories identified in the preceding chapter see their origin precisely in the lack of adequate mathematics, such as: the reconstruction of an algebra in the brackets of the time evolution of the analytic equations with external terms, Eq. (1.3), clearly requires the development of a suitable new algebra other than Lie algebra; the classical and operator treatment of nonlinear, nonlocal and nonHamiltonian interactions for extended particles at short mutual distances (figure 1.3) clearly requires a mathematics broader than that effective for conventional Hamiltonian and quantum theories; the insufficiencies of curvature to represent gravitation, combined with the inabilities by general relativity to reach a grand unification and a quantum version of gravity dating back to Einstein's times without solution, are a clear manifestation of the need for a more appropriate geometry for gravitational events; and the same occurs for other problems whose solution is impossible with the mathematics of the 20th century.
Santilli has also stated repeatedly in his writings that: There cannot be a really new theory without a really new mathematics, and there cannot be a really new mathematics without new numbers.
Hence, as a theoretical physicist, he devoted the majority of his time to the search of new numbers, and then to the construction of new mathematics based on them. Discoveries in physics, chemistry, biology, astrophysics and engineering required the minority of his time.
To understand the mathematical discoveries outlined below, one should keep in mind the main problem investigated by Santilli. Recall from Section 1.1 the legacy of Lagrange and Hamilton according to which the representation of nature requires the knowledge of two quantities, a Lagrangian or a Hamiltonian and the external force F(t, r, p, ...). Recall also from Eq. (1.3) that the presence of the external forces causes the loss of all algebras in the brackets of the time evolution of physical quantities, thus preventing the construction of physically meaningful covering theories.
Hence, Santilli set his research goal to identify an identical reformulation of Hamilton's equation (1.2) depending on a second quantity, besides the Hamiltonian, capable of restoring an algebra in the brackets of the time evolution and that algebra had to be a covering of the Lie algebra.
After extensive research and the systematic investigations of all possible alternatives, Santilli finally assumed the representation of Lagrange's and Hamilton's external forces via a generalization of the basic unit into a form explicitly dependent on local variables generally used in physics, hereon denoted I^{*}(t, r, p, E, ...). All other alternatives failed because of their lack of invariance over time, that is, the inability to predict the same numerical values under the same conditions at different times, thus being physically inconsistent.
By comparison, the unit is the most fundamental invariant of all theories, thus being the best solution for the preservation of the same time invariance as that of the truncated analytic equations. However, the generalization of the basic unit requires a corresponding, progressive, and systematic generalization of the totality of the mathematics of the 20th century, and this explains the dimension as well as novelty of Santilli's mathematical discoveries.
In this chapter we outline the rudiments of Santilli mathematics, at times also called hadronic mathematics to indicate the mathematics underlying hadronic mechanics, namely, we shall outline the formulation of numerical fields, vector and metric spaces, geometries, algebras and groups, etc., when characterized by a basic unit I^{*}(t, r, p, E, ...) that, besides being nowhere singular, has otherwise an unrestricted functional dependence on all needed local variables. The application of Santilli mathematics for the resolution of Lagrange's and Hamilton's legacy is outlined in Chapter 3.
Mathematicians should be aware that all mathematical discoveries outlined in this chapter originated from specific physical needs following clear insufficiencies of the preexisting mathematical and physical methods. Mathematicians should also keep in mind that Santilli has been a member of the Department of Mathematics of Harvard University from 1978 to 1983 under DOE financial support, thus having all qualifications for mathematical discoveries even while being a theoretical physicist. Nevertheless, mathematicians should keep in mind that, except a number of papers written in pure mathematics language for mathematicians, numerous mathematical discoveries were presented by Santilli in papers intended for physicists and published in physics journals, the understanding being that their reelaboration in the language of pure mathematics is elementary.
n this mathematical chapter, the conventional associative multiplication "ab" of two generic quantities a, b (such as numbers, matrices, operators, etc.) will be denoted with the symbol "a x b" in order to differentiate it from various new multiplications discovered by Santilli that are still associative, yet more general than the trivial multiplications ab.
A main criticism ventured in academia interested in preserving old mathematics (but not in the industry using new mathematics) is that {\it Santilli's new mathematics is trivial because it boils down to putting a "hat" on symbols without major changes of preexciting axioms.} As we shall; see, the statement is technically correct because Santilli's various different mathematics can indeed be presented by putting "hats" or other indices in preexisting mathematical symbols.
However, on mathematical grounds the statement is dishonest when proffered by expert mathematicians because the implications are far from being trivial. For example:
A) The placement of Santilli's "hat" (meaning isotopy) on the symbols of Lie's theory allows the extension of the applications from linear systems to all possible (well behaved) nonlinear systems, a discovery far from being trivial;
B) The placement of Santilli's "hat" in the symbols of the symplectic geometry renders the same universal for the characterization of all possible (well behaved) nonHamiltonian system directly in the frame of the experimenter and without any sue of the transformation theory, a result with historical mathematical significance (since the symplectic geometry is rendered directly universal) and industrial implications (since for the first time the optimal control theory is applicable to the real nonHamiltonian systems of our real environment);
C) The placement of Santilli's "hat" in the symbols of special relativity extend its its applicability from dynamics in vacuum to dynamics within physical media with implications such as the elimination of "dark matter" and "dark energy", the development of basically new, environmentally clean and cost competitive new fuels and energies (Chapter 9), a discovery of clear historical proportion and implication rather than of trivial character.
In reality, the greatness of Santilli's mathematical discoveries, ignored by equivocal mathematicians under the illusion of personal gains, has been precisely that of achieving a formulation of his various new mathematics that coincides at the abstract realizationfree level with conventional mathematics, thus essentially discovering {\it new realizations} of preexisting abstract mathematical axioms.
2.2. DISCOVERY OF NEW NUMBERS
2.2A. Discovery of isonumbers (1983)
Numbers are at the foundation of all quantitative sciences since, by definition, the latter require mathematical elaborations predicting numbers that can be verified with experiments. For various topological and other technical reasons, experimental measurements requires the adopted "ordinary numbers" (hereon referred to those with characteristic zero and denoted with the letter n) to verify the axioms of a numerical field F(n, x, I) with associative multiplication n x m, (left and right) multiplicative unit I, I x n = n x I = n, addition n + m = p ε F and additive unit 0, 0+n = n+0 = n, ∀ n, m ε F.
The achievement of the modern number theory required
contributions from the best scientific minds in history, including Gauss, Legendre, Jacobi, Cauchy, Lebesgue, Diriclet, Hamilton, Cayley, and many others.
A major historical effort was dedicated to the classification of all possible numbers, that is, all possible sets verifying the axioms of a numerical field. By the middle of the 20th century, it was universally believed in mathematics that the classification of all ordinary numbers (again, those with characteristic zero) had been achieved with the results that all possible ordinary numbers are given by real numbers, complex numbers and quaternionic numbers. Octonions do not qualify as numbers because they violate the associativity of the multiplication (m x n ) x p = m x ( n x p).
As part of his Ph. D. in theoretical physics in the late 1960s at the University of Torino, Italy, Santilli set up his research goal of achieving a generalizationcovering of quantum mechanics for which, to avoid the illusion of a real generalization, he needed numbers more general than those used in quantum mechanics, such as the real and complex numbers.
The difficulty of Santilli's task was that, on one side, very authoritative mathematicians claimed emphatically that all ordinary numbers verifying the axioms of a numerical field had been classified while, on the other side, Santilli needed new numbers verifying indeed said axioms to avoid physical inconsistencies identified later on.
With great scientific audacity, and based on the conviction that mathematics will never admit final formulations, Santilli ignored all authoritative claims and set himself up to review the foundations of number theory. His position at the Department of Mathematics of Harvard University proved to be instrumental, not because of any help by departmental colleagues, but because their skepticisms reinforced his determination.
In this way, Santilli first discovered that the axioms of a numerical field do not require that the multiplicative unit be necessarily the number I = +1 dating back to prebiblical times but used in pure mathematics up to the 20th century, since the left and right multiplicative unit can be an arbitrary positivedefinite quantity I^{*} = 1/T > 0 generally outside the original set F(n, x, I), provided that the multiplication is suitably redefined in the form n x^{*} m = n x T x m, under which I^{*} remains indeed the correct left and right unit, I^{*} x^{*} n = n x^{*} I^{*} = n for all elements of the set.
Santilli then proved that, under the above assumptions for the multiplication and its unit while keeping the conventional addition and its unit, all axioms of a field were verified even when the new unit I^{*} is not an element of the original field F, in which case the new numbers are written n^{*} = n x I^{*}. We reach in this way new numbers and fields for which Santilli suggested the name of isonumbers and isofields. from the Greek meaning of preserving the original axioms. They are known today as Santilli isoreal, isocomplex and isoquaternionic numbers, or generically isonumbers, the new unit I^{*} = 1/T > 0 is called Santilli isounit, its inverse T is called the isotopic element, and the new multiplication a x^{*} b between two generic quantities a, b is called isomultiplication. The new sets F^{*} are called Santilli isofields and are generally written in the form
(2.1) F^{*}(n^{*},x^{*}, I^{*}): I^{*} = 1 / T > 0, n^{*} = n x I^{*}, n^{*} x^{*} m^{*} = (n x I^{*}) x T x (m x I^{*}) = (n x m) x I^{*} .
(2.2) F^{*}(n^{*},x^{*}, I^{*}) ≈ F(n, x, I),
In short, Santilli discovered a new realization of the conventional axioms of a field permitting new physical, chemical and other applications identified in subsequent chapters. When the new unit I^{*} is outside the original set F, F^{*}(n^{*},x^{*}, I^{*}) is called an isofield of the first kind, and the numbers n^{*} are called isonumbers of the first kind. When I^{*} is an element of the original field F, that is, an ordinary number, F^{*}(n^{*},x^{*}, I^{*}) is called an isofield of the second kind, in which case the isonumbers of the second kind are often assumed to be the original numbers n without the multiplication by I^{*}.
Santilli's isofields of the second kind are primarily used in mathematics, particularly to show the insufficiency of contemporary number theory, and its various notions such as that of prime (see the example below). Santilli's isofields of the first kind are of primary use for the new nonHamiltonian classical and operator mechanics and their applications. Therefore, unless otherwise stated, in this book we shall always use Santilli's isofields of the first kind and simply refer them as "isofields."
Even though isofields are isomorphic to conventional fields, as indicated by their very name and Eq. (2.2), their differences are by far nontrivial, and their scientific implications beyond our imagination at this time. For instance, fo the general isofields (of the first kind) we have expressions of the type
(2.3) I^{*} = 3: 2 x^{*} 3 = 18; 4 = prime number.
These results signaled one of the biggest mathematical discoveries of the 20th century because it gave rise to momentous advances in physics, chemistry, biology and other quantitative sciences reviewed in the subsequent chapters.
Quite symptomatically, Santilli published for the first time his isonumbers in his two historical papers of 1985 on the isotopies of Lie's theory, particularly for the structural lifting of the fundamental symmetry of physics, the rotational symmetry that, in turn, is the basis for his lifting of Galilei's and Einstein's relativities:
Lieisotopic liftings of Lie symmetries, I: General considerations
R. M. Santilli,
Hadronic J. Vol. 8, 2535 (1985)
Lieisotopic liftings of Lie symmetries, II: Lifting of rotations
R. M. Santilli,
Hadronic J. Vol. 8, 3651 (1985)
A mathematically rigorous presentation of isonumbers and isofields was then given in the 1993 paper
Isonumbers and genonumbers of dimension 1, 2, 4, 8, their isoduals
and "hidden numbers" of dimension 3, 5, 6, 7.
R. M. Santilli, Algebras, Groups and Geometries Vol. 10, 273322 (1993).
A comprehensive study was then presented in the monograph
"Elements of Hadronic Mechanics", Vol. I: "Mathematical Foundations"
R. M. Santilli, Ukraine Academy of Sciences, 1st edition (1993).
Numerous independent papers and books have been written on Santilli isonumbers and isofields (see the General Bibliography). We here merely quote the monograph written in 2001 by the Chinese mathematician ChunXuan Jiang that remains a significant general study in the field to this day
"Foundations of Santilli Isonumber Theory"
ChunXuan Jiang, International Academic Press (2001).
A readable presentation of Santillis isonumbers in Italian is given by the paper
I nuovi numeri santilliani,
Parte I: I nuovi numeri isotopici e loro applicazioni
Michele Sacerdoti, to appear.
2.2B. Discovery of genonumbers (1993)
Despite the dimension and implications of the preceding discovery, Santilli remained dissatisfied because his main objective was to reach a structural generalization of quantum mechanics suitable for the representation of energy releasing processes, such as nuclear fusions, that are irreversible over time (that is, the time reversal images violate causality laws). Isonumbers could not allow such a generalization because they have no "time arrow."
Hence, Santilli went back to work and reexamined the foundations of his own isotopic number theory. He discovered in this way that, in addition not to require the value I = +1 for the multiplicative unit, the axioms of a field do not require that the unit for the multiplication to the right be equal to the unit for then multiplication to the left, provided that all multiplications are correspondently ordered to the right and to the left, respectively.
This discovery gave rise to a broader class of new numbers (again with characteristic zero, the sole known to have physical applications at this writing), also verifying the axioms of a field, called by Santilli genonumbers in the Greek meaning that they induce a new structure. The new numbers are known today as Santilli's genoreal, genocomplex and genoquaternionic numbers to the right and to the left or, generically, as genonumbers..
By using the symbols I^{f} and n x^{f} m for the genounit and genomultiplication to the right (physically interpreted as forward in time) and the symbols ^{b}I and n ^{b}x m for the genounit and genomultiplication to the left, (physically interpreted as backward in time) we can write the genofields in the form
(2.4) F^{r}(n^{f}, x^{f}, I^{f}): I^{f} = 1 / S, n^{f} = n x I^{f}, n^{f} x^{f} m^{f} = n^{f} x S x m^{f} = ( n x m) x I^{f}, (2.5) ^{b}F(^{b}n, ^{b}x, ^{b}I): ^{b}I = 1 / R, ^{b}n = ^{b}I x n, ^{b}n ^{b}x ^{b}m = ^{b}n x R x ^{b}m = ^{b}I x (n x m), (2.6) I^{f} = (^{b}I)^{c}
where c is a conjugation depending on the desired application (such as Hermitean conjugation. complex conjugation, inverse, transpose, etc.) needed for the interconnection between the right and left genofields.
Again, genofields are isomorphic to conventional fields by conception and construction. Nevertheless, the implications are by far nontrivial. For instance, by using the inverse for the conjugation c, a generic realization of F^{f}(n, x^{f}, I^{f}) and ^{b}F(^{b}, ^{b}n, ^{b}I) is given by
(2.7) I^{f} = 3, ^{b}I = 1/3, 2 x^{f} 3 = 18, 2 ^{b}x 3 = 2,
namely, not only the product of 2 times 3 does not yield the usual number 6, but the product to the right is different than that to the left, all in a way fully compatible with the axioms of a numerical field.
This discovery carries scientific implications greater than those originating from isonumbers, because genonumbers have permitted the construction of mathematically rigorous methods for the invariant treatment of irreversibility, including the study of new energies, that are not treatable with the mathematics of the 20th century, because the latter has no "time arrow". In fact, Santilli genotheories represent irreversibility via the most basic mathematical quantity, the unit, with the physical interpretation that genounits and genomultiplication to the right (left) represent motion forward (backward) in time.
Santilli presented the discovery of genonumbers in his historical mathematical paper of 1993
Isonumbers and genonumbers of dimension 1, 2, 4, 8, their isoduals
and "hidden numbers" of dimension 3, 5, 6, 7.
R. M. Santilli, Algebras, Groups and Geometries Vol. 10, 273322 (1993).
and applied the new numbers in his monographs
"Elements of Hadronic Mechanics", Vol. I: "Mathematical Foundations"
R. M. Santilli, Ukraine Academy of Sciences, 1st edition (1993).
"Elements of Hadronic Mechanics"
Vol. II: "Theoretical Foundations"
R. M. Santilli, Ukraine Academy of Sciences (1994).
A readable presentation of Santilli's genonumbers in Italian is given by the paper
I nuovi numeri santilliani,
Parte II: I nuovi numeri genotopici ed iperstrutturali\\ e loro applicazioni
Michele Sacerdoti, to appear.
2.2C. Discovery of hypernumbers (1994)
Despite the above momentous discoveries, Santilli continued to remain dissatisfied because, as he stated in his works and correspondence, I cannot accept the idea that the DNA code can be understood with genonumbers because, even though they do represent the irreversibility of biological processes, they cannot possibly represent how two atoms of a DNA can produce an entire organ with a very large number of constituents.
In this way, the genonumbers were extended to yet new numbers today known as Santilli's hyperreal, hypercomplex and hyperquaternionic numbers to the right and to the left, or generically as hypernumbers that are multivalued, namely, not only the units and products to the right and to the left are different, but the hyperunit has an ordered set of values and, consequently, the multiplication yields an ordered set of results. For instance, the hyperlifting of example (2.7) would yield expressions of the type
(2.8) I^{f} = {3, 1/2, 1/3, ...} , 2 x^{f} 3 = {18, 3, 2, ...},
= ^{b}I = (I^{f}) ^{1} = {1/3, 2, 3, ...}, 2 ^{b}x 3 = {2, 12, 18, ...}.
It should be indicated that Santilli's hypernumbers are different than those belonging to hyperstructures because the former use conventional operations while the latter use abstract operations. Also, Santilli's hypernumbers verify all axioms of a field, while conventional hyperstructures do not generally admit any unit at all, thus not being generally formulated over a field, with consequential severe restrictions in applications.
Santilli published his hypernumbers for the first time in the monograph
"Elements of Hadronic Mechanics", Vol. I: "Mathematical Foundations"
R. M. Santilli, Ukraine Academy of Sciences, 2nd edition (1995).
and then in the mathematical memoir
Nonlocalintegral isotopies of differential calculus,
mechanics and geometries
R. M. Santilli, Rendiconti Circolo Matematico Palermo, Suppl. Vol. 42, 782 (1996).
with applications to biology presented in the subsequent monograph
"Isotopic, Genotopic and Hyperstructural Methods in Theoretical Biology"
R. M. Santilli, Ukraine Academy of Sciences (1997).
A recent mathematical presentation of hypernumbers, including its formulation via hyperstructural methods, can be found in Chapter 5 of the monograph
Hadronic Mathematics, Mechanics and Chemistry, Volume III:
Iso, Geno, HyperFormulations for Matter
R. M. Santilli, International Academic Press (2008)
2.2D. Discovery of isodual numbers (1993)
Despite all the above discoveries, each being quite significant, Santilli remained dissatisfied because, as he puts it in his works and correspondence: When I look at the stars, I feel very frustrated as a physicist for my complete inability to study whether a faraway star or quasar is made up of matter or of antimatter.
As indicated in Section 1.4, mathematical and physical methods of the 20th century were insufficient to allow any consistent classical description of antimatter. The new iso, geno and hypernumbers were insufficient to reach the needed classical description of antimatter precisely because of their isomorphism to conventional numbers. In fact, charge conjugation is an antiautomorphism. Hence, a classical representation of antimatter admitting an operator image compatible with charge conjugation needs a mathematics that is antihomomorphic or, better, antiisomorphic to the conventional mathematics, as well as to its iso, geno, and hyperliftings.
When at the Department of Mathematics of Harvard University, he conducted in the early 1980s a comprehensive search in the Cantabridgean mathematics libraries and concluded that the mathematics needed for a classical representation of antimatter did not exist in the form needed by physicists, such as to yield under quantization an image equivalent to charge conjugation. Consequently, the needed new mathematics had to be built.
A day in February 1982 Santilli invited one of his mathematics colleagues to visit the mathematics library (located in the ground floor of Harvard's Science Center) and suggested him to select any desired volume by opening it at any desired page. He would prove that the mathematics in that arbitrary volume in that arbitrary page would not allow a physically consistent classical representation of antimatter. He was indeed right.
As a physicist, Santilli was forced, again, to study yet new mathematics and, for that scope, he was forced to study yet new numbers. In this way, he discovered that the axioms of a numerical field admit negative units and the resulting fields are antiisomorphic to conventional fields as desired. More generally he introduced a new map he called isoduality (denoted with an upper index d) consisting of an antiHermitean operation given for an arbitrary quantity Q(n, ...) by
(2.9) Q^{d}(n^{d}, ...) =  Q^{†}(n^{†}, ...),
provided that the above map is applied to the totality of the elements of a given theory and all its operations. This gave rise to: Santilli's isodual conventional, iso, geno and hypernumbers; negative definite units called isodual conventional, iso, geno, and hyperunits; and corresponding multiplications called isodual conventional, iso, geno, and hypermultiplications.
As the simplest possible illustration, consider the conventional field F(n, x, I). Then, Santilli isodual field is given by
(2.10) F^{d}(n^{d}, x^{d}, I^{d}): I^{d} = I, n^{d} =  n^{†}, (n x m)^{†} =  (m^{†} x n^{†}).
The isoduals of iso, geno and hypernumbers can be similarly constructed via isoduality (2.9).
Even though seemingly trivial, isodual numbers have their own rather deep implications requiring attention to prevent inconsistencies. For instance, the statement of having +1,000 dollars in the bank, in reality means for isodual numbers that the account is 1,000 dollars in the red because the number +1,000 is now referred to the basic unit 1, the isodual norm of 1,000 is negative, etc.
To illustrate the mathematical novelty, we can report the following episode quoted by Santilli in footnotes of some of his books. In June 1996, Santilli and his wife Carla went to Palermo, Sicily, to pay their tribute to the Circolo Matematico Palermo for the publication of a special issue of its famous mathematics journal entirely dedicated to Santilli's isotopies. During that occasion, as a gesture of appreciation, the Editor in Chief of the journal, Prof. P. Vetro, found a 20 minutes opening at a mathematics conference going on in Palermo at that time and suggested Santilli to present there his new mathematics just appeared in the Rendiconti.
Santilli accepted the offer and elected to present his recently discovered isodual number theory and related mathematics by initiated his lecture with the projection in the big screen of a transparency with only the number "1" in it and the indication that he assumed that quantity as the basic unit of his mathematics. At that view and statement, the audience went into great agitation with numerous questions from all sides, often repeated various times, to such a disarray that 20 minutes passed without Santilli being able to present any additional transparency.
Mathematicians are accustomed to write structures in an abstract, realizationfree form. For this purpose Santilli suggests the use of the conventional symbol F(n, x, I) for the abstract unification of all his new numbers, provided one has a knowledge of all possible realizations, not only of the unit, but also of the related multiplications.
The above abstract unification would cause serious problems if used in physics because, e.g., it could cause the inadvertent mixing of particles and antiparticles. This is the reason that in physics it is much better to have different specific formulations for fields, isofields, genofields, hyperfields and their isoduals, since the identification of the assumed numbers and their unit identifies the level of treatment and related applications.
We cannot close this section without an indication of yet another mathematical discovery by Santilli given by iso, geno, hyperfields and their isoduals when the related generalized unit singular (or divergent), namely, are admitted to have a functional. dependence with null (or infinite) value, an occurrence simply impossible for the 20th century mathematics, since fields are assumed to have the trivial unit + 1. As we shall see in Chapter 3, this case is of particular physical relevance since Lim I^{*} → 0 represents gravitational singularities.
To the Foundation's best knowledge, Santilli published for the first time his isodual numbers in the historical mathematics papers and monograph of 1993
Isonumbers and genonumbers of dimension 1, 2, 4, 8, their isoduals and "hidden numbers" of dimension 3, 5, 6, 7.
R. M. Santilli, Algebras, Groups and Geometries Vol. 10, 273322 (1993).
"Isodual Theory of Antimatter, with Application to Antigravity, Grand Unification and Cosmology",
R. M. Santilli,Springer (2006).
"Elements of Hadronic Mechanics", Vol. I: "Mathematical Foundations"
R. M. Santilli, Ukraine Academy of Sciences, 1st edition (1993).
A readable presentation of Santilli's isodual numbers in Italian is given by the article
I nuovi numeri santilliani,
Parte III: I nuovi numeri isoduali e loro applicazioni
Michele Sacerdoti, to appear.
2.3. DISCOVERY OF ISO, GENO, HYPERDIFFERENTIAL CALCULI, FUNCTIONAL ANALYSIS AND THEIR ISODUALS (1996)
Santilli's main scientific objective has been the study of Lagrange's and Hamilton's legacy (Section 1.1), namely, the study of contact nonHamiltonian interactions at all possible levels, from Newtonian mechanics to second quantization. Besides the need for new numbers, Santilli faced another major technical obstacle, that of achieving the representation of all possible (well behaved) nonHamiltonian forces via an action principle, because such a principle is necessary for quantization. As a matter of fact, the lack of achievement of any quantum formulation of nonHamiltonian interactions during the 20th century was precisely due to the lack of any action principle for their classical representation, with consequential lack of any consistent method for their quantization.
Again, as a theoretical physicist, Santilli was forced to study pure mathematics as a condition to formulate consistent physical theories. After decades of trials and errors, Santilli recalled in the mid 1990s that Newton had to invent (with Leibnitz) the differential calculus before he was in a position to write his celebrated equations.
In this way, Santilli inspected the differential calculus and discovered that, contrary to a deeply rooted belief in pure mathematics for over about centuries, the differential calculus is indeed dependent on the assumed basic unit. Let r be the coordinate of a Newtonian particle and dr its differential. Assume the isotopic lifting of r into an isocoordinate r^{*} = r x I^{*}, with isounit I^{*} = 1 / T > 0. In this case, Santilli proved that the isodifferential and isoderivative are given by
(2.11) d^{*}r^{*} = Txd(rxI^{*}), d^{*}/d^{*}r^{*} = I^{*}xd/dr^{*}.
If the isounit is independent from the local variable of the calculus, the differential is indeed independent from the local valuable because
(2.12) I^{*} = 1/T = const., d^{*}r^{*} = TxI^{*}xdr = dr, d^{*}/d^{*}r^{*} = d/dr,
thus recovering the indicated belief in pure mathematics. However, when the isounit depends on local variable, I^{*} = I^{*}(r, ...), the above simplification is no longer possible because we have for the differential
(2.13) d^{*}r^{*} = Txd[rxI^{*}(r, ...)] = dr + TxrxdI^{*}(r, ...).
with a corresponding complex relation for the isoderivative. The geno hyper and isodual versions are evidently characterized by the use of the corresponding generalized units. Note that the geno, and hyperdifferential calculi for matter and their isoduals for antimatter are particularly important for the correct treatment of irreversible processes, as we shall see in Chapters 3,4,5.
The above studies marked the discovery of a structural generalization of the differential calculus that, as illustrated by the momentous implications outlined in this presentation, is indeed yet another mathematical discovery of clear historical proportions, today known as Santilli's so, geno, hyperdifferential calculi for matter and their isoduals for antimatter.
It should be noted that the conventional differential calculus has only one formulation, the conventional one. By contrast, Santilli's generalized differential calculi have two formulations expressed in Eqs. (2.11)(2.13) for the isotopic case, namely, a first formulation on isospaces over isofields, and a second one given by its projection on conventional spaces over conventional fields. Note that at the abstract representationfree level, isodifferential and conventional calculi coincide, and the same holds for the other calculi.
As we shall see in Section 3.12, Santilli's generalized differential calculi did permit the representation, for the first time, of all (well behaved) nonHamiltonian Newtonian systems via a generalized action principle, thus permitting the identification, also for the first time, of their rigorous map into operator forms constituting the foundations of hadronic mechanics.
These various new calculi were first published in the 1995 second edition of the monograph
"Elements of Hadronic Mechanics", Vol. I: "Mathematical Foundations"
R. M. Santilli, Ukraine Academy of Sciences, 2nd edition (1995).
and in all subsequent works
Nonlocalintegral isotopies of differential calculus,
mechanics and geometries
R. M. Santilli, Rendiconti Circolo Matematico Palermo, Suppl. Vol. 42, 782 (1996).
The functional isoanalysis was initiated by Santilli with the isotopies of basic functions such as exponential, logarithm, trigonometric and other functions, see Chapter 6 of the monograph
"Elements of Hadronic Mechanics", Vol. I: "Mathematical Foundations"
R. M. Santilli, Ukraine Academy of Sciences, 1st edition (1993).
The construction of the new isoanalysis was continued by the physicist J. V. Kadeisvili, e.g. in the papers
Elements of Functional Analysis
J. V. Kadeisvili,
Algebras, Groups and Geometries Vol. 9, pages 283318 (1992)
Elements of FourierSantilli isotransforms
J. V. Kadeisvili,
Algebras, Groups and Geometries Vol. 9, pages 310347 (1992)
Additional work was done by the physicists A. K. Aringazin et al, see Appendix 4.B of the monograph
"Elements of Hadronic Mechanics"
Vol. II: "Theoretical Foundations"
R. M. Santilli, Ukraine Academy of Sciences (1994).
2.4. DISCOVERY OF ISO, GENO, HYPER, SPACES AND THEIR ISODUALS (1983).
2.4A. Santilli's generalized spaces
As it is well known, all quantitative studies are defined on a representation space, such as the Euclidean Minkowski Riemannian, Finslerian or other (metric or pseudometric) space that, in turn, is defined over a field of numbers. It is evident that the generalization of ordinary numbers produced a corresponding lifting of conventional spaces, today's known as Santilli's iso, geno, and hyper Euclidean, Minkowskian, Riemannian, Finslerian and other spaces for matter and their isoduals for antimatter.
The implications of these broader spaces are far reaching, as we shall see. Consider the conventional, (3+1)dimensional Minkowski space M(r, m, I) with spacetime coordinates r = (r^{k}), k = 1, 2, 3, 4, r^{4} = t, metric m = Diag. (1, 1, 1,  c^{2}) and invariant r^{2} = (r^{i} x m_{ij} x r^{j}) x I where I is the unit of the Lorentz symmetry SO(3.1), I = Diag. (1, 1, 1, 1). Then, the isounit, isometric, and isoline element on MinkowskiSantilli isospace are given by
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(2.14) M^{*}(r^{*}, m^{*}, I^{*}): I^{*} = 1 / T = Diag. ( 1/T_{1}^{2}, 1/T_{2}^{2}. 1/T_{3}^{3}, 1/T_{4}^{4}) = = Diag. ( n_{1}^{2}, n_{2}^{2}, n_{3}^{2}, n_{4}^{2}) > 0, (2.15) m^{*} = T x m = (T_{i}^{j} x m_{jk}) = Diag. (T_{1}^{2}, T_{2}^{2}, T_{3}^{3},  c^{2} x T_{4}^{4}) = = Diag. (1/ n_{1}^{2}, 1/n_{2}^{2}, 1/ n_{3}^{2},  c^{2}/n_{4}^{2}), (2.16) r^{*2*} = ( r^{*i} x^{*} m^{*}_{ik} x^{*} r^{*k}) x I^{*} = (r^{12} x T_{1}^{2} + r^{22}x T_{2}^{2} + r^{32} x T_{3}^{3}  t^{2} x c^{2} x T_{4}^{4}) x I^{*} = = ( r^{12}/n_{1}^{2} + r^{22}/n_{2}^{2} + r^{32}/n_{3}^{2}  t^{2} x c^{2}/n_{4}^{2}) x I^{*} ε F^{*}(n^{*}, x^{*}, I^{*}),
where: we have shown the most general possible, diagonal realization of Santilli's isounit and its realization in physics via the socalled characteristic quantities n_{k}, k = 1, 2, 3, 4; r^{4} = t x c; and the notation r^{k2}, k = 1, 2, 3, 4, denotes the square of r^{k}.
Figure 2.1: Santilli illustrates the far reaching implications of his isogeometries via the "isobox" consisting of a box inspected by two observers: an observer in the exterior verifying the conventional Minkowski spacetime, and an observer in the interior verifying the MinkowskiSantilli isospacetime. Santilli isonumbers and isospaces can be claimed to be understood if one understands that: the interior observer can be in the infinite future or past time with respect to the exterior observer; if the exterior observer sees a cube with 2 m side, the interior observer can see a room of dimension arbitrarily bigger or smaller compared to the exterior view; and if the exterior observer see a cube, the interior observer can see a cathedral. For explanations, interested readers should study the quoted literature.
2.4B. Preliminary implications
The implications of the above discovery can only be qualified as historical, as shown in the rest of this presentation. We only mention the achievement, for the first time in scientific history via a metric, of:
1) The representation of arbitrary speed of light C = c /n_{4}, where n_{4} is the familiar index of refraction, with values C smaller (bigger) than c for physical media of low (high) density;
2) The representation of the actual dimension and shape of particles via the spacecomponent n^{2}_{k}, k = 1, 2, 3 (normalized to the value 1 for the vacuum);
3) The representation of the density of the particles (or medium) considered via n_{4}^{2} (also normalized to the value 1 for the vacuum);
4) The representation of the inhomogeneity of the physical medium considered via, e.g., a dependence of the characteristic quantities on the distance and other variables, n_{k} = n_{k}(r, ...);
5) The representation of the anisotropy of physical medium considered via a different value of the characteristic quantities.
A fundamental property of Santilli isospaces is that, by conception and construction, they are isomorphic to the original spaces for all positivedefinite isounits. In fact, at the abstract, realizationfree level, there is no difference between the conventional Minkowski space and the MinkowskiSantilli isospace, to such an extent that they can be expressed via the same symbols, only subjected to different interpretations. As we shall see in the subsequent chapters, this feature has very important implications for numerous aspects of scientific knowledge.
Another important property is that Santilli's isospaces unify all possible spaces with the same dimension. In fact, isoline element (2.16) clearly includes as particular cases the Minkowskian, Riemannian, Finslerian, nonDesarguesian and other line elements. Hence, M^{*}(r^{*}, m^{*}, I^{*}) unifies all possible spacetimes in (3.1)dimensions. In the event the positivedefiniteness of the isounit is relaxed, M^{*}(r^{*}, m^{*}, I^{*}) unifies all possible 4dimensional spaces, including the Euclidean one, the differentiation between one space and the other being set by the unit.
As we shall see, the above unification alone has far reaching implications, such as the achievement of the first and only known, axiomatically consistent grand unification of electroweak and gravitational interactions that had escaped the best minds of the 20th century, including Albert Einstein.
Santilli's geno and hyper spaces have implications perhaps more intriguing than those of the isospaces, because the former provide the first known geometric representation of irreversibility by embedding the direction of time in the geno and hypermetric itself, while the new spaces remain isomorphic to the original space even though, quite remarkably, the geno and hypermetrics are not necessarily symmetric.
Isospaces were first presented in two historical papers of 1983 on the structural generalization of the Minkowski space, the Lorentz symmetry and special relativity, with classical representation in the paper
Lieisotopic lifting of special relativity
for extended deformable particles
R. M. Santilli, Lettere Nuovo Cimento Vol. 37, 545555 (1983),
and operator counterpart
Lieisotopic lifting of unitary symmetries and
of Wigner's theorem for extended deformable particles
R. M. Santilli, Lettere Nuovo Cimento Vol. 38, 509521 (1983).
A more detailed mathematical treatment via the isotopies of the Euclidean space was presented in the two papers of 1985 quoted earlier
Lieisotopic liftings of Lie symmetries, I: General considerations
R. M. Santilli,
Hadronic J. Vol. 8, 2535 (1985)
Lieisotopic liftings of Lie symmetries, II: Lifting of rotations
R. M. santilli,
Hadronic J. Vol. 8, 3651 (1985)
In reality Santilli wrote first the latter two papers on the isoEuclidean space and then wrote the paper on the isotopies of Minkowski space. Unfortunately the former ended up being published two years following the publication of the latter due to incredible editorial obstructions Santilli felt obliged to report in the first 1985 paper.
Iso, geno and hyperspaces for matter and their isoduals for antimatter were systematically presented in the historical memoir published by the Rendiconti in 1996, with the initiation of their topology
Nonlocalintegral isotopies of differential calculus,
mechanics and geometries
R. M. Santilli, Rendiconti Circolo Matematico Palermo, Suppl. Vol. 42, 782 (1996).
Comprehensive studies were then published in Santilli's various books, including
"Elements of Hadronic Mechanics", Vol. I: "Mathematical Foundations"
R. M. Santilli, Ukraine Academy of Sciences (1993),
"Elements of Hadronic Mechanics"
Vol. II: "Theoretical Foundations"
R. M. Santilli, Ukraine Academy of Sciences (1994)
Systematic mathematical studies on the new spaces and the the resulting new topology were conducted in the following monograph among various others studies (see the general bibliography)
Fundamentos de la Isoteoria de LieSantilli
Raul M. Falcon Ganfornina and Juan Nunez Valdes International Academic Press (2001)
Figure 2.2: Santilli's illustration of his mathematical hyperisogeometric locomotion based on the directional alteration (increase or decrease) of the distance via the use of sequential, extremely high concentrations of positive energies referred to positive units and negative energies referred to negative units. The understanding of this basically new locomotion requires a technical knowledge of Santilli's hypergeometries and their isoduals. Under the above conditions, an observer from the outside (Figure 2.1) can see the vehicle performing geometric singularities, such as instantaneous, 90^{o} changes of directions, instantaneous accelerations to extreme speeds, and the like, while the vehicle and its content are essentially at rest when inspected from the inside, because motion is caused by the change of the geometry (isolocomotion via positive energy), rather than the conventional Newtonian locomotion. But alterations of the space geometry cause inevitable alterations of time. Hence, for the vehicle to control its position in both space and time, there is the additional need of using negative energies (hyperisolocomotion with both positive and negative energies). Additionally, the vehicle requires no fuel tank because the only possibility of achieving the needed extreme energy densities is to obtain them from space conceived as a universal medium characterized by a superposition of extremely large values of positive and negative energies coexisting with each other, thus resulting in the conventional vacuum, because defined in different spaces (twovalued hyperisogeometry). The technical understanding of the latter aspect requires a technical knowledge of Santilli representation of the synthesis of the neutron from a proton and an electron as occurring in stars, that requires 0.782 MeV whose most plausible origination is precisely that from space, otherwise a star would never initiate producing light due to the initial need, for a small star, of at least 10^{30} MeV per second (see Chapter 5). Santilli stresses that his hyperisolocomotion is computerizable, but it is a purely mathematical model due to lack of knowledge on how to extract from space the needed very large amounts of positive and negative energies.
2.4C. Iso, geno, hypertopologies and their isoduals
The central mathematical tools of 20th century quantitative sciences have been the conventional differential calculus and topology which tools, being strictly localdifferential, are the ultimate reason for the abstraction of particles as dimensionless points moving in empty space (conditions characterizing the {\it exterior dynamical problem in vacuum})\index{Exterior dynamical problem}
and related dramatic limitations, such as those originating from Theorem 1.1.
Since the initiation of his research in the late 1960s, Santilli's primary objective has been the study of extended, nonspherical and deformable particles in conditions of partial or complete mutual penetration (Figures 1.3, 1.4) or for extended particles moving within a physical medium (conditions characterizing the {\it interior dynamical problem within physical media})\index{Interior dynamical problem}.
It is evident that such a study could not be conducted in a consistent way without the prior lifting of the localdifferential character of the 20th century mathematics into a suitable nonlocalintegral form. This problem occupied Santilli's mind for decades due to the lack of the new mathematics in the form needed for physical applications, that is, as a covering of conventional localdifferential methods. Said covering character was needed to recover 20th century physics when particles exit physical media and return to move in vacuum. Following extensive search in Italian and American mathematics libraries, Santilli was unable to locate the indicated new mathematics in the needed covering form. Therefore, the new mathematics had to be built prior to any quantitative physical, chemical or bio;logical study.
After decades of attempts, Santilli finally succeeded to lift the localdifferential calculus in the covering nonlocalintegrodifferential form (Section 2.3) as needed for physical applications, by embedding all nonlocalintegral terms in the isounit $\hat I(t, r, p, ...) > 0$ as in Eq. (2.11), so as to recover the conventional differential calculus as a trivial particular cases when $\hat I = I$ [loc. cit.].
Being primarily interested in physical applications, Santilli contacted various mathematicians and suggested them to study the isotopy of the conventional topology, such a the isotopy the conventional {\it Euclidean topology}\index{Euclidean topology} at the foundation of both the Galilean and special relativities and quantum mechanics. The mathematicians Gr. T. Tsagas (then Chairman of the Department of Mathematics of Aristotle University, Thessaloniki, Greece) and D. S. Sourlas (also from Greece) answered Santilli's call and initiated the construction of the proposed {\it isotopology}\index{Isotopology} in the early 1990s
"Mathematical Foundation of the LieSantilli Theory"
D. S. Sourlas and G. T. Tsagas,
Ukraine Academy of Sciences 91993)
However, these initial versions were formulated on a conventional field, thus preventing consistent applications to physics (due to the Theorems of Catastrophic Inconsistencies of Section 3.7). Consequently, Santilli addressed the problem in 1996
Nonlocalintegral isotopies of differential calculus,
mechanics and geometries
R. M. Santilli, Rendiconti Circolo Matematico Palermo, Suppl. Vol. 42, 782 (1996).
and achieved the first known formulation of the isotopology on an isospace over an isofield for the characterization of matter interior problem, as well as its isodual for antimatter interior problem. In memoir [loc. cit.], Santilli also proved the intrinsic capability of the isotopology to characterize extended, nonspherical and deformable particles via realizations of the isounit of type (2.14), thus completing the construction of the new mathematics needed for basic physical, chemical and biological advances.
Subsequently, comprehensive studies on the isotopology were conducted in early 2000s by the mathematicians R. M. Falcon Ganfornina and J. Nunez Valdes of the Department of Mathematics of the University of Seville, Spain, who achieved the final form of the nonlocalintegrodifferential topology used by Santilli in all applications for matter, and today known as the TsagasSourlasSantilliGanforninaValdes isotopology or TSSGN isotopology for short.\index{Isotopology.
The best and most comprehensive mathematical reference on the TSSGN isotopology, including a review and quotation of all preceding literature, is the monograph of 2001 by R. M. Falcon Ganfornina and J. Nunez Valdes
Fundamentos de la Isoteoria de LieSantilli
Raul M. Falcon Ganfornina and Juan Nunez Valdes International Academic Press (2001)
that essentially reports the Ph. D. Thesis of the first author under the guidance of the second author. Ironically, during the defense of the Ph. D. Thesis, both the student and his teacher were attacked for apparent triviality of the results, because the covering isotopology can be expressed by putting a "hat" on all symbols of the conventional topology.
However, as indicated in the Foreword of this chapter, besides momentous mathematical advances, the new isotopology allows the extension of Einstein's axioms, from their pointlike abstraction of particles moving in vacuum, to extended particles in conditions of total or partial mutual penetration, as occurring in the structure of hadrons, nuclei and stars, with consequential prediction and quantitative treatment of much needed new clean energies that are inconceivable with the old topology (Chapter 9).
for which reason we included comments in the Foreword of this chapter.
The best known presentation of the isodual TSSGN isotopology for antimatter is that in Santilli's monograph
"Isodual Theory of Antimatter, with Applications to Antigravity, Grand Unification and Cosmology,"
R. M. Santilli,
Springer (2006)
A recent readable presentation of these new topologies for physicists is that available in Volumes II and III of the 2008 monographs
Hadronic Mathematics, Mechanics and Chemistry, Volumes I, II, III, IV and V:
R. M. Santilli, International Academic Press (2008)
At this writing (Springs 2009), only the TSSGN isotopology for matter and its isodual for antimatter are known, while their geno and hyperformulations and related isoduals are unknown, although expected to be a natural extension of the indicated isotopology and its isodual. Their study by interested mathematicians is solicited and financially supported by the Santilli Foundation due to their dramatic implications, such as setting mathematical foundations for the first known rigorous connection between irreversible genomechanics and thermodynamics.
2.5. DISCOVERY OF ISO, GENO, HYPERSYMPLECTIC GEOMETRIES AND THEIR ISODUALS (1996)
As it is well known, the symplectic geometry allows one of the most rigorous studies of classical Hamiltonian systems, as well as their quantization. Hence, Santilli could not escape a reinspection of the symplectic geometry because his main physical objective was to represent the most general possible (sufficiently smooth) nonHamiltonian systems.
Consider the conventional canonical symplectic structure on a cotangent bundle with local charts r and p on the reals, ω = dp ∧ dr. It was easy for Santilli to formulate its isotopic covering on an isocotangent bundle by first showing that local isochart is given by r^{*} = rxI^{*}, p^{*} = pxI^{*1} on the isoreals, with consequential isocanonical two isoform
(2.17) ω^{*} = d^{*}p^{*} ∧ d^{*}r^{*},
that, as one can see, coincides with the conventional canonical twoform for all constant isounits, but possesses otherwise dramatic differences with the conventional version because it does allow the desired representation of all wellbehaved equations of motion with all possible potential and nonpotential forces.
These studies lead to what are today known as Santilli's iso, geno, and hypersymplectic geometries for matter and their isoduals for antimatter. Their most salient feature is that of coinciding with the conventional symplectic geometry at the abstract level to such an extent that Santilli insists in writing the equations for his covering geometries via the symbols of the conventional geometry, and merely subjects them to a broader interpretation.
The above coverings of the symplectic geometry were first published in the monograph
"Elements of Hadronic Mechanics", Vol. I: "Mathematical Foundations"
R. M. Santilli, Ukraine Academy of Sciences (1993),
as well as in his mathematical memoir
Nonlocalintegral isotopies of differential calculus,
mechanics and geometries
R. M. Santilli, Rendiconti Circolo Matematico Palermo, Suppl. Vol. 42, 782 (1996).
2.6. ISOTOPIC UNIFICATION OF MINKOWSKIAN, RIEMANNIAN AND FINSLERIAN GEOMETRIES (1998)
A very special feature of Santilli's isotopies is that of unifying seemingly different structures into a covering form that enjoys the basic property of invariance. Following the achievement in 1983 of his isoMinkowski spaces (Section 2.4), Santilli realized that there is no difference between his isoMinkowski metric m^{*}(r, ...) and a conventional Riemannian metric g(r) or a Finslerian metric g(r, v, ...)since the explicit form of the characteristic quantities n_{i} is unrestricted by the isotopies (only their positivedefinite character is requested for isotopies).
But Santilli knew at that time (early 1990s) that the isoMinkowski spaces are isomorphic to the conventional space. Hence, his isotopic methods offered a unique possibility of unifying the Minkowskian and the Riemannian geometries, with far reaching implications, such as the first axiomatically consistent grand unification of electroweak and gravitational interactions reviewed in the physics chapters.
In this way, Santilli achieved a new geometry on isospaces over isofields, today called MinkowskiSantilli isogeometry equipped with all the machinery of the Riemannian geometry (such as covariant derivative, Christoffel's symbols, etc.), that does unify the Minkowskian and Riemannian geometries, while admitting both as particular cases depending on the selected isounit.
This additional historical achievement was published by Santilli in various works, with primary presentation in the memoir
Isominkowskian geometry for the gravitational treatment of matter
and its isodual for antimatter
R. M. Santilli, Intern. J. Modern Phys. D Vol. 7, 351407 (1998).
Numerous papers then appeared showing the so called "direct universality" of the MinkowskiSantilli isogeometry, that is, the capability of admitting as particular cases all infinitely possible (non singular) geometries on a (3+1)dimensional space (universality), directly in the selected coordinates without any need for the transformation theory (direct universality). Among numerous papers on this aspect, we quote the following
Universality of Santilli's isoMinkowskian geometry
A. K. Aringazin and K. M. Aringazin, in "Frontiers of Fundamental Physics"
M. Barone and F. Selleri, Editors, Plenum 91995)
2.7. LIEISOTOPIC COVERING OF LIE'S THEORY AND ITS ISODUAL (1978)
As it is well known, Lie's theory has been the fundamental mathematical tool of the 20th century quantitative sciences, thus having been the subject of vast attention and having achieved a vast diversifications into various branches, such as:
1) The universal enveloping associative algebra U over a field F(n, x, I) as a vector space whose elements are: the unit matrix I = Diag. (1, 1, ..., 1) with the dimension of the selected representation; the N (Hermitean) generators G_{k}, k = 1, 2, 3, ..., N, with conventional associative products G_{i} x G_{j}; and the infinitedimensional basis characterized by the Poincare'BirkhoffWitt theorem via the ordered monomials
(2.18) U: I, G_{k}, G_{i} x G_{j} i ≤ j, G_{i} x G_{j} x G_{k} i ≤ j ≤ k, ...
which basis is necessary for the definition of exponentiation W = exp (G x w x i), where w ε F, and other operations on U;
2) The Ndimensional Lie algebra L which is the antisymmetric algebra U^{} attached to U with Lie product and closure relations
(2.19) L: [G_{i}, G_{j}] = G_{i} x G_{j}  G_{j} x G_{i} = C_{ij}^{k} x G_{k},
where the Cs are the structure constants of L;
3) The Lie group g whose realization most important in physics is that of Lie's transformation groups of a quantity Q(w) that can be written in the following finite and related infinitesimal forms
(2.20) g: Q(w) = W(w) x Q(0) x W(w)^{†} = exp (G x w x i) x Q(0) x exp(  i x w x G), (2.21) i x dQ / dw = [Q, G] = Q x G  G x Q;
plus the representation theory generally constructed either on a right acting module u or, equivalently for Lie's theory, the left acting module u,
It should be recalled that Lie's theory characterizes the fundamental dynamical equations of quantum mechanics, those of the time evolution via Eqs. (2.21) with w representing time t. Lie's theory also characterizes all fundamental symmetries in physics, such as the Lorentz and Poincare' symmetries at the foundation of special relativity, the SU(3) symmetry for the classification of particles, etc.
Immediately following the study of Lie's theory during his graduate studies in physics in Torino, Italy, in the late 1960s, Santilli realized the excessive limitations of the theory, since Lie's theory solely applies for systems that are linear, localdifferential and Hamiltonian (canonical at the classical level and unitary at the operator level), while the systems of the real world are generally nonlinear, nonlocalintegral and admit both Hamiltonian and nonHamiltonian interactions. Hence, Santilli set in the late 1960s his goal to reach a structural generalization of Lie's theory applicable to a broader class of systems.
As a main result of his Ph. D. thesis, Santilli published in 1967 the first formulation in physics records of the Lieadmissible covering of Lie's theory studied in the next section that is ideally suited for rigorous formulations of irreversibility as we shall see in Chapter 3.
However, Santilli knew of the existence in nature of systems that are nonHamiltonian, yet verify all conventional conservation laws (see Section 3.12), whose characterization requires an algebra which is a covering of Lie algebras (to exit from the class of Hamiltonian equivalence), yet it is characterized by an antisymmetric product (to characterize the conservation of total quantities). Lieadmissible algebras do not verify this requirement because, as we shall see in the next section, their product is neither totally antisymmetric nor totally symmetric, thus being particularly suited to represent time rate of variations of physical quantities, but not their conservation.
Following decades of silent research, Santilli released in 1978 a particular case of the broader Lieadmissible theory consisting of an isotopic (axiompreserving) generalization (he called lifting) of all branches of Lie's theory, today known as the LieSantilli isotheory,, that constitutes one of the biggest mathematical and physical discoveries of the 20th century, not only because of fundamental mathematical novelty, but also because of its predictable far reaching implications for all quantitative sciences, as shown in Chapters 3, 4, 5.
Santilli's Lieisotopic theory is based on all preceding mathematical discoveries, that is, its correct formulation requires isofields, isospaces, isodifferential calculus, isofunctional analysis, etc., to such an extent that the lack of isotopic lifting of only one methodological aspect of Lie's theory causes catastrophic inconsistencies (lack of invariance of the theory under its own action, etc.). In fact, the mixing of Lie and LieSantilli's methods would be like formulating the conventional Lie's theory on Santilli's isofields, resulting in evident inconsistencies.
Under the above understanding, the presentation of Santilli's Lieisotopic theory is now (following its discovery) rather elementary and its main branches can be summarily presented for applied mathematicians as follows:
1^{*}) The universal enveloping isoassociative algebra U^{*} over an isofield F^{*}(n^{*}, x^{*}, I^{*}) as a vector space whose element are: the isounit I^{*} = 1/T > 0 (where the positivedefiniteness is assumed to preserve Lie's axioms and the dimension is that of the used isorepresentation); the same (Hermitean) generators G_{k}, k = 1, 2, 3, ..., N, of Lie's theory with isoassociative product and related isounit
(2.22) G_{i} x^{*} G_{j} = G_{i} x T x G_{j} (2.23) I^{*} x^{*} G^{*} = G^{*} x^{*} I^{*} = G^{*} ∀ G ε U^{*}
and the infinite dimensional isobasis characterized by the Poincare'BirkhoffWittSantilli isotheorem with ordered isomonomials
(2.24) U^{*}: I^{*}, G_{k}, G_{i} x^{*} G_{j} i ≤ j, G_{i} x^{*} G_{j} x^{*} G_{k} i ≤ j ≤ k, ...
permitting the definition of isoexponentiation
(2.25) W^{*}(w^{*}) = I^{*} + i^{*} x^{*} w^{*} x^{*} G/1 ! + ... = [exp(G x T x w x i)] x I^{*} = I^{*} x [exp(i x w x T x G)],
and other operations on U^{*};
2^{*}) The Ndimensional LieSantilli isoalgebra L^{*} which is the antisymmetric isoalgebra U^{*} attached to U^{*} with LieSantilli isoproduct and closure relations
(2.26) L^{*}: [G_{i}, G_{j}]^{*} = G_{i} x^{*} G_{j}  G_{j} x^{*} G_{i} = G_{i} x T x G_{j}  G_{j} x T x G_{i} = C_{ij}^{*k} x^{*} G_{k},
where C_{ij}^{*k} characterizes the isostructure isofunctions of LC_{ij}^{k} with constant particularizations;
3^{*}) The LieSantilli isogroups g^{*} whose realization most important in physics is that of Santilli's isotransformation isogroups of a generic quantity Q^{*}(w^{*}) on U^{*} over F^{*} that can be written in the following finite and infinitesimal forms each in a dual way, the formulation on U^{*} over F^{*} and its projection on U over F
(2.27) g^{*}: Q^{*}(w^{*}) = W^{*}(w^{*}) x^{*} Q^{*}(0^{*}) x^{*} W^{*}(w^{*})^{†} = exp (G xT x w x i) x Q(0) x exp(  i x w x T x G), (2.28) i^{*} x^{*} d^{*}Q^{*} / d^{*}w^{*} = [Q, G]^{*} = Q x T x G  G x T x Q;
plus the isorepresentation isotheory generally constructed either on a right acting isomodule u^{*} or, equivalently, the left acting isomodule u^{*}.
As one can see, Santilli's isotheory causes the emergence of a generally nonlinear, nonlocal and nonHamiltonian operator T in the exponent of the isotransformations, as well as in the broader isobrackets of the infinitesimal transforms, thus permitting indeed the originally desired extension of Lie's theory to nonlinear, nonlocal and nonHamiltonian systems with far reaching implications indicated in the subsequent chapters.
Remarkably, Santilli proved the "direct universality" of his isotheory for all well behaved nonlinear, nonlocal and nonHamiltonian systems via the following:
THEOREM 2.7.1: All sufficiently smooth nonlinear, nonlocalintegral and nonHamiltonian systems (whether classical noncanonical or operator nonunitary) on conventional spaces over a conventional field always admit an isounit for which they can be identically reformulated on isospaces over isofields where they are isolinear, isolocal and isocanonical or isounitary (verify the axioms of linearity, locality and Hamiltonian character on isospaces over isofields).
The reconstruction of linearity, locality and canonicity or unitarity is merely done by embedding all nonlinear, nonlocal and nonHamiltonian terms in the isounit. This important property is the conceptual essence of Santilli's isotheory in both its mathematical meaning and physical applications. Recall that Lie's theory is strictly linear, localdifferential and Hamiltonian on conventional spaces over ordinary fields. In the event Santilli had not preserved at the covering level these fundamental properties, his theory could not be called an "isotopy" (axiompreserving) lifting of Lie's theory. In turn, the loss of the isotopic character would have caused serious physical problems.
Recall from the preceding sections that the isotopies have the important capabilities of unifying seemingly different mathematical structures. In the original proposal of 1978, Santilli proved that the Lieisotopic algebra O^{*}(3) unifies all simple (compact and noncompact) Liealgebra of dimensions three, and then formulated the following:
CONJECTURE 2.7.1: All simple Liealgebras of dimension N can be unified into one single simple Lieisotopic algebra of the same dimension.
The mathematician Gr. Tsagas proved the above conjecture to be correct for all simple Lie algebras with the exclusion of the exceptional algebras (see the reference below). The Foundation is interested in funding the completion of the proof of Conjecture 2.7.1 by qualified mathematicians.
The axiomatic unity of the conventional Lie theory and its isotopic covering is such that Santilli insists in presenting the latter with the same symbols of the former, only subjected to a broader realization, as it is the case for isonumbers, isospaces, isogeometries, etc.
As we shall see, the above property also differentiates Santilli's studies from a variety of other attempts to generalize Lie's theory, all known today to verify the Theorems of Catastrophic Mathematical and Physical Inconsistencies recalled in Section 3.9, precisely because of the latter theories are based on the broadening of Lie's theory, on one side, combined with the preservation of the conventional mathematics, on the other side.
The above isotopic lifting of Lie's theory was constructed by Santilli for the sole treatment of matter. For the classical treatment of antimatter in such a way to achieve compatibility with the operator formulations, Santilli needed an antiisomorphic image of the above Lieisotopic theory that he constructed via his isodual map (2.9) applied to the totality of quantities and their operations of the Lieisotopic theory. This resulted in two new coverings of Lie's theory today known as Santilli isodual Lie theory and isodual Lieisotopic theory, that are not reviewed here for brevity.
The discovery of the isotopic covering of each branch of Lie's theory was published for the first time in 1978 when Santilli was at Harvard University under DOE support via the two hundred pages historical memoir
On a possible Lieadmissible covering of Galilei's relativity
in Newtonian mechanics for nonconservative and Galilei formnoninvariant systems
R. M. Santilli, Hadronic J. Vol. 1, 223423 (1978).
The theory was then expanded in the series of volumes published by the most prestigious scientific house of the time, Springer Verlag, in its most prestigious series of Text and Monograph in Theoretical Physics,
"Foundations of Theoretical Mechanics, I: The Inverse Problem in Newtonian Mechanics"
R. M. Santilli, SpringerVerlag (1978)
"Foundations of Theoretical Mechanics, II: Birkhoffian Generalization of hamiltonian Mechanics"
R. M. Santilli, SpringerVerlag (1982).
These original presentations were based on isospaces, but defined on conventional fields. Subsequently, Santilli discovered the lack of completion of this formulation and, following the availability of the isonumbers, reached a mathematically consistent formulation in various works, such as in the monograph that included a treatment of isodual Lie theory and isodual Lieisotopic theory
"Elements of Hadronic Mechanics", Vol. I: "Mathematical Foundations"
R. M. Santilli, Ukraine Academy of Sciences, 1st edition (1993),
"Elements of Hadronic Mechanics"
Vol. II: "Theoretical Foundations"
R. M. Santilli, Ukraine Academy of Sciences. 1st edition (1993)
The discovery of the isodifferential calculus permitted Santilli to achieve the final formulation of his Lieisotopic theory that was published in the second edition of 1995 of the above two volumes. A comprehensive presentation of the isodual Lie theory and the isodual Lieisotopic theory is available in the monograph
"Isodual Theory of Antimatter, with Application to Antigravity, Grand Unification and Cosmology",
R. M. Santilli, Springer (2006)
Santilli's most recent presentation is available in the monograph with a treatment of the Lie, Lieisotopic theories and their isoduals
Hadronic Mathematics, Mechanics and Chemistry, III:
Iso, Geno, HyperFormulations for Matter
and Their Isoduals for Antimatter
R. M. Santilli, International Academic Press (2008)
Due to its historical importance, the LieSantilli isotheory has been the subject of numerous independent studies, among which we can quote the review papers
An introduction to the LieSantilli isotopic theory
J. V. Kadeisvili, Mathematical Methods in Applied Sciences Vol. 19, 13491395 (1996)
Foundations of the LieSantilli isotheory
J. V. Kadeisvili, Rendiconti Circolo Matematico Palermo, Suppl. Vol. 42, 83135 (1996).
and the monographs
"Mathematical Foundation of the LieSantilli Theory"
D. S. Sourlas and G. T. Tsagas, Ukraine Academy of Sciences 91993).
"Santilli's Isotopies of Contemporary Algebras
Geometries and Relativities",br>
J. V. Kadeisvili, Ukraine Academy of Sciences, Second edition (1997)
Fundamentos de la Isoteoria de LieSantilli
Raul M. Falcon Ganfornina and Juan Nunez Valdes International Academic Press (2001)
The proof of Conjecture 2.7.1 for all simple Lie algebra with the exclusion of the exceptional algebras can be found in the paper
Studies on the classification of LieSantilli algebras,
G. Tsagas, Algebras, Groups and Geometries Vol. 13, pages 129148 (1996).
For a comprehensive list of all contributions on the LieSantilli isotheory, the interested scholar is suggested to consult the
General Bibliography on Santilli Discoveries
2.8. LIEADMISSIBLE COVERING OF THE LIEISOTOPIC THEORY AND ITS ISODUAL (1967)
Remarkably, Santilli remained dissatisfied with his own Lieisotopic theory for physical and not mathematical reasons. Due to its structure and underlying topology, Lie's theory is ideally suited to represent a closedisolated system of particles that, being necessarily abstracted as pointlike, have no collisions, thus characterizing a Hamiltonian system (namely, a system entirely described by the Hamiltonian). This is typically the case for the atomic structure and other systems. In these cases, the antisymmetric character of Lie's brackets [A, B] = A x B  B x A =  [B, A] permits the representation of the conservation of the total quantities (represented in physics by the generators), as it is the case for the Hamiltonian
(2.29) i x dH/dt = [H, H] = H x H  H x H = 0.
Santilli's Lieisotopic theory does enlarge the class of represented systems into particles that are extended (see Section 3.12), thus experiencing collisions with both Hamiltonian and nonHamiltonian interactions, yet the systems are isolated from the rest of the universe, thus also verifying total conservation laws. This is the case at a classical level of the structure of a planet such as Jupiter, or a nucleus at the operator level. Since Santilli's isotopic product is also antisymmetric, [A, B]^{*} = A x T x B  B x T x A =  [B, A]^{*}, it allows total conservation laws, such as
(2.30) i x dH/dt = [H, H]^{*} = H x T x H  H x T x H = 0,
where H is the conventional Hamiltonian and T represent all contact nonHamiltonian interactions and effects (see Chapter 3).
Hence, the Lieisotopic theory cannot be a final theory because the systems of the physical reality are, in general, open, nonconservative and irreversible, as it is the case for a constituent of Jupiter or a proton in the core of a star when considering the rest of the system as external.
Santilli then searched for a covering of Lie's isotopic theory with a product (A, B) that is neither totally antisymmetry not totally symmetric, (A, B) ≠ ± (B, A) as a condition to characterize timerateofvariations f(t) of physical quantities,
(2.31) i x dH / dt = (H, H) = f(t) ≠ 0,
since conservation laws are a trivial particular case.
While doing his Ph. D. studies at the University of Torino,. Italy, in the late 1960s, Santilli conducted for years a comprehensive search at European mathematical libraries to identify the desired covering of Lie's theory. He was finally rewarded with the identification of a paper of 1947 by the American mathematician A. A. Albert who introduced, without any specific realization or elaboration, the notion of Lieadmissible algebras as a (generally nonassociative) algebra U with abstract elements a, b, c, ... and abstract (generally nonassociative) product ab such that the attached algebra U^{} given by the same vector space as U but equipped with the product [a, b] = ab  ba, is Lie.
Albert also introduced the notion of Jordanadmissible algebra as the same algebras U when such that the attached algebra U^{+} with product {a, b} = ab + ba is Jordan. Following additional extensive library search, Santilli could only identify in European mathematics libraries a second note in the field by M. Tomber, although without realizations or elaborations.
Inspired by Albert's paper, Santilli published in 1967 the following paper on the embedding of Lie algebras in Lieadmissible algebras verifying central condition (2.31)
Embedding of Liealgebras in nonassociative structures
R. M. Santilli, Nuovo Cimento Vol. 51, 570576 (1967).
where he presented for the first time a specific realization of Lieadmissible and Jordanadmissible algebras with product (A, B) = p x A x B  q x B x A identified in more details in Section 3.8.
To understand the novelty of this paper (and others by Santilli written in 19671968 not quoted here for brevity), we recall that in 1967 Santilli moved from the University of Torino, Italy, to the University of Miami, Coral Gables, Florida, for a one year stay. During that time, he applied for a job to virtually all Departments of Physics in the U. S. A. by presenting with pride his discovery of Lieadmissible and Jordanadmissible algebras and their applications for the characterization of the time rate of variation of physical quantities. To his demise, no physicist in the U. S. A. knew the existence or meaning of these algebras at that time.
Numerous applications for a job at various U. S. Departments of Mathematics turned out also to be sterile because of the general lack of knowledge by mathematicians of the time of the algebras herein considered. In fact, the above 1967 paper was the very first on Lieadmissible and Jordanadmissible algebras in the physic literature and it was the mere third paper in the field in the mathematics literature, including the two preceding papers by Albert and Tomber duly quoted in the above listed reference.
In this way, Santilli understood that there was no possibility to secure an academic job in the USA with so advanced a research. He then turned his attention to orthodox lines of studies and soon got a position at Boston University. During the subsequent ten years, Santilli published excellent but fully aligned papers at Phys. Rev. MIT Annals of Physics and orthodox journals of that nature.
It was only in 1978 that Santilli decided to return to his "first scientific love" and released his studies on Lieadmissible algebras in the historical 200 pages long memoir quoted in Section 2.7. He subsequently developed further these studies resulting in a covering of his Lieisotopic theory today known as Santilli's Lieadmissible theory (or genotheory), that is based on the preceding discoveries of genofields, genospaces, genodifferential calculus, etc. and can be outlined via the following branches:
^{b}1^{f}) An enveloping genoassociative algebra to the right (forward) U^{f} with ordered product to the right over the genofield to the right F^{f}(n^{f}, x^{f}, I^{f}) with elements given by: the genounit to the right I^{f} = 1/S; the generators G_{k}, k = 1, 2, ...,. N (as for the original Lie algebra), ordered genoassociative products to the right and related genounit
(2.32) I^{f} = 1/S, G_{i} x^{f} G_{j} = G_{i} x S x G_{j}, (2.33) I^{f} x^{f} G_{k} = G_{k} x^{f} I^{f} = G_{k} ∀ G_{k} ε U^{f};
the infinitedimensional genobasis acting on a genomodule to the right u^{r}
(2.34) U^{f}: I^{f}, G_{k}, G_{i} x^{f} G_{j}, i ≤ j G_{i} x^{f} G_{j} x^{f} G_{k}, i ≤ j ≤ k, , ....
and related genoexponentiation to the right
(2.35) W^{f}(w^{f}) = I^{f} + i^{f} x^{f} w^{f} G/1! + ... = [exp(G x S x w x i)] x I^{f},
plus an enveloping genoassociative algebra to the left (backward) ^{b}U on the genofield to the left ^{b}F(^{b}n, ^{b}x, ^{b}I) with elements: the genounit to the left ^{b}I = 1 / R; the generators _{k}G (ordered to the left), k = 1, 2,..., N, with genoproduct and genounit
(2.36) ^{b}I = 1/R, _{i}G ^{b}x _{j}G = _{i}G x S x _{j}G, (2.37) ^{b}I ^{b}x _{k}G = _{k}G ^{b}x ^{b}I = _{k}G ∀ _{k}G ε ^{l}U
the infinite dimensional genobasis acting on a genomodule to the left ^{b}u (where now u^{b} ≠ ± ^{l}u)
(2.38) ^{b}U: ^{b}I, _{k}G, _{i}G ^{b}x _{b}G, i ≤ j, _{i}G ^{b}x _{j}G ^{b}x _{k}G, i ≤ j ≤ k, ...
genoexponentiation to the left
(2.39) ^{b}W(^{b}w) = ^{b}I x [ exp(i x w x R x G)],
and subsidiary condition
(2.40) ^{b}I = (I^{f})^{†},
that are important in physical applications, e.g., to connect consistently motions forward and backward in time. The combination of the two genoenvelopes ^{b}U x U^{f} then acts on the genobimodule ^{b}u x u^{f} which is the representation genospace of the theory.
^{b}2^{f}) Santilli's Lieadmissible algebras ^{b}L^{f} as the bimodular algebra attached to ^{b}U x U^{f} characterized by the new product and closure rules
(2.41) ^{b}L^{f}: (_{i}G, G_{j}) = _{i}G ^{b}x _{j}G  G_{j} x^{f} G_{i} = G_{i} x R x G_{j}  G_{j} x S x G_{i} = ^{b}C^{fk}_{ij} x G_{k},
where the last expression is the projection of the algebra in the space of the original Lie algebra. As one can see, the resulting new product, here generically written (A, B) = A x R x B  B x S x A, is indeed jointly Lieadmissible and Jordan admissible although in Santilli's isotopic sense because
(2.42) [A, B]^{*} = (A, B)  (B, A) = A x L x B  B x L x A, L = R + S, (2.43) {A, B}^{*} = (A, B) + (B, A) = A x J x B + B x J x A, J = R  S, ^{b}3^{f}) Santilli genotransformation groups ^{b}g^{f} characterized by the left and right genoexponentiations (here written for simplicity in the representation space of the original Lie algebra)
,p>
(2.44) ^{b}g^{f}: W^{f} x^{f} Q(0) ^{b}x ^{b}W = exp(G x S x t x i) x Q(0) x exp(i x t x R x G).
with infinitesimal version characterized precisely by Santilli Lieadmissible brackets in the following simplified form
(2.45) i x dQ/dw = (Q, G) = A x R x G  G x S x Q,
where G and w are the same generator and parameter as those of the attached Lieisotopic algebra.
Unexpectedly, Santilli proved that his Lieadmissible covering of his Lieisotopic theory is isomorphic to the Lieisotopic and conventional Lie theory despite the lack of totally antisymmetric character of the product. This property was proved by noting that each of the two genoassociative algebras ^{b}U and U^{f}, when defined on the respective modules ^{b}u and u^{f} over the corresponding genofields ^{b}F and F^{f} are isomorphic to the conventional associative enveloping algebra U over F.
The understanding of this important mathematical property, rather crucial for quantitative representations of irreversibility, can be seen by noting that, when the two products A x R x B and B x S x A are considered with respect to the conventional unit I of Lie's theory, the two algebras with products (A, B) = A x R x b  B x S x A and [A, B] = A x B  B x A are manifestly nonisomorphic.
However, when the product A x R x B is computed with respect to the genounit I^{f} = 1/R, the result is equivalent to that of the product B x S x A represented with respect to the genounit ^{b}I = 1/S, and both products are equivalent to the product A x B with respect to the unit I.
Santilli decomposed his Lieadmissible product into a totally antisymmetric and a totally symmetric forms,.
(2.46) (A, B) = [A, B]^{*} + {A, B}^{*} = = (A x Y x B  B x Y A) + (A x Z x B + B x Z x A), R = Y + Z, S = Y  Z,
and proved the following important
THEOREM 2.8.1: Lieadmissible algebras with product (A, B) are "directly universal," in the sense of admitting as particular cases all possible algebras on a field of characteristic zero (universality) without use of the transformation theory (Direct universality).
In fact Santillis Lieadmissible algebras contain as particular cases all known or otherwise possible algebras with a bilinear composition law, such as: associative, flexible, alternative, Lie, Jordan, Lieisotopic, Jordanisotopic, supersymmetric and any other possible algebra.
Figure 2.3: Another "vignette" presented by Santilli to his colleagues at the Lyman laboratory of physics of Harvard University at the initiation of his stay there in September 1978 under DOE support, illustrating the universality of Lieadmissible algebras because containing as particular cases all known or otherwise possible algebras. The study of Lieadmissible algebras encountered extreme opposition at Harvard University because of their consequential generalization of Einsteinian doctrines, quantum mechanics, quantum chromodynamics, and all that, as indeed achieved by hadronic mechanics thanks to Santilli's resilience to academic vexations (see the Ethical Notes of Sections 3.10, 3.11, 3.12)
The above presentation is solely intended for Lieadmissible treatment of matter in irreversible conditions. For the corresponding treatment of antimatter we have Santilli's isodual Lieadmissible theory, that can be constructed via the application of the isodual map (2.9) to the totality of the quantities and their operation of the Lieadmissible theory.
The Lieadmissible covering of the Lieisotopic theory was presented in the historical 200 pages long memoir quoted above
On a possible Lieadmissible covering of Galilei's relativity
in Newtonian mechanics for nonconservative and Galilei formnoninvariant systems
R. M. Santilli, Hadronic J. Vol. 1, 223423 (1978)
with physical applications presented in the joint memoir submitting hadronic mechanics as a covering of quantum mechanics
Need of subjecting to an experimental verification the validity within a hadron of Einstein special relativity and Pauli exclusion principle
R. M. Santilli, Hadronic J. Vol. 1, 574901 (1978).
Santilli then developed further his Lieadmissible theory in the two volumes
"LieAdmissible Approach to the Hadronic Structure, I: Non applicability of the Galilei and Einstein Relativities,"
R. M. Santilli, Hadronic Press (1978)
LieAdmissible Approach to the Hadronic Structure, II: Coverings of the Galilei and Einstein Relativities"
R. M. Santilli, Hadronic Press (1982)
Additional presentations were made by Santilli in the two monographs published by SpringerVerlag in 1978 and 1982; a recent update is available in the 2008 monograph
"Foundations of Theoretical Mechanics, I: The Inverse Problem in Newtonian Mechanics"
R. M. Santilli, SpringerVerlag (1978)
"Foundations of Theoretical Mechanics, II: Birkhoffian Generalization of Hamiltonian Mechanics"
R. M. Santilli, SpringerVerlag (1982).
Subsequently, Santilli wrote a comprehensive presentation of his Lieadmissible theory and its isodual in the two monographs
"Elements of Hadronic Mechanics", Vol. I: "Mathematical Foundations"
R. M. Santilli, Ukraine Academy of Sciences. 1st edition (1993),
"Elements of Hadronic Mechanics"
Vol. II: "Theoretical Foundations"
R. M. Santilli, Ukraine Academy of Sciences, 1st edition (1994)
A more recent presentation is available in the monograph
Hadronic Mathematics, Mechanics and Chemistry, Vol. III: Iso, Geno, HyperFormulations for Matter
and Their Isoduals for Antimatter
R. M. Santilli, International Academic Press (2008),
and the following memoir published by the Italian Physical Society where Santilli applies his Lieadmissible theory for the first and only known invariant representation of irreversibility for matter and, separately, for antimatter, originating at the most elementary level of nature, that of elementary particles and antiparticles
Lieadmissible invariant representation of irreversibility for matter and antimatter at the classical and operator level
R. M. Santilli, Nuovo Cimento B Vol. 121, p. 443595 (2006)
The most active mathematician in the study of Santilli's Lieadmissible algebras has been H. C. Myung,, e.g., with the monographs
"Lie Algebras and Flexible LieAdmissible Algebras"
H. C. Myung,
Hadronic Press (1982)
"MalcevAdmissible Algebras"
H. C. Myung,
Birkhauser (1986)
"Mutations of Alternative Algebras"
A. Elduque and H. C. Myung,
Kluwer academic Publisher (1994)
Numerous mathematical works on lieadmissibility can be located in the
General Bibliography on Santilli Discoveries
2.9. INTEGRABILITY CONDITIONS FOR THE EXISTENCE OF A LAGRANGIAN
2.9A. Integrability conditions in Newtonian mechanics (1978).
Virtually the entire scientific production of the 20th century was based on the use of Lagrangian or Hamiltonian representations of Newtonian systems, then extended to operator formulations. Beginning with his graduate studies, Santilli set himself up to broaden these representations so as to avoid excessive abstractions and simplifications of reality, since said representations apply for pointlike approximation of particles with sole actionatadistance interactions.
Hence, as part of his program, Santilli conducted a comprehensive study of the broadest possible systems representable via a Lagrangian or a Hamiltonian, and conducted this study via a systematic analysis of the integrability conditions for the existence of a Lagrangian or a Hamiltonian for given dynamical systems, called conditions of variational selfadjointness.
By indicating with the symbol SA (NSA) forces or equations verifying (violating) the conditions of variational selfadjointness, Santilli writes Newton's equations in a form decomposing forces into a component derivable from a potential (SA) and a second term representing all forces not derivable from a potential )(NSA)
(2.47) m x dv/dt  F^{SA}(r, v)  F^{NSA}(t, r, v, ...) = 0.
and writes Hamilton's equations (1.2) in the corresponding form
(2.48) ( dr/dt  ∂ H(r, p)/∂ p )^{SA} = 0, (2.49) [ (dp/dt + ∂ H(r, p)/∂ r )^{SA}  F(t, r, p, ...) ]^{NSA} = 0.
As we shall see in Chapter 3, Santilli's mathematics allows the reformulation of Newton's equation into an identical selfadjoint form merely formulated on generalized spaces over generalized fields. The regaining of selfadjointness permits the recovering an action function with consequential means for a rigorous map to contact nonHamiltonian interactions into operator forms, with endless applications.
These studies resulted in the two monographs indicated earlier publisher by SpringerVerlag, written when he was at Harvard University, with the most comprehensive references in the field up to 1982 that required Santilli one full year of search in the Cantabridgean libraries (an impeccable ethical conduct that is per se a great rarity in the contemporary widespread plagiarisms in science),
"Foundations of Theoretical Mechanics, I: The Inverse Problem in Newtonian Mechanics"
R. M. Santilli, SpringerVerlag (1978)
"Foundations of Theoretical Mechanics, II: Birkhoffian Generalization of Hamiltonian Mechanics"
R. M. Santilli, SpringerVerlag (1982).
2.9B. Integrability conditions in field theory (1975)
As indicated earlier, Santilli conceived first in 1967 his Lieadmissible theory and then studied its Lieisotopic particularization. He did the same for the conditions of variational selfadjointness. In fact, when he was at the Center for Theoretical Physics of the Massachusetts Institute of Technology in the mid 1970s, Santilli first studied the integrability conditions for the existence of a Lagrangian or a Hamiltonian for the most general possible tensorial field equations, and published their simplification for Newtonian systems reviewed above, written subsequently when he was a Harvard University.
These studies produces the following three memoirs that constitute the most serious scholar works in the field ands remain grossly unsurpassed to this day in their essential results and mathematical rigor
Necessary and sufficient conditions for the existence of
a Lagrangian in field theory, I: Variational approach to
selfadjointness for tensorial field equations
R. M. Santilli, (MIT) Ann. Phys. Vol. 103, 354408 (1977)
Necessary and sufficient conditions for the existence of
a Lagrangian in field theory, II: Direct analytic representation of tensorial field equations
R. M. Santilli, (MIT) Ann. Phys. Vol. 103, 409468 (1977)
Necessary and sufficient conditions for the existence of
a Lagrangian in field theory, III: Generalized analytic representations of tensorial field equations
