Ruggero Maria Santilli

P. O. Box 1577, Palm Harbor, FL 34682

Curriculum Vitae

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Last revision March 11, 2003

ISOPARTICLES, THE NEW HADRONIC CONSTITUENTS AS CHARACTERIZED BY THE POINCARE'-SANTILLI ISOSYMMETRY AND RELATIVISTIC HADRONIC MECHANICS

The new physics of strongly interacting particles permitted by hadronic mechanics is based on the following main assumptions formulated at the time of the original proposal (38) of 1978 to build hadronic mechanics, that have remained essentially unchallenged through the subsequent years:

**
POSTURALE I: **

*
POSTULATE II: Hadronic mechanics permits a new structure model of each individual hadron based on physical massive constituents that can be produced free in the spontaneous decays, generally those with the lowest mode; and *

*
POSTULATE III: Quarks are mathematical quantities permitting the the new structure models of hadrons to be compatible with the SU(3)-color classifications.*

The beauty and validity of SU(3)-color models for the Mendeleev-type classification of hadrons into families is now part of history and needs not to be repeated here. It is sufficient to recall the truly incredible capability of these models to predict new hadrons that were subsequently experimentally confirmed, much along the predictive capacities of the Mendeleev classification for new yet undetected atoms.

The new structure models of mesons with ordinary massive constituents permitted by hadronic mechanics will be studied later on in this web page.

In this section we point out some of the numerous **insufficiencies of the conjecture that quarks are the ultimate, "elementary" and "physical" constituents of hadrons.**

To begin, it is important to recall that, if a physics professor at the beginning of the 20-th century had requested one of his graduate students to prepare his thesis on the use of the Mendeleev classification of atoms to provide a joint structure model of each atom of a given family, that professor would have lost his/her tenure because * the history of science has established the general need of TWO different, yet compatible models, one for the CLASSIFICATION and a separate model for the STRUCTURE of each element of a given family.*

Regrettably, this historical teaching was abandoned in the hadron physics of the 20-th century and one single model, the SU(3)-color model, was assumed for BOTH the CLASSIFICATION of hadron into family AND the STRUCTURE of each individual hadron of a given SU(3) family.

In reality, the SU(3) classification did prove to be a solid scientific achievement, while the joint assumption that quarks are the ultimate, elementary and physical constituents of hadrons has been faced with an ever increasing number of controversies, and will not resist the test of time.

The history of science has also thought that *mathematical and physical methods so effective for the "classification" are not generally applicable to the different problem of "structure," the latter generally requiring a generalization of the former.* For instance, after decades of failed attempts in the study of the structure of atoms via the method so effective for their Mendeleev classification, the need for broader mathematics and mechanics became compelling, thus mandating the construction of quantum mechanics and its underlying infinite-dimensional spaces as it is now part of history.

Unfortunately, the hadron physics of the 20-th century abandoned this additional historical teaching and *tacitly* assumed that the same mechanics so effective for the SU(3) classification of hadrons (quantum mechanics) is also exactly valid for the interior of hadrons, without any serious scrutiny of the credibility of such a basic assumption and by actually ignoring a large number of evidence to the contrary.

The physical conditions used for the classification of hadrons are those of isolated particles moving in vacuum under long range electromagnetic interactions, as typically expressed by particle accelerators, under which conditions the exact validity of quantum mechanics is beyond doubt. By comparison, the assumption that the same mechanics is also *exactly* valid in the hyperdense medium in the interior of hadrons does not constitute a scientific position, because under so drastic differences of physical conditions the only possible scientific issue is "which generalization" of quantum mechanics applies.

Ironically, the insistence on the exact validity of quantum mechanics in the interior of hadrons has been one of the origins of the controversies on quark conjectures, such as the **lack of an exact confinement of quarks,** i.e., a confinement establishing via proved theorems the inability of quarks to pass through a potential barrier in violation of Heisenberg's uncertainty principle and other basic axioms of quantum mechanics. As well known, when examined on tigour grounds, all available quark models do admit a finite probability of tunnel effects into free conditions, which prediction is contrary to all available experimental evidence. form.

The above occurrence is ironic because the assumption of a generalization of quantum mechanics in the interior of hadrons while continuing to assume quantum mechanics as valid in the exterior does indeed imply the above indicated "true confinement" of quarks, trivially, because of the incoherence of the external and internal Hilbert spaces (see Kalnay (216), Kalnay and Santilli (217), and Santilli (227)).

The latter studies have resulted in a new quark theory in which quarks are called **isoquarks** and are defined as the fundamental (regular) representation of the iso-SU(3)-color.

Recall that the exterior-classification of hadrons is defined on the ordinary Hilbert space with inner product (s|x|s) and expectation value (s|xPx|s), while, for the isoquark theory the interior-structural problem is defined on the iso-Hilbert space with isoinner product (s^|xTx|s^) and isoprobability (s^|xTxPxTx|s^). It is then easy to see that, if |s) represents a free quark and |s^) represents an isoquark, the incoherence of the interior and exterior hilbert spaces prohibits any finite probability for |s^> to become free, that is, *to preserve its isotopic character while moving in the exterior vacuum.* In fact, under the indicated assumptions we have the universal and rigorously proved property

(1.1) (s^|x|s) = ((s^|xTxP)x|s) = 0, |s^) = Ux|s), T = UxU^{+} ≠ 1,

This assures the absolutely null probability that an isoquark can propagate in the outside of hadrons, that is, can be free, as *necessary* for the conjecture that quarks are physical particles to acquire the very premises of credibility.

The case is even more ironic if one understands that **all isotopic symmetries are locally isomorphic to the original symmetries** because of the positive-definite character of the isounit (55). In fact, the isotopic SU^(3) symmetry is indeed locally isomorphic to the conventional SU(3), as first proved by Mignani (180). Therefore, **isotopic SU(3)-color model are mathematically and physically equivalent to the conventional models yet they have a rigorously proved identically null probability of tunnel effects of free isoquarks.**

The irony becomes even greater if one understands that **all perturbative series that are divergent for SU(3) become converged under isotopy ** (see Jannussis and Mignani (175), Santilli (55) and others). This is due to the fact that *all isounits (isotopic elements) are bigger (smaller) than one in absolute value,* thus implying the following isorenormalization of convergent series

(1.2a) A(k) = A(0) + kx(AxH - HA)/1! + ... -> divergent,

(1.2b) A(k) = A^(0) + kx(AxTxH - HxTxA)/1! + ... -> N = convergent

(1.2c) k > 1, | T | << 1.

Since mathematics is a rigorous science beyond personal beliefs, the above feature of hadronic mechanics expresses a realistic possibility of achieving, in due time, a **convergent perturbative theory for strong interactions.**

Despite the possibility of alleviating some of the inconsistencies or insufficiencies of the hadron physics of the 20-th century via the use of hadronic mechanics, the conjecture that quarks or isoquarks are the ultimate, elementary physical constituents of hadrons * cannot* resist the test of time for numerous reasons, such as:

I) **Quarks and isoquarks are purely mathematical objects,** since, according to their own definition, they are **mathematical representations of a mathematical symmetry defined in a mathematical unitary-complex space.** In fact quarks (isoquarks) are the fundamental-regular representation of the symmetry (isosymmetry) SU(3) defined in their unitary (isounitary) carrier space.

II) **Quarks cannot be defined, let alone detected, in our spacetime** in view of the O'Rafearthaigh Theorem (that prohibits mixing of spacetime and internal symmetries), the fact that quarks * are not* be admitted by the fundamental symmetry of spacetime, the Lorentz-Poincare' symmetry (due to their fractional charges), and other reasons.

III) ** The so called "quark masses" are purely mathematical parameters that cannot be defined in our spacetime.** A well known fundamental condition for any mathematical parameter to acquire the physical meaning of actual inertial masses is that of being the eigenvalue of the second order Casimir invariant of the Poincare' symmetry,

(1.3) p^{2}x|s) = m^{2}xc^{2}x|s).

But quarks * cannot* be characterized by the Poincare' symmetry, as well known. Therefore, the idea that quark masses have inertia is a pure figment of academic imagination deprived of real scientific credibility.

**Since quarks are purely mathematical quantities by their own definitions, quarks cannot be even defined, let alone detected in our spacetime for numerous technical reasons and quarks cannot possess inertial masses, HOW CAN ANYBODY ASSUME WITHOUT CRITICAL SCRUTINY THAT QUARKS ARE THE ACTUAL, ELEMENTARY PHYSICAL CONSTITUENTS OF HADRONS AND EXPECT ACCEPTANCE BY THE PHYSICS COMMUNITY AT LARGE?**

The above are only some of the inconsistencies, insufficiencies and problematic aspects of quark conjectures as expressed back in 1978 (38). Since that time, numerous additional problematic aspects have occurred, such as the inability of quark conjectures to permit a rigorous interpretation of hadron spins, and other problematic aspects that are here ignored because they are redundant over the basic structural ones indicated earlier.

This illustrates some of the reason for assuming Postulate III (quarks are purely mathematical objects, as) a necessary condition for new advances, including new clean energies. In reality, as we shall see, quark conjectures are the biggest obstruction against industrial applications of hadron physics, that is, of turning hadron physics, from, its purely academic contemporary character, to a discipline as industrially meaningful as that of electromagnetic interactions.

**2. CONCEPTUAL FOUNDATIONS OF THE NEW STRUCTURE MODEL OF HADRONS.**

Let us recall the historical distinction between **exterior and interior dynamical problems** that was dominant in physics up to the beginning of the 20-th century and then abandoned.

**Exterior problem** can be defined as being constituted of particles at sufficiently large mutual distances to permit their effective point-like approximation, said particles moving in vacuum without collisions under long range electromagnetic interactions. Majestic illustrations of exterior dynamical problems are the planetary systems (see Figure 1 below).

**Interior problems** can be defined as being constituted of extended particles moving within physical media, thus experiencing both long range as well as contact zero-range interactions. A majestic illustration of interior problem was usually assumed to be given by the structure of Jupiter (see Figure 2 below).

Exterior and interior problems admit numerous additional differentiations. For instance, exterior problems are reversible in time (invariant under time reversion), while interior problems are generally irreversible in time. Also, exterior problems can be completely described via the sole knowledge of a potential. By comparison, this is basically insufficient for interior problems because the contact zero-range resistive interactions cannot be described with a potential or a Hamiltonian.

The above differentiations are sufficient to mandate two different mathematics and mechanics for the description of exterior and interior problems. In fact, the conventional Hamiltonian mechanics (that with the "truncation" of Hamilton's external terms - see below) is fully sufficient to describe exterior problems. By comparison, a correct description of interior problems requires at least *two* quantities, the Hamiltonian for long range potential interactions, and an additional quantity for nonpotential interactions.

Iso-Hamiltonian mechanics has already been proved to be effective for interior dynamical problems because it is the only known mechanics capable of providing an *invariant* description of *all* possible Newtonian interior problems directly in the frame of the experimenter (invariant direct universality).

The conceptual foundations of the new structure model of hadrons permitted by hadronic mechanics are:

*1) The SU(3)-color classification constitutes an exterior dynamical problems, in which case quantum mechanics is exactly valid.
*

*
2) The structure of each given hadron of a given SU(3) multiplet constitutes an interior dynamical problem for which the covering hadronic mechanics applies.*

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For these reasons, Santilli (52-55) built the **isotopic covering of the Galilean and special relativity** with contact interactions invariantly represented by the isounit precisely for the treatment of interior structural problems such as that of Jupiter. Intriguingly, rather than "breaking" the fundamental Galilean and Poincare'
symmetries, Santilli (loc. cit.) proved their **exact** character also for interior problems, when formulated on isospaces over isofields.

*
The classical illustrations of this figure also provides the conceptual foundations of the new physics permitted by hadronic mechanics. In fact, hadrons, nuclei and molecules are indeed interior dynamical problems since none of them admits a Keplerian center, being all composed of constituents in contact with each others. The lifting of the Galilean and Poincare' symmetries via a generalization of the basic unit representing precisely contact forces assures a correct interpretation of the interior structure while admitting exterior conditions at the limit when the isounit I^ recovers the trivial value I.,*

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The distinction between exterior and interior systems was abandoned at the beginning of the 20-th century due to the successes of Hamiltonian and quantum mechanics, thus implying their reduction of the entire universe to massive points that, being dimensionless, cannot experience resistive forces. In this way, the structure of Jupiter as well as of hadrons was reduced to an ensemble of massive points moving in vacuum. A main argument was that "nonconservative Newtonian systems are illusory" because macroscopic nonconservative forces disappear when matter is reduced to its point-like constituents, in which case a conventional Hamiltonian is sufficient for the classical and quantum description of structural problems.

However, visual observation established by the picture of Figure 1 establishes the presence in Jupiter's structure of **vortices with varying angular momenta and other clearly nonconservative internal effects.** The possibility of reducing Jupiter's structure to dimensionless points has been disproved by the following:

**THEOREM 2.1 (224):** *A classical irreversible system cannot be consistently decomposed into a finite number of elementary constituents all in reversible conditions and, vice-versa, a finite collection of elementary constituents all in reversible conditions cannot yield an irreversible macroscopic ensemble.*

<.i>
The occurrence established by the above theorems dismiss as nonscientific
20-th century views on the reduction of Jupiter to point-like constituents,
and identify the real scientific needs, the construction of formulations
that are ** structurally irreversible,** that is, irreversible for all
possible Hamiltonians, and are applicable at all levels of study,
from Newtonian mechanics to second quantization.

It should be indicated that the above scientific imbalance existed only in the
20-th century because **Newton's equations (1) are generally irreversible** since Newton's force F(t, x, r) can be decomposed into the sum of two forces, the first that is of potential type (technically called *variationally selfadjoint,* SA) and the second that is of nonpotential type (technically called *variationally
nonselfadjoint,*NSA) (48,51)

(2.1a)) mxdv/dt = F^{NSA}(t, r, v).

,

(2.1b) F^{SA}(r) = - DV/Dr, F^{NSA} ≠ -DV/Dr.

where D/Dv represents partial derivative (see Section I.1.1 for notations). It is evident that, since all
known F^{SA} are reversible, ** irreversibility in Newtonian mechanics
originates in the contact nonpotential forces** F^{NSA}.

In a way fully aligned with Newton's teaching, Lagrange (2) and Hamilton (3)
formulated their celebrated analytic equations in terms of a function, today called
the Lagrangian L(r, v) and the Hamiltonian H(r, p), representing all F^{SA},
plus ** external terms** representing precisely the contact nonpotential
forces F^{NSA},

(2.2a) (d/dt) DL/Dv - DL/Dr = F^{NSA},

(2.2b) dr/dt = DH/Dp, dp/dt = - DH/Dr + F^{NSA}

<
(2.2c) H = p^{2}/2m + V(r),>br>

Regrettably, at the beginning of the 20-th century the above analytic equations were **truncated** with the removal of external terms, by acquiring the form of virtually universal use

(2.3a) (d/dt) DL/dv - DL/Dr = 0,

(2.3b) dr/dt = DH/Dp, dp/dt = - DH/Dr.

Since all known Lagrangians and Hamiltonians are reversible in time,
** according to the teaching of Lagrange and Hamilton,
exterior dynamical systems can be represented with the truncated equations (2.3), while interior dynamical systems necessarily require rather complete analytic equations (2.2) with external terms.**

It should be noted that the analytic equations with external terms cannot be used nowadays because they suffer from the catastrophic inconsistencies of Theorem I.2.1. To see this occurrence, it is sufficient to note that **the brackets of the time evolution characterized by Hamilton's equations with external terms**

(2.4) dA/dt = [A, H] + (DA/Dp)xF^{NSA},

**violate the right scalar and distributive laws and, therefore, they do not characterize any algebra, let alone any Lie algebra.** This implies the loss of ALL algebras, thus prohibiting the construction of a consistent covering theory (23). In any case, Eqs. (2.2) are manifestly noncanonical, thus manifestly activating Theorem I.2.1.

The above feature was identified in the original proposal to build hadronic mechanics (23) and constitutes the very foundations of the generalization of the truncated equations (2.3) that, on one side, is also directly universal for all possible interior systems and., on the otehr side, admits a consistent Lie algebra in the brackets of the time evolution and, above all, it is invariant. Iso-Hamilton's equations (I.3.9) are the result of the laborious search that lasted about two decades, and required the publication of monographs (48-55).

**The central classical assumption of these studies is the use of the truncated analytic equations for exterior dynamical problems and the use of their isotopic covering for the invariant treatment of interior problems.** In fact, Eqs. (I.3.9) are directly universal for all possible Newtonian systems; they can represent Jupiter as an ensemble of extended, nonspherical and deformable particles thanks to isounit (I.3.7c); and can indeed represent irreversible systems via an explicit time dependence of the isounit (although the broader genomechanics is suggested in this case (226)). Moreover, **the brackets of the isotopic time evolution**

(2.5) dA/dt = [A, H]* = (D^A/D^r)x(D^H/D^p) - (D^H/D^r)x(D^A/D^p),

**verify all right and left distributive and scalar laws, thus characterizing an algebra, and that algebra turns out to be a Lie-Santilli isoalgebra** (23,62-68), namely, an algebra verifying the Lie axioms on isospaces over isofields.

Note the crucial role of the isodifferential calculus for the achievement of the above results. In fact, the recovering of a Lie algebra in the brackets of the time evolution laws under nonselfadjoint forces is due precisely to the embedding of the latter forces in the differential. In turn, this illustrates the very meaning of isotopies since isobrackets (2.5) coincide at the abstract level with the conventional Lie brackets.

Another widespread belief is that "Newtonian systems with nonconservative internal forces do not permit total conservation laws." This belief is disproved by the visual observation of Jupiter that, when considered as isolated from the rest of the universe, it verifies all ten Galilean or Poincare' conservation laws (of the total energy, total linear momentum, total angular momentum and uniform motion of the center of mass). Yet Jupiter exhibits in its interior manifestly nonconservative internal effects, as indicated earlier.

The origin of the internal nonconservative effects is due precisely to the fact that* jupiter's structure is composed of extended constituents (atoms and molecules) because point-particles cannot have contact nonpotential effects.* As a result, **the structure of Jupiter was suggested as the classical image of the structure of hadrons since the original proposal (*23,38).**

In reality, whenever a system is isolated from the rest of the universe, internal nonconservative effects are indeed possible under the mere condition that they cancel each other. In particular, the ten Galilean total conservation laws are verified by nonconservative systems under the following conditions (see monograph (51) for details)

(2.6a) S_{k} F_{k}^{NSA} = 0,

(2.6b) S_{k}r_{k}(scalar product)F_{k}^{NSA} = 0,

(2.6c) S_{k}r_{k}(vector product)F_{k}^{NSA} = 0.

where S stands for sum. We reach in this way a basically new conception of systems called **isolated non-Hamiltonian systems** namely, systems that verify all total conservation laws (including that of the Hamiltonian). Yet they cannot be represented via the sole knowledge of the Hamiltonian, thus requiring a lifting of the unit as the second needed quantity. In preparation for the new structure model of hadrons herein considered, these new systems were extensively studied first at the classical level (51) and then at the operator level (55).

The new structure model of hadrons cannot be technically understood without a technical notion of the indicated classical and operator, closed non-Hamiltonian systems.

**3. THE NEW NOTION OF HADRONIC CONSTITUENTS.**

Our main assumption for the characterization of hadronic constituents, first submitted in memoir (38) of 1978, is that **in the transition from exterior to interior conditions (that is, from motion in vacuum to motion within hyperdense media in the interior of hadrons) conventional quantum mechanical particles experience a deformation of their "intrinsic" characteristics, called "mutation."**

The best illustration and first experimental verification is given by magnetic moments. In fact, it is well known that the deformation of a rotating charged sphere implies the necessary alteration (mutation in our terminology) of its magnetic moment. Until particles are abstracted as being point-like, no such mutation is evidently possible. However, mutations become inevitable the moment particles and/or their wavepackets are admitted as they are in the physical reality, that is, extended and, therefore, deformable. Once the deformability of particle, that is, the deformability of their wave packets is admitted, mutations are simply inevitable.

As an example, the magnetic moment of the electron has been measured with incredible accuracy countless times, but always in exterior conditions in vacuum under long range electromagnetic interactions. When the same electron is "compressed inside the proton" (according to Rutherford's original conception of the neutron of 1920 - see Part III), the assumption that the electron magnetic moment necessarily remains unchanged has no scientific value or credibility, because deformations of the electron wave structure are simply unavoidable when the electron is immersed within hyperdense media, with consequential mutation of its magnetic moment.

Similar mutations of all other intrinsic characteristics then follow either from mere compatibility arguments or via the use of the applicable isotransformation theory (see next section). Experimental verifications supporting hadronic mechanics are presented in Section 5.

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Mutations are also inevitable because of a number of other occurrences. As it is well known, * all interactions cause renormalizations of numerical values.* Again, these quantum mechanical renormalizations cannot affect intrinsic characteristics of particles because they are of potential-Hamiltonian type, as well known.

Along similar lines, **contact nonpotential interactions cause new renormalizations, called "isorenormalizations" that affect intrinsic characteristics of particles because characterized by the lifting of the basic unit,** as studied in detail in the literature (see, e.g. monograph (55) and references quoted therein). As a result, **isorenormalizations are synonym of mutations.**

It should be indicated that **an isolated bound system of isoparticles must constitute an ordinary quantum mechanical particle.** Therefore internal mutations of hadronic constituents will always be under the constrains of cancelling each other and result in no mutation for the state as a whole, much along the nonconservative effect of Jupiter (Figure 1) that are purely internal and cannot possibly result in the nonconservation of Jupiter's total energy.

Finally, we should indicate that **the probability for an isoparticle to become a particle coincides with the probability for that particle to be free in the exterior of hadrons.** In fact, isoparticles can only be defined on iso-Hilbert spaces that can only occur within the hyperdense medium inside hadrons. By comparison, a particle can only be defined on a conventional Hilbert space that can only occur in the exterior vacuum.

As we shall see, the notion of mutation is absolutely crucial for the identification of hadronic constituents with physical particles. In turn, the capability for hadronic constituents to be produced free is absolutely crucial for industrial applications of hadron physics.

**4. CHARACTERIZATION OF HADRONIC CONSTITUENTS VIA ISOSPECIAL RELATIVITY**

Recall that "particles" have been best characterized in the physics of the 20-th century via special relativity and, more specifically, as unitary irreducible representations (called "irreps") of the Poincare' symmetry defined in our exterior Minkowskian spacetime over the field of conventional real numbers.

Along similar lines, **"isoparticles" and their mutations can be best characterized via isospecial relativity and, more particularly, as isounitary irreducible representations (called "isoirreps") of the Poincare'-Santilli isosymmetry defined on the internal iso-Minkowskian spacetime over the internal fields of isonumbers.**

*4.1. ISOMINKOWSKIAN GEOMETRY.*

Let M(q,m,R) be the conventional *Minkowski spacetime* with local coordinates

(4.1) q = (q^{k}) = (r, t), k = 1, 2, 3, 4,

(4.2)
metric

(4.2)
m = Diag. (1, 1, 1, -1)

and invariant

(4.3) q^{2} = q^{i}xm_{ij}xq^{j}

defined on the field R of real numbers R.

The **iso-Minkowski space,** also called **Minkowski-Santilli isospace** M^(q^,m^,R^), first introduced in Ref. (26) (see also Refs. (29,226), monographs (55,63-68) and references quoted therein) of 1983, is characterized by the **spacetime isocoordinates (isospacetime)**

(4.4) q^ = (r^, t^) = qxI^,

**isometric **

(4.5) K^ = m^xI^ = (Txm)xI^,M^ = Txm, T = 4x4 positive-definite matrix,

(namely, a 4x4-matrix whose elements are isonumbers) and **isoinvariant**

(4.6) q^^{2}^ = q^^{i}*(m^_{ij}xI^)*q^^{j} =
[q^{i}x(Txm)_{ij}xq^{j}]xI^,

defined on the field R^ of **isoreal numbers,** all isotopic structures having the same 4x4-dimensional, nowhere singular, positive-definite **isounit **

(4.7) I^ = Diag. (g_{11}^{-1}, g_{22}^{-1}, g_{33}^{-1}, g_{44}^{-1}) =

= Diag (n_{1}^{2}, n_{2}^{2}, n_{3}^{2}, n_{4}^{2}) =

= Diag (b_{1}^{-2}, b_{2}^{-2}, b_{3}^{-2}, b_{4}^{-2}) = 1/T > 0,

and **isotopic element**

(4.8) T = Diag. (g_{11}, g_{22}, g_{33}, g_{44}) =

Diag (n_{1}^{-2}, n_{2}^{-2}, n_{3}^{-2}, n_{4}^{-2}) =

= Diag (n_{1}^{2}, b_{2}^{2}, b_{3}^{2}, b_{4}^{2})
> 0,

where the quantities g_{kk} = 1/n_{k}^{2} = b_{k}^{2} > 0 are expressed according to the main notations used in the literature (see Section 5), are sufficiently smooth and positive-definite, and have an unrestricted functional dependence on local variables, such as time t, space coordinates r, density d of the medium, local electric field E, local magnetic field B, etc., and are called **characteristic functions of the iso-Minkowski space.**.

By using the isodifferential calculus, the **isotopic line element** on M^ is given by

(4.9a) d^q^^{2}^ = (d^r^_{1}^{2}^ + d^r^_{2}^{2}^ +
d^r^_{3}^{2}^ - c_{o}xd^t^^{2}^)xI^ =

= (g_{11}xdr_{1}^{2} + g_{22}xdr_{2}^{2} +
g_{33}xdr_{3}^{2} - c_{o}xg_{44}xdt^{2})xI^

= (dr_{1}^{2}/n_{1}^{2} + dr_{2}^{2}/n_{2}^{2} +
dr_{3}^{2}/n_{3}^{2} - dt^{2}xc_{o}^{2}/n_{4}^{2})xI^ =

= (b_{1}^{2}xdr_{1}^{2} + b_{2}^{2}xdr_{2}^{2} +
b_{3}^{2}xdr_{3}^{2} - b_{4}^{2}xc_{o}xdt^{2})xI^

(4.9b) g_{kk}(q, v, d, E, B, ...) = 1/n_{k}^{2}(q, v, d, E, B, ...).

As we shall see at the end of this section, the g_{kk} elements have a direct gravitational meaning. However, this presentation is intended for the structure of hadrons. Therefore, we shall restrict the characteristic functions g_{kk} = 1/n_{k}^{2} = b_{k}^{2} to provide the **first known geometric representation of the density and shape of hadrons.**

It is easy to see that the isotopy M -> M^ permits **a direct geometrization of the locally varying speed of electromagnetic waves when propagating within physical media,**

(4.10) c_{o} -> c = c_{o}/n_{4} = b_{4}c_{o},

where n_{4} = b_{4}^{-1} is the familiar **index of refraction. ** Therefore, the characteristic function n_{4} = b_{4}^{-1} provides **a direct geometrization of the density of the medium inside hadrons** that is normalized to n_{4} = 1 for the case of the vacuum.

Note that the iso-Minkowskian geometry implies no restriction on the numerical value of n_{4} = b_{4}^{-1} except for not being null (to avoid a gravitational singularity, see below). Therefore, n_{4} = b_{4}^{-1} can be
smaller, equal or bigger than 1. It then follows that **the iso-Minkowskian geometry predicts that, depending on the density of the medium, electromagnetic waves can propagate at speeds smaller, equal or bigger than their speed in vacuum.** In particular, speeds

(4.11) c = c_{o}/n_{4} = b_{4}xc_{o} < 1, n_{4} > 1, b_{4} < 1,

represents the well known propagation of electromagnetic waves in media of relatively low density in our macroscopic environment, such as air, water, oil, etc. On the contrary,

(4.12) c = c_{o}/n_{4} = b_{4}xc_{o} > 1, n_{4} < 1, b_{4} > 1,

represents the propagation of electromagnetic waves within special guides, as experimentally established (for brevity,see Ref. (120) and experimental papers quoted therein) as well as, most importantly for new clean energies, the propagation of electromagnetic waves within hyperdense media as existing in the interior of hadrons, nuclei and stars. As we shall see in Section 5, ** ALL fits of experimental data in hadron physics via hadronic mechanics confirms that the causal speed in the INTERIOR of hadrons is bigger than the speed of light in vacuum.** In turn, this feature has far reaching implications, such as the **prediction of new clean energies, elimination of the need for dark matter,** and others.

By no means causal speeds c > c_{o} within hadrons imply that the hadronic constituents are **tachyons** because they are ordinary massive particles simply immersed within a medium whose causal speed is bigger than that in vacuum. In fact, to be tachyons under these conditions, the hadronic constituents should have speeds bigger than c = c_{o}/n_{4}, in which case acausal events can be proved by graduate students.

Note that the existence of speeds c > c_{o} is inevitable from isorenormalizations or mutations. As we shall see in Parts III and IV, speeds c > c_{o} in the interior of hadrons are crucial for the prediction and industrial development of new clean energies.

Note also that **the maximal causal speed on isospace over isofields remains the speed of light "in vacuum."** This is due to the fact that, on one side, the speed of light is lifted into the value

(4.13) c_{o}^{2} -> c^{2} = c_{o}/n_{4}^{2},

while, on the other side the related **unit** is lifted by the **opposite** amount

(4.14) 1_{4} -> n_{4}^{2},

thus resulting in an invariance of the original value c_{o} in view of the fact that line elements have the structure one can see from Eq. (4.5) (see also Figure 3 for details)

(4.15) Invariant = (distance)^{2}x(unit)^{2}

The space components n_{1}^{-2}, n_{2}^{-2}, n_{3}^{-2} represent the **actual, extended, nonspherical and deformable shape **of the particle considered, also normalized to 1 for the case of the perfect and rigid sphere.

The **space inhomogeneity** of the physical medium within hadrons is represented by the locally varying character of the n's. The **spacetime anisotropy** of said medium for the case of cylindrical symmetry along the third axis is represented by the difference between n_{3} and n_{4}.

(4.16) Anisotropy: n_{3} ≠ n_{4}.

However, again, **the iso-Minkowski space over isofields is fully homogeneous and isotropic.** Therefore, the indicated inhomogeneity and anisotropy solely occur in the **projection** of the iso-Minkowski space into our conventional spacetime.

We should finally mentioned that **the entire iso-minkowskian geometry can be uniquely and unambiguously constructed via a noncanonical or nonunitary transform of the conventional geometry according to rules (I.3.6) as the reader is encouraged to verify.**

*4.2. ISOPOINCARE' SYMMETRY.*

Let

.
(4.17) P(3.1) = SO(3.1) x T(a)

be the Poincare' symmetry, where SO(3.1) is the (connected) Lorentz symmetry and T(4) represents the (Abelian) symmetry under spacetime translations.

The isotopies

(4.18) P^(3.1) = L^(3.1)xT^(4),

provide the universal invariant of isotopic line element (4.9). The new isosymmetry P^(3.1) was first submitted by Santilli (26) in 1983, and then developed in various works (see, e.g., monographs (29,55)), and it is called today the **Poincare'-Santilli isosymmetry**, SO^(3.1) is called the **Lorentz-Santilli isosymmetry,** and T^(4) characterizes **isotranslations in isospacetime** (see monographs by mathematicians (64,66,67) and literature quoted therein).

To avoid structural inconsistencies that result in the loss of invariance (Theorem I.2.1), the construction of the Poincare'-Santilli isosymmetry requires the isotopies of the totality of the conventional symmetry, including the isotopies of numbers, Euclidean topologies, carrier spaces, functional analysis, differential calculus, Euclidean and symplectic geometries, universal enveloping associative algebras, Lie algebras, Lie transformation groups, symmetries and the representation theory (14). Clearly these aspects are beyond the limited scope of this presentation. Therefore, we can present here only a rudimentary review.

The isotopies of the Poincare' algebra can be essentially expressed by preserving the original ten generators J_{ij}, P_{k}, i, j, k = 1, 2, 3, 4, (although they are now defined on isospaces over isofields in which case we add the symbol "^" as per our notations) and by lift instead the Lie product from the familiar form [A, B] = AxB - BxA (where AxB is the usual associative product of matrices) into the **Lie-Santilli isoproduct **

(4.19) [A, B]* = A*B - B*A = AxTxB - BxTxA.

The transition from the isoalgebra to the isotransformation group is done via the **isoexponentiation**

(4.20) e^^{A} = I^x(e^{TxA}) = (e^{AxT})xI^,

where the isounit I^= 1/T is fixed for the entire algebras and constitutes the basic invariant of the new symmetry (the iso-Casimir invariant of order zero).

Note the appearance of the isotopic element directly in the exponent of the isotheory. This is strong evidence of the nontriviality of the isotopies because the original Lie algebras is always linear, local and canonical, while its isotopic image is generally nonlinear, nonlocal and noncanonical/nonunitary. Nevertheless, the isotheory reconstructs linearity, locality and canonicity/unitarity on isospaces over isofields, thus permitting, for the first time on scientific record, the invariant treatment of broader theories.

Under the condition for I^ to be positive-definite, the **Poincare'-Santilli isoalgebra** is given by (26)

(4.21a) [J^_{ij}, J^_{rs}]* = ix(m^_{jr}xJ^_{is} - m^_{ir}xJ^_{js}
- m^_{js}xJ^_{ir} + m^_{is}xJ^_{jr}),

(4.21b) [J^_{ij}, P^_{k}]* = ix(m^_{ik}xP^_{j} - m^_{jk}xP^_{i}),

(4.21c) [P^_{i}, P^_{j}]* = 0, i, j, r, s = 1, 2, 3, 4,

where m^ is the iso-Minkowskian metric (4.5).

Since T is positive-definite (that is, m^ has the same signature as m), it is easy to prove that **the Poincare'-Santilli isoalgebra is isomorphic to the conventional algebra,** and the same occurs for groups (see below). Therefore, **the isotopies render the Poincare' symmetry directly universal for all infinitely possible spacetimes with signature (+, +, +, -).**

Since the generators of any symmetry constitute conserved quantities, the above results are sufficient to guarantee that **the Poincare'-Santilli isosymmetry verifies all ten "conventional" total relativistic conservation laws.** Yet the symmetry is not purely Hamiltonian since it requires the knowledge of TWO quantities, the Hamiltonian P^_{4} and the isounit I^. Therefore, isosymmetry P^(3.10 is the best formulation of the new structure model of hadrons as "closed non-Hamiltonian systems."

The **iso-Casimir invariants** of P^(3.1) are

(4.22) I^, P^^{2}^ = P^*P^, W^^{2}^ = W^*W^, W^_{i} = e^_{ijrs}*J^^{jr}*P^^{s}.

By isoexponentiating the above isoalgebras via the use of Eq. (4.20), we reach the following explicit form of isotransforms:

*4.3. ISOROTATIONS*

(4.23a) r^{1}' = r^{1}xcos[a(g_{11} x g_{22})^{1/2}] -
r^{2}x g_{22}xg_{11}^{-1} x sin[a(g_{11} x g_{22})^{1/2}]

(4.23b) r^{2}' = r^{1}xg_{11}xg_{22}^{-1}xsin[a(g_{11}x
g_{22})^{1/2}] + r^{2}xcos[a(g_{11}xg_{22})^{1/2}],

where a is the angle of rotation in the 1-2 plane. For the general case in three dimensions, see Eqs. (6,3,21), p. 232 oof Ref. (55).

**4.4. LORENTZ-SANTILLI ISOTRANSFORMS**

(4.24a) r^{3}' = r^{3}x
cosh[v(g_{33}xg_{44})^{1/2}] -
q^{4}x g_{44}x(g_{33}xg_{44})^{-1/2}x sinh[v(g_{33}xg_{44})^{1/2}],

(4.24b) q^{4}' = r^{3}xg_{33}x(g_{33}g_{44})^{-1/2}xsinh
[v(g_{33}xg_{44})^{1/2}] +
q^{4}xcosh[v(g_{33}xg_{44})^{1/2}].

where v is the relative velocity in our spacetime, and can be exemplified in the form

(4.25a)
r^{3}' =g^x(r^{3} - bxq^{4}),

(4.25b)
q^{4}' = g^x(q^{4} -
b^xr^{3}),

(4.25c) q^{4} = c_{o}xt,
b^^{2} =
r^{k}xg_{kk}xr^{k}/
c_{o}xg_{44}xc_{o},
g^ = 1/(1 - b^^{2})^{1/2}.

For a comprehensive study on the Lorentz-Santilli isotransforms, interested visitors may consult monograph (55).

*4.5. ISOTRANSLATIONS*

(4.26) q^{k}' = q^{k} + A^{k}(q, ...),

(4.26b) A^{k} = u^{k}x(g_{kk} + u^{i}x[g_{ii}, P^{k}]* + ...).

*4.6. ISOTOPIC INVARIANCE*

(4.27) m^ -> m^' = nxm^, I^ -> I^' = I^/n,

where n is a positive scalar.

As one can see, **the Poincare'-Santilli isosymmetry is eleven-dimensional,** where the 11-th dimensionality is given by the evident invariance of line element (4.9) under isotopy (.4.27). Note that, contrary to popular belief throughout the 20-th century, **the conventional Poincare' symmetry is also eleven dimensional.** The 11-th dimension was not discovered because it required the prior discovery of **new numbers,** those with an arbitrary unit.

************************************
**

***************************
**

*********************************

*4.6. ISORELATIVITY*

Special relativity is generally presented by contemporary academia as
providing a description of all infinitely possible relativistic
systems existing in the universe. Such a view is purely political and without scientific foundation, because a scientifically serious approach requires the identification of the *limitations* of special relativity, as a pre-requisite for further advances.

To begin, special relativity *cannot* provide a consistent *classical*
description of point antiparticles moving in vacuum. The mere change of the sign of the charge is grossly insufficient and lead to gross inconsistencies. In fact, due to the existence of only one quantization channel, a classical "antiparticle" according to special relativity admits as quantum image a <"particle" with the wrong sign of the change. This is the reason why the isodual theory of antimatter was developed [5] including the construction of the novel **isodual special relativity for point antiparticles** we cannot possibly review here.

The content of Part I establishes that special relativity *cannot*
be exactly valid for extended particles (and antiparticles) moving
within physical media. Additionally,
special relativity *cannot* describe irreversible processes
for both matter and antimatter, trivially, because all known Hamiltonians are reversible and the axiomatic structure of special relativity is reversible too. Therefore, studies of structurally irreversible processes via a relativity that is totally reversible have no known value or credibility.

Finally, the complexity of biological systems is immensely beyond
the rather limited descriptive capacity of special relativity. As an illustration, by recalling that a pillar of special relativity (the rotational symmetry) solely applies to perfectly rigid bodies and the relativity is reversible, any insistence in biological applications of special relativity would imply that our bodies are perfectly rigid and fully eternal, thus resulting in a nontechnical nonscientific nonsense.

Particularly misleading is the widespread statement throughout the 20-th century of the
"universal constancy of the speed of light" because contrary to
known experimental evidence that ** the speed of light is a a
local variable depending on the medium in which it propagates,**
with well known expression (4.10), where the familiar index of refraction
possesses a rather complex functional dependence on frequencies, the density of the medium, and other variables.

For the evident intent of salvaging the desired universality of
special relativity, speeds c < c_{o} have been interpreted until
recently by reducing the propagation of light within a physical
medium to the propagation of photons scattering from
atom to atom. Since propagation in this case occurs in vacuum, supporters have the impression of salvaging their beloved relativity. However, such a reduction is not evidently
applicable to the propagation within physical media of radio waves
with wavelength of the order of one meter. The same reduction also
fails to provide a quantitative interpretation of the dependence
of the speed on the frequency, as visible by the naked eye in
Newton's spectral decomposition of light. In any case, the
reduction of light to photons scattering among atoms has been
definitely disproved by the recent experimental evidence of speeds
c > c_{o} occurring within special guides or within media of high
density (see Ref. (120) and literature quoted therein).

An illustration of the **inapplicability** (and not of the
"violation") of special relativity within physical media is given
by the** propagation of light and particles in water,** where the
speed of light is of the order of two thirds that in vacuum while
electrons can propagate with speeds * bigger* than the local value c,
resulting in the emission of the Cerenkov light.

In this case, if the local
speed of light c is assumed as the universal invariant, the
propagation of electrons at speeds v > c is a violation of the
principle of causality. If the speed of light "in vacuum"
c_{o} is assumed as the universal invariant "in water," there
is the violation of the relativistic law of addition of speeds because the sum of two speeds
of light c *does not* yield the local speed of
light c, as any student can easily verify, and there is the violation of other basic axioms of
special relativity (see monograph (55) for additional problematic
aspects).

It should be also indicated that, ** when applied to the
propagation of light and particles within physical media, special
relativity activates the catastrophic inconsistencies of Theorem
I.2.1.** This is due to the fact that ** the transition from the
speed of light in vacuum to that within physical media requires a
noncanonical or nonunitary transform.** This point can be best
illustrated by using the metric originally proposed by Minkowski,
that can be written $\eta = Diag. (1, 1, 1, -c_{o}^{2}). Then, the
transition from c_{o} to c = c_{o}/n_{4} in the metric can *only* be
achieved via a noncanonical or nonunitary transform

(4.28) m = Diag. (1, 1, 1, -c_{o}^{2}) -> m^ = Diag. (1, 1, 1, -c_{o}^{2}/n_{4}^{2}) =
UxmxU^{++} ≠ I.

An invariant resolution of the above inconsistencies and
limitations has been provided by the lifting of special relativity
into a new formulation today known as ** isorelativity,** or **
Lorentz-Poincare'-Einstein-Santilli isorelativity,** where the
term "isorelativity" stands to indicate that the principle of
relativity applies on isospacetime over isofields, and not on its
projection on ordinary spacetime. Also, the additional
characterization of "special" is redundant because, as review
below, ** isorelativity achieves a geometric unification of
special and general relativities.** In this section we outline the
isotopies of special relativity, while the inclusion of classical
and quantum gravity is only indicated below.

Isorelativity was first proposed by R. M. Santilli in Ref. (26) of
1983 via the first invariant formulation of the iso-Minkowskian
spaces and related iso-Lorentz symmetry. The studies were
then continued in: Ref. (11) of 1985 with the first isotopies of
the rotational symmetry; Ref. (28) of 1993 with the first
isotopies of the SU(2)-spin symmetry; Ref. (29) of 1993) with the
first isotopies of the Poincare' symmetry; and Ref. (33) of 1998
with the first isotopies of the SU(2)-isospin symmetries, Bell's
inequalities and local realist. The studies were then completed
with memoir (15) of 1998) presenting a comprehensive formulation
of the iso-Minkowskian geometry, including its formulation via the
mathematics of the Riemannian geometry (such iso-Christoffel's
symbols, isocovariant derivatives, etc.).

Numerous independent studies on isorelativity are available in the
literature (see, e.g., Refs. (63-68) and [8-11]), such as:
Aringazin's proof (192) of the direct universality of the
Lorentz-Poincar\'e-Santilli isosymmetry for all infinitely
possible spacetimes with signature (+,+,+,-); Mignani's exact
representation (118) of the large difference in cosmological
redshifts between quasars and galaxies when physically connected;
the exact representation of the anomalous behavior of the
meanlives of unstable particles with speed by Cardone et al
(110,11); the exact representation of the experimental data on the
Bose-Einstein correlation by Santilli (112) and Cardone and
Mignani (113); the invariant and exact validity of the
iso-Minkowskian geometry within the hyperdense medium in the
interior of hadrons by Arestov et al. (120); the first exact
representation of molecular features by Santilli and Shillady
(125,126); and numerous others.

Evidently we cannot review isorelativity in the necessary details
to avoid a prohibitive length. Nevertheless, to achieve minimal
self-sufficiency of this presentation, it is important to outline
at least its main structural lines.

Isorelativity can be constructed via the method of Section
I.3, namely, by assuming that the basic noncanonical or
nonunitary transform characterizing the new metric m^
coincides with the isounit (where the
diagonalization is permitted by its Hermiticity). Isorelativity then follows by applying said noncanonical/nonunitary transforms to the totality of mathematical and physical structure of the ordinary special relativity.

This procedures leads to the isotopies of: the basic unit; numbers; metric spaces; functional analysis; Minkowskian and symplectic geometries; algebras, symmetries, conservation laws, representation theory, etc. These isotopies have been outlined above. We are therefore left with the identification of the basic axioms of isorelativity.

As indicated earlier, **by conception and construction, the basic axioms of isorelativity coincide with those of special relativity when formulated on iso-Minkowski space M^ over the field of isonumbers R^.** Therefore, all differences occur in the **projection** of isorelativity in the conventional spacetime M.

Stated differently, special relativity has only **one** interpretation, the conventional one. By contrast, isorelativity has **two** interpretations depending on whether any given quantity, axiom or law is referred to the conventional unit of special relativity, I = Diag. (1, 1, 1, 1) or to the isounit. In the former case rather profound deviations occur. In the latter case there is no distinction whatsoever from isospecial and special relativity.

Along these lines, the first application of isorelativity is to provide an **invariant representation of locally varying speeds of electromagnetic waves** merely permitted by isometric m^ that is a particular case of the general line element (4.9).

Next, isorelativity has been constricted for the** invariant
description of systems of extended, nonspherical and deformable
particles under Hamiltonian and non-Hamiltonian interactions.**
Practical applications then require the knowledge of the actual
shape of the particles considered, here assumed for simplicity as
being spheroidal ellipsoids.

Note that the minimum number of constituents of a closed
non-Hamiltonian system is **two. ** In this case we have shapes
represented with the assumed isounit. .
Applications also require the identification of the nonlocal
interactions, e.g., whether occurring on an extended surface
or volume. Specific realizations of the isounit for these systems will be considered in the applications.

A third important part of isorelativity is given by the
following isotopies of conventional relativistic axioms when projected in our spacetime that, for
the case of motion along the third axis, can be expressed as follows (29):

**ISOAXIOM I.** The projection in our spacetime of
the maximal causal invariant speed is given by:

(4.29) V_{Causal-Max} = c_{o}xn_{3}/n_{4} = c_{o}xb_{4}/b_{3}.

This isoaxiom resolves the inconsistencies of special relativity
recalled earlier for particles and electromagnetic waves
propagating in water. In fact, water is homogeneous and isotropic,
thus requiring that n_{3} = n_{4}.
In this case the maximal causal speed for a
massive particle moving within homogeneous and isotropic water is c_{o} as experimentally established for
electrons, while the local speed of electromagnetic waves is c =
c_{o}/n_{4}< c_{o} as also experimentally established.

Note that the above resolution and Isoaxioms I require the **abandonment of the speed
of light as the maximal causal speed for motion within
physical media, and its replacement with the maximal causal speed
of particles.** It happens that in vacuum these two maximal
causal speeds coincide. However, even in vacuum the correct
maximal causal speed remains that of particles and * not* that
of light, as popularly believed. At any rate, physical media are
generally opaque to light but generally not to particles.
Therefore, the assumption for maximal causal speed as that of
light that cannot propagate within the medium considered would be
evidently vacuous.

* ISOAXIOM II. The projection in our spacetime of the isorelativistic
addition of speeds within physical media is given by:*<>br>

(4.30) V_{Tot} = (v_{1} + v_{2})/(1 +
v_{1}^{k}xg_{kk}xv_{2}^{k}/
c_{o}xg_{44}xc_{o}).

We have again the correct occurrence that ** the sum of two
maximal causal speeds in water yields the maximal causal speed in water,** as the reader is
encouraged to verify. Note that such a result is impossible
for special relativity. Note also that the isorelativistic
sum of two speeds of light in water does not
yield the speed of light in water, thus confirming that the
speed of light within physical media (assuming that they are
transparent to light) is not the fundamental maximal causal speed.

** ISOAXIOM III.** The projection in our spacetime
of the isorelativistic laws of dilation of time T_{o} and
contraction of length L_{o} and the variation of mass m_{o}
with speed are given by:

(4.31a) T = T_{o}xg^,

(4.31b) L = L_{o}/g^,

(4.31c) m = m_{o}xg^,

(4.31d)g' = 1/(1 - b^^{2})^{1/2}.

(4.32) b^^{2} =
r^{k}xg_{kk}xr^{k}/
c_{o}xg_{44}xc_{o}.

Note that in homogeneous and isotropic media (such as water) these values coincide with the relativistic ones
as it should be since particles have in
these media the maximal causal speed c_{o}. Note again the necessity of
avoiding the interpretation of the local speed of light as the
maximal local causal speed. Note finally that the mass diverges at the
maximal local causal speed, but * not* at the local speed of
light.

**ISOAXIOM IV.** The projection in our
spacetime of the iso-Doppler law is given by (for 90^{o} angle of
aberration):

(4.33) w = w_{o}xg^,

The above isorelativistic axiom has rather deep cosmological implications. In fact, Isoaxiom IV permits an exact, numerical and
invariant representation of the large differences in cosmological
redshifts between quasars and galaxies when physically connected.
In this case light simply exits the huge quasar chromospheres
already redshifted due to the decrease of the speed of light,
rather than the speed of the quasars (118).

Isoaxiom IV also permits a numerical interpretation of the
internal blue- and red-shift of quasars due to the dependence of
the local speed of light on its frequency. Finally, Isoaxiom IV
predicts that * a component} of the predominance toward the red
of sunlight at sunset is of iso-Doppler nature* in view of the
bigger decrease of the speed of light at sunset as compared to the
same speed at the zenith (evidently
because of the travel within a
comparatively denser atmosphere).

(i>Ib> ISOAXIOM V.
The projection in our spacetime of the isorelativistic law of
equivalence of mass and energy is given by:

(4.34) E = mxc^{2} = mxc_{o}^{2}/n_{4}^{2} =
mxc_{o}^{2}/b_{4}^{2}.

Among various applications, ** Isoaxiom V removes any need for
the "missing mass" in the universe.** This is due to the fact that
all isotopic fits of experimental data agree on values n_{4} < 1 within the hyperdense media in the interior of hadrons, nuclei
and stars (55,120). As a result, Isoaxiom V yields a value of the
total energy of the universe dramatically bigger than that it has been
believed until now under the assumption of the universal validity
of the speed of light in vacuum. For other intriguing
applications, e.g., for the rest energy of hadronic constituents,
we refer the interested reader to monographs (55,61).

**4.7. ISOGRAVITATION.**

Isorelativity has a number of implications for gravity that are interesting per se and also have a direct relevance for the structure of hadrons. In fact, the assumption of the isotopic line element (4.9) as the basic invariant for the hadronic structure directly implies the inclusion of gravitation, since the g_{kk} elements can be arbitrary Riemannian elements. The following aspects are, therefore, significant for the structure of hadrons.

There is no doubt that the classical and operator
formulations of gravitation on a *curved* space has been the most
controversial theory of the 20-th century because of an ever
increasing plethora of problematic aspects that have remained
basically unresolved due to the lack of their acknowledgment, let
alone their resolution, by leading research centers in the field
(see, for instance, H. E. Wilhelm (220) and references quoted
therein).

One of the reasons that special relativity in vacuum has a majestic
axiomatic consistence is its ** invariance** under the
Poincare' symmetry. Recent studies have shown that the
formulation of gravitation on a curved space or, equivalently,
the formulation of gravitation based on "covariance," is
necessarily noncanonical at the classical level and nonunitary at
the operator level, thus suffering of all catastrophic
inconsistencies of Theorem I.2.1 (45,46). These catastrophic
inconsistencies can only be resolved via a new conception of
gravity based on a ** universal invariance,** rather than
covariance.

Additional studies have identified profound axiomatic
incompatibilities between gravitation on a curved space and
electroweak interactions. These incompatibilities have resulted in being responsible for the lack of achievement of an axiomatically
consistent grand unification since Einstein's times (32,35,37),
among which we mention:

1) Electroweak theories are based on invariance while
gravitation is not;

2) Electroweak theories are flat in their axioms while gravitation
is not; and

3) Electroweak theories are bona fide field theories, thus
admitting positive and negative energy solutions, while
gravitation can only admit positive energies.

No knowledge of isotopies can be claimed without a knowledge that
isorelativity has been constructed also to resolve at least some
of the controversies on gravitation. The fundamental requirement
is ** the abandonment of the formulation of gravity on a
Riemannian space and its formulation instead on an iso-Minkowskian
space** (15)} via the following basic steps:

I) Factorization of any given Riemannian metric g(q) into a
nowhere singular and positive-definite 4x4-matrix
T_{grav}(q) times the Minkowski metric m,

(4.35) g(q) = T_{grav}(q) x m,

II) Assumption of the inverse of T_{grav}as the
fundamental isounit of gravitation,

(4..36) I^_{grav}(q) = 1/T_{grav},

III) Submission of the totality of the Minkowski space and
relative symmetries to the noncanonical/nonunitary transform

(4.37) U(q)xU(q)^{+} = I^_{grav}.

The above procedure yields the iso-Minkowskian spaces and related
geometry (15), resulting in a
new conception of gravitation, called ** isogravity,** with the
following main features (15,32,35,37,55):

i) Isogravity is characterized by a universal ** symmetry** (and
not a "covariance"), the Poincare'-Santilli isosymmetry
P^(3.1) for the gravity of matter with isounit I^_{grav}(q)
(with the isodual isosymmetry P^^{d}(3.1) for the gravity of
antimatter);

ii) All conventional field equations, such as the Einstein-Hilbert
and other field equations, can be identically formulated via the
Minkowski-Santilli isogeometry since the latter preserves all the
tools of the conventional Riemannian geometry, such as the
Christoffel's symbols, covariant derivative, etc. (15);

iii) Isogravitation is isocanonical at the classical level and
isounitarity at the operator level, thus resolving the
catastrophic inconsistencies of Theorem I.2.1;

iv) An axiomatically consistent operator version of gravity always
existed and merely crept in un-noticed throughout the 20-th century
because gravity is embedded where nobody looked, in the **
unit** of relativistic quantum mechanics, and it is given by
isorelativistic hadronic mechanics as in the iso-Dirac equation of the next section (3.90).

v) The basic feature permitting the above advances is** the
abandonment of curvature for the characterization of gravity
(namely, curvature characterized by metric g(q) referred to the
unit I) and its replacement with isoflatness (namely, the
verification of the axioms of flatness in isospace with unit I^ _{grav}(q), while
preserving conventional curvature in its projection on
conventional spacetime).** Equivalently, when curvature is characterized
by the Riemannian metric g(q) = T

A resolution of numerous controversies on classical formulations
of gravity then follow from the above main features, such as: the
resolution of the century old controversy on the lack of existence
of consistent total conservation laws for gravitation on a
Riemannian space, which controversy is resolved under the
universal P^(3.1) symmetry by mere visual verification that
the generators of the conventional and isotopic Poincare'
symmetry are the same, since they represent conserved quantities; the controversy on
the fact that gravity on a Riemannian space admits a well defined
"Euclidean," but not ""Minkowskian" limit,which controversy is
trivially resolved by isogravity via the limit I^_{grav}(q) -> I; and other controversies.

A resolution of the controversies on quantum gravity can be seen
from the property that relativistic hadronic mechanics **is** a
quantum formulation of gravity whenever T = T_{grav}(q). The resulting operator gravity is as axiomatically consistent as
conventional relativistic quantum mechanics because the two
formulations coincide, by construct, at the abstract,
realization-free level. Thus, isogravity resolves the now vexing controversies on "quantum gravity" that have afflicted the physics for most of the 20-th century without any true resolution in sight.

Once curvature is abandoned in favor of the broader isoflatness,
the axiomatic incompatibilities existing between gravity and
electroweak interactions are resolved because: isogravity
possesses, at the abstract level, the * same* Poincare'
invariance of electroweak interactions; isogravity can be
formulated on the * same* flat isospace of electroweak
theories; and isogravity admits positive energies for matter and negative energies for antimatter in
the * same* way as it occurs for electroweak theories. An
axiomatically consistent ** iso-grand-unification of gravitation and electroweak interaction** was then proposed by R. M. Santilli at the VIII Marcel Grossman Meeting on gravitation held in Jerusalem, in June 1997 (32,35).

Note that the above grand-unification requires the prior **
geometric unification of the special and general relativities,**
that is achieved precisely by isorelativity and its underlying
iso-Minkowskian geometry. In fact, special and general
relativities are merely differentiated in isospecial relativity by
the explicit realization of the unit.

Intriguingly, ** black
holes are now characterized by the zeros of the isounit** (55)

(4.38) I^_{grav}(q) = 0.

The above formulation recovers all conventional results on
gravitational singularities, such as the singularities of the
Schwarzschild's metric, since they are all described by the
gravitational content T_{grav}(q) of the Riemannian metric g(q) = T_{grav}(q)xm, since m is flat.

This illustrates again that **all conventional results of
gravitation, including experimental verifications, can be
reformulated in invariant form via isorelativity.**

Moreover, the problematic aspects of general relativity mentioned
earlier refer to the ** exterior gravitational problem.** Perhaps
greater problematic aspects exist in gravitation on a Riemannian
space for ** interior gravitational problems,** e.g., because of
the lack of characterization of basic features, such as the
density of the interior problem, the locally varying character of
the speed of light, etc. These additional problematic aspects are also
resolved by isospecial relativity due to the unrestricted
character of the functional dependence of the isometric that
therefore permits a direct geometrization of the density, local;
variation of the speed of light, etc.

The cosmological implications are also intriguing. In fact,
isorelativity permits a new conception of cosmology based on the
{\it universal invariance} $\hat P(3.1)\times \hat P^d(3.1)$ in
which there is no need for the "missing mass" (as indicated
earlier), time and the speed of light become local variables, and
the detected universe has a dimension considerably * smaller*
than that currently believed (because some of the cosmological
redshift is due to the decrease of the speed of light in
chromospheres, rather than to the speed of quasars). Also, at the limit
case of equal distribution of matter and antimatter in the
universe, ** isocosmology predicts that the universe has
identically null total energy, identically null total time, and
identically null other physical characteristics, thus permitting
mathematical studies of its creation} (because of the lack of
singularities at its formation**.

** 5. CHARACTERIZATION OF HADRONIC CONSTITUENTS VIA THE ISO-DIRAC EQUATION.**

All elementary hadronic constituents considered in these studies have spin 1/2. Therefore, the best quantitative characterization of the hadronic constituents is via the **isotopies of the Dirac equation** first proposed by Santilli in Ref. (30) in a fully invariant form (see monograph (55) for a comprehensive treatment and literature).

The iso-Dirac equation is particularly important to characterize the mutations of the intrinsic characteristics of particles into isoparticles. In fact, **the conventional Dirac equation represents an electron under the external electromagnetic field of the proton** (as well known, a consistent extension of Dirac's equation to the two-body system constituted by the H-atom has not been achieved to this day). In this case, all conventional intrinsic characteristics of particles are preserved and, therefore, there are no mutations, namely, they are ordinary "particles" as characterized by special relativity and the Poincare' symmetry of Minkowskian spacetime.

By comparison, **the iso-Dirac equation represents an isoparticle under external electromagnetic AND contact nonpotential interactions,** as necessary for hadronic constituents (since their wavepackets has essentially the same dimension as a hadron). In this case the additional contact nonpotential interactions cause a renormalization, in general, of ALL intrinsic characteristics, thus yielding isoparticles.

Let |e> be the eigenstates on the conventional Hilbert space over the field of complex numbers for the representation of an **electron** via the conventional Dirac equation. Let

(5,1) |e^> = Ux|e>, UxU^{+} ≠ 1,

be the isostate on iso-Hilbert space over the isofield C^ representing the **isoelectron.**
The simplest possible version of the **Dirac-Santilli isoequation** on M^ over R^ for the characterization of |e^> is given by (30)

(5.2a) [G^^{k}*(p^_{k} - ixe^*A^_{k}) - ixm^]*|e^>,

(5.2b) G^^{k} = n_{k}^{-1}xG^{k}xI^, k = 1, 2, 3,
G^^{4} = n_{4}^{-1}xG^{4}xI^, m^ = mxI^,

where the G^s are the **isogamma matrices** and the Gs are the conventional Dirac gamma matrices.

<>p>
Note that the above realization is the simplest possible in the sense that the isotopy solely occurs in spacetime without an isotopy of the SU(2) spin, that is not necessary for a first relativistic study of our structure model of hadrons. For the most general possible form of the Dirac-Santilli isoequation one may consult monograph (55) and various contributions quoted therein.

The **orbital isosymmetry** SO^(3) of the isoelectron is characterized by

(5.3a) L^_{1} = (r^_{2})*(p^_{3}),
L^_{2} = (r^_{3})*(p^_{1}),
L^_{3} = (r^_{1})*(p^_{2}),

(5.3b) [L^_{1}, L^_{2}]* = n_{3}^{2}xL^_{3},
[L^_{2}, L^_{3}]* = n_{1}^{2}xL^_{1},
[L^_{3}, L^_{1}]* = n_{2}^{2}xL^_{2},

(5.3c) (L^)^{2}^*|e^> = (n_{1}^{2}xn_{2}^{2} + n_{2}^{2}xn_{3}^{2} + n_{3}^{2}xn_{1}^{2})x|e^>,

(5.3d) (L^_{3})*|e^> = (+/-)n_{1}xn_{2})x|e^>,

where [A, B]* = A*B - B*A = AxTxB - BxTxA is the isocommutator. Note that the above particular realization of the isogroup SO^(3) is also locally isomorphic to the conventional SO(3) group (because the n's are positive-definite).

The **isotopic formulation of SU^(2)-spin** is given by

(5.4a) J^_{1} = (G^_{2})*(G^_{3})/2,
J^_{2} = (G^_{3})*(G^_{1})/2,
J^_{3} = (G^_{1})*(G^_{2})/2,

(5.4b) [J^_{1}, J^_{2}]* = n_{3}^{-2}xJ^_{3},
[J^_{2}, J^_{3}]* = n_{1}^{-2}xJ^_{1}, [J^_{3}, J^_{1}]* = n_{2}^{-2}xJ^_{2},

(5.4c) (J^^{2}^*|e^> = (1/4)x(n_{1}^{-2}xn_{2}_{-2} +
n_{2}^{-2}xn_{3}^{-2} + n_{3}^{-2}xn_{1}^{-2})x|e^>,

(5.4d) (J^_{3})*|e^> = (+/-)(1/2)(n_{1}^{-1}xn_{2}^{-1})x|e^>.

Note again that SU^(2) is locally isomorphic to the conventional SU(2),. Nevertheless, the eigenvalues of the spin are not generally constant to represent the electron when in the core of a collapsing star, or other extreme internal conditions under which the preservation of the quantum value 1/2 is a pure nonscientific-political belief.

The realization of the Poincare'-Santilli isosymmetry P^(3.1) = SL^(2.c)xT^(4) permitted by the iso-Dirac equation are characterized by the generators

(5.5) P^(3.1): J^_{k}, K^_{k} = (G^_{k})*(G^_{4})/2, P^, k = 1, 2, 3, p^_{i}, i = 1, 2, 3, 4.

Their isocummutation rules are given by Eqs. (4.21), thus confirming the local isomorphiism between P^(3.1) and P(3.1).

The **isoelectron** e^ characterized by the iso-Dirac equation is an irreducible isorepresentation of the P^(3.1). One can see in this way the **mutation (isorenormalization) of the intrinsic characteristics of the electron,** as desired.

Additional mutations characterized by the iso-Dirac equation are those of the magnetic and electric dipole moments, whose derivation has been worked out in Ref. (30) as a simple isotopy of the conventional derivation, resulting in the isolaws valid for the case of an axial symmetry along the third axis

(5.7a) m^ = mxn_{4}/n_{3},

(5.7b) d^ = dxn_{4}/n_{3}.

The above laws provide a technical representation of the well known semiclassical property recalled earlier that the deformation of a charged and spinning sphere necessary implies an alteration of its magnetic moment. In particular, we have a decrease (increase) of the magnetic moment when we have a prolate (oblate) deformation.

It is an instructive exercise for the interested reader to verify that the above realization of the iso-Dirac equation *cannot* be constructed via a nonunitary transform of the conventional Dirac theory, but requres special maps (technically we have an irregular representation of a Lie-Santilli isosymmetry, the regular one being achievable via nonunitary transforms of conventional representations.

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Last revision February 8, 2003

**6. EXPERIMENTAL VERIFICATIONS.**

**7. THE NEW STRUCTURE MODEL OF MESONS WITH PHYSICAL CONSTITUENTS.**

**8. COMPATIBILITY OF THE NEW STRUCTURE MODEL OGF MESONS WITH THE SU(3) CLASSIFICATION.**

***********************************

[1]

(1) I. Newton, * Philosophiae Naturalis Principia Mathematica* (1687),
translated and reprinted by Cambridge Univ. Press. (1934).

(2) J. L. Lagrange, * Mechanique Analytique* (1788), reprinted by
Gauthier-Villars, Paris (1888).

(3) W. R. Hamilton, * On a General Method in Dynamics* (1834), reprinted
in {\it Hamilton's Collected Works,} Cambridge Univ. Press (1940).

(4) S. Lie, * Over en Classe Geometriske Transformationer,* English translation by E.
Trell, Algebras Groups and Geometries {\bf 15}, 395 (1998).

(5) A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. {\bf 47}, 777 (1935).

(6) P. A. M. Dirac, * The Principles of Quantum
Mechanics,* Clarendon Press, Oxford, fourth edition (1958).

(7) A. A. Albert, Trans. Amer. Math. Soc. {\bf 64}, 552 (1948).

[2] ** BASIC MATHEMATICAL PAPERS:**

(8) R. M. Santilli, Nuovo Cimento {\bf 51}, 570 (1967).

(9) R. M. Santilli, Suppl. Nuovo Cimento {\bf 6}, 1225 (l968).

(10) R. M. Santilli, Hadronic J. {\bf 3}, 440 (l979).

(11) R. M. Santilli, Hadronic J. {\bf 8}, 25 and 36 (1985).

(12) R. M. Santilli Algebras, Groups and Geometries {\bf 10}, 273 (1993).

(13) R. M. Santilli and T. Vougiouklis, contributed paper in {\it New Frontiers in
Hyperstructures,} T., Vougiouklis, Editor, Hadronic Press, p. 1 (1996).

(14) R. M. Santilli, Rendiconti Circolo Matematico di
Palermo, Supplemento {\bf 42}, 7 (1996).

(15) R. M. Santilli, Intern. J. Modern Phys. D {\bf 7}, 351 (1998).

[3] ** ISODUAL FORMULATIONS:**

(16) R. M. Santilli, Comm. Theor. Phys. {\bf 3}, 153 (1993).

(17) R. M. Santilli, Hadronic J. {\bf 17}, 257 (1994).

(18) R. M. Santilli, Hadronic J. {\bf 17}, 285 (1994).

(19) R. M. Santilli, Communication of the JINR, Dubna, Russia,. No. E2-96-259 (1996).

(20) R. M. Santilli, contributed paper in {\it New Frontiers of Hadronic Mechanics,}
T.L.Gill, ed., Hadronic Press (1996).

(21) R. M. Santilli, Hyperfine Interactions, {\bf 109}, 63 (1997).

(22) R. M. Santilli, Intern. J. Modern Phys. A {\bf 14}, 2205 (1999).

[4] ** ISOTOPIC FORMULATIONS:**

(23) R.M.Santilli: Hadronic J. {\bf 1}, 224 (1978).

(24) R. M. Santilli, Phys. Rev. D {\bf 20}, 555 (1979).

(25) H. C. Myung and R. M. Santilli, Hadronic J. {\bf 5}, 1277 (1982).

(26) R. M. Santilli, Novo Cimento Lett. {\bf 37}, 545 (1983).

(27) R. M. Santilli, Hadronic J. {\bf 8}, 25 and 36 (1985).

(28) R. M. Santilli, JINR Rapid. Comm. {\bf 6}, 24 (1993).

(29) R. M. Santilli, J.Moscow Phys.Soc. {\bf 3}, 255 (1993).

(30) R. M. Santilli, Communication of the JINR, Dubna, Russia, # E4-93-352, 1993, published in Chinese J.Syst.Ing. \& Electr.{\bf 6}, 177 (1996).

(31) R. M. Santilli, Found. Phys. {\bf 27}, 635 (1997).

(32) R. M. Santilli, Found. Phys. Letters {\bf 10}, 307 (1997).

(33) R. M. Santilli, Acta Appl. Math. {\bf 50}, 177 (1998).

(34) R. M. Santilli, contributed paper to the {\it Proceedings of the International Workshop on
Modern Modified Theories of Gravitation and Cosmology,} E. I. Guendelman, Editor, Hadronic Press, p. 113 (1998).

(35) R. M. Santilli, contributed paper to the {\it Proceedings of the VIII M. Grossmann Meeting
on General Relativity,} Jerusalem, June 1998, World Scientific, p. 473 (1999).

(36) R. M. Santilli, contributed paper in {\it Photons: Old Problems in Light of New Ideas,} V. V.
Dvoeglazov, Editor, Nova Science Publishers, pages 421-442 (2000).

(37) R. M. Santilli, Found. Phys. Letters {\32}, 1111 (2002).

[5] ** GENOTOPIC FORMULATIONS:**

(38) R. M. Santilli: Hadronic J. {\bf 1},574 and 1267 (1978).

(39) R. M. Santilli, Hadronic J. {\bf 2}, 1460 (l979) and {\bf 3}, 914 (l980).

(40) R. M. Santilli, Hadronic J. {\bf 4}, 1166 (l981).

(41) R. M. Santilli, Hadronic J. {\bf 5}, 264 (l982).

(42) H. C. Myung and R. M. Santilli, Hadronic J. {\bf 5}, 1367 (l982).

(43) R. M. Santilli, Hadronic J. Suppl. {\bf 1}, 662 (l985).

(44) R. M. Santilli, Found. Phys. {\bf 27}, 1159 (1997).

(45) R. M. Santilli, Modern Phys. Letters {\bf 13}, 327 (1998).

(46) R. M. Santilli, Intern. J. Modern Phys. A {\bf 14}, 3157 (1999).

[6] ** HYPERSTRUCTURAL FORMULATIONS:**

(47) R. M. Santilli, Algebras, Groups and Geometries {\bf 15}, 473 (1998).

[7] ** MONOGRAPHS:**

(48) R. M. Santilli, {\it Foundations of Theoretical Mechanics}, Vol. I,
Springer--Verlag, Heidelberg--New York (1978).

(49) R. M. Santilli, {\it Lie-admissible Approach to the Hadronic Structure,}
Vol.I, Hadronic Press, Palm Harbor, Florida (1978).

(50) R. M. Santilli, {\it Lie-admissible Approach to the Hadronic Structure,} Vol.
II, Hadronic Press, Palm Harbor, Florida (1981).

(51) R. M. Santilli, {\it Foundations of Theoretical Mechanics}, Vol. II,
Springer--Verlag, Heidelberg--New York (1983).

(52) R. M. Santilli, {\it Isotopic Generalizations of Galilei and Einstein Relativities,}
Vol. I, Hadronic Press, Palm Harbor, Florida (1991).

(53) R. M. Santilli, {\it Isotopic Generalizations of Galilei and Einstein Relativities,}
Vol. II, Hadronic Press, Palm Harbor, Florida (1991).

(54) R. M. Santilli, {\it Elements of Hadronic Mechanics},
Vol I, Ukraine Academy of Sciences,
Kiev, Second Edition (1995).

(55) R. M. Santilli, {\it Elements of Hadronic Mechanics},
Vol II, Ukraine Academy of Sciences,
Kiev, Second Edition (1995).

(56) C. R. Illert and R. M. Santilli, {\it Foundations of Theoretical Conchology,}
Hadronic Press, Palm Harbor, Florida (1995).

(57) R. M. Santilli {\it Isotopic, Genotopic and Hyperstructural Methods
in Theoretical Biology}, Ukraine Academy of Sciences, Kiev (1996).

(58) R. M. Santilli, {\it The Physics of New Clean Energies and Fuels According to
Hadronic Mechanics,} Special issue of the Journal of New Energy, 318 pages (1998).

(59) R. M. Santilli, {\it Foundations of Hadronic Chemistry with Applications to New
Clean Energies and Fuels,} Kluwer Academic Publishers, Boston-Dordrecht-London
(2001).

(60) R. M. Santilli, {\it Ethical Probe of Einstein's Followers in the USA: An insider's view,} Alpha Publishing, Newtonville, MA (1984).

(61) R. M. Santilli, {\it Documentation of the Ethical Probe,} Volumes I, II and III, Alpha Publishing, Newtonville, MA (1985).

(62) H. C. Myung, {\it Lie Algebras and Flexible
Lie-Admissible Algebras,} Hadronic Press (1982).

(63) A. K. Aringazin, A. Jannussis, D.F. Lopez, M. Nishioka, and B. Veljanoski,
{\it Santilli's Lie-isotopic Generalization of Galilei's and Einstein's
Relativities,} Kostarakis Publishers, Athens (1991).

(64) D. S. Sourlas and G. T. Tsagas, {\it Mathematical
Foundations of the Lie-Santilli Theory,} Ukraine Academy of Sciences,
Kiev (1993).

(65) J. Lohmus, E. Paal and L. Sorgsepp, {\it
Nonassociative Algebras in Physics}, Hadronic Press, Palm Harbor, FL,
USA (1994).

(66) J. V. Kadeisvili, {\it Santilli's Isotopies of
Contemporary Algebras, Geometries and Relativities},
Second Edition, Ukraine Academy of Sciences, Kiev , Second Edition (1997).

(67) R. M. Falcon Ganfornina and J. Nunez Valdes, {\it Fondamentos de la Isoteoria de
Lie-Santilli,} (in Spanish) International Academic Press, America-Europe-Asia,
(2001), also available in the pdf file
http://www.i-b-r.org/docs/spanish.pdf

(68) Chun-Xuan Jiang, {\it Foundations of Santilli's Isonumber Theory,}
with Applications to New Cryptograms, Fermat's Theorem and Goldbach's Conjecture,
International Academic Press, America-Europe-Asia
(2002) also available in the pdf file
http://www.i-b-r.org/docs/jiang.pdf

[8] ** CONFERENCE PROCEEDINGS AND REPRINT VOLUMES:**

(69) H. C. Myung and S. Okubo, Editors, {\it Applications of Lie-Admissible Algebras
in Physics,} Volume I, Hadronic Press (1978).

(70) H. C. Myung and S. Okubo, Editors, {\it Applications of Lie-Admissible Algebras
in Physics,} Vol. II, Hadronic Press (1978).

(71) H. C. Myung and R. M. Santilli, Editor, {\it Proceedings of the Second Workshop on
Lie-Admissible Formulations,} Part I, Hadronic J. Vol. 2, no. 6, pp. 1252-2033 (l979).

(72) H. C. Myung and R. M. Santilli, Editor, {\it Proceedings of the Second Workshop on
Lie-Admissible Formulations,}Part II, Hadronic J. Vol. 3, no. 1, pp. 1-725 (l980.

(73) H. C. Myung and R. M. Santilli, Editor, {\it Proceedings of the Third Workshop on
Lie-Admissible Formulations,}Part A, Hadronic J. Vol. 4, issue no. 2, pp. 183-607 (l9881).

(74) H. C. Myung and R. M. Santilli, Editor, {\it Proceedings of the Third Workshop on
Lie-Admissible Formulations,} Part B, Hadronic J. Vo. 4, issue no. 3, pp. 608-1165 (l981).

(75) H. C. Myung and R. M. Santilli, Editor, {\it Proceedings of the Third Workshop on
Lie-Admissible Formulations,} Part C, Hadronic J. Vol. 4, issue no. 4, pp. 1166-1625
(l981).

(76) J. Fronteau, A. Tellez-Arenas and R. M. Santilli, Editor, {\it Proceedings of the First
International Conference on Nonpotential Interactions and their Lie-Admissible
Treatment,} Part A, Hadronic J., Vol. 5, issue no. 2, pp. 245-678 (l982).

(77) J. Fronteau, A. Tellez-Arenas and R. M. Santilli, Editor, {\it Proceedings of the First
International Conference on Nonpotential Interactions and their Lie-Admissible
Treatment,} Part B, Hadronic J. Vol. 5, issue no. 3, pp. 679-1193 (l982).

(78) J. Fronteau, A. Tellez-Arenas and R. M. Santilli, Editor, {\it Proceedings of the First
International Conference on Nonpotential Interactions and their Lie-Admissible
Treatment,} Part C, Hadronic J. Vol. 5, issue no. 4, pp. 1194-1626 (l982).

(79) J. Fronteau, A. Tellez-Arenas and R. M. Santilli, Editor, {\it Proceedings of the First
International Conference on Nonpotential Interactions and their Lie-Admissible
Treatment,} Part D, Hadronic J. Vol. 5, issue no. 5, pp. 1627-1948 (l982).

(80) J.Fronteau, R.Mignani, H.C.Myung and R. M. Santilli, Editors, {\it Proceedings of the
First Workshop on Hadronic Mechanics,} Hadronic J. Vol. 6, issue no. 6, pp. 1400-1989
(l983).

(81) A. Shoeber, Editor, {\it Irreversibility and Nonpotentiality in Statistical Mechanics,}
Hadronic Press (1984).

(82) H. C. Myung, Editor, {\it Mathematical Studies in Lie-Admissible Algebras,} Volume I,
Hadronic Press (1984).

(83) H. C. Myung, Editor, {\it Mathematical Studies in Lie-Admissible Algebras,} Volume II,
Hadronic Press (1984).

(84) H. C. Myung and R. M. Santilli, Editor, {\it Applications of Lie-Admissible Algebras
in Physics,} Vol. III, Hadronic Press (1984).

(85) J. Fronteau, R. Mignani and H. C. Myung, Editors, {\it Proceedings of the Second Workshop on
Hadronic Mechanics,} Volume I Hadronic J. Vol. 7, issue no. 5, pp. 911-1258 (l984).

(86) J.Fronteau, R.Mignani and H.C.Myung, Editors, {\it Proceedings of the Second Workshop on
Hadronic Mechanics,} Volume II, Hadronic J. Vol. 7, issue no. 6, pp. 1259-1759 (l984).

(87) D. M. Norris et al, {\it Tomber's Bibliography and Index in Nonassociative
Algebras,} Hadronic Press, Palm Harbor, FL (1984).

(88) H. C. Myung, Editor, {\it Mathematical Studies in Lie-Admissible Algebras,} Volume III,
Hadronic Press (1986).

(89) A. D. Jannussis, R. Mignani, M. Mijatovic, H. C. Myung B. Popov and A. Tellez Arenas,
Editors, {\it Fourth Workshop on Hadronic Mechanics and Nonpotential Interactions,}
Nova Science, New York (l990).

(90) H. M. Srivastava and Th. M. Rassias, Editors, {\it Analysis Geometry and Groups:
A Riemann Legacy Volume,} Hadronic Press (1993).

(91) F. Selleri, Editor, {\it Fundamental Questions in Quantum Physics and Relativity,}
Hadronic Press (1993).

(92) J. V. Kadeisvili, Editor, {\it The Mathematical Legacy of Hanno Rund}, Hadronic Press
(1994).

(93) M. Barone and F. Selleri Editors, {\it Frontiers of Fundamental Physics,} Plenum, New
York, (1994).

(94) M. Barone and F. Selleri, Editors, {\it Advances in Fundamental Physics,} Hadronic Press
(1995).

(95) Gr. Tsagas, Editor, {\it New Frontiers in Algebras, Groups and Geometries
,} Hadronic Press (1996).

(96) T. Vougiouklis, Editor, {\it New Frontiers in Hyperstructures,}
Hadronic Press, (1996).

(97) T. L. Gill, Editor, {\it New Frontiers in Hadronic Mechanics,} Hadronic Press (1996).

(98) T. L. Gill, Editor, {\it New Frontiers in Relativities,} Hadronic Press (1996).

(99) T. L. Gill, Editor, {\it New Frontiers in Physics,}, Volume I, Hadronic Press (1996).

(100) T. L. Gill, Editor, {\it New Frontiers in Physics,}, Volume II, Hadronic Press (1996).

(101) C. A. Dreismann, Editor, {\it New Frontiers in Theoretical Biology,} Hadronic Press
(1996).

(102) G. A. Sardanashvily, Editor,{\it New Frontiers in Gravitation,} Hadronic Press (1996).

(103) M. Holzscheiter, Editor, {\it Proceedings of the
International Workshop on Antimatter Gravity,} Sepino, Molise, Italy,
May 1996, Hyperfine Interactions, Vol. {\bf 109} (1997).

(104) T. Gill, K. Liu and E. Trell, Editors, {\it Fundamental Open Problems in Science at the
end of the Millennium,}} Volume I, Hadronic Press (1999).

(105) T. Gill, K. Liu and E. Trell, Editors, {\it Fundamental Open Problems in Science at the
end of the Millennium,}} Volume II, Hadronic Press (1999).

(106) T. Gill, K. Liu and E. Trell, Editors, {\it Fundamental Open Problems in Science at the
end of the Millennium,}} Volume III, Hadronic Press (1999).

(107) V. V. Dvoeglazov, Editor {\it Photon: Old Problems in Light of New Ideas,} Nova Science
(2000).

(108) M. C. Duffy and M. Wegener, Editors, {\it Recent Advances in Relativity Theory} Vol. I,
Hadronic Press (2000).

(109) M. C. Duffy and M. Wegener, Editors, {\it Recent Advances in Relativity Theory} Vol. I,
Hadronic Press (2000).

[9] ** EXPERIMENTAL VERIFICATIONS:**

(110) F. Cardone, R. Mignani and R. M. Santilli, J. Phys. G: Nucl. Part. Phys.
{\bf 18}, L61 (1992).

(111) F. Cardone, R. Mignani and R. M. Santilli, J. Phys. G: Nucl. Part. Phys. {\bf 18}, L141
(1992).

(112) R. M. Santilli, Hadronic J. {\bf 15}, Part I: 1-50 and Part II: 77134 (l992).

(113) Cardone and R. Mignani, JETP {\bf 88}, 435 (1995).

(114) R. M. Santilli, Intern. J. of Phys. {\bf 4}, 1 (1998).

(115) R. M. Santilli Communications in Math. and Theor. Phys. {\bf 2}, 1 (1999).

(116) A. O. E. Animalu and R. M. Santilli, Intern. J. Quantum Chem. {\bf 26},175 (1995).

(117) R. M. Santilli, contributed paper to {\it Frontiers of Fundamental Physics,} M. Barone and
F. Selleri, Editors Plenum, New York, pp 4158 (1994).

(118) R. Mignani, Physics Essays {\bf 5}, 531 (1992).

(119) R. M. Santilli, Comm. Theor. Phys. {\bf 4}, 123 (1995).

(120) Yu. Arestov, V. Solovianov and R. M. Santilli, Found. Phys. Letters {\bf 11}, 483 (1998).

(121) R. M. Santilli, contributed paper in the {\it Proceedings of the International Symposium on
Large Scale Collective Motion of Atomic Nuclei,} G. Giardina, G. Fazio and M. Lattuada,
Editors, World Scientific, Singapore, p. 549 (1997).

(122) J. Ellis, N. E. Mavromatos and D. V. Napoulos in {\sl Proceedings of the
Erice Summer School, 31st Course: From Superstrings to the Origin of Space--Time},
World Sientific (1996).

(123) C. Borghi, C. Giori and A. Dall'OIlio Russian J. Nucl. Phys. {\bf 56}, 147 (1993).

(124) N. F. Tsagas, A. Mystakidis, G. Bakos, and L. Seftelis, Hadronic J. {\bf 19}, 87 (1996).

(125) R. M. Santilli and D. D. Shillady, Intern. J. Hydrogen Energy {\bf 24}, 943 (1999).

(126) R. M. Santilli and D., D. Shillady, Intern. J. Hydrogen Energy {\bf 25}, 173 (2000).

(127) R. M. Santilli, Hadronic J. {\bf 21}, pages 789-894 (1998).

(128) M.G. Kucherenko and A.K. Aringazin, Hadronic J. {\bf 21}, 895 (1998).

(129) M.G. Kucherenko and A.K. Aringazin, Hadronic Journal {\bf 23}, 59 (2000).

(130) R. M. Santilli and A. K. Aringazin, "Structure and Combustion of
Magnegases", e-print http://arxiv.org/abs/physics/0112066, to be published.

,
[10] ** MATHEMATICS PAPERS:**

(131) S. Okubo, Hadronic J. {\bf 5}, 1564 (1982).

(132) J. V. Kadeisvili, Algebras, Groups and Geometries {\bf 9}, 283 and 319 (1992).

(133) J. V. Kadeisvili, N. Kamiya, and R. M. Santilli, Hadronic J. {\bf 16}, 168 (1993).

(134) J. V. Kadeisvili, Algebras, Groups and Geometries {\bf 9}, 283 (1992).

(135) J. V. Kadeisvili, Algebras, Groups and Geometries {\bf 9}, 319 (1992).

(136) J. V. Kadeisvili, contributed paper in the {\it Proceedings of the
International Workshop on Symmetry Methods in Physics,} G. Pogosyan et al., Editors,
JINR, Dubna, Russia (1994).

(137) J. V. Kadeisvili, Math. Methods in Appl. Sci. {\bf 19} 1349 (1996).

(138) J. V. Kadeisvili, Algebras, Groups and Geometries, {\bf 15}, 497 (1998).

(139) G. T. Tsagas and D. S. Sourlas, Algebras, Groups and Geometries {\bf 12}, 1 (1995).

(140) G. T. Tsagas and D. S. Sourlas, Algebras, Groups and Geometries {\bf 12}, 67 (1995).

(141) G. T. Tsagas, Algebras, Groups and geometries {\bf 13}, 129 (1996).

(142) G. T. Tsagas, Algebras, Groups and geometries {\bf 13}, 149 (1996).

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*
(254) P. Marmet, IEEE Trans. Plasma Sci. {\bf 17}, 238 (1988)
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