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The nontechnical presentation below was written in 1999. As of today (January 2003) the best technical presentation on hadronic mechanics and chemistry is available in the monograph
R. M. Santilli,
Kluwer Academic Publisher
December 2001
ISBN 1-4020-0087-1
Order by e-mail at Kluwer Academic Publishers.

An 83 pages memoir on the foundations of hadronic mechanics and chemistry can be printed out from the pdf file
<R. M. Santilli

in press at the Journal of Dynamical Systems and Geometric Theories.

Additional technical presentations are available in Scientific Works



Original content uploaded February 15, 1997. First revisions and expansions uploaded on April 9,1997. Current version dated October 9, 1999. The IBR wants to thank various visitors for critical comments. Additional critical comments sent to would be appreciated. Please note that, due to the current limitations of the html format, formulae could not be written in their conventional symbols, and had to be simplified.

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Prepared by the IBR staff.

I.1. HISTORICAL NOTES. There is little doubt that the first half of this century will enter in history as the triumph of the special relativity, quantum mechanics and quantum electrodynamics. The second half of this century will be likely remembered as characterized by a variety of attempts aimed at a generalization of pre-existing formulations.

     The physical motivations for seeking a broadening of well established theories are numerous. Those considered in this Web Page are due to the limitations of the theories as established by their mathematical structure.

     The classical and operator versions of established theories are linear, local-differential and Hamiltonian. As such, they are exactly valid for all physical conditions in which particles can be well approximated as being point-like moving in the homogeneous and isotropic vacuum, under action-a-distance, potential interactions. These are systems historically referred to as those of the "exterior dynamical problems". Examples of exact validity of established theories are the planetary and atomic structures, as well as the electroweak interactions at large.

     A broadening of established theories is studied for more general systems, historically referred to as those of the "interior dynamical problems" and consisting of extended, nonspherical and deformable particles moving within generally inhomogeneous and anisotropic physical media. These conditions prevent any effective point-like approximation of particles, and admit the most general known interactions of linear and nonlinear, local-differential and nonlocal-integral as well as potential-Hamiltonian and nonpotential-nonhamiltonian type.

     A classical example of interior dynamical systems is given by the space-shuttle during re-entry in atmosphere, whose equations of motion are arbitrarily nonlinear (in the velocity and other variables), integro-differential (in the sense that the center-of-mass trajectory is local-differential with corrective integral terms due to the shape), and definitely nonhamiltonian (in the sense of violating the topology and integrability conditions for the existence of a Hamiltonian). The latter conditions are manifestly outside any hope of quantitative treatment via the conventional Hamiltonian mechanics, thus establishing the need for more adequate mechanics.

     An operator example of interior dynamical systems is given by a neutron in the core of a neutron star, or any interior astrophysical problems. In fact, the neutron is not a point, but an extended hyperdense object with a charge radius of about 1 fm. When immersed within the equally hyperdense inhomogeneous and anisotropic core of a neutron star, the neutron is expected to experience interactions essentially equivalent to those of the space- shuttle during re-entry, i.e., of nonlinear (this time in the wavefunction and possibly its derivatives), nonlocal-integral (this time with integrals over the volume of overlapping of the charge distribution of the neutron within the medium of the star) and of nonhamiltonian type (in the sense of having contact effects for which the notion of potential has no meaning), plus conventional action-a-distance interactions. These systems are outside any hope of effective and quantitative treatment via conventional quantum mechanics, thus establishing the need for more adequate covering mechanics.

     The historical origin of the distinction between exterior and interior systems and of their respective treatments can be traced back to Lagrange. Hamilton and other founders of analytic mechanics. In essence, the virtual entirety of the physics developed during this century, all the way to quantum chromo-dynamics, is based on the assumption that one single quantity, the Lagrangian or the Hamiltonian, can represent the totality of the physical reality.

     This assumption is however against the teaching by Lagrange and Hamilton's themselves. In fact, the reading of their original works (rather than of contemporary reviews), reveals the insistence by Lagrange and Hamilton on the insufficiency of one single quantity, today called the Lagrange or the Hamiltonian, to represent the entire physical reality and. For this reason, they formulated their celebrated equations with "external terms". In this historical setting, exterior dynamical problems in vacuum are represented with a Hamiltonian without external terms, while interior dynamical problems are represented with the Hamiltonian (for all action-at-a--distance interactions), plus external terms (for all nonhamiltonian effects).

     The external terms in the analytic equations have been truncated in this century, essentially because of the successes of the special relativity and quantum mechanics. Nevertheless, the legacy of Lagrange and Hamilton persists in its entirety and has seen its most comprehensive studies precisely in the second part of this century.

     The most salient distinctions between the equations without and with external terms are those of algebraic, geometric and analytic character. Analytic equations without external terms are known to characterize a Lie algebra with the brackets [A, B] of their time evolution; they possess a symplectic geometric structure; and are derivable from a (first-order) canonical action principle.

     In 1967, during his Ph.D. studies in theoretical physics at the University of Turin, Italy, the Italian-American Physicist Ruggero Maria Santilli proved that, as conventionally written, the brackets of the time evolution of Hamilton's equations with external terms do not verify an algebra as commonly understood in mathematics (because they violate the right scalar and distributive laws). However, when properly written, these brackets characterize a covering of Lie algebras first identified in 1948 by the American mathematician A. A. Albert under the name of "Lie-admissible algebras" (i.e., the brackets (A, B) are non-Lie yet they admit Lie brackets as particular cases and their antisymmetric part [A, B] = (A, B) - (B, A) is Lie (see the preceding Web Page 18 for mathematical details). The emerging analytic equations are generally called in the literature ≥Hamilton-Santilli Lie-admissible equations≤. This year (1997) represents the 30-th anniversary of the appearance of Lie-admissible algebras in physics, that is a fundamental notion of of this Web Page.

     In subsequent studies Santilli also proved that the underlying geometry is no longer symplectic but of a new type called symplectic-admissible. The lack of derivability of the analytic equations with external terms from a variational principle in its conventional formulation is well known, e.g., because of the general loss for integro-differential systems of the differential and variational calculus of the conventional equations. This establishing the need for a broadening of conventional mathematics studied in the preceding Web Page 18, whose physical realizations will be outlined later on.

     In the late 1920's G. D. Birkhoff introduced a structural generalization of Hamilton's equations that is derivable from the most general possible first-order Pfaffian action. In the early 1980's Santilli proved that Birkhoff's equations: possess a generalized yet still Lie structure in the brackets of the time evolution; are characterized by the conventional symplectic geometry, although in its most general possible exact but noncanonical realization; and can represent all analytic, local-differential Hamilton's equations with external terms.

     In summary, a step-by step generalization of the conventional Hamiltonian mechanics (that without external terms) was known by the mid 1980 and it is given by the Hamilton-Santilli Lie-admissible mechanics, whose analytic equations are Hamilton's equations with external terms rewritten in an algebraically consistent form with a Lie-admissible structure. This new mechanics is "directly universal", i.e., capable of representing in the frame of the observer the most general known nonlinear, nonlocal and nonhamiltonian Newtonian systems of the interior dynamical problem. Since Lie-admissible algebras are coverings of Lie algebras the Hamilton-Santilli Lie-admissible mechanics admits as particular cases the Birkhoffian mechanics as well as any other known generalization of conventional Hamiltonian mechanics, including the most recent generalizations of the so-called supersymmetric type.

     A comprehensive presentation and bibliography on these historical profiles can be found in the monographs:
R. M. Santilli, "Foundations of Theoretical Mechanics", Vol.l (1978) and II (1983), Springer-Verlag, Heidelberg, Heidelberg, Germany; and "Lie-Admissible Approach to the Hadronic Structure", Vol. I 91978) and II (1982), Hadronic Press, Nonantum, Ma.

I.2. THE OPEN PROBLEM OF THE ORIGIN OF IRREVERSIBILITY. When faced with interior dynamical systems, a rather general trend is that of reducing them to a collection of exterior problems, i.e., to a collection of elementary point-like particles moving in vacuum, in the evident hope of preserving old knowledge unchanged. This hope has been shattered by a number of recent studies.

     To begin, exterior and interior problems are "inequivalent" on numerous different grounds, e.g., the former are linear, local and Hamiltonian, while the latter are nonlinear, nonlocal and nonhamiltonian, thus requiring a new mathematics for their effective treatment.

     This prevents simplistic attempts at "closing" open-nonconservative systems into a conservative form inclusive of their environment, that imply the suppression of the very physical characteristics to be represented, such as internal nonlocal, nonhamiltonian and irreversible effects.

     Moreover, exterior problems are "irreducible" to exterior ones, as established by the following:

"NO REDUCTION THEOREM" I.1: A classical system in nonconservative and irreversible conditions within a physical medium (such as the space-shuttle during re-entry in our atmosphere) cannot be consistently reduced to a finite collection of elementary constituents all in stable and reversible trajectories in vacuum (as necessary to apply quantum mechanics). Vice-versa, a finite collection of point-particles in stable and reversible orbits cannot consistently yield a macroscopic system in nonconservative and irreversible conditions.

     As it is well known, numerous attempts have been conducted during this century to reconcile the physical evidence of our macroscopic irreversibility with the intrinsically reversible character of quantum mechanics, without evidently achieving a resolution due to the irreconcilable character of the two profiles.

     The above theorem establishes that quantum mechanics simply cannot be used any longer for a scientific and effective study on the origin of irreversibility, thus stimulating the construction of a covering of quantum mechanics specifically built for particles in interior conditions. After all, the same conclusion can be reached via the search of operator images of classical interior systems as well as in a number of other ways (see later on).

     The problem of the origin of irreversibility is a most effective mean for the selection of the appropriate classical analytic equations and then the study of their operator image, that is a main aspect of this Web Page. In fact, Hamilton's equations without external terms are "structurally reversible", in the sense that they are reversible for reversible Hamiltonians. On the contrary, Hamilton's equations with external terms in SantilliĻs Lie-admissible re-formulation are "structurally irreversible" in the sense that they are irreversible even for reversible Hamiltonians. As such, they constitute a solid analytic foundations for scientific studies of interior systems.

     The latter distinction is important to understand that irreversibility is ultimately beyond the notion of Hamiltonian for various reasons. First, all action-at-a-distance interactions are known to be reversible; second, irreversibility originates from "contact" effects for which the notion of potential has no meaning; and, therefore, an effective representation of irreversibility requires exiting the class of equivalence of classical Hamiltonian mechanics and, inevitably, of quantum mechanics, thus confirming the "No Reduction Theorem " I-1.

     Intriguingly, Birkhoff's equations are also structurally irreversible, thus indicating the possibility of reconciling irreversibility with Lie's algebras, although in their most general possible (regular) realization. The equations however apply only for the local-differential approximation of generally nonlocal-integral interior systems. Hamilton's equations with external terms written in their Lie-admissible form therefore remain preferable, if nothing else, because of their "direct universality" indicated earlier.

     In summary, the studies herein considered have established that irreversibility is a physical characteristic structurally beyond the representational capabilities of a Hamiltonian. In fact, all action-at-a-distance, potential interactions are notoriously reversible. Irreversibility therefore originates with ≥contact≤ effects among the constituents of the system such as mutual overlappings of wavepackets or charge distributions of particles that, as such, have ≥zero range≤. As a result, the representation of irreversibility via the additional of irreversible terms in the Hamiltonian is ultimately misleading because it assigns a potential to a characteristic for which the notion of potential has no meaning of any known nature.

     Contrary to the attempts conducted until recently in reconciling the physical evidence of classical irreversibility with a conjectural, reversible, particle world, irreversibility originates at the most ultimate level of physical reality, the interior problem of particles, as assumed by recent models of interior astrophysical systems by J. Ellis, N. E. Mavromatos and D. V. Nanopoulos (see the Proceedings of Erice 31-st Course, ≥From Superstrings to the Origin of Space-Time≤, World Scientific, Singapore, 1996), and others.

     A scientific representation of irreversibility therefore mandates a broadening of classical Hamiltonian mechanics with a corresponding broadening of quantum mechanics for the representation of nonlinear, nonlocal and nonpotential effects at all levels. Among a variety of possibilities, that adopted in this Web Page is: 1) selection of the Hamilton-Santilli Lie-admissible mechanics because it is ≥directly universal≤ for all possible interior systems and ≥structurally irreversible≤ in the sense indicated earlier; 2) identification of the new mathematics needed for the correctly formulation of classical irreversibility in its most general known, nonlinear, nonlocal and nonhamiltonian form; and 3) identification of the operator image with related novel applications to hadron physics, nuclear physics, astrophysics and other fields.

     Comprehensive studies and references on the above studies on the origin of irreversibility at the ultimate level of elementary particles in interior conditions can be found in the monographs:
R. M. Santilli, "Elements of Hadronic Mechanics", Vols. I and II (Ukraine Academy of Sciences, Kiev, Second Edition, 1995).

I.2. PROBLEMATIC ASPECTS OF CLASSICAL AND QUANTUM DEFORMATIONS. As indicated earlier, besides the historical analytic equations with external terms, numerous enlargements of quantum mechanics have been attempted during the second half of this century. The latter are known under the generic name of "deformations" and include the so-called q-deformations, k-deformations, quantum groups and other models. In this sense, the classical analytic equations with external terms can be re-interpreted as ≥deformations≤ of the conventional equations without external terms.

     Studies conducted by R. M. Santilli, S. Okubo, D. Schuch, A. Jannussis, R. Mignani, D. Skaltsas, D. Lopez and others [I-1], have indicated that, despite an undeniable mathematical beauty, both the classical and quantum deformations do not preserve the axiomatic consistency of the original formulations and are therefore afflicted by a number of physical shortcomings.

     In a few words, a necessary condition for any broader operator (classical) theory to be nontrivial is that it must be outside the class of unitary (canonical) equivalence of quantum mechanics (classical Hamiltonian mechanics) and, therefore, must have a nonunitary (noncanonical) time evolution UxU› ā I. The latter nonunitary (noncanonical) character then implies the loss of the original axiomatic consistency with a consequential number of physical shortcomings. Essentially the same case occurs for classical noncanonical images.

     Even though not generally identified as such in the literature, quantum deformations are a simple operator image of HamiltonĻs equations with external terms because they both share the fundamental character of the brackets (A, B) of the time evolution of being Lie-admissible in the sense identified in Sect. I.1. As a result, the operator and classical deformations under consideration here have deep inter-relations.

     For example, the q-deformations with time evolution

(I.1) idA/dt = (A, H) = AxH - qxHxA , (AxB)xC = Ax(BxC),

were first proposed by R. M. Santilli in 1967 as part of his Ph. D. studies precisely as realizations of Albert's Lie-admissible algebras (see [I- 1] for details and references). In fact, the brackets (A, B) admit Lie brackets as particular cases for q = 1 and the attached antisymmetric brackets [A, B] = (A, B) - (B, A) = (1 + q)x(AxB - BxA) are Lie, as it occurs for the Hamilton-Santilli Lie-admissible equations.

     It is now important to know that all the above deformations have rather serious physical shortcomings as current formulated, that is, formulated with the same mathematics of the original theories. In fact, the q-deformations, k-deformations, quantum groups, and any other operator theory with nonunitary time evolution, have the following physical shortcomings [I-1]:

     1) Lack of invariance of the fundamental unit (e.g., the space-time unit I = diag. (1, 1, 1, 1) of the Poincare' symmetry), because under nonunitary transforms we have I ->I'; = UxIxU› = UxU› =/ I. This implies lack of invariance of the basic units of space and time (that are instead conserved in quantum mechanics), with consequential lack of unambiguous application to experiments, because it is not possible to conduct meaningful measurements, say, of a length, with a stationary meter changing in time.

     2) Lack of conservation of the Hermiticity in time, with consequential lack of physically acceptable observables, as one can verify by inspecting the behavior of Hermiticity under nonunitary transforms when defined on a conventional Hilbert space.

     3) Lack of invariance of physical laws, e.g., because of the lack of invariance of the basic brackets (A, B) under the nonunitary transform UxU› ā I of their own time evolution, as one can verify.

     4) Lack of uniqueness and invariance of numerical predictions, because of the lack of uniqueness (e.g., for the exponentiation) and invariance of special functions and transform needed for data elaboration (for instance, the "q-parameter" becomes a "Q-operator" under a nonunitary transform, Q = qxUx(U›), with consequential evident loss of all original special functions and transforms constructed for the q-parameter, and consequential loss of physical meaning of data elaborated with the latter).

     5) Loss of the axioms of the special relativity,a rather serious occurrence that is suffered by all generalizations under consideration, evidently because deformed spaces and symmetries are no longer isomorphic to the original ones. This creates the sizable problems of: first, identifying new axioms capable of replacing Einstein's axioms; second, proving their axiomatic consistency; and, third, establishing them experimentally.

     Other generalizations-broadening of quantum mechanics are afflicted by other shortcomings that also deserve attention. For instance, nonlinear theories (i.e., theories nonlinear in the wavefunctions here denoted | s >) of the type

(I.2) H(t, r, p, |s >, ...) x |s > = E x |s >,

lose the superposition principle. As a result, they cannot be applied in a physically acceptable way to the study of composite systems, such as the hadronic or nuclear structures, besides generally having a nonunitary structure, with consequential additional problematic aspects indicated above. This occurrence is known as "Schuch's No Quantization Rule" [I-1] (see D. Schuch, Phys. Rev. A, Vol. 55, 1997, in press).

     Similarly, A. Jannussis, R. Santilli, D. Skaltsas and others (see Ref. [I-1] for details and references) have shown that the traditional form of representing dissipation in nuclear physics via the addition of an imaginary potential to a Hermitean Hamiltonian,

(I.3) idA/dt = (A, H'›, H') = Ax(H'›) - H'xA

H = H+, H' = H + iV /not= H'›,

(here written as the infinitesimal version of the time evolution in finite form), or any broadening of the Liouville equation in statistical mechanics via the addition of external collision terms,

(I.4) idR/dt = (R, H, C) = [R, H] + C = RxH - HxR + C, H = H›,

have physical problematic aspects more serious than those of the deformations considered above. This is due to the fact that the transition from the quantum mechanical brackets [R, H], H = H› to the triple systems [R, H'›, H], H'› ā H' there is the loss of all algebras in the time evolution, let alone the loss of all Lie algebras. As a result, statements such as "protons and neutrons of spin 1/2" have no known meaning for nuclear models characterized by triple brackets (A,H'›,H') or plasma models characterized by triple brackets (R, H, C), because of the impossibility of even defining the SU(2) spin symmetry.

     Similarly, R. Mignani, A. Jannussis, R. Santilli and others [loc. cit.] have shown that all broadening of quantum mechanical time evolutions via a nonassociative envelope, i.e., all time evolutions of the type

(I.5) idA/dt = AoH - HoA , ((AoB)oC) /not= (Ao(BoC)),

where AoH and HoA are nonassociative, such as Weinberg's nonlinear theory and other models, are afflicted by problems of axiomatic consistency more serious than those of all preceding theories. In fact, the preceding theories do have a unit that then results to be noninvariant. On the contrary, theories of type (5) have no unit all all (i.e., there is no element e such that eoA = Aoe = A for all possible elements A of the considered set). Also, Models (1)-(3) do admit an exponentiated version of the time evolution which then results to be nonunitary and, therefore, noninvariant. On the contrary, nonassociative theories (5) lose consistent exponentiations (because of the general loss of the Poincare'-Birkhoff-Witt theorem for nonassociative envelopes).

     Moreover, nonassociative envelopes do not permit a consistent broadening of quantum mechanics due to the loss of equivalence between the Heisenberg-type and the Schroedinger-type formulations (this is due to the difference between the nonassociative character of the operator envelope as compared to the associative character of the modular action in Schroedinger's representation, an occurrence known as "Okubo's No-Go Theorem on Quantization"; see [I-1] for a review and references). Similar occurrences hold for all other nonunitary broadening of quantum mechanical formulations.

     Studies [I-1] have also indicated a similar occurrence for the classical counterpart. In fact, when formulated via conventional mathematics (i.e., on conventional spaces over conventional fields), Hamilton-Santilli Lie-admissible equations and Birkhoff's equations do not possess invariant units of space and time, therefore lacking unambiguous applications to measurements.

     The above shortcomings have requested a re-inspection of the mathematical foundations of contemporary theoretical physics for the achievement of axiomatically consistent classical and operator representations of interior dynamical systems.

     In summary, by keeping into account the above shortcomings, an important suggestion of memoir [I-1] is to exercise caution before abandoning the majestic axiomatic consistency of the special relativity and its underlying classical and quantum relativistic mechanics, with particular reference to their invariant basic unit, canonical-unitary structure, unambiguous measurement theory, preservation of observability at all times, unique and invariant numerical predictions, and other well known physically consistent characteristics.

     The research outlined in this Web Page has been therefore conceived for the specific purpose of preserving the axioms of the special relativity at both classical and operator levels, yet possessing broader noncanonical and nonunitary structures.

I.4. SANTILLI'S ISOTOPIC, GENOTOPIC, HYPERSTRUCTURAL LIFTINGS AND THEIR ISODUALS. The only solution of the physical shortcomings of the preceding section known at this writing is that based on SantilliĻs iso-, geno-, hyper-mathematics and their isoduals outlined in the preceding Web Page 18 [M-1.1].

     Recall that the isotopies are nonlinear, nonlocal (integral) and nonpotential (noncanonical or nonunitary) maps of any given linear, local and potential (canonical or unitary) structure that preserve the original axioms because they reconstruct linearity, locality and canonicity or unitarity in certain generalized spaces and fields.

     The main idea of the isotopies is the lifting of the basic space unit I = diag. (1, 1, 1, 1) of conventional mechanics into a 3x3 positive-definite matrix E = 1/T called isounit, while all conventional (associative) products are lifted by the "inverse" amount, according to the general rules

(I.6a) I = diag. (1, 1, 1) -> E = E(t, r, v, a, ...) =

    = Diag. {n_1^2, n_2^2, n_3^2}xF = 1/T,

(I.6b) AxB -> A*B = AxTxB,

where v = dr/dt, a = dv/dt, the nĻs and F are well behaved, real-valued and positive-definite functions, under which rules E is the correct left and right unit of the new theory, E*A = A*E = A for all possible A. The relativistic extension is straighforward and will be ignored at this time (see Sect. I-19).

     The most important physical meaning of the isounit is that of representing extended, nonspherical and deformable shapes that, for the case of spheroidal ellipsoids, is done via the semiaxes nk-squared of the diagonal isounit (7a) with nondiagonal forms for more general shapes. The contact, nonlinear, nonlocal and nonhamiltonian interactions are represented via the factor term F = F(t, r, v, ...). Note that both these characteristics are manifestly outside the representational capabilities of a Hamiltonian. As such, they should be represented with anything possible, ≥except≤ the Hamiltonian. Santilli suggested their representation via the unit of the theory because effective on various physical and mathematical grounds.

     The genotopies occurs when the isounit is no longer symmetric although its real-valuedness is generally assumed in physical applications. The multivalued hyperlifting occur when the isounit is characterized by an ordered "set" of generally nonsymmetric elements. The isodual images of conventional, isotopic, genotopic and hyperstructural formulations are given by the map A -> IsodA = -A› applied to the totality of the original structures thus yielding theories with ≥negative-definite≤ units, e.g., IsodI = -I = - Diag. (1, 1, 1) or IsodE = -E.

     As indicated in Web Page 18, for consistency the above liftings and their isodual must be applied to the totality of the original mathematical structures, including numbers and angles, conventional and special functions and transforms, differential calculus, algebras, geometries, topologies, etc., yielding the so-called iso-, geno-, hyper-mathematics and their isoduals.

     The isomathematics is used for an axiomatically consistent and invariant re-formulations of Birkhoff's equations, while the genomathematics is used for an axiomatically consistent and invariant formulation of Hamilton-Santilli Lie-admissible equations. The hypermathematics characterizes a new type of analytic mechanics outlined below.

     It should be stressed that, if the above liftings are applied only to some structures (such as the Euclidean or Hilbert spaces), but not to others (such as the base fields of numbers, or vice-versa), serious axiomatic inconsistencies follow. As an illustration, the deformations change only "part" of the original formalism, the product of the envelope AxB -> A*B = qxAxB, while preserving the original unit I of AxB unchanged. The loss of its invariance of the entire theory is then consequential. On the contrary, the isotopies are based on lifting the associative product into the axiom-reserving form AxB -> A*B = AxTxB, where T is an invertible operator admitting of the parameter q as a particular case. Jointly the basic unit is deformed by the inverse amount I -> E = 1/T, that is then the correct left and right unit of the new theory, E*A = A*E = A. preservation of the original axiomatic properties, including invariance, is then consequential, as we shall see.

    To reach an invariant formulation, the q-deformations of type (1) must then be reformulated with the mathematics of the preceding Web Page 18 with the basic unit E = 1/q, thus requiring the use of new numbers, new spaces, new geometries, etc.

     Similarly, quantum groups generally deform an original quantum symmetry, while preserving the original Hilbert space. Lack of preservation of Hermiticity in time under nonunitary time evolutions is then consequential. By comparison, the isotopies lift symmetries while jointly lifting the underlying Hilbert space in such a way to preserve Hermiticity at all times despite the nonunitary character of the time evolution.

     One known way for quantum groups to achieve a physically acceptable axiomatic structure, is that they should also be reformulated via the new mathematics of the preceding Web Page 18 (the communication to the IBR headquarters at;of any other known alternative with related references would be appreciated).

     Similarly, nonlinear theories of type (2) violate an important structural axiom of quantum mechanics, the linearity of its operator theory. On the contrary, the isotopies represent a class of nonlinear system much greater than those of Eq.s (2), with an arbitrary nonlinearity also in the derivatives of the wavefunction,

(I.7) H(t, r, p)xT(t, r, p, |s>, D|s>/Dr, ...)x|s> = Ex|s>,

(where D/Dr stands for partial derivative) yet they reconstruct linearity in isospaces over isofields (see the preceding mathematical Web Page and ref. [I-1]) because all nonlinear terms are embedded in the invariant unit. As a result, nonlinear isotopic theories of type (6) do preserve the superposition principle and they can indeed be consistently applied to the study of composite systems with nonlinear internal effects, such as hadronic or nuclear structures.

     To achieve a physically acceptable axiomatic structure, nonlinear theories (2) should be identically reformulated in the isotopic structure, i.e., via the factorization of all nonlinear terms in the Hamiltonian and their embedding in the isounit, H(t, r, p, |s>, ...) = H'(t, r, p)xT(|s>, ...). A similar resolution of all other shortcomings also occurs (see [I-1] for brevity).

     All preceding formulations are solely recommended for the characterization of matter. Their isodual images are used for a novel characterization of antimatter beginning at the classical level which then persists at all subsequent levels. In fact, the isodual map is anti-isomorphic as it is the case for charge conjugation, the difference being that the latter is solely applicable on Hilbert spaces, while the former is applicable at all levels of study, whether classical or quantum mechanical. In particular, when formulated on Hilbert spaces, isoduality has been proved to be equivalent to charge conjugation and been able to recover existing experimental data on antiparticles, such as those of electroweak character (see [I-1] and R. M. Santilli, Hyperfine Interactions, 1997, in press).

     To avoid a prohibitive length, classical profiles will be only briefly outlined while devoting more attention to operator formulations.

I-5. SEVEN NEW TYPES OF NEWTON'S EQUATIONS. They are given by the isotopic, genotopic and hyperstructural liftings of the conventional equations for matter and the isoduals of all four of them for the representation of antimatter, submitted for the first time in memoir [M-I-1].

     In essence, Newton's equations have remained unchanged since their inception in the 1600's. As such, they can only represent point-particles with local-differential forces, as necessary in order not to violate the topology of the underlying Euclidean space. The isotopies of Newton's equations represent instead particles with their actual, extended, nonspherical and deformable shape under action-at-a-distance, potential as well as contact nonpotential interactions of generally nonlocal type (e.g., with surface integrals). The genotopic equations represent the latter system under the additional condition of irreversibility, while the hyperstructural equations are multivalued.

     The above iso-, geno- and hyper-liftings of NewtonĻs equations are reached via: A) lifting the conventional field of real numbers into their iso-, geno- and hyper-forms outlined in Web Page 18; B) Lifting accordingly the carrier space of NewtonĻs equations, the seven-dimensional Euclidean space E(t)xE(r)xE(v); and, above all, C) lifting the derivatives used in the formulation of the equations into the corresponding iso-, geno- and hyper-derivatives Ź*/Ź*r = TxŹ/Źr, where T = 1/T is symmetric, nonsymmetric and multi-valued, respectively (see [M-I-1] for details).

     Unlike the trend in the literature of this century of considering antimatter only at the level of "second" quantization, the "new physics of antimatter" begins at the purely classical and Newtonian level and it is characterized by the anti-automorphic isodual image of the theory for matter. This yields four additional equations for the characterization of antimatter given by the isodual image of the conventional, isotopic, genotopic and hyperstructural equations.

     An understanding of the new equations requires the knowledge that the conventional, isotopic, genotopic and hyperstructural equations coincide at the abstract, realization-free level, and the same happens for their isodual images.

     One should note that we are referring here to one of the very few structural generalization of Newton's equations in Newtonian mechanics since Newton's time (because other generalizations are of relativistic or gravitational character). These advances were possible thanks to the prior discovery of new mathematics, specifically conceived to be applicable beginning at the Newtonian level.

I-6. SEVEN NEW TYPES OF LAGRANGIAN AND HAMILTONIAN MECHANICS. They are also given by isotopic, genotopic and hyperstructural liftings of the conventional mechanics, plus the isoduals of all four of of them, first submitted in Ref. [M-I-1]. As it is the case for the preceding Newtonian profile, the generalizations are based on iso-, geno-, hyper-liftings of the mathematics of the conventional mechanics (fields, spaces, geometries differential calculus etc.) and their isoduals.

     A primary purpose of the above novel forms of classical mechanics is that of being "directly universal" in Newtonian mechanics, that is, of representing via a first-order variational principle all infinitely possible exterior and interior Newtonian systems of extended, nonspherical and deformable particles under the most general known linear and nonlinear, local-differential and nonlocal-integral and potential as well as nonpotential forces (universality), directly in the frame of the experimenter (direct universality).

     These advances therefore permit studies simply impossible with conventional methods, such as the representation of the trajectory of the space-shuttle during re-entry in our atmosphere as an integro-differential system verifying Tsagas-Sourlas topology of the preceding Web Page, where the differential part characterizes the center-of-mass motion with corrective integral terms due to the shape represented via the generalized unit and calculus.

     The new mechanics also permit the optimization of actual, extended shapes within resistive media (such as a wing moving in atmosphere or a body moving in water) via the optimal control isotopic theory, and similar integro- differential problems that cannot evidently be treated via a local-differential mechanics.

     Let us consider the case of the isotopies in 6-dimensional phase space with local coordinates b* = {r*, p*} = {rxE, pxT}, where the rĻs are contravariant and the pĻs are covariant. The isodifferentials are then given by d*r = Exdr and d*p = Txdp with total six-dimensional unit E-tot = Diag. (E, T), E = 1/T. The background geometry is then the isosymplectic geometry on isocotangent bundle over isofields all with the same six-dimensional isounit E-tot in which :the conventional canonical one form S1 = pxdr is lifted into the iso-one-form S*1 = (p*)*d*r*; the exact nowhere degenerate symplectic two-form S2 = d(S1) = drVdp (where V is the exterior product) is lifted into the isoexact, nowhere degenerate isosymplectic iso=two-form S*2 = d*(S*1) = (S2)xExtot; with corresponding liftings of the remaining aspects of the symplectic geometry.

     The above lifting of the symplectic geometry has the implied a corresponding lifting of Hamiltonian mechanics [M-I-1] based on analytic equations called ≥Hamilton-Santilli isoequations≤ that: 1) are ≥direct universal≤ for the representation of Hamilton's equations with nonlocal-integral external terms (this is a class of systems broader than that of Birkhoff's equations); 2) preservation of the abstract axioms of Lie's theory, although realized in a broader form; and 3) preservation of all axiomatic properties of the truncated Hamilton's equations, including invariance of the basic units. The invariant reformulation of BirkhoffĻs equations for the simpler case of local-differential and nonhamiltonian systems is characterized by the mere factorization of the isounit E-tot from their general, exact, symplectic two-form á = S2xE-tot.

     The Hamilton-Santilli genoequations, first proposed in Ref. [M-I-1], are essentially given by the isoequations in which the unit E-tot is real-valued and nowhere singular, but no longer symmetric. as pointed out in the preceding Web Page 18, this permits an axiomatic representation of irreversibility, that is, its representation via the fundamental unit. In fact, the theory now admits ≥two≤ generalized nonsymmetric units, E-tot for motion forward in time, and its transpose for motion backward in time, with corresponding ordered products to the right and to the left. Due to the difference between E-tot and its transpose, the equations are structurally irreversible even for reversible Hamiltonians as desired. the reformulation of Hamilton-Santilli Lie-admissible equations in the above genotopic form is given by factorizing in the Lie-admissible tensor the genounit for motion forward in time.

     The hyperstructural equations [M-I-1] are a still broader, multi-valued irreversible formulations in which the units for motions forward and backward in time are characterized by an order set of nonsymmetric matrices.

     All the preceding equations are suggested for the sole characterization of matter, The isodual conventional, iso-, geno- and hyper-analytic equations are the anti-automorphic images of the preceding ones under isoduality and are recommended for the characterization of antimatter.

I-7. SEVEN NEW TYPES OF QUANTIZATION. In the mid-1980's the U. S. mathematician E. B. Lin presented the first isotopy of the symplectic quantization. This first study was followed by the isotopies of the naive quantization by the physicists A. O. E. Animalu and R. M. Santilli. Recall that the conventional naive quantization is the map of the canonical action A into the expression -ixIxLn|s>, where I is Planck's constant, under which the Hamilton-Jacobi equations yield the familiar Schroedinger's equations. The naive isoquantization is the map of the isoaction A* into the expression -ixExLn|s>, where E is the iso-, geno-, or hyper-unit. The Hamilton-jacobi equations of the iso-, geno-, and hyper-mechanics then yield the fundamental equations of hadronic mechanics as identified in the next section (see [I-1] for details and references).

     The recent advent of the isodifferential calculus and isosymplectic geometry then permitted the achievement of maturity of formulation in isosymplectic quantization and in its isodual [M-I-1,I-1]. The geno- and hyper-quantization have not been studied to date, although their existence is expected as a natural lifting of the preceding formulations (see Sect. 4 below).

I-8. NONRELATIVISTIC HADRONIC MECHANICS. In 1978 Santilli (Hadronic J. Vol. 1, pages 228, 574 and 1267) proposed the construction of a generalization of quantum mechanics under the name of (nonrelativistic) hadronic mechanics, based on the following fundamental dynamical equations of isotopic type,

(I.8) idA/dt = AxT(t, r, |s>, ...)xH - HxT(t, r, |s>, ...)xA, T = T› > 0,

as well as of the broader genotopic type

(I.9) idA/dt = AxR(t, r, |s>, ...)xH - HxS(t, r, |s>,...)xA , R /not= R›,

with the recent addition [M-I-1, I-1] of the dynamical equations of multivalued-hyperstructural type

(I.10) idA/dt = Ax{R_1, R_2, ... }xH - Hx{S_1, S_2, ...}xA , {R} = {S›},

plus their isoduals for antimatter here omitted for brevity, where d/dt represent total derivative of the iso-, geno- and hyper-type, respectively and | s > represent the wavefunction.

     The above Heisenberg-type equations were complemented with the Schroedinger-type versions by Myung and Santilli and, independently, by Mignani, in the early 1980's via conventional differential calculus, and more recently by Santilli with the form of differential calculus, Hilbert spaces and fields necessary for invariance [M-I,1, I-1]. The latter are given by the iso-Schroedinger's equations

(I.11a) iD*|s>/D*t = iI*_txD|s>/Dt H(t, r, p)xT(t, r, |s>, ...)x|s> = Ex|s>

(I.11b) p_kxT(r,...)x|s>, ... )x|s> = -iD_k*|s>/D*r = -iT_k^ixD_i|s>/Dr , T = T›,

where: D*/D*t = T(t)xD/Dt, D/Dt is ordinary partial derivative, E(t) = 1/T(t) is the one-dimensional isounit of time, and D*/D*r = T(r,...)xD/Dt, E(r,...) = 1/T(r,...) is the three-dimensional isounit of space; the geno-Schroedinger equations

(12a) iD*|s>/D*t = H(t, r, p)xR(t, r, |s>, ...)x|s> = Ex|s>

(I.12b) pxR(r, |s>, ... )x|s> = -iD*|s>/D*r, R =/ R›,

where now the derivatives are of genotopic type; and the hyper-Schroedinger equations (13a) iD*|s>/D*t = H(t, r, p)x{R_1, R_2, ...}x|s> = Ex|s>

(I.13b) px{R_1, R_2, ...}x|s> = -iD*|s>/D*r,

where now the derivatives are of hyper-type, and where all variables are tacitly assumed for simplicity of notation to be of the appropriate iso-, geno- and hyper type. Of paramount importance for mathematical and physical consistency is the use of the applicable mathematics, i.e., the use of iso-, geno-, hyper-fields, spaces, algebras, functional analysis, etc. for the corresponding iso-, geno- and hyper-equations.

     Needless to say, the above equations admit as particular cases the conventional ones when all units acquire the simplest possible value 1. In that sense, hadronic mechanics is a covering of quantum mechanics.

     Eqs. (8)-(13) are solely recommended for the description of "particles" in interior conditions of increasing complexity. Nonrelativistic hadronic mechanics admits the isoduals of the conventional and generalized equations for the characterization of "antiparticles"..

     In addition to the mathematicians listed in the preceding Web Page 18, the 1978 proposal to build hadronic mechanics was studied by numerous physics, including: Mignani, Okubo, Kadeisvili, Ellis, Nanopoulos, Adler, Schuch, Jannussis, Animalu, Aringazin, Gill, Mavromatos, Callebaut, Arestov, Mills, Rauch, Tsagas, Mystakidis, Balkos, Sewftelis, Nishioka, Brodimas, Skaltzas, Lohmus, Paal, Sorgsepp, Caldirola, Cardone, Eder, Smith, Fronteau, Gasperini, Vacaru, Ioannidou, Kalnay, Karayannis, Kliros, Klimyk, Kobussen, Lopez, Mijatovic, Ntibashirrakandi, Papadoupoulos, Papaloukas, Papatheou, Streclas, Veljanoski, Vlahos, Tellez Arenas, (N) Tsagas, and others.

     Axiomatic consistency of the new mechanics was reached only recently in memoir [I-1] thanks to the advent of the iso- geno- and hyper-calculus and their isoduals. In fact, the dynamical equations as initially written with the conventional differential calculus have not resulted to be invariant. Also, as pointed out in the preceding Web Page, the proof of the axiom-preserving character of the genotopic and hyperstructural formulations was reached only recently, thanks to the background geno- and hyper-mathematics.

     Hadronic mechanics was constructed to provide a more realistic representation of the structure of hadrons and their interactions. The atomic structure is linear, local-differential and potential owing to the large mutual distances among its constituents, under which conditions quantum mechanics is exactly valid. By comparison, hadrons are some of the densest objects measured by mankind in laboratory until now, thus being composed by "extended wavepackets" in conditions of deep mutual penetration, that result in expected nonlinear, nonlocal-integral and nonpotential-nonunitary effects even for constituents with "point-like charges", as established for the Cooper pair in superconductivity (see Sect. IX).

     The visitor of this Web Page should be aware that nowadays the terms "hadronic mechanics" are referred to a rather vast discipline that includes the following eight branches, all uniquely characterized by the assumed basic unit,

     1-2) conventional quantum mechanics and its isodual.
     3-4) isotopic mechanics and its isodual.
     5-6) genotopic mechanics and its isodual
     7-8) hyperstructural mechanics and its isodual,

each branch admitting a unique and unambiguous classical formulation, as outlined earlier, with a unique and unambiguous map interconnecting classical and operator formulations.

     Quantum mechanics is assumed as exact for the conditions of its original conception, the atomic structure and the electroweak interactions at large for matter, while its isodual is under study for a novel treatment of antimatter.

     The isotopic branch of hadronic mechanics is under study to attempt a more realistic representation of the hadronic structure and the strong interactions at large for matter, while its isodual is studied for a novel representation of corresponding antiparticles. The isotopic branch is selected for all stable, and therefore time-reversal invariant, hadronic structures that, in first approximation, can be considered to be the case for all mesons, baryons and other hadrons with a sufficiently long meanlife in particle standards.

     The genotopic branch is under study for an axiomatically consistent identification of the origin of macroscopic irreversibility in the ultimate layer of the particles world. In fact, the genotopic equations are unique and unambiguous operator images of the classical Hamilton-Santilli genoequations assumed in Sect. I.2 as the fundamental representation of macroscopic irreversibility. The same equations are also used for consistent studies of dissipative nuclear processes, irreversible biological structures, irreversible black-hole structure, and similar problems. The isodual genotopic branch is under study for the corresponding antimatter systems in physics or bifurcations in biology.

     Finally, the multivalued hyperstructural branch and its isodual are under study for novel cosmological conceptions of the universe or for novel applications in theoretical biology (see the next Web Page).

     By conception and construction, the conventional, isotopic, genotopic and hyperstructural mechanics coincide at the abstract-realization-free level, and the same occurs for the corresponding isodual images. The above property has the paramount importance of guaranteeing the axiomatic consistency of hadronic mechanics "ab initio". In fact, criticisms on its structure are de facto criticisms on the axiomatic structure of ordinary quantum mechanics.

     For this reason, hadronic mechanics ≥cannot≤ be claimed to be a ≥new mechanics≤. In fact, the latter can be claimed only for ≥new axioms≤, that are carefully ≥avoided≤ in the studies herein reported, the ultimate objective being that of preserving the axioms of the special relativity as outlined in the next section. In fact, hadronic mechanics merely provides ≥new realizations≤ of the abstract axioms of conventional quantum mechanics.

     Also, it is evident that the isoeigenvalue equation H*| > = Ex| > provides an explicit and concrete realization of the theory of "hidden variables" (because the two right, associative, modular actions "Hx| >" and "H*| >" = "HxTx| >" coincide at the abstract level for all T > 0), for which von Neumann's theorem and Bell's inequalities do not apply (because of the nonunitary structure of the new theory). It is also evident that hadronic mechanics constitutes a form of "completion" of quantum mechanics much along the historical argument by Einstein, Podolsky and Rosen and, for this reason, the word "completion" appears in the title of Ref. [I-1].

     It is at times believed that hadronic mechanics has a complicated mathematical structure that, as such, dampens possible applications. In reality, the entire mathematical formalism of hadronic mechanics can be uniquely and unambiguously constructed via very simple maps of the corresponding formalism of quantum mechanics. For instance, the isotopic branch of hadronic mechanics can be uniquely and unambiguously constructed in its entire explicit form via the systematic application of a nonunitary map (see [I-1] for all details)

(I.14) UxU› = E = 1/T /not= I ,

to the conventional aspects of quantum mechanics without any known exception, as necessary for consistency. In fact, under the application of said nonunitary map we have:

a) the lifting of the conventional unit I of quantum mechanics into the isounit and of conventional numbers n into the isonumbers

(I.15a) I -> I' = UxIxU› = E = 1/T;

(I.15b) n -> UxnxU› = nx(UxU›) = nxE;

b) the lifting of the conventional associative product of operators AxB into the isoassociative product of the isotopic branch of hadronic mechanics

(I.16) AxB -> Ux(AxB)xU› = A'*B' = A'xTxB', T = (UxU›)^{-1} = T›,

    A' = UxAxU›, B' = UxBxU›

thus E = 1/T = E› as desired with the correct Hermiticity property. The iso-Heisenberg equation (8) then follows;

c) the lifting of the conventional inner product of the Hilbert space of quantum mechanics into that of the Myung-Santilli isohilbert space and related isoexpectation values

(I.17a) < R |xTx| S > , | R, S > = Ux| r, s > , T = (UxU›)to -1,

(17b) < A > = < S |xTxAxTx| S > / < S |xTx| S >

and the same occurs for the construction of "all" aspects of the isotopic branch of hadronic mechanics, including isoexponentiation, isosymmetries, special isofunctions and isotransforms, etc.

     The genotopic branch can be constructed via a more general transform of quantum mechanics with the "dual" nonunitary structure [I-1]

(I.18) MxN› /not= I and NxM› /not= I, MxM› /not= I, Nx N› /not= I, M /not= N.

In fact, the application of transforms MxN› yields: the lifting of the unit and numbers of quantum mechanics into the forward and backward genounits and genonumbers of Web Page 18,

(I.19a) E^{>} = MxIxN› = 1/S,

(I.19b) n^{>} = nx(MxN›) = nxE^{>},

for the characterization of motion forward in time. The application of the conjugate transform NxM› then characterizes the corresponding backward quantities for motion backward in time. All other aspects of the genotopic branch of hadronic mechanics can be derives via the same elementary method. The hyperstructural branch can be constructed via the above dual transforms of the genotopic branch under the condition that they are multivalued, e.g., thy are the tensorial product of an ordered set of nonunitary transforms.

     The isodual branches of hadronic mechanics can be easily constructed via the systematic application of the isodual map Q -> isodQ = -Q› to the totality of the quantities and operations of any given branch, from the unit to special functions and transforms.

     Thanks to the availability of a consistent axiomatic structure and the above simple means for its construction, specific applications and experimental verifications of hadronic mechanics are today rather simple and can be done via the following general rules:

A) Identify first whether the system considered is external or internal and reversible or irreversible. If the system is external (i.e., there is no plausible expectation for nonlocal internal effects due to sufficiently short mutual distances of the constituents), conventional quantum mechanics applies exactly, as stressed earlier. If the system is internal (i.e., nonlocal effects are indeed expected), then one must identify first whether the system considered is time-reversal invariant or not. For instance, the structure of hadrons is expected to be an internal problem because the size of the wavepackets of all massive particles is of the same order of magnitude of the hadrons themselves, thus resulting in total mutual penetration of the wavepackets of the constituents. In this case hadronic mechanics is expected to apply. If the meanlives of the hadrons considered are sufficiently long for particle standards, then the isotopic branch of hadronic mechanics can be selected, otherwise the use of the genotopic branch is preferable.

B) Represent all conventional action-a-distance interactions (e.g., photons or gluons exchanges) via a conventional Hamiltonian.

C) Represent all non-Hamiltonian characteristics and effects via the iso-, geno or hyper-units with general structure

(I.20a) E_{space} = {Diag. ( n_1^2, n_2^2, n_3^2)} x F_s(t, r, v, a, ...) = UxU›

(I.20b) E_{time} = {n_4^2}xF_t(t, r, v, a, ...) = WxW›,

where: the space n's represents the extended, nonspherical and deformable shape of the charge distributions of the hadron considered (in the above case, spheroidal ellipsoids for spinning hadrons), thus assuming the values 1 for perfectly spherical shapes; n4 represents the density of the physical medium considered normalized to the value 1 in vacuum; and the F's represents all non-Hamiltonian interactions, e.g., the "contact interactions" among wavepackets that, as such, have "zero-range", cannot be represented with any potential and, therefore, cannot be mediated by any exchange.

     In all cases studied until now, it has been possible to identity the iso-, geno- or hyper-unit in a unique and unambiguous way. The only difficulties are conceptual and due to the lack of familiarity with the problems intended for study. In fact, due to protracted use during this century, physicists are accustomed to represent features of the exterior problem of point-particles in vacuum, without a familiarity with the interior problem of extended particles in conditions of deep mutual penetration, as represented by hadronic mechanics.

     A typical illustration is given by the extended, nonspherical and deformable shapes of the charge distribution of hadrons and their densities. These important physical characteristics of hadrons are absent in contemporary hadron physics because outside the representational capabilities of conventional, quantum mechanics and, as such, they are somewhat unfamiliar. On the contrary, the same characteristics are fundamental for the study of the same particles via hadronic mechanics because they characterize the fundamental unit according to Eqs. (21).

     It is also at times believed that hadronic mechanics is ≥too general≤ because it admits an infinite number of possible units. Such a view is equivalent to the statement that quantum mechanics is too general because it admits an infinite number of different Hamiltonians. In reality, one of the conditions for physical effectiveness of quantum mechanics is precisely its capability to admit all infinitely possible Hamiltonians. Along the same lines, one of the reason for the physical effectiveness of hadronic mechanics is precisely that of preserving all possible Hamiltonians and adding an infinite number of possible units for each Hamiltonian.

     This is due to the fact that the same number of particles in exterior conditions in vacuum admit a large number of different interior conditions, as established, say, by the existence of a large number of different astrophysical bodies all with the same mass and elements. All these bodies have the same Hamiltonian (because they have the same constituents, thus having the same action-a-a-distance interactions), and their physical differences under the same Hamiltonian are represented precisely by the generalized units. A somewhat similar situation occurs for hadrons of the same family that may be treated via a unified Hamiltonian. The physical differences in the experimentally established variation of the density from hadron to hadron is then represented via different generalized units for different hadrons.

     A most important property of hadronic mechanics is that of preserving unchanged the conventional physical laws of quantum mechanics, such as Heisenberg's uncertainties, Pauli's exclusion principle, the superposition principle, causality, etc. For instance, the uncertainties are numerically preserved unchanged under the isotopies, because (for Planck's constant/2-pi =1) [I-1]

(I.21) dr x dp </= [r, p]*>/2 = < E >/2 =

    = < S |xTxExTx| S >/2< S |xTx| S > = 1/2

(where d represents delta, p is covariant and r contra-variant, see [M-I-1] for other cases and the isodifferential calculus has been used) and the same happens for other laws.

     This occurrence has rather deep epistemological, theoretical and experimental implications that remain to be investigated in great part. At this moment we can say that, by no means,the experimental validity of conventional quantum laws permits a scientific claim on the exact validity of the conventional quantum mechanics, because exactly the same laws are admitted by much broader covering mechanics with a nonunitary structure.

     These remarkable occurrences are due to the novel invariance law of the Hilbert space, first presented in [I-1] and here expressed for an isotopic element T independent from the variable of integration,

(I,22) < r |x| s > x I = < r |xTx| s > x E, E = 1/T

that confirms the admission by the axioms of the conventional quantum mechanics of the isotopic, genotopic and hyperstructural realizations. Note that the new invariance law (23) has remained undetected since Hilbert's time because its identification required the prior discovery of new numbers, SantilliĻs iso-, geno-, and hyper-numbers with arbitrary units. In fact, the transition from quantum to hadronic mechanics is nothing else that a re-scaling of the basic units.

     With the understanding, again, that conventional quantum mechanics is here assumed to be exactly valid for physical conditions without appreciable nonlinear, nonlocal and nonunitary effects, and that the use of hadronic mechanics is recommended for quantitative studies of the latter effects whenever expected, nowadays hadronic mechanics has achieved mathematical maturity [M-I-1], axiomatic consistency [I-1], and experimental verifications in nuclear physics, hadron physics, astrophysics, superconductivity, theoretical biology and other disciplines (see the outline in [I-1]).

I.6 RELATIVISTIC HADRONIC MECHANICS. A comprehensive and diversified effort was conducted by Santilli and his group over the years for a step-by-step axiom-preserving lifting of the special relativity and relativistic quantum mechanics (see the latest presentation in Ref. [I-1]), that consists of:

     i) The isotopies and isodualities of the Minkowski space M(y,m,R) with space-time coordinates y, and metric m = Diag. (1, 1, 1, -1) over the reals R, into the isominkowski space (see the preceding Web Page) M*(y*,G*,R*) with isocoordinates y* = yxE, isometric M* = TxmxE = (m*)xE, m* = Txm and isoinvariant

(I.23) [y* - z*]^2* = (y* - z*)^t*(G*)*(y* - z*) = [(y - z)^txm*x(y - z)]xE,

over the isoreals, where t denotes transpose, x denotes conventional product, and the basic isounit E = 1/T is now a 4x4 nowhere singular, real-valued and positive-definite matrix. As a result, the isotopic element T can be diagonal and the above isoseparation can be explicitly written

(I.24) {[ y_1 - z_1)/n_1]^2 + [(y_2 - z_2)/n_2]^2 +

    + [(y_3 - z_3)/n_3]^2 - [t_1 - t_2)c/n_4]^2}xE,

(1,24b) C = c/n_4

where c is the speed of light in vacuum.

     ii) The isotopies and isoduality of relativistic quantum mechanics, centered on the universal symmetry of isoinvariant (24) and its isodual, that are given by the isotopies P*(3.1) and isodualities IsodP*(3.1) of the Poincare' symmetry P(3.1), including the liftings of SO(3), SU(2)-spin,SO(3.1), SL2.C), and discrete symmetries.

     iii) The isotopies and isoduality of the special relativity, called SantilliĻs isospecial relativity, including the lifting of its main axioms, such as those on the maximal causal speed, relativistic addition of speeds, Lorentz contraction, time dilation, Doppler's law, etc.

     The main outcome of these studies can be summarized as follows. The current formulation of the special relativity is exactly valid only for the conditions of its original conception, the exterior dynamical conditions of point-particles in vacuum. Lorentz was the first to identify the lack of applicability of his own symmetry for interior problems and, in fact, as quoted by Pauli's on his historical book on relativities, Lorentz initiated himself the study of the symmetry of electromagnetic waves propagating in our atmosphere that were known already at that time to propagate with speed C = c/n < c.

     Nowadays, the speed of light in vacuum is no longer a barrier because of the experimental detection of photons traveling in certain guides faster than c, and the confirmed astrophysical measures of masses expelled in supernova or other explosions at speed greater than c.

     The preservation of the special relativity in its current formulation for arbitrary maximal causal speeds C = c/n, n > or < 1, is prevented by numerous problems of consistency, such as violation of causality or, in case of its preservation, violation of the relativistic law of addition. For instance, under the causal speed c in water, the relativistic sum of two speeds of light in water does not yield the speed of light in water or the speed of light in vacuum. Even more evident inconsistencies occurs when one believes in the exact validity of the special relativity within inhomogeneous and anisotropic physical media.

     The implications of the isospecial relativity on the isominkowskian space M*(y*,G*,R*) under the universal Poincare'-Santilli isosymmetry P*(3.1) are the following:

     1) Extending the arena of applicability of the special relativity, from the sole case of the constant speed c in vacuum to all infinitely possible speeds within physical media C = c/n. In fact, as one can see in Eq. (25), the isominkowskian geometry was conceived by Santilli in 1983 (Lett. Nuovo Cimento Vol. 37, p. 545, 1983) to provide a direct geometrization of the speed C = c/n, n = n4 in the most general case of inhomogeneous and anisotropic physical media, such as our atmosphere. In the same studies of 1983, the isolorentz symmetry was first constructed to provide the universal invariance for arbitrary speeds C, a property proved to be correct by subsequent studies, and the isotopies of the basic axioms of the special relativity were first introduced.

     We should note that the special and isospecial relativity coincide at the abstract, realization-free level, as it is the case for all isotopies, to such an extent that even the maximal causal speed in isospace M* is c and not C. In fact, in the lifting M -> M*, c is lifted by the amount c -> c/n, but the corresponding unit is lifted by the inverse amount 1 -> n, thus preserving the value c as the ,maximal causal speed (because the invariant is [Length-squared]x[Unit-squared]). By keeping in mind the content of the preceding mathematical Web Page, the characterization of the locally varying speed C = c/n occurs in the ≥projection≤ of the isospecial relativity in our space-time.

     2) The isospecial relativity provides a unification of the special and general relativities into one single structure in which the two relativities are differentiated by different values of the unit. The functional dependence of the isotopic element T remains unrestricted in the isotopies. As a result. they can also depend on the local space-time coordinates y. It then follows that the isominkowskian metrics m* = Txm can indeed admit, as a particular case, all possible (regular) Riemannian metrics g(x) in (3+1)-dimensions according to the factorization [I-1]

(I.25) g(x) = T(x)xm = m*, E = 1/T.

     Thus, for E = 1/T where T = g/m, we have the general relativity, while for the simplest possible case E = Diag. (1, 1, 1, 1) we have the special relativity.

     It then follows that the isominkowskian geometry permits a symbiotic unification of the Minkowskian and Riemannian geometries. In fact, it preserve all Minkowskian axioms, including that of isoflatness, that is, flatness in isospace over isofields, while admitting all mathematical tools of the Riemannian geometry (connection, covariant derivative, etc.), although formulated with respect to the isodifferential calculus (see the specialized literature on this point).

     The above occurrence may appear paradoxical to non-experts in the field, but its technical origin is elementary. In fact, curvature is in reality contained in the term T(y) of the factorization g(y) = T(y)xm, and certainly not in the Minkowskian term m. It then follows that when the Riemannian geometry for a given metric g(y) is reformulated with respect to new unit E = 1/T, curvature evidently disappears.

     To restate in a different way this important point, the isominkowskian geometry verifies the axiom of flatness in isospace over isofields and, in this sense it provides an isoflat representation of gravity (read: when computed with respect to the generalized unit E = 1/T). On the contrary, the same geometry recovers the Riemannian geometry identically, including the ordinary curvature, when projected in our conventional space-time over conventional fields (read: when computed with respect to the conventional unit E = I). In particular, the isominkowskian geometry admits Einstein's field equations identically (except for inessential factors), by therefore preserving its experimental verifications.

     Despite the preservation of the conventional gravitational content, Santilli's isominkowsian formulation of gravity is not trivial. To begin, the transition from the Minkowskian to the Riemannian metric is known to be noncanonical. Recent studies [I-1] have shown that, being noncanonical, the Riemannian formulation of gravity does not possess an invariant unit of space and time. In other words, the Riemannian geometry is afflicted by of the same physical shortcomings identified in Sect. I.3. The visitor of this Web Page can therefore decide for himself/herself whether the same gravitational field equations and related experimental verifications should be formulated in a geometry with a noninvariant basic unit, Riemann's geometry, or they should be identically formulated in another geometry with the invariant basic unit, Santilli's isominkowskian geometry.

     iv) From the invariance of isoseparation (I.23) for the gravitational value of T, one can see that the isominkowskian formulation of gravity permits the achievement of a universal symmetry for gravitation, the isopoincare' symmetry P*(3.1), that turns out to be locally isomorphic to the conventional Poincare' symmetry P(3.1). This removes the historical disparity whereby the special relativity is equipped with a universal symmetry, while the general relativity is not.

This occurrence is not a mere mathematical curiosity because it resolves problems debated over this entire century, such as the compatibility of gravitational and relativistic conservation laws (that is resolved in the isominkowskian treatment by the fact that the generators of Poincare' symmetry remains the same for all possible isounits, that is, the conserved quantities of the special and general relativities coincide under their isominkowskian formulation and are only considered in different spaces); it imposes for gravitation the same rigid guidelines of covariance imposed by the Poincare' symmetry in special relativity, with the resolution of other problematic aspects, e.g., that of the source, compliance with the forgotten Freud identity, and other important aspects.

     v) The isominkowskian formulation of gravity implies a basically novel form of quantum gravity, that is simply given by relativistic hadronic mechanics for the case in which the isounit has the gravitational value (I.25). This resolves a number of additional problems of quantum gravity that are similar to those of quantum deformations, such as the lack of invariance of the units of space and time, lack of preservation in time of Hermiticity-observability, and others [I-1]. Since the isopoincare' symmetry is locally isomorphic to the conventional symmetry, the structure constants are the same and only realized in different spaces. This implies that the linear momenta isocommute, [p, p]* = 0, a property that confirms the isoflat representation of gravity.

     By recalling the abstract identity of relativistic quantum and hadronic mechanics, it appears that a consistent quantum form of gravity always existed. It simply remained undetected until recently because embedded where nobody looked for: in the unit.

     vi) The isominkowskian formulation of gravity also implies novel possibilities of grand unification, currently under investigation by various scholars, according to which gravitation is embedded in the unit of unified gauge theories in their isotopic form initiated by Gasperini in the mid 1980's (see [I-1] for details and references).

     vii) The restriction of the isounit of relativistic hadronic mechanics to a dependence on the space-time coordinates is un-necessary because the isounit has an arbitrary, generally nonlinear dependence on coordinates, wavefunction and their derivatives or arbitrary orders. As a result, being structurally broader than Riemann, the isominkowskian geometry permits the representation of interior conditions simply beyond any possibility of quantitative treatment via the Riemannian geometry. For instance, the extension of the Schwarzschild metric to elm waves within physical media with speed C = c/n(t, r, ...) is notoriously impossible with Riemann, but it becomes elementary with Santilli's isogeometry, and it is given by the isotopy g -> g* = Txg, with T in the diagonal form as in Eq. (25).

Similarly, relativistic hadronic mechanics permits, apparently for the first time, quantitative studies of interior gravitational problems with a more realistic dependence of the metric in interior non-first-order Lagrangian effects.

     Note that none of the above possibilities 1)-vii) is permitted by the Riemannian formulation of gravity.

     As it was the case for the mathematical profile, despite the considerable efforts by numerous scholars, the studies herein reported are also in their first infancy and so much remains to be done. Needless to say, this is the aspect which should render the field attractive to researchers interested in basic advances.

     In the preceding mathematical Web Page, following Ref. [M-1-1], we placed priority on novel numbers because the rest of the novel mathematics can be uniquely constructed via mere compatibility arguments.

     In this Web Page we follow the guidelines of ref. [I-1] and place utmost priority, this time, on Newtonian mechanics because the remaining analytic and operator profiles can also be uniquely constructed via mere compatibility arguments.

     The fundamental physical notion of this Web Page is that of ≥hyperparticles≤ in nonconservative and irreversible conditions that admits all other simpler notions of particles as particular cases. The fundamental mathematical notion is the characterization of hyperparticles via irreducible bi-isomodular representation of Lie-admissible groups and algebras. The emerging structure of hadronic mechanics includes as particular cases all existing models, not only those of genotopic, isotopic and conventional type, but also string, supersymmetric and other models. These notions are evidently too complex for the limited objectives of this Web Page, and simpler forms of particles and their representations will be used.

     As Prof. A. Jannussis of the University of Patras stated in his opening talk of the International Conference on the Frontiers of Physics (held in Olympia, Greece, in 1993)

"Hadronic mechanics supersedes all theories to date."

For hadronic mechanics, see the memoir and the large literature quoted therein:
[I-1] R. M. Santilli, Relativistic Hadronic Mechanics: Nonunitary Axiom-Preserving Completion of Relativistic Quantum Mechanics,
Foundations of Physics, Vol. 27, pages 635-729, 1997.

For the inconsistencies of generalized theories treated with conventional mathematics see the memoir:
[I-2] R. M. Santilli, Origin, problematic aspects and invariant formulation of classical and quantum deformations,
Intern. J. Modern Phys. A, Vol. 14, pages 3157-3239, 1999

For Santilli's isospecial relativity and related unification of special and general relativities, see the memoir:
[I-3] R. M. Santilli, Isominkowskian geometry for the gravitational treatment of matter and its isodual for antimatter,
Intern. J. Modern Phys. D, Vol. 7, pages 351-407 (1998)

For the Iso-Grand-Unification including electroweak and gravitational interactions for matter and antimatter in exterior and interior conditions, see the Proceedings of the VII M. Grossmann Meeting on General Relativity and Gravitation, Jerusalem, June 1997, and the letter:
[I-4] R. M. Santilli, Isotopic grand unification with the inclusion of gravity,
Founnd. Phys. Letters, Vol. 10, pages 307-327 , 1997.

For the explicit realization of hidden variables provided by hadronic mechanics, related lifting of Pauli's matrices and Bell's inequalities, see:
[1-5] R. M. Santilli, Isorepresentations of the Lie-isotopic SU(2) algebras, with applications to nuclear physics, and local realism:
Acta Applicandae Mathematicae, Vol. 50, pages 177-190, 1998.

For experimental evidence supporting Santilli's isominkowskian geometry, isopoincare' symmetry, and isospecial relativity within the hyperdense media inside hadrons, see
[I-6] Yury Arestov, Evidence on the isominkowskian character of hAdronic structure,
Foundations of Physics Letters, Vol. 11, pages 483-493, 1998.

For the Lie-Santilli isotheory, see the monograph: [16] Gr. Tsagas and D. Sourlas, Mathematical Foundations of the Lie-Santilli theory, Ukraine Academy of sciences, Kiev (1991).

For the Poincare'-Santilli isosymmetry, see the memoir: [1-7] J. V. Kadeisvili, An introduction to the Lie-Santilli isotheory,
Mathematical Methods in Applied Sciences, Vol. 19, pages 1349-1395, 1996.

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Prepared by the IBR staff.

PROPOSED RESEARCH II-1: Represent specific systems of extended, nonspherical and deformable particles under nonconservative external forces and in irreversible conditions via the iso-, geno- and hyper-Newton equations [M-1-1]. In particular, identify the representation of the shape of the particles via the generalized units and study their degrees of freedom. Study the connection between the representation of a system via a set of geno-equations or only one hyper-equation. Study the Hamilton-Jacobi theory for each mechanics. Study the isodual images of the systems for a Newtonian representation of antimatter and verify its compatibility with available classical experimental data.

TECHNICAL ASSISTANCE: Contact the IBR staff at or

Prof. T. L. Gill
Howard University
Research Center ComSERC
Washington, D. C. 20059, USA

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Prepared by the IBR staff.

PROPOSED RESEARCH III-1: Study the following aspects of the iso-, geno- and hyper-mechanics [M-I-1]: the transformation theory of each of these three mechanics; the invariance of the basic units; the representation of the Newtonian systems of the preceding problems via a first-order isocanonical action; the preservation at the analytic level of all main characteristics of said Newtonian representation, including the representation of the actual, nonspherical and deformable shape of the particles; study the isodualities of these generalized mechanics and verify their compatibility with classical experimental evidence on antiparticles.

TECHNICAL ASSISTANCE: Contact the IBR staff at or

Prof. T. L. Gill
Howard University
Research Center ComSERC
Washington, D. C. 20059, USA

Prof. R. Miron
Faculty of Mathematics
University "Al. I. Cuza"
6600 Iasi, Romania



. H. Q. Placido and A. E. Santana
Instituto de Fisica
Universidad Federal de Bahia
Campus de Ondina
40210-340 Salvador, Bahia, Brasil
Hadronic J. Vol. 19, p. 319 (1996)

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Prepared by the IBR staff.

PROPOSED RESEARCH IV-1: Study the iso-, geno- and hyper-symplectic quantization [M-I-1], i.e., the maps of the Hamilton-Santilli iso-, geno- and hyper-mechanics into the corresponding branch of hadronic mechanics. Above all, verify that the main characteristics at the Newtonian level, such as the representation of extended, nonspherical and deformable charge distributions under nonlinear integro-differential and nonhamiltonian forces, are preserved in their entirety at the operator level. Study the isodual images of the above generalized quantization and prove their equivalence to charge conjugation, as already established at the isotopic level. Verify that the emerging operator isodual theories are compatible with available experimental data on antiparticles.

Contact the IBR staff at or

Prof. D. Schuch
Institute fur Theoretische Physik
J. W. Goethe-universitat
Robert-Mayer-Strasse 8-10
D-60054 Frankfurt am Main, Germany
Fax +49-69-31 95 23

Prof. E. B. Lin
Department of Mathematics
University of Toledo
Toledo, OH 43606

Prof. T. L. Gill
Howard University
Research Center ComSERC
Washington, D. C. 20059, USA
REFERENCES [M-I-1] and [I-1].


E. B. Lin
Department of Mathematics
University of Toledo
Toledo, OH 43606
'New Frontiers in Algebras, Groups and Geometries"
Gr. Tsagas, Editor, Hadronic Press (1996), p. 161.

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