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Elements of Iso-, Geno-, Hyper-Mathematics for Matter, Their Isoduals for Antimatter, and Their Applications in Physics, Chemistry, and Biology Foundations of Physics   September 2003, vol. 33, no. 9,   pp. 1373-1416(44) Santilli R.M.[1] [1] Institute for Basic Research, P.O. Box 1577, Palm Harbor, Florida 34682. ibr@gte.net, http://www.i-b-r.org Abstract:

Pre-existing mathematical formulations are generally used for the treatment of new scientific problems. In this note we show that the construction of mathematical structures from open physical, chemical, and biological problems leads to new intriguing mathematics of increasing complexity called iso-, geno-, and hyper-mathematics for the treatment of matter in reversible, irreversible, and multi-valued conditions, respectively, plus anti-isomorphic images called isodual mathematics for the treatment of antimatter. These novel mathematics are based on the lifting of the multiplicative unit of ordinary fields (with characteristic zero) from its traditional value +1 into: (1) invertible, Hermitean, and single-valued units for isomathematics; (2) invertible, non-Hermitean, and single-valued units for genomathematics; and (3) invertible, non-Hermitean, and multi-valued units for hypermathematics; with corresponding liftings of the conventional associative product and consequential lifting of all branches of mathematics admitting a (left and right) multiplicative unit. An anti-Hermitean conjugation applied to the totality of quantities and their operation of the preceding mathematics characterizes the isodual mathematics. Intriguingly, the emerging formulations preserve the abstract axioms of conventional mathematics (that based on the unit +1). As such, the new formulations result to be new realizations of existing abstract mathematical axioms. We then show that the above mathematical advances permit corresponding liftings of conventional classical and quantum theories with a resolution of basic open problems in physics, chemistry, and biology, numerous experimental verifications, as well as new industrial applications.

Keywords: antimatter; nonlocality; irreversibility; multivaluedness

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http://www.i-b-r.org/docs/Iso-Geno-Hyper-paper.pdf

Extended version of invited plenary talks at the Conference of the International Association for Relativistic Dynamics, Washington, D.C., June 2002 International Congress of Mathematicians, Hong Kong, August 2002 International Conference on Physical Interpretation of Relativity Theories, London, September 2002 ISO-, GENO-, HYPER-MECHANICS FOR MATTER, THEIR ISODUALS, FOR ANTIMATTER, AND THEIR NOVEL APPLICATIONS IN PHYSICS, CHEMISTRY AND BIOLOGY Ruggero Maria Santilli Institute for Basic Research P. O. Box 1577 Palm Harbor, FL 34682, U.S.A. E-address ibr@gte.net Web Site http://www.i-b-r.org December 22, 2002 In press at the Journal of Dynamical Systems and Geometric Theories ISO-, GENO-, HYPER-MECHANICS FOR MATTER, THEIR ISODUALS, FOR ANTIMATTER, AND THEIR NOVEL APPLICATIONS IN PHYSICS, CHEMISTRY AND BIOLOGY Ruggero Maria Santilli TABLE OF CONTENT ABSTRACT 3 1. INTRODUCTION 4 2. CONSTRUCTION OF ISODUAL MECHANICS FROM CLASSICAL ANTIMATTER 5 2.1: The scientific unbalance caused by antimatter 2.2: Elements of isodual mathematics 2.3: Isodual spaces and geometries 2.4: Isodual Lie theory 2.5: Isodual Newtonian Mechanics 2.6: Isodual Hamiltonian Mechanics 2.7: Isodual Quantum Mechanics 2.8: Isodual special relativity 2.9: Origin of the isodual theory in Dirac’s equation 2.10: Experimental verifications and applications 3. CONSTRUCTION OF ISOMECHANICS FROM NONLOCAL, NONLINEAR AND NONPOTENTIAL INTERACTIONS 14 3.1: The scientific unbalance caused by nonlocal interactions 3.2: Catastrophic inconsistencies of noncanonical-nonunitary theories 3.3: Elements of isomathematics 3.4: Isotopologies, isospaces and isogeometries 3.5: Lie-Santilli isotheory and its isodual 3.6: Iso-Newtonian Mechanics and its isodual 3.7: Iso-Hamiltonian Mechanics and its isodual 3.8: Isotopic Branch of nonrelativistic Hadronic Mechanics and its isodual 3.9: Invariance of isotopic theories 3.10: Simple construction of isotheories 3.11: Isorelativity and its isodual 3.12: Isorelativistic Hadronic Mechanics and its isodual 3.13: Isogravitation, iso-grand-unification and isocosmology 1 3.14: Experimental verifications and scientific applications 3.15: Industrial applications to new clean energies and fuels 4. CONSTRUCTION OF GENOMECHANICS FROM IRREVERSIBLE PROCESSES 52 4.1: The scientific unbalance caused by irreversibility 4.2: The forgotten legacy of Newton, Lagrange and Hamilton 4.3: Catastrophic inconsistencies of formulations with external terms 4.4: Initial versions of irreversible mathematics 4.5: Elements of genomathematics 4.6: Lie-Santilli genotheory and its isodual 4.7: Geno-Newtonian Mechanics and its isodual 4.8: Geno-Hamiltonian mechanics and its isodual 4.9: Genotopic Branch of Hadronic Mechanics and its isodual 4.10: Invariance of genotheories 4.11: Simple construction of genotheories 4.12: Genorelativity and its isodual 4.13: Experimental verifications and applications 5. CONSTRUCTION OF HYPERMECHANICS FROM BIOLOGICAL SYSTEMS 68 5.1: The scientific unbalance caused by biological systems 5.2: The multivalued complexity of biological systems 5.3: Elements of hypermathematics 5.4: The complexity of hypertime and hyperrelativity for biological systems 5.5: Eric Trell’s hyperbiological structures 5.6: The lack of final character of all scientific theories ACKNOWLEDGMENTS 72 REFERENCES 73 2 ABSTRACT Pre-existing numbers, related mathematics and consequential physical theories are generally used for the treatment of new scientific problems. In this memoir we outline the research conducted by various mathematicians, physicists and chemists over the past two decades who have shown that the inverse approach, the construction of new numbers, related new mathematics and consequential new physical theories from open physical, chemical and biological problems, leads to new intriguing formulations of increasing complexity called iso-, geno- and hyper-mathematics for the treatment of matter in reversible, irreversible and multi-valued conditions, respectively, plus anti-isomorphic images called isodual mathematics for the treatment of antimatter. These novel formulations are based on new numbers characterized by the lifting of the multiplicative unit of ordinary fields (with characteristic zero) from its traditional value +1 to: (1) invertible, Hermitean and single-valued units for isomathematics; (2) invertible, non-Hermitean and single-valued units for genomathematics; and (3) invertible, non-Hermitean and multi-valued units for hypermathematics; with corresponding liftings of the conventional associative product and consequential lifting of all branches of mathematics admitting a (left and right) multiplicative unit. An anti-Hermitean conjugation applied to the totality of quantities and their operation of the preceding mathematics characterizes the isodual mathematics. The above new mathematics are then used for corresponding liftings of Newtonian, Hamiltonian and quantum mechanics, today known as iso-, geno-, hyper-mechanics for the description of matter and their isoduals for antimatter, with compatible liftings of geometries and symmetries, and, inevitable of contemporary relativities. The above new body of knowledge is also known as hadronic mechanics, superconductivity and chemistry, wherein conventional Hamiltonians represent conventional, linear, local and potential interactions among point particles, while generalized units provide an invariant representation of extended, nonspherical and deformable particles under additional nonlinear, nonlocal and nonpotential interactions due to deep mutual penetration of wavepackets at short distances. Whenever the latter effects are ignorable due to large distances, conventional units, mathematics and relativities are recovered identically. We finally outline the novel scientific and industrial verifications and applications permitted by the new mathematics and relativities in physics, chemistry and biology, including numerous experimental verifications, and applications for new clean energies and fuels which are prohibited by contemporary mechanics, special relativity and related mathematics. 3 1. INTRODUCTION Customarily, new problems in physics, chemistry, biology and other quantitative sciences are treated via pre-existing mathematics. Such an approach is certainly valuable at the initiation of new studies. However, with the advancement of scientific knowledge such an approach historically lead to serious limitations and controversies due to the insufficiency of the used mathematics for the problem at hand. On historical grounds, the above occurrence is illustrated by the fact that the mathematics so effective for the study of planetary systems (Hamiltonian vector fields, Hamilton- Jacobi equations, etc.) resulted to be inadequate for the study of the atomic structure. In fact, the latter mandated the use of a new mathematics, that based on infinite dimensional Hilbert spaces over a field of complex numbers, which has no application for planetary mechanics. Similar occurrences exist in contemporary science due to the continued use for new scientific problems of pre-existing mathematics proved to be so effective in preceding scientific problems. This is the case for: (1) the lack of a classical formulation of antimatter, due to the inapplicability of conventional mathematics so effective for the classical treatment of matter; (2) the lack of quantitative studies of nonlocal-integral interactions as occurring in chemical valence bonds, due to the inapplicability of conventional mathematics because of its strictly local-differential character; (3) the lack of representation of the irreversible and multi-valued nature of biological systems, due to insufficiencies of both conventional mathematics and hypermathematics as currently formulated; and other cases. In this memoir we outline research conducted by numerous mathematicians, physicists and chemists over the past two decades showing that the construction of new mathematics from open scientific problems does indeed permit new, intriguing scientific horizons with far reaching implications in mathematics as well as quantitative science in general. As we shall see, the emerging new mathematics are based on progressive generalizations of the multiplicative unit +1 into everywhere invertible and sufficiently smooth, but otherwise arbitrary quantities (such as numbers, matrices or integro-differential operators) with corresponding generalizations of the associative product, thus implying corresponding generalizations of all branches of conventional mathematics (hereinafter defined as the mathematics based on the multiplicative left and right unit +1 over a field of characteristic zero). In this memoir we also show that the above new mathematics imply certain liftings of Newtonian, Hamiltonian and quantum mechanics with corresponding liftings of geometries and symmetries and, inevitably, of contemporary relativities. In fact, all efforts outlined in this memoir were aimed at the construction of new mechanics today known as iso-, geno-, and hyper-mechanics for the description of matter in conditions of increasing complexity, and corresponding isodual mechanics for the description of antimatter. All new scientific and industrial applications can then be reduced to a few primitive mechanical or, more properly, relativity axioms. 4 Alternatively, an objective of this memoir is to show that no truly novel scientific advance is possible without truly novel mechanics. In turn, no mechanics can be considered as truly new without new mathematics. Finally, no mathematics can possibly be truly new without new numbers. This illustrates the reason why, out of the rather vast scientific studies over three decades reviewed in this memoir, primary efforts were devoted to the search of new numbers from which new ,mathematics, new relativities and new scientific and industrial applications uniquely follow. The reader should be aware that the literature in the topic of this memoir is rather vast because it encompasses numerous studies in pure mathematics, applications in various quantitative sciences, several experimental verifications as well as rapidly expanding new industrial applications. As a result, in this paper we can only review the most fundamental aspects of the new formulations. A technical knowledge of the new advances can only be achieved via the study of the quoted literature. To avoid a prohibitive length, references have been restricted to contributions specifically based on the lifting of the unit with a compatible lifting of the product. Regrettably, we have to defer to the specialized literature the treatment of connections with numerous other studies. References are grouped by main fields indicated with square brackets (e.g., [5]), while individual references are indicated with curved brackets (e.g., (201)). Except for monographs, proceedings and reprint volumes, the titles of the individual contributions are not provide to avoid a prohibitive length, as well as because of their lack of general availability in the physics literature without an extensive library search. The reader should be aware that the new mathematics and their applications are still in their initial stages and so much remains to be done. The author would be grateful for any comment, as well as for the indication by interested colleagues of mathematical or or other references in the origination of the new formulations which have escaped his knowledge. 2. CONSTRUCTION OF ISODUAL MECHANICS FROM CLASSICAL ANTIMATTER 2.1: The scientific unbalance caused by antimatter. One of the biggest scientific unbalances of the 20-th century has been the treatment of matter at all possible levels, from Newtonian to quantum mechanics, while antimatter was solely treated at the level of second quantization. In particular, the lack of a consistent classical treatment of antimatter left fundamental open problems, such as the inability to study whether a far away galaxy or quasar is made up of matter or of antimatter. It should be indicated that classical studies of antimatter simply cannot be done by merely reversing the sign of the charge, because of inconsistencies due to the existence of only one quantization channel. In fact, the quantization of a classical particle with the reversed sign of the charge leads to a particle (rather than a charge conjugated antiparticle) with the wrong sign of the charge. The origin of this scientific unbalance was not of physical nature, and was instead due to the lack of a mathematics suitable for the classical treatment of antimatter in such a 5 Figure 1: An illustration of the first scientific unbalance of the 20-th century, the inability due to the lack of adequate mathematics to conduct quantitative studies as to whether far away galaxies and quasars are made up of matter or of antimatter. way to be compatible with charge conjugation at the quantum level. Charge conjugation is an anti-homomorphism. Therefore, a necessary condition for a mathematics to be applicable for the classical treatment of antimatter is that of being anti-homomorphic, or, better, anti-isomorphic to conventional mathematics. The absence of the needed mathematics is confirmed by the fact that classical treatments of antimatter require fields, functional analysis, differential calculus, topology, geometries, algebras, groups, etc. which are anti-isomorphic to conventional formulations. The absence in the mathematics of the 20-th century of such a mathematics then mandated its construction as requested by the physical reality here considered (rather than adapting physical reality to pre-existing mathematics). 2.2: Elements of isodual mathematics. A novel mathematics verifying the above conditions was proposed by R. M. Santilli in Ref. (11) of 1985 and then developed in various works (see Refs.(12, 14,15, 54,55,160,161) and [3]). The fundamental idea is the assumption of a negative-definite, left and right multiplicative unit, called isodual unit, and denoted Id, where I denotes the conventional positive-definite unit, I > 0, Id = ¡I < 0; (2:1) with corresponding reformulation of the conventional associative product A £ B among generic quantities A, B (such as numbers, vector fields, operators, etc.) into the form A£dB = A £ (Id)¡1 £ B; (2:2) under which Id is the correct left and right multiplicative unit of the theory, A£dId = Id£dd A = A; (2:3) 6 for all elements A of the considered set. More generally, isodual mathematics is given by the image of a given mathematics admitting a left and right multiplicative unit under the following isodual map A(x; :::) ! Ad(xd; :::) = ¡Ay(¡xy; :::): (2:4) when applied to the totality of conventional quantities and their operations, with no exception of any type. In this paper we cannot possibly review the entire isodual mathematics, and must restrict ourselves to an elementary review of only the foundations. DEFINITION 2.1: Let F = F(a;+;£) be a field of characteristic zero representing real numbers F = R(n;+;£); a = n, complex numbers F = C(c;+;£); a = c, or quaternionic numbers F = Q(q;+;£); a = q, with conventional associative, distributive and commutative sum a + b = c 2 F, associative and distributive product a £ b = c 2 F, left and right additive unit 0, a + 0 = 0 + a = a 2 F, and left and right multiplicative unit I > 0, a £ I = I £ a = a; 8a; b 2 F. The isodual fields (first introduced in Refs. (11,12)) are rings Fd = Fd(ad;+d;£d) with isodual numbers ad = ¡ay; (2:5) associative, distributive and commutative isodual sum ad +d bd = ¡(a + b)y = cd 2 Fd; (2:6) associative and distributive isodual product ad £d bd = ad £ (Id)¡1 £ bd = cd 2 Fd; (2:7) additive isodual unit 0d = 0; ad +d 0d = 0d +d ad = ad; (2:8) and isodual multiplicative unit Id = ¡Iy; ad £d Id = Id £d ad = ad; 8ad; bd 2 Fd: (2:9) LEMMA 2.1 (12): Isodual fields are fields (namely, isodual field verify all axioms of a field with characteristic zero). The above lemma establishes the property (first identified in Refs. (11,12)) that the axioms of a field do not require that the multiplicative unit be necessary positive-definite, because it can also be negative-definite. The proof of the following property is equally simple. LEMMA 2.2 (12): Fields (of characteristic zero) and their isodual images are antiisomorphic to each other. 7 Lemmas 2.1 and 2.2 illustrate the origin of the name ”isodual mathematics.” In fact, the needed mathematics must constitute a ”dual” image of conventional mathematics, while the prefix ”iso” is used in its Greek meaning of preserving the original axioms. It is evident that for real numbers we have nd = ¡n, while for complex numbers we have cd = (n1 +i£n2)d = ¡n1 +i£n2 = ¡¯c, with a similar formulation for quaternions. DEFINITION 2.2 (12): A quantity is called isoselfdual when it coincides with its isodual. It is easy to verify that the imaginary unit is isoselfdual because id = ¡¯i = i: (2:10) As we shall see, isoselfduality is a new symmetry with rather profound implications, e.g., in cosmology. It is evident that, for consistency, all operations of numbers must be subjected to isoduality. This implies: the isodual powers (ad)nd = ad £d ad £d ad::: (2:11) (n times with n an integer); the isodual square root ad(1=2)d = ¡q¡ay y = ad; 1d(1=2)d = ¡i; (2:12) the isodual quotient ad=dbd = ¡(ay=by) = cd; bd £d cd = ad; etc: (2:13) An important property for the characterization of antimatter is that isodual fields have a negative–definite norm, called isodual norm (12) jadjd = jayj £ Id = ¡(aay)1=2 < 0; (2:14) where j:::j denotes the conventional norm. For isodual real numbers nd we have the isodual norm jndjd = ¡jnj < 0, the isodual norm for for isodual complex numbers jcdjd = ¡(n21 + n22 )1=2, etc. Recall that functional analysis is defined over a field. Therefore, the lifting of fields into isodual fields requires, for necessary condition of consistency, the formulation of the isodual functional analysis (54). We here merely recall that sind µd = ¡sin(¡µ); cosd µd = ¡cos(¡µ); (2:15a) cosd 2d µd +d sind 2d µd = 1d = ¡1; (2:15b) the isodual hyperbolic functions sinhd wd = ¡sinh(¡w); coshd wd = ¡cosh(¡w); (2:16a) 8 coshd 2d wd ¡d sinhd 2d wd = 1d = ¡1; (2:16b) the isodual logarithm logd nd = ¡log(¡n): (2:17) Particularly important is the isodual exponentiation which can be written edAd = Id + Ad=d1!d + Ad £d Ad=d2!d + ::: = ¡eAy ; (2:18) Other properties of the isodual functional analysis can be easily derived by the interested reader (see also Refs. (14,21,22). It is little known that the differential and integral calculi are indeed dependent on the assumed basic unit. In fact, the lifting of I into Id and of F into Fd implies the isodual differential calculus, first introduced in Ref. (14), which is characterized by the isodual differentials ddxd = dx; (2:19) with corresponding isodual derivatives @dfd(xd)=d@dxd = ¡@ ¯ f(¡¯x)=@(¡¯x); (2:20) and other isodual properties interested readers can easily derive. Note that the differential is isoselfdual. 2.3: Isodual spaces and geometries. Conventional vector and metric spaces are defined over a field. It is then evident that the isoduality of fields requires, for consistency, a corresponding isoduality of vector, metric and all other) spaces. DEFINITION 2.3: Let S = S(x; g;R) be an N–dimensional metric space with realvalued local coordinates x = fxkg, k = 1; 2; :::;N, nowhere degenerate, sufficiently smooth, real–valued and symmetric metric g(x; :::) and related line element x2 = (xt £ g £x)£I = (xi £gij £xj)£I 2 R. The isodual spaces, first introduced in Refs. (11,14), are vector spaces Sd(xd; gd;Rd) with isodual coordinates xd = ¡xt where t denotes transposed, isodual metric gd(xd; :::) = ¡gt(¡xt; :::), and isodual line element (xd)2d = (x2)d = (¡xt)2d = (xd) £d (gd) £d (xtd) £d Id = = [(¡xj)(¡£)(¡gji)(¡£)(¡xi)](¡£)(¡I) = ¡x2 2 Rd: (2:21) The isodual Euclidean space Ed(xd; ±d;Rd) is a particular case of Sd when gd ij = ±d ij : The isodual distance on Ed is negative definite and it is given by Dd = ¡D, where D is the conventional (positive-definite) distance on E. The isodual sphere on a 3-dimensional isodual space Ed is the perfect sphere with negative radius and expression rd2d = [(xd2d 1 +d x2d 2 +d x2d 3 ]£d Id = ¡r2 2 Rd. The isodual Minkowskian, isodual Riemannian and isodual symplectic geometries can be defined accordingly (14,15). 2.4: Isodual Lie theory. Lie’s theory in its conventional formulation in mathematics or physics can only characterize matter at the classical level, thus preventing the study 9 of antimatter via fundamental tools so familiar for matter, such as symmetries and conservation laws. To overcome such an unbalance, R. M. Santilli proposed in the isodual Lie theory in the original proposal of isoduality (11), whose explicit form is left to the interested reader. We merely indicate that the isoduality of Lie’s theory were proposed also for the classification of all possible realizations of abstract simple Lie algebras. From the above rudiments interested readers can construct the rest of the isodual mathematics, including: isodual topologies, isodual manifolds, etc. Particularly important for physical applications is the isodual Lie theory (first introduced in Ref. (11) (see also (14,22)), including isodual universal enveloping associative algebras, isodual Lie algebras, isodual Lie groups, isodual symmetries, and isodual representation theory, which we cannot review here for brevity. The main physical theories characterized by isodual mathematics can be outlined as follows. 2.5: Isodual Newtonian Mechanics. To resolve the scientific unbalance between matter and antimatter indicated earlier, the isodual mathematics has first permitted a Newtonian characterization of antimatter consistent with all available experimental data (14,22). Then, isodual mathematics has identified a new quantization channel (which is distinct from conventional symplectic quantization) leading to an operator formulation which is equivalent to charge conjugation (14,16,21). We first have the isodual Newton equations md £d ddvd ddtd = Fd(td; xd; vd); vd = ddxd ddtd : (2:22) and related formulations known under the name of isodual Newtonian Mechanics. 2.6: Isodual Hamiltonian Mechanics. The direct analytic representation of the above equations are permitted by the following isodual action functional (14) (for the case when the Newtonian force F is representable with a potential, see below for nonhamiltonian interactions) ±dAd(td; xd) = ±d Z d (pd £d ddxd ¡d Hd £d ddtd) = 0; (2:23) which characterizes the following the isodual Hamilton equations (14) ddxd ddtd = @dHd @dpd = pd; ddpd ddtd = ¡ @dHd @dxd ; (2:24) with corresponding isodual Hamilton-Jacobi equations @dAd @dtd +d Hd = 0; @dAd @dxdk ¡d pd k = 0; k = 1; 2; 3: (2:25) Again, interested readers are encouraged to verify that the above isodual Hamiltonian Mechanics does indeed represent correctly all classical experimental data on antimatter. . 10 2.7: Isodual Quantum Mechanics. The isoduality of the naive (or symplectic) quantization can be expressed via the elementary map (14,16,21,55) based on the isodual Planck’s constant ¯hs = Id = ¡1 Ad(td; xd) ! ¡id £d Id £d LndÃd(td; xd); (2:26) which can be applied to the isodual Hamilton-Jacobi equations yielding the expressions @dAd @dtd +d Hd = 0 ! ¡id £d Id £d @dÃd @dtd +d Ãd £d Hd = 0; (2:27a) @dAd @dxdk ¡d pd k = 0 ! ¡id £d Id @Ãd @dtd ¡d Ãd £d pd k = 0; (2:27b) Therefore, isodual mathematics characterizes the novel isodual quantum mechanics, also known as the isodual branch of hadronic mechanics, which can be expressed on the isodual Hilbert space Hd with isodual states jÃ >d= ¡ < Ãdj and isodual inner product < Ãdj £d jÃd > over the isodual field Cd with basic isodual Schroedinger equations (for a Hermitean Hamiltonians H) id£d < Ãdj@d=d@dtd = ¡ < Ãdj £d Hd = ¡ < Ãdj £d Ed; (2:28a) ¡ < Ãdj £d pd k = ¡id£d < Ãdj@d=d@dxdk; (2:28b) and corresponding isodual Heisenberg equations (for a Hermitean observable A) id £d ddAd=dddtd = [A;H]d = ¡dHd £d Ad +d Ad £d Hd = ¡[A;H]; (2:29a) [pd i ; xdj ]d = ¡id £d ±di j ; [xdi; xdj ]d = [pd i ; pd j ]d = 0: (2:29b) The equivalence of the above operator formulation of antimatter with charge conjugation has been proved in Refs. (16,21). 2.8: Isodual special relativity. The vast scientific literature on special relativity throughout the 20-th century is silent on the fact that, classically, it can solely represent matter, under the belief that classical antimatter can be represented via a mere change of the sign of the charge without an inspection of the various consequential inconsistencies identified earlier. To resolve this impasse, R. M. Santilli proposed in Refs. [3] the isodual special relativity which is based on the isodual topology (14), the isodual Minkowski space (15), the isodual Poincar´e symmetry (29) and the isodualities of relativistic dynamics and physical laws (55). The explicit form of these structures can be easily constructed by the interested reader via the isodual map (2.4). We only note that that the change of the sign of elementary charges or, more appropriately, charge conjugation, are anti-homomorphic maps, while isoduality is an antiisomorphic map. Therefore, according to special relativity antiparticles exist in the same spacetime of particles, while, according to isodual special relativity, antiparticles exist in the isodual spacetime which coexists with, yet it is independent from our spacetime. 11 2.9: Origin of the isodual theory in Dirac’s equation. On historical grounds it should be indicated that the isodual mathematics and related theory of antimatter originated from an inspection of the celebrated Dirac equation (6). In fact, its basic unit displays precisely the negative-definite unit for the antiparticle component, namely, the isodual unit, °o = i £ Ã I2£2 02£2 02£2 ¡I2£2 ! = i £ Ã I2£2 02£2 02£2 Id 2£2 ! (2:31) Similarly, by recalling that Pauli’s matrices are Hermitean, the space components of Dirac’s gamma matrices exhibit precisely the isodual Pauli’s matrices of antimatter precisely for the antimatter component of the equation, (°k) = Ã 02£2 ¾2£2 ¡¾2£2 02£2 ! = Ã 02£2 ¡¾d 2£2 ¡¾2£2 02£2 !: (2:32) It then follows that Dirac’s gamma matrices have the new symmetry of being isoselfdual (Definition 2.2) °¹ = ¡°¹y = °¹d; (2:33) and, when interpreted via the isosual mathematics, Dirac’s equation directly describes the Kronecker product of an electron and a positron f°¹[p¹ ¡ 2 £ A¹)x)=co] + i £ mg £ dÁ(x) = = [Ã 0 ¡¾kd ¡¾k 0 !£ (pk ¡ e £ Ak=co)¡ ¡i £ Ã I 0 0 Id !£ (p4 ¡ e £ A4=co) + i £ m] £ Ã Á(x) Ád(x) ! = 0: (2:34) Finally, the true invariance of Dirac’s equation is not that under the Poincar´e symmetry alone, as popularly believed until now, but rather under the Poincar´e symmetry and its isodual STot = P(3:1) £ Pd(3:1): (2:35) Note the elimination of the controversial ”hole theory” and second quantization, since the isodual theory of antimatter holds at the classical level,, let alone that in first quantization, without excluding, of course, its applicability to second quantization. In particular, as indicated earlier, negative-energy solutions of Dirac’s equations mandated the ”hole theory” because referred to positive units, thus being unphycical, while the negativeenergy solutions of Eq. (2.34) are referred in isodual mathematics to negative units, thus being fully physical. Note also the elimination of the additional controversy on the ”four-dimensional irreducible representation of spin 1/2” because, under the proper interpretation, Dirac’s gamma matrices solely characterize a two-dimensional representation of spin 1/2, as a Kronecker product of one representation for matter and its isodual for antimatter. Note 12 the need to eliminate second quantization to admit only two-dimensional irreducible representations of spin 1/2, as mandated by Lie’s theory. Regrettably, Dirac was unaware of the fact that a negative unit can indeed be the correct unit of an appropriate mathematics and, as a consequence, he developed the ”hole theory” restricting the treatment of antiparticles to the sole level of second quantization. It is an easy prediction that, in the event the isodual numbers had been discovered prior at the beginning of the 20-th century (rather than in 1985 (11)), Dirac would never have proposed his ”hole theory. 2.10: Experimental verifications and applications. It is easy to see that the isodual theory represents correctly all available classical experimental evidence on antimatter. For instance, the Coulomb laws for matter with charge q and antimatter with charge qd at mutual distance r are given by FMatterObserver = q £ qd=r2; FAntimatterObserver = qd £d q=drd2; (2:30a) FMatterObserver = q £ q=r2; FAntimatterObserver = qd £d qd=drd2; (2:30b) and they correctly represent mutual attraction (mutual repulsion) for matter-antimatter, (matter-matter and antimatter-antimatter) for both matter and antimatter observers, where F is attractive when having negative value for a matter observer on R, as conventional, and a positive value for an antimatter observer on Rd, the opposite occurring for attraction. For additional details, interested reader can inspect Ref. (22). The equivalence of the isoduality on Hilbert spaces with charge conjugation proved in Refs. (16,21) establishes that the isodual theory of antimatter with available experimental data at the operator level too. Despite its simplicity, the physical, astrophysical and cosmological implications of isodual mathematics are rather deep. To begin, the isodual map (2.4) implies the change of the sign not only of the charge, but also of all other physical quantities of matter, including mass, energy, time, etc. For instance, the energy eigenvalue E of Eqs. (2.9) has negative values since it is positive in Eq. (2.9), yet computed on Rd. Note that the measurement of physical quantities with respect to negative definite units resolves the traditional inconsistencies for negative mass and energy. In particular, the isodual theory of antimatter recovers the old hypothesis that antiparticles move backward in time (since they have a negative-definite time) by resolving the inherent violation of causality which lead to its abandonment in the second half of the 20-th century. In fact, motion backward in time measured with respect to a negative unit of time is as causal as the conventional motion forward in time referred to a positive unit of time. Most importantly, the isodual theory of antimatter mandates the existence of antigravity defined as a gravitational repulsion experienced by antimatter in the field of matter and vice-versa (16,21,22), while resolving the historical objections against antigravity. As an illustration, the creation of an electron-positron pair is represented by an isoselfdual state on H£Hd over C£Cd, whose eigenvalues can only be defined over the field of the observer 13 (that is, R for a matte observer and Rd for an antimatter observer), as studied in detail in Refs. (21,60). Such an occurrence prevents configurations of correlated electron-positron systems which, for the case of antigravity without the isodual mathematics, could violate the principle of conservation of the energy (e.g., by obtained blueshifts without applied force). Other objections against antigravity for antimatter-matter systems are resolved in similar ways. J. P. Mills has shown in Ref. (160) that the experimental verification of antigravity proposed in Ref. (17) can be conducted in a resolutory form with available technology. The proposed experiment essentially consists of a horizontal vacuum tube of about one meter in diameter and ten meters in length with internal collimators at one end and a scintillator at the other end. The release of photons through the collimators would establish on the scintillator the point of no gravity; the release of electrons would show at the scintillator a downward shift due to gravity ; and the release of positrons would show an upward or downward shift on the scintillator depending on whether they are experiencing antigravity or gravity, respectively. The experiment would be resolutory because, when the electrons and positrons have very small energy (of the order of electron Volts), the downward or upward shift of their impact on the scintillator would be visible to the naked eye. This low energy experiment has been ignored by experimentalists in favor of other high energy experiments due to the inconsistencies of the prediction of antigravity when treated with conventional mathematics. It is hoped that, in view of the resolution of these inconsistencies thanks to isodual mathematics, experimentalists will reconsider their view and conduct indeed such a fundamental experiment. Above all, isodual mathematics has fulfilled the primary scope for which it was constructed, the initiation of quantitative classical studies as to whether far away galaxies and quasars are made-up of matter or of antimatter. In fact, isodual mathematics predicts that antimatter emits a new photon, called the isodual ;photon (21), which has experimentally detectable characteristics different than those of the ordinary photon emitted by matter, e.g., the isodual photon is repelled by the gravitational field of our Earth, and has a parity different than that of the ordinary photon. The experimental resolution on Earth whether light from a far away galaxy or quasar is made up of photons or of isodual photons would then resolve the open cosmological problem whether the universe is only made up of matter, or antimatter galaxies and quasars are equally present. An important application of the isodual theory of antimatter has been developed by J. Dunning Davies (161) who has constructed the first (and only) known thermodynamics for antimatter stars. In this way, quantitative cosmological studies on the antimatter component of the universe are already under way. 3. CONSTRUCTION OF ISOMECHANICS FROM NONLOCAL, NONLINEAR AND NONPOTENTIAL INTERACTIONS 3.1: The scientific unbalance caused by nonlocal interactions. Another large scientific unbalance of the 20-th century has been the adaptation of nonlocal-integral systems to the pre-existing mathematics which is notoriously local-differential.This approach 14 has created serious limitations and controversies which have remained unsolved despite attempts conducted over three quarter of a century, such as: (1) the lack of numerically exact representations of chemical valence bonds in molecular structures; (2) the historical inability to achieve an exact representation of nuclear magnetic moments; (3) the absence of an exact representation of the Bose-Einstein correlation without ad hoc free parameters to fit data; and other unresolved problems. In all these cases we have the mutual overlapping/penetration of particles and/or their wavepackets at distances of the order of 10¡13cm, which conditions are strictly nonlocal-integral and, as such, they are beyond the exact applicability of the mathematical structure, let alone physical laws of quantum mechanics. It is appropriate here to recall that quantum mechanics and its underlying mathematics permitted a numerically exact representation of all experimental data of the Hydrogen atom. By contrast, the same mathematics and quantum laws have not permitted an equally exact representation of the experimental data of the Hydrogen molecule, since a historical 2% of molecular binding energy has been missed for about one century under the rigorous applicability of quantum axioms (thus excluding screenings of the Coulomb law which imply noncanonical/nonunitary transforms, thus exiting the class of equivalence of the original theory). Since the sole difference between one isolated Hydrogen atom and two atoms coupled into the Hydrogen molecule is given by the electron valence bonds, the above occurrence illustrates the exact validity of quantum mechanics and related mathematics when the systems can be effectively appropriate as point particles at sufficiently large mutual distances (as it is the case for the structure of the Hydrogen atom), while the same theory and related mathematics have a merely approximate character when systems contain interactions at short distances (as it is the case for the mutual overlapping of the wavepackets of valence electrons with antiparallel spins). The insufficiency is due to the fact that conventional mathematics is local-differential in its structure, thus solely permitting the representation of valence bonds as occurring between point particles interacting at a distance. This representation is evidently valid in first approximation because electrons have indeed a point charge. Nevertheless, such a local-differential representation is insufficient because of the lack of treatment of the mutual penetration of the electron wavepackets which cannot be consistently reduced to a finite set of isolated points. It is then evident that a more adequate treatment of valence bonds in chemistry, as well as all nonlocal-integral interactions in general, requires a new mathematics which is partly local-differential (to represent conventional Coulomb interactions) and partly nonlocal-integral (to represent the overlapping of the wavepackets). Additional physical requirements establish that the needed mathematics must be nonlinear in the wavefunction, thus preventing the use of conventional quantum mechanics because nonlinear Schroedinger’s equations violate the superposition principle with consequential inapplicability to composite systems (45,46). 15 Figure 2: An illustration of the second scientific unbalance of the 20-th century, the lack of adequate mathematics for quantitative studies of nonlocal-integral interactions occurring the in deep overlappings of the wavepackets of electron valence bonds in molecules, as well as, in general, all interactions of particles ar short distances. It should be indicated that numerous ”nonlocal interactions” exist in the physical literature, but they have generally been adapted to be representable with a local potential in a Hamiltonian (for theoretical and experimental studies on this type of nonlocality nonlocality, see, e.g., C. A. C. Dreismann (220-223) and references quoted therein). The interactions occurring in valence bonds are dramatically more general than the above inasmuch as they occur in the finite volume of wave overlappings which is not reducible to a finite number of isolated points and they are of nonpotential type because of contact, zero-range nature, thus prohibiting a consistent representation with a Hamiltonian, let alone a potential. More specifically, the granting of a potential energy to the deep wave-overlappings of valence bonds would be the same as granting potential energy to the resistive force experienced by an extended object moving within a medium. All in all, one century of failed attempt to achieve an exact representation of molecular data establish beyond credible doubts that the nonlocal, nonlinear and nonpotential interactions occurring in valence bonds are beyond the axiomatic structure of quantum mechanics, thus demanding nonunitary theories as a necessary condition to exit from the class of unitary equivalence of quantum mechanics. 3.2: Catastrophic inconsistencies of noncanonical-nonunitary theories. Recall that a crucial feature of quantum mechanics, which permitted its dominance in physical applications, is the invariance, namely, the preservation of numerical values, physical laws and mathematical axioms under the time evolution of the theory. Since all quantum time 16 evolutions are expressible via a Hermitean Hamiltonian, the above feature is essentially expressed by the invariance of the mathematical structure of quantum mechanics under unitary transforms. In fact, quantum theories are expressible via a Hermitean Hamiltonian H = p2=2m+ V = Hy defined on a Hilbert space H over the field of complex number C with a unitary time evolution U(t) £ jÃ(to) >= jÃ(t) >;U £ Uy = Uy £ U = I; U(t) = ei£H£t; (3:1) which, as such, leaves invariant all basic units by definition, I ! I0 = U £ I £ Uy = I; (3:2) the conventional associative product A £ B between two arbitrary quantities A and B (numbers, matrices, operators, etc.) A £ B ! U £ (A £ B) £ Uy = (U £ A £ Uy) £ (U £ B £ Uy) = A0 £ B0; (3:3) and all numerical predictions. To illustrate this important property, suppose that a given quantum model; predicts the energy of 5 eV according to the familiar equation H(to) £ jÃ(to) >= 5eV £ jÃ(to) > : (3:4) Then the value 5 eV is preserved at all subsequent times in view of the unitary invariance U(t) £ H(to) £ jÃ(to) >= (U £ H £ Uy) £ (U £ Uy)¡1 £ (U £ jÃ >) = = H(t) £ jÃ(t) >= U(t) £ (5eV £ jÃ(to) >) = 5eV £ jÃ(t) > : (3:5) The invariance of physical laws and mathematical axioms then follows. It is evident that no representation of, the interactions occurring the valence bonds, and other mutual penetrations of particles at short distances, is physically acceptable unless it achieves the same invariance enjoyed by quantum mechanics. But the interactions herein considered are assumed to be non-Hamiltonian, thus beingnonunitary by conception. The difficulties of the task at hand can then be expressed by the following THEOREM 3.1 (45,46): All formulations with classical noncanonical and operator nonunitary time evolutions do not have time-invariant numerical predictions, physical laws and mathematical axioms when formulated via the mathematics of classical and quantum Hamiltonian mechanics, thus having no known physical or mathematical value. To illustrate the above theorem, note that, by assumption, the time evolution needed to represent nonlocal, nonpotential and nonlinear valence bonds has the nonunitary structure W(t) £ jÁ(to) >= jÃ(t) >;W £Wy 6= I;W(t) 6= ei£H£t: (3:6) 17 It is then easy to see that, when the nonunitary theory is formulated via the mathematics of unitary theories, it does not admit time-invariant numerical predictions, thus having no known physical meaning or value (46). To illustrate this important occurrence, note that nonunitary time evolutions do not preserve the basic units by conception, I ! I0 = W £ I £Wy 6= I; (3:7) where one should keep in mind that a realization of I is given by the basic units of the Euclidean space I = Diag:(1cm; 1cm; 1cm) expressed in the usual dimensionless form I = diag:(1; 1; 1):. Similarly, nonunitarity transforms do not preserve the conventional associative product, A £ B ! W £ (A £ B) £Wy = (W £ A £Wy) £ (W £Wy)¡1 £ (W £ B £Wy) = = A0 £ (W £Wy)¡1 £ B0 6= A0 £ B0: (3:8) It is then easy to verify the lack of invariance of numerical, predictions. In fact, suppose that the given nonunitary theory also predicts the energy of 5 eV at the initial time, H(to) £ jÃ(to) >= 5eV £ jÃ(to) >; (3:9) and that 15 seconds later (W £Wy)t=15sec = 1=3: (3:10) We then have W £ H(to) £ jÃ(to) >= (W £ H £Wy) £ (W £Wy)¡1 £ (W £ jÃ >) = = W(t) £ (5eV £ jÃ(to) >); (3:11) namely, H(t) £ jÃ(t) >= (W £Wy) £ (5eV jÃ(t) >) = 15eV £ jÃ(t) > : (3:12) The lack of preservation in time of numerical predictions then implies consequences today known as ”catastrophic physical inconsistencies,” such as: the lack of applicability of the theory to measurements (because of the loss of invariant basic units); Lopez’s Lemma (172,173) (loss of Hermiticity in time with consequential lack of acceptable observables); violation of causality and probability laws; etc. (45,46,171-175). The use of conventional mathematics for broader nonunitary theories leads to equally serious mathematical problems today known as ”catastrophic mathematical inconsistencies.” Suppose that a nonunitary theories is formulated at a given initial time to on a metric space defined over the field of real numbers R with basic unit I. But, by their very definition, nonunitary time evolutions do not leave invariant the basic unit, I ! I0 = U £ I £ Uy 6= I. Therefore, nonunitary time evolutions do not admit the original basic unit I at all times t > to, with consequential loss of the base field R. In turn, 18 the loss of the base field implies the inability to properly define the metric space acting on it, with consequential catastrophic collapse of the entire mathematical structure. Classical noncanonical theories are afflicted by similar catastrophic mathematical and physical inconsistencies (45,46,171-175). It should be recalled that, when facing non-Hamiltonian systems, a rather general tendency is that of transforming such systems into a form which is (at least locally) Hamiltonian via the use of Darboux’s theorem of the symplectic geometry or the Lie- Koening theorem of analytic mechanics (51). Unfortunately, these transformations cannot be used for the study of valence bonds because: (i) the system considered are nonlocal (thus implying the inapplicability a priori of the topologies needed for the quoted theorems); (ii) Darboux’s transformations are nonlinear, thus implying the impossibility of placing measuring apparata in the new Darboux’s coordinates; and (iii) in view of their nonlinearity, Darboux’s transformations cause the loss of the inertial character of reference frames with consequential loss of Galileo’s and Einstein’s relativities. In view of these shortcomings, the only physically acceptable representations are those occurring in the fixed coordinates of the observer, called direct representations (51). Only after such a representation is consistently achieved the transformation theory may have value. In summary, the representation of chemical bonds as well as other nonlocal interactions at short distances requires the abandonment of Hamiltonian theories which, in turn, implies the necessary use of theories whose time evolution is noncanonical at the classical level and nonunitary at the operator level. Still in turn, such noncanonical/nonunitary theories require the necessary construction of a new mathematics capable of resolving the catastrophic inconsistencies reviewed above. 3.3: Elements of isomathematics. The new mathematics specifically constructed for quantitative invariant treatment of nonlocal, nonpotential and nonlinear interactions among extended particles under mutual penetration at short distance is today known under the name of isomathematics, (where the prefix ”iso” also denotes the preservation of conventional axioms). Isomathematics was first proposed by R. M. Santilli in Ref. (23) of 1978 and subsequently studied by several mathematicians, theoreticians and experimentalists (see Refs. [2,4-11]). The main idea is that, as indicated earlier, valence bonds include conventional localdifferential Coulomb interactions, plus nonlocal, nonlinear and nonpotential interactions due to wave-overlappings. The former interactions can be effectively represented with the conventional Hamiltonian, while the latter interactions can be represented via a generalization of the basic unit as a condition to achieve invariance (since the unit is the basic invariant of any theory). Therefore, the main assumption of isomathematics is the lifting of the conventional unit of current formulations, generally given by an N-dimensional unit matrix I = Diag.(1, 1, ..., 1) > 0, into a quantity ˆI, called isounit, which possesses all topological properties 19 of I (such a positive-definiteness, same dimensionality, etc.), while having an arbitrary functional dependence on coordinates x, velocities v, wavefunctions Ã, their derivatives @xÃ, and any other needed variable (23,51), I = Diag:(1; 1; :::; 1) > 0 ! ˆI(x; v; Ã; @xÃ; :::) = 1= ˆ T(x; v; Ã; @xÃ; :::) > 0: (3:13) The conventional associative and distributive product A£B among generic quantities A, B (such as numbers, vector fields, operators, etc.) is jointly lifted into the more general form A £ B ! Aˆ£B = A £ ˆ T £ B; (3:14) which remains associative and distributive, thus being called isoproduct, under which the left and right character of I is preserved, i.e., I £ A = A £ I = A ! ˆI ˆ£A = Aˆ£ ˆI = A; (3:15): for all elements A of the set considered. An example of isounits (hereon assumed to be multiplicative) representing the interactions due to wave-overlappings of valence bonds is given by ˆI = eN(Ã)£R dx£Ãy(x)£Ã(x); (3:16) and illustrates the desired representation of nonlocal, nonlinear and nonpotential interactions. Another example of isounits is given by ˆI = Diag:(n21 ); (3:17) and illustrates the representation of the actual, extended, generally nonspherical and deformable shape of the particle considered, in this case, a spheroidal ellipsoid. The following is another example of isounit representing a relativistic system of extended, nonspherical and deformable particles under nonlinear, nonlocal and nonpotential interactions ˆITot = ¦k=1;2;:::;nDiag:(n2 k1; n2 k2; n2 k3; n2 k4) £ ¡(x; v; Ã; @Ã; (3:18):::); where ¡ is a function characterized by the considered nonlinear, nonlocal and nonpotential interactions, and nk4 represents the density of the medium in the interior of the particle k with the value nk4 = 1 for the vacuum, or, equivalently, the local speed of light within physical media c = co=nk4, in which case nk4 is the familiar index of refraction and co is the speed of light in vacuum. Numerous additional examples of isounits exist in the literature [4-11]. Note that the features represented by the isounits are strictly outside any representational capability by the Hamiltonian. 20 DEFINITION 3.1: Let F = F(a;+;£) be a field as per Definition 2.1. The isofields, first introduced in Ref. (23) of 1978 (see Ref. (12) for a mathematical treatment) are rings ˆ F = ˆ F(ˆa; ˆ+; ˆ£) whose elements are the isonumbers ˆa = a £ ˆI; (3:19) with associative, distributive and commutative isosum ˆa ˆ+ ˆb = (a + b) £ ˆI = ˆc 2 ˆ F; (3:20) associative and distributive isoproduct ˆa ˆ£ ˆb = ˆa £ ˆ T £ˆb = ˆc 2 ˆ F; (3:21) additive isounit ˆ0 = 0; ˆa ˆ+ ˆ0 = ˆ0 ˆ+ ˆa = ˆa; (3:22) andmultiplicative isounit ˆI = 1= ˆ T > 0; ˆa ˆ£ ˆI = ˆI ˆ£ˆa = ˆa; 8ˆa; ˆb 2 ˆ F; (3:23) where ˆI is not necessarily an element of F. Isofields are called of the first (second) kind when ˆI = 1= ˆ T > 0 is (is not) an element of F. LEMMA 3.1: Isofields of first and second kind are fields (namely, isofields verify all axioms of a field with characteristic zero). The above property establishes the fact (first identified in Ref. (12) that, by no means, the axioms of a field require that the multiplicative unit be the trivial unit +1, because it can be a negative-definite quantity as for the isodual mathematics, as well as an arbitrary positive-definite quantity, such as a matrix or an integrodifferential operator. Needless to say, the liftings of the unit and of the product imply a corresponding lifting of all conventional operations of a field. In fact, we have the isopowers ˆaˆn = ˆa ˆ£ˆa ˆ£:::; ˆ£ˆa(ntimes) = an £ ˆI; (3:24) with particular case ˆI ˆn = ˆI; (3:25) the isosquare root ˆa ˆ 1=2 = a1=2 £ ˆI1=2; (3:26) the isoquotient ˆaˆ= ˆb = (ˆa=ˆb) £ ˆI = (a=b) £ ˆI; (3:27) the isonorm ˆj ˆaˆj = jaj £ ˆI; (3:28) 21 where jaj is the conventional norm; etc. Despite their simplicity, the above liftings imply a complete generalization of the conventional number theory particularly for the case of the first kind (in which ˆI 2 F) with implications for all aspects of the theory. As an illustration, the use of the isounit ˆI = 1=3 implies that ”2 multiplied by 3” = 18, while 4 becomes a prime number. An important contribution has been made by E. Trell (143) who has achieved a proof of Fermat’s Last Theorem via the use of isonumbers, thus achieving a proof which is sufficiently simple to be of Fermat’s time. A comprehensive study of Santilli’s isonumber theory of both first and second kind has been conducted by C.-X. Jiang in monograph (68) with numerous novel; developments and applications. Additional studies on isonumbers have been done by N. Kamiya et al. (156) and others (see mathematical papers (10) and proceedings (8)). The lifting of fields into isofields implies a corresponding lifting of functional analysis into a form known as isofunctional analysis studied by J. V. Kadeisvili (132-133), A. K. Aringazin et al. (144) and other authors. A review of isofunctional analysis up to 1995 with various developments has been provided by R. M. Santilli in monographs (54,55). We here merely recall the isofunctions ˆ f(ˆx) = f(x £ I) £ ˆI; (3:29) the isologarithm ˆ logˆea = ˆI £ logea; ˆ logˆeˆe = ˆI; ˆ logˆe ˆI = 0; (3:30) the isoexponentiation, ˆe ˆ A = ˆI ˆ+ ˆ Aˆ= ˆ 1! ˆ+ ˆ Aˆ£ ˆ Aˆ= ˆ 2! ˆ+::: = (e ˆ A£ˆ T ) £ ˆI = ˆI £ (e ˆ T£ ˆ A): (3:31) The conventional differential calculus must also be lifted, for consistency, into the isodifferential calculus first identified by R. M. Santilli in memoir (14) of 1996, with isodifferential ˆ dˆx = ˆ T £ dˆx = ˆ T £ d(x £ ˆI); (3:32) which, for the case when ˆI does not depend on x, reduces to ˆ dˆx = dx; (3:33) the isoderivatives ˆ@ ˆ f(ˆx)= ˆ@ˆx = ˆI £ [@f(ˆx)=@ˆx]; (3:34) and other similar properties. The indicated invariance of the differential under isotopy, ˆ dˆx = dx, illustrates the reason why the isodifferential calculus has remained undetected since Newton’s and Leibnitz’s times. 3.4: Isotopologies, isospaces and isogeometries. Particularly important for these notes is the isotopy of the Euclidean topology independently identified by G. T. Tsagas and D. S. Sourlas (139) and R. M. Santilli (14), as well as the isotopies of the Euclidean, 22 Minkowskian, Riemannian and symplectic geometries, first identified by Santilli in various works (see Refs. (14,15,29,54,55) and references quoted therein). We cannot possibly review here these advances for brevity. We merely mention that any given n-dimensional metric or pseudometric space S(x; m;R) with basic unit I = Diag:(1; 1; :::; 1), local coordinates x = (xi); i = 1; 2; ::; n; n £ n- dimensional metric m and invariant x2 = xi £ mij £ xj 2 R is lifted into the isospaces ˆ S(ˆx; ˆ m; ˆR) with isocoordinates, isometric and isoinvariant respectively given by I = Diag:(1; 1; :::; 1) ! ˆIn£n(x; v; :::) = 1= ˆ T(x; v; :::); (3:35a) x ! ˆx = x £ ˆI;m ! ˆm = ˆ T(x; v; :::) m; (3:35b) x2 = xi £ mij £ xj £ I 2 R ! ˆxˆ2 = ˆxi ˆ£ ˆmij ˆ£ ˆxj £ ˆI = = fxi £ [ ˆ T(x; v; :::) £ m] £ xjg £ ˆI 2 ˆR: (3:35c) where one should note that ˆm is an isomatrix, namely, a matrix whose elements are isonumbers (thus being multiplied by ˆI to be in ˆR) and all operation are isotopic (in this way the calculation of the value of an isodeterminant cancels out all multiplications by ˆI except the last, thus correctly producing an isonumber). An inspection of the functional dependence of the isometric ˆm = ˆ T(x; v; :::) £m then reveals that isospaces ˆ S(ˆx; ˆmˆR) unify all possible spaces with the same dimension and signature. As an illustration, the isotopy of the 3-dimensional Euyccldiean space includes as particular case the 3-dimensional Riemannian, Finslerian as well as any other space with the same dimension and signature (+;+; +) (in view of the positive-definiteness of ˆI. Broader unifications are possible in the event such positive-definiteness is relaxed. Since the isotopies preserve the original axioms, the unification of the Euclidean and Riemannian geometry implies the reduction of Riemannian geometry to Euclidean axioms on isospaces over isofields. In turn, such a geometric unification has far reaching implications, e.g., for relativities, grand unifications and cosmologies (see later on). It should be mentioned that ”deformations” of conventional geometries are rather fashionable these days in the physical and mathematical literature. However, these deformations are generally afflicted by the catastrophic inconsistencies of Theorem 3.1 because, when the original geometry is canonical, the deformed geometry is noncanonical, thus losing the invariance needed for consistent applications. The isotopies of conventional geometries were constructed precisely to avoid such inconsistencies by reconstructing invariance on isospaces over isofield while having a fully noncanonical structure, as shown below. Therefore, for ”deformations” the generalized metric ˆm = ˆ T £m and related invariant are referred to conventional units and fields R, while for ”isotopies” the same generalized metric ˆm = ˆ T £ m is referred to a isounit which is the inverse of the deformation of the metric, ˆI = ˆ T¡1. While the deformed geometry verify axioms different then the original ones, the lifting of the original metric m by the matrix ˆ T while the basic unit is lifted by the inverse amount implies the preservation of the original axioms, with consequential unifications of different geometries. 23 Particularly intriguing are the isotopies of the symplectic geometry, known as isosymplectic geometry (14) which are based on the following fundamental isosymplectic twoisoform ˆ dˆpˆ^ ˆ dˆr = ˆ! = dp ^ dr = !; (3:36) due to the fact that, for certain geometric reasons, the isounit of the variable p in the cotangent bundle (phase space) is the inverse that of x (i.e., when ˆI = 1= ˆ T is the isounit for x, that for p is ˆ T = 1=ˆI). The invariance ˆ! = ! provide a reason why the isotopies of the symplectic geometry have escaped identification by mathematicians for over one century. Despite their simplicity, the isotopies of the symplectic geometry have vast implications, e.g., a broader quantization leading to a structural generalization of quantum mechanics known as hadronic mechanics, as outlined below. 3.5: Lie-Santilli isotheory and its isodual. As well known, Lie’s theory (4) is based on the conventional (left and right) unit I = Diag:(1; 1; :::; 1) of the universal enveloping associative algebra. The lifting I ! ˆI(x; :::) implies the lifting of the entire Lie theory, first proposed by R. M. Santilli in Ref. (23) of 1978 and then studied in numerous works (see, e.g., memoir (14) and monographs (51,54,55)). The isotopies of Lie’s theory are today known as the Lie-Santilli isotheory following studies by numerous mathematicians and physicists (see the monographs by D. S. Sourlas and Gr. Tsagas (64), J. V. Kadeisvili (66), R. M. Falcon Ganfornina and J. Nunez Valdez (67), proceedings [8] and contributions quoted therein). Let »(L) be the universal enveloping associative algebra of an N-dimensional Lie algebra L with (Hermitean) generators X = (Xi), i = 1, 2, ..., n, and corresponding Lie transformation group G over the reals R. The Lie-Santilli isotheory is characterized by: (I) The universal enveloping isoassociative algebra ˆ» with infinite-dimensional basis characterizing the Poincar´e-Birkhoff-Witt-Santilli isotheorem ˆ» : ˆI; ˆXi; ˆXi ˆ£ ˆX j ; ˆXi ˆ£ ˆX j ˆ£ ˆXk; :::; i · j · k; (3:37) where the ”hat” on the generators denotes their formulation on isospaces over isofields; (II) The Lie-Santilli isoalgebras ˆL ¼ (ˆ»)¡ : [ ˆXiˆ; ˆXj ] = ˆXi ˆ£ ˆX j ¡ ˆXj ˆ£ ˆX i = ˆ Ck ij ˆ£ ˆX k; (3:38) (III) The Lie-Santilli isotransformation groups ˆG
 * ad(1=2)d £d ad(1=2)d
 * n22
 * n23
 * ˆ A( ˆ w) = (ˆeˆi

ˆ£ ˆX ˆ£ ˆ w) ˆ£ ˆ A(ˆ0) ˆ£(ˆe¡ˆi ˆ£ ˆ wˆ£ ˆX ) = (ei£ ˆX £ˆ T£w) £ A(0) £ (e¡i£w£ˆ T£ ˆX ); (3:39) where ˆ w 2 ˆR are the isoparameters; the isorepresentation theory; etc. The non-triviality of the above liftings is expressed by the appearance of the isotopic element ˆ T(x; :::) at all levels (I), (II) and (III) of the isotheory. The arbitrary functional dependence of ˆ T(x; :::) then implies the achievement of the desired main features of the isotheory which can be expressed by the following: 24 LEMMA 3.2 (14): Lie-Santilli isoalgebras on conventional spaces over conventional fields are generally nonlocal, nonlinear and noncanonical, but they verify locality, linearity and canonicity when formulated on isospaces over isofields. To illustrate the Lie-Santilli isotheory in the operator case, consider the eigenvalue equation on H over C, H(x; p; Ã; :::) £ jÃ >= E £ jÃ >. This equation is nonlinear in the wavefunction, thus violating the superposition principle and preventing the study of composite nonlinear systems, as indicated earlier. However, under the factorization H(x; p; Ã; :::) = H0(x; p) £ ˆ T(x; p; Ã; :::); (3:40) the above equation can be reformulated identically in the isotopic form H(x; p; Ã; :::) £ jÃ >= H0(x; p) £ ˆ T(x; p; Ã; :::) £ jÃ >= H0 ˆ £jÃ >= E £ jÃ >= ˆE ˆ £jÃ >; (3:41) whose reconstruction of linearity on isospaces over isofields (called isolinearity (14)) is evident and so is the verification of the isosuperposition principle with resulting applicability of isolinear theories for the study of composite nonlinear systems. Similar results occur for the reconstruction on isospace over isofields of locality (called isolocality) and canonicity (called isocanonicity). A main role of the isotheory is then expressed by the following property: LEMMA 3.3 (29): Under the condition that ˆI is positive-definite, isotopic algebras and groups are locally isomorphic to the conventional algebras and groups, respectively. Stated in different terms, the Lie-Santilli isotheory was not constructed to characterize new Lie algebras, because all Lie algebras over a field of characteristic zero are known. On the contrary, the Lie-Santilli isotheory has been built to characterize new realizations of known Lie algebras generally of nonlinear, nonlocal and noncanonical character as needed for a deeper representation of valence bonds or, more generally, systems with nonlinear, nonlocal and noncanonical interactions. The mathematical implications of the Lie-Santilli isotheory are significant. For instance, Gr. Tsagas (142) has shown that all simple non-exceptional Lie algebras of dimension N can be unified into one single Lie-Santilli isotope of the same dimension, while studies for the inclusion of exceptional algebras in this grand unification of Lie theory are under way In fact, the characterization of different simple Lie algebras, including the transition from compact to noncompact Lie algebras, can be characterized by different realizations of the isounit while using a unique form of generators and of structure constants (see the first examples for the SO(3) algebra in Ref. (23) of 1978 and numerous others in the quoted literature). The physical implications of the Lie-Santilli isotheory are equally significant. We here mention the reconstruction as exact at the isotopic level of Lie symmetries when believed to be broken under conventional treatment. In fact, R. M. Santilli has proved: the exact reconstruction of the rotational symmetry for all ellipsoidical deformations of 25 the sphere (12); the exact SU(2)-isospin symmetry under electromagnetic interactions (28,33); the exact Lorentz symmetry under all (sufficiently smooth) signature-preserving deformations of the Minkowski metric (26); and the exact reconstruction of parity under weak interactions (55). R. Mignani (180) has studied the exact reconstruction of the SU(3) symmetry under various symmetry-breaking terms. In all these cases the reconstruction of the exact symmetry has been achieved by merely embedding all symmetry breaking terms in the isounit. The construction of the isodual Lie-Santilli isotheory for antimatter is an instructive exercise for interested readers. The main physical theories characterized by isomathematics are given by: 3.6: Iso-Newtonian Mechanics and its isodual. As it is well known, Newton (1) had to construct the differential calculus as a necessary pre-requisite for the formulation of his celebrated equations. Today we know that Newton’s equations can only represent point-particles due to the strictly local-differential character of the underlying Euclidean topology. The fundamental character of Newtonian Mechanics for all scientific inquiries is due to the preservation at all subsequent levels of study (such as Hamiltonian mechanics, quantum mechanics, quantum chemistry, quantum field theory, etc.) of: 1) The underlying Euclidean topology; 2) The differential calculus; and 3) The notion of point particle. By keeping in mind Newton’s teaching, the author has dedicated primary efforts to the isotopic lifting of the conventional differential calculus, topology and geometries (14) as a necessary pre-requisite for a structural generalization of Newton’s equations into a form representing extended, nonspherical and deformable particles under action-at-adistance/ potential as well as contact/nonpotential forces. The need for such a lifting is due to the fact that point particles cannot experience contact-resistive forces. This feature has lead to subsequent theories, such as Hamiltonian and quantum mechanics, which solely admit action-at-a-distance/potential forces among point particles. Such a restriction is indeed valid for a number of systems, such as planetary systems at the classical level and atomic systems at the operator level, because the large distances among the constituents permit an effective point–like approximation of particles. However, when interactions occur at short distances, as in the case of electron valence bonds (Figure 2) or the mutual penetration of the wavepackets of particles in general, the point-like approximation is no longer sufficient and a representation of the actual, extended, generally nonspherical and deformable shape of particles is a necessary prerequisite to admit contact nonpotential interactions. By recalling the fundamental character of Newtonian mechanics for all of sciences, the achievement of a consistent representation of the contact interactions of valence electron bonds at the operator level requires the prior achievement of a consistent Newtonian representation. To outline the needed isotopies, let us recall that Newtonian mechanics is formulated 26 on the Kronecker product Stot = St£Sx£Sv of the one dimensional space St representing time t, the tree dimensional Euclidean space Sx of the coordinates x = (xk ®) (where k = 1; 2; 3 are the Euclidean axes and ® = 1; 2; :::; n represents the number of particles), and the velocity space Sv; v = dx=dt. It is generally assumed that all variables t; x; and v are defined on the same field of real numbers R. However, the unit of time is the scalar I = 1, while the unit of the Euclidean space is the matrix I = Diag:(1; 1; 1). Therefore, on rigorous grounds, the representation space of Newtonian mechanics Stot = S1 £ Sx £ Sv must be defined on the Kronecker product of the corresponding fields Rtot = Rt £ Rx £ Rv with total unit ITot = 1 £ Diag:(1; 1; 1)x £ Diag:(1; 1; 1)v. Newtonian systems requested for the isotopies are given by the so-called closed-isolated non-Hamiltonian systems (51), namely, systems which are closed-isolated from the rest of the universe, thus verifying all ten Galilean total conservation laws, yet they admit internal non-Hamiltonian forces due to contact interactions. A typical illustration is given by the structure of Jupiter which, when considered as isolated from the rest of the universe, does indeed verify all Galilean conservation laws, yet its internal structure is clearly non-Hamiltonian due to vortices with varying angular momentum and similar internal dissipative effects. In essence, contact nonpotential forces produce internal exchanges of energy, linear and angular momentum but always in such a manner to verify total conservation laws. A Newtonian representation of closed-isolated non-Hamiltonian systems of extended particles is given by(Ref. (51), page 236) m® £ ak® = m® £ dvk® dt = F®(t; x; v) = FSA ® (x) + FNSA ® (t; x; v); (3:42a) X ®=1;:::;n FNSA ® = 0; (3:42b) X ®=1;:::;n x®KFNSA ® = 0; (3:42c) X ®=1;:::;n x®^FNSA ® = 0; (3:42d) where: SA (NSA) stands for variational selfadjointness (variational nonselfadjointness), namely, the verification (violation) of the integrability conditions for the existence of a potential, and conditions (3.9b), (3.9c) and (3.9d) assure the verification of all ten Galilean conservation laws (for the total energy, linear momentum, angular momentum, and uniform motion of the center of mass). The restrictions to FNSA verifying the above conditions is tacitly assumed hereon. The isotopies of Newtonian mechanics, also called Newton-Santilli isomechanics (63- 68), requires the use of the isotime ˆt = t £ ˆIt with isounit ˆIt = 1= ˆ Tt and related isofield ˆR t, the isocoordinates ˆx = (ˆxk ®) = x £ ˆIx; with isounit ˆIx = 1= ˆ Tx and related isofield ˆRx, and the isospeeds ˆv = (vk®) = v £ ˆIv with isounit ˆIv = 1= ˆ Tv and related isofield ˆRv. 27 IsoNewtonian Mechanics is then formulated on the Kronecker product of isospaces ˆ STot = ˆ St £ ˆ Sx £ ˆ Sv over the Kronecker product of isofields ˆRt £ ˆRx £ ˆRv. The isospeed is the given by ˆv = ˆ dˆx ˆ dˆt = ˆIt £ d(x £ ˆIx) dt = v £ ˆIt £ ˆIx + x £ ˆIt £ dˆIx dt = v £ ˆIv; (3:43a) ˆIv = ˆIt £ ˆIx £ (1 + x £ ˆ Tx £ dˆIx dt
 * (3:43b)

The Newton-Santilli isoequation and its isodual, first proposed in memoir (14) of 1996 (where the isodifferential calculus was first achieved) can be written ˆm® ˆ£ ˆ dˆvk® ˆ dˆt = ¡ ˆ@ ˆ V (ˆx) ˆ@ˆxk ®
 * (3:44)

namely, the equations are conceived in such a way to formally coincide with the conventional equations for selfadjoint forces, FSA = ¡@V=@x; while all nonpotential forces are represented by the isounits or, equivalently, by the isodifferential calculus. Such a conception is the only one known which permits the representation of extended particles with contact interactions which is invariant (thus avoiding the catastrophic inconsistencies of Theorem 3.1) and achieves closure, namely, the verification of all ten Galilean conservation laws. An inspection of Eqs. (3.10) is sufficient to see that iso-Newtonian mechanics reconstructs canonicity on isospace over isofields, thus avoiding Theorem 3.1. Note that this would not be the case if nonselfadjoint forces appear in the right hand side of Eqs. (3.10) as in Eqs. (3.9a). The verification of all Galilean conservation laws is equally established by a visual inspection of Eqs. (3.10) since their symmetry, the iso-Galilean symmetry with structure (3.8), is the Galilean symmetry, only formulated on isospace over isofields (53). By recalling that conservation laws are represented by the generators of the underlying symmetry, conventional total conservation laws then follow from the fact that the generator of the conventional Galilean symmetry and its isotopic lifting coincide. When projected in the conventional Newtonian space STot, Eqs. (3.10) can be explicitly written ˆmˆ£ ˆ dˆv ˆ dˆt = m £ ˆIt £ d(v £ ˆIv) dt = = m £ a £ ˆIt £ ˆIv + m £ v £ ˆIt £ dˆIv dt = ¡ ˆ@ ˆ V (ˆx) ˆ@ˆx = ¡ˆIx £ @V @x that is m £ a = ¡ˆ Tt £ ˆ Tv £ ˆIx £ @V @x ¡ m £ v £ ˆ Tv £ dˆIv dt with necessary and sufficient conditions for the representation of all possible SA and NSA forces ˆIt £ ˆIv £ ˆIx = I; ˆIx = 1= ˆ Tt £ ˆ Tx; (3:48a) 28 m £ v £ ˆ Tv £ dˆIv dt = FNSA(t; x; v); (3:48b) which always admit a solution, since they constitute a system of 6n algebraic (rather than differential) equations in the 6n + 1 unknowns given by ˆIt, and the diagonal ˆIx and ˆIv. As an illustration, we have the following equations of motion of an extended particle with the ellipsoidal shape experiencing a resistive force FNSA = ¡° £ v because moving within a physical medium m £ a = ¡° £ v (3:49a) ˆIv = Diag:(n21 ) £ e°£t=m: (3:49b) Interested readers can then construct the representation of any desired NSA forces (see also memoir (14) for other examples). Note the natural appearance of the velocity dependence, as typical of resistive forces. Note also that the representation of the extended character of particles occurs only in isospace because, when Eqs. (3.10) are projected in the conventional Newtonian space, all isounits cancel out and the point characterization of particles is recovered. Note finally the direct universality of the Newton-Santilli isoequations, namely, their capability of representing all infinitely possible Newton’s equations in the frame of the observer. As indicated earlier, Eqs. (3.42) can only describe a system of particles. The construction of the isodual Newton-Santilli isoequations for the treatment of a system of antiparticles is left to the interested reader. We finally indicate that the invariance of closed non-Hamiltonian systems (3.42) is given by the Galilei-Santilli isosymmetry ˆG(3:1) and their isoduals by ˆGd(3:1) (see Refs. (52,53) for brevity). 3.7: Iso-Hamiltonian Mechanics and its isodual. Eqs. (3.10) admit the analytic representation in terms of the following isoaction principle (14) ˆ± ˆ A(ˆt; ˆx) = ˆ± ˆ Z (ˆpk® ˆ£p ˆ dˆxk ®) ¡ ˆH ˆ£ t ˆ dˆt = = ± ˆ Z [pk® £ ˆ Tx(t; x; p; :::) £ d(xk ® £ ˆIx) ¡ H £ ˆ Tt(t; x; p; :::)d(t £ ˆIt) = 0: (3:50) Note the main result permitted by the isodifferential calculus, consisting in the reduction of an action functional of arbitrary power in the linear momentum (arbitrary order) to that of first power in p first order. Since the optimal control theory and the calculus of variation depend on the first order character of the action functional, the above reduction has important implications, such as the treatment of extended objects moving within resistive media apparently for the first time via the optimal control theory, since a first order conventional action is impossible for the systems considered.. Note that when the isounits are constant, isoaction the isoaction coincides with the conventional action. This illustrates the apparent reason why the isotopies of the action principle creeped in un-noticed for over one century. 29 It is easy to prove that the above isoaction principle characterizes the Hamilton-Santilli isoequations (14) ˆ dˆx ˆ dˆt = ˆ@ ˆH ˆ@ ˆp = ˆp ˆm = ˆp ˆm ˆ dˆp ˆ dˆt = ¡ ˆ@ ˆH ˆ@ˆx = ˆ FSA + ˆ FNSA; ; (3:51a); ˆH = X ®=1;:::;n ˆpk® ˆ£p ˆpk ® ˆ2 ˆ£ ˆm® (3:51b) ˆIt = 1; ˆIx = I + FNSA=FSA; ˆIp = ˆ Tx; (3:51c) where one should note the real-valued, symmetric and positive-definite character of all isounits, and corresponding Hamilton-Jacobi-Santilli isoequations ˆ@ ˆ A ˆ@ˆt + ˆH = 0; ˆ@ ˆ A ˆ@ˆxk ® ¡ ˆpk® = 0: (3:52) As it was the case for Eqs. (3.10), iso-Hamiltonian mechanics has been conceived to coincide at the abstract level with the conventional formulation. Nevertheless, the following main differences occur: 1) Hamiltonian mechanics can only represent point particles while its isotopic covering can represent the actual, extended, nonspherical and deformable shape of particles via the simply identification of isounits (3.11c); 2) Hamiltonian mechanics can only represent a rather restricted class of Newtonian systems, those with potential forces, while its isotopic covering is directly universal for all possible (sufficiently smooth) SA and NSA Newtonian systems; 3) All NSA forces are represented by the isounits or, equivalently, by the isodifferential calculus, thus permitting their invariant description, since iso-Hamiltonian mechanics clearly reconstructs canonicity on isospaces over isofields. Iso-Hamiltonian mechanics as outline above can only described closed non-Hamiltonian systems of particles. The construction of its isodual for antiparticles is an instructive exercise for interested readers. 3.8: Isotopic Branch of nonrelativistic Hadronic Mechanics and its isodual. The preservation of the form of Newton’s and Hamilton’s equations has far reaching implications, since it permit a simple lifting of quantization, resulting in a generalization of quantum mechanics known under the name of isotopic branch of Hadronic Mechanics (see memoir (31) for a general review), which permits, apparently for the first time, an axiomatically consistent and invariant representation of extended particles under linear and nonlinear, local and nonlocal, and potential as well as nonpotential interactions. Recall that the conventional naive or symplectic quantization A ! ¡i £ ¯h £ LnÃ is solely applicable for first-order action functionals A(t; x) and, as such, it is not applicable to the isoaction ˆ A(ˆt; ˆx) = ˆ A(t; x; p; :::) due to its higher order when formulated on conventional spaces. Nevertheless, it is easy to show that the following naive isoquantization 30 holds (for ˆ¯h replaced by ˆIx) ˆ A(ˆt; ˆx) ! ¡ˆi ˆ£ ˆ Tx ˆ£ ˆLn ˆ Ã(ˆt; ˆx); (3:53) which, when applied to Eqs. (3.12) permits the map here expressed for the case when ˆIx is a constant (see Ref. (55) for the general case) ˆ@ ˆ A ˆ@ˆt + ˆH = 0 ! ¡i £ ˆIt £ @ ˆ Ã @t + ˆH £ ˆ Tx £ ˆ Ã = 0; (3:54a) ˆ@ ˆ A ˆ@ˆxk ® ¡ ˆpk® = 0 ! ¡i £ ˆIx £ @ ˆ Ã @x ¡ ˆp £ ˆ Tx £ ˆ Ã = 0: (3:54b) The above equations can be more properly formulated over the iso-Hilbert space (25) ˆH with isostates j ˆ Ã(ˆt; ˆx) > and isoinner product < ˆ Ãj ˆ £j ˆ Ã > £ˆI over the isofield ˆ C (see memoir (31) for a review). The new mechanics is characterized by the iso-Schroedinger equations (first derived in Refs. (25,179) with ordinary mathematics and first formulated via the isodifferential calculus in Ref. (14)) ˆi ˆ£ ˆ@ ˆ@ˆt j ˆ Ã >= ˆH ˆ £j ˆ Ã >= ˆH (ˆx; ˆp)£ ˆ T(ˆx; ˆp; ˆ Ã; ˆ@ ˆ Ã; ::::)£j ˆ Ã >= ˆE ˆ £j ˆ Ã >= E£j ˆ Ã >; (3:55a) ˆpk ˆ £j ˆ Ã >= ¡ˆi ˆ£ ˆ@kj ˆ Ã >= ¡i £ ˆIi k £ @ij ˆ Ã >; ˆI ˆ £j ˆ Ã >= j ˆ Ã >; (3:55b) and the iso-Heisenberg equations (first derived in Ref. (38) via conventional mathematics and first formulated via the isodifferential calculus in Ref. (14)) ˆi ˆ£ ˆ d ˆ A ˆ dˆt = [ ˆ Aˆ; ˆH ] = ˆ Aˆ£ ˆH ˆ¡ ˆHˆ£ ˆ A = = ˆ A £ ˆ T(ˆt; ˆx; ˆp; ˆ Ã; ˆ@ ˆ Ã; :::) £ ˆH ¡ ˆH £ ˆ T(ˆt; ˆx; ˆp; ˆ Ã; ˆ@ ˆ Ã; :::) £ ˆ A; (3:56a) [ˆxiˆ;ˆpj ] = ˆi ˆ£ ˆ±i j = i £ ±i j £ ˆI; [ˆxi; ˆxj ] = [ˆpi; ˆpj ] = 0: (3:56b) A first important property the reader can easily prove is that iso-Hermiticity coincides with conventional Hermiticity. Consequently, all quantities which are observable for quantum mechanics remain observable under isotopies. In particular, it is equally easy to prove that all Hermitean quantities which are conserved for quantum mechanics remain conserve under isotopies, again, because the symmetries of Schroedinger’ and iso-Schroedinger’s equations are isomorphic and their generators coincide. The above results implies the existence of a new notion of bound state of particles as the operator image of closed non-Hamiltonian systems (3.9), namely, a bound state admitting internal Hamiltonian as well as nonlinear, nonlocal and nonpotential interactions while preserving conventional total conservation laws. Note that these are precisely the characteristics needed for quantitative studies of electron valence bonds, as well as, more generally, bound states of particles at shot mutual distances. 31 Another important property of hadronic mechanics is that, in view of the lack of general commutativity between ˆH and ˆ T, the iso-Schroedinger and iso-Heisenberg’s equations have a nonunitary time evolution when formulated on conventional Hilbert spaces over conventional fields, j ˆ Ã(t) >= (ei£H£ˆ T£t) £ j ˆ Ã(0) >= U(t) £ j ˆ Ã(0) >;U £ Uy 6= I; (3:57) However, all nonunitary transforms admit an identical reformulation as isounitary transform on iso-Hilbert spaces, U £ Uy 6= I; U = ˆU £ ˆ T1=2; (3:58a) ˆU ˆ£ ˆU y = ˆU y ˆ£ ˆU = ˆI: (3:58b) The above property is a necessary condition to exit from the class of equivalence of quantum mechanics, thus illustrating the nontriviality of the lifting. Yet another property is that nonlinear Schroedinger’s equations cannot represent composite systems because of the violation of the superposition principle, while hadronic mechanics resolves this limitation. In fact,, all nonlinear Schroedinger’ s equations can be identically rewritten in the isotopic form with the embedding of all nonlinear terms in the isotopic element, H(x; p; Ã)jÃ >= H0(x; p) £ ˆ T(x; p; Ã); :::) £ jÃ >= E £ jÃ >; (3:59) under which linearity, and, therefore, the superposition principle, are trivially reconstructed in isospace over isofields. The isoexpectation values of an observable ˆ A on ˆH over ˆ C are given by < ˆ Ãj ˆ£ ˆ Aˆ £j ˆ Ã > < ˆ Ãjˆ £j ˆ Ã > £ ˆI 2 ˆ C: (3:60) It is easy to prove that the isoexpectation values coincide with the isoeigenvalues, as in the conventional case. In particular, the isoexpectation value of the isounit recovers Planck’s unit < ˆ Ãjˆ£ ˆI ˆ £j ˆ Ã > < ˆ Ãjˆ £j ˆ Ã > = ¯h = 1: (3:61) Note also the following invariance of Hilbert’s inner product under isotopy (for the case when the isotopic element does not depend on the integration variable) < Ãj £ jÃ > £I ´< Ãj £ ˆ T £ jÃ > £ˆI; (3:62) which invariance explains why the isotopies of Hilbert spaces remained un-discovered since Hilbert’s time even though they have the important implication of causing a structural generalization of the conventional formulation of quantum mechanics. Note that, despite its simplicity, invariance (3.62) required the prior identification of new numbers, those with arbitrary unit ˆI. 32 The latter properties establish that the isotopic branch of hadronic mechanics coincides with quantum mechanics at the abstract, realization-free level. This feature is important to assure the axiomatic consistency of hadronic mechanics, as well as to clarify the fact that hadronic mechanics is not a new theory, but merely a novel realization of the abstract axioms of quantum mechanics. The isodual isotopic branch of hadronic mechanics for the treatment of antimatter is given by the image of the theory under map (2.4) and its outline is here omitted for brevity. For additional intriguing features of hadronic mechanics, interested readers can inspect memoir (31) and monograph (55). 3.9: Invariance of isotopic theories. The invariance of the basic axioms and numerical predictions pf isotopic theories under time as well as other transforms was first achieved in Refs. (14,31). It can be proved on isospaces over isofields by reformulating any given, nonunitary transform in the isounitary form, W £Wy = ˆI;W = ˆW £ ˆ T1=2;W £Wy = ˆW ˆ£ ˆW y = ˆW y ˆ£ ˆW = ˆI; (3:63) and then showing that the basic isoaxioms are indeed invariant, i.e., ˆI ! ˆI0 = ˆW ˆ£ ˆI ˆ£ ˆW y = ˆI; (3:64a) ˆ Aˆ£ ˆB ! ˆW ˆ£ ( ˆ Aˆ£ ˆB ) ˆ£ ˆW y = = ( ˆW £ ˆ T £ A £ ˆ T £ ˆW y) £ ( ˆ T £ ˆW y)¡1 £ ˆ T £ ( ˆW £ ˆ T)¡1 £ ( ˆW £ ˆ T £ ˆB £ ˆ T £ ˆW y) = = ˆ A0 £ ( ˆW y £ ˆ T £ ˆW )¡1 £ ˆB 0 = ˆ A0 £ ˆ T £ ˆB0 = ˆ A0 ˆ£ ˆB 0; etc: (3:64b) The invariance is ensured by the numerically invariant values of the isounit ˆI and of the isotopic element ˆ T under nonunitary-isounitary transforms, namely, ˆI ! ˆI0 = ˆI; ˆ T ! ˆ T0 = ˆ T; ˆ£ ! ˆ£0 = ˆ£: (3:65) The resolution of the catastrophic inconsistencies of Theorem 3.1 is then consequential. The achievement of invariant for classical noncanonical formulations is equivalent to the preceding nonunitary one and its explicit form is left to the interested reader for brevity. 3.10: Simple construction of isotheories. A simple method has been identified in Refs. (14,31) for the construction of the entire isomathematics and its physical applications. It consists in: (i) representing all conventional interactions with a Hamiltonian H and all nonhamiltonian interactions and effects with the isounit ˆI; (ii) identifying the latter interactions with a nonunitary transform U £ Uy = ˆI 6= I; (3:66) 33 and (iii) subjecting the totality of conventional mathematical and physical quantities and all their operations to said nonunitary transform, I ! ˆI = U £ I £ Uy = 1= ˆ T; a ! ˆa = U £ a £ Uy = a £ ˆI; (3:67a) a £ b ! U £ (a £ b) £ Uy = (U £ a £ Uy) £ (U £ Uy)¡1 £ (U £ b £ Uy) = ˆa ˆ£ˆb; (3:67b) eA ! U £ eA £ Uy = ˆI £ e ˆ T£ ˆ A = (e ˆ A£ˆ T ) £ ˆI; (3:67c) [Xi;Xj ] ! U £ [XiXj ] £ Uy = [ ˆXiˆ; ˆXj ] = U £ (Ck oj £ Xk) £ Uy = ˆ Ck ij ˆ£ ˆX k = = Ck ij £ ˆXk; (3:67d) < Ãj £ jÃ >! U£ < Ãj £ jÃ > £Uy = =< Ãj £ Uy £ (U £ Uy)¡1 £ U £ jÃ > £(U £ Uy) =< ˆ Ãjˆ £j ˆ Ã > £ˆI; (3:67e) H £ jÃ >! U £ (H £ jÃ >) = (U £ H £ Uy) £ (U £ Uy)¡1 £ (U £ jÃ >) = = ˆH ˆ £j ˆ Ã >; etc: (3:67f) It should be indicated that not all Lie-Santilli isoalgebras can be constructed via nonunitary transforms of conventional Lie algebras. As an illustration, the classification of all possible isotopic ˆ SU(2) algebras exhibits eigenvalues different then the conventional ones, while only conventional eigenvalues are admitted under nonunitary transforms (see Refs. (28,33) for brevity). 3.11: Isorelativity and its isodual. Special relativity is generally presented by contemporary academia as providing a descriptions of all infinitely possible relativistic systems existing in the universe. In Section 2 we have shown that special relativity cannot provide a consistent classical description of point antiparticles moving in vacuum. The content of this section establishes that special relativity cannot be exactly valid for extended particles and antiparticles moving within physical media. In the next section we shall show that special relativity cannot describe irreversible processes for both matter and antimatter. Finally, in Section 5 we shall show that the complexity of biological systems is immensely beyond the rather limited descriptive capacity of special relativity. Particularly misleading is the widespread statement 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 c = co=n, where co is the speed of light in vacuum and n is the familiar index of refraction possessing a rather complex functional dependence on frequencies !, the density of the medium d, and other variables, n = n(!; d; :::). For the evident intent of salvaging the desired universality of special relativity, speeds c < co have been interpreted until recently by reducing the propagation of light within a physical medium to the propagation of photons in vacuum scattering from atom to atom. 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 34 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 > co 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 c = 2£co=3 while electrons can propagate with speeds bigger than c, resulting in the emission of the Cerenkov light. If the local speed of light c is assumed as the universal invariant, then the propagation of electrons at speeds v > c is a violation of the principle of causality. If the speed of light in vacuum co is assumed as the universal invariant in water, there is the violation of the relativistic law of addition of two speeds of light c because it does not yield the local speed of light c, 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 3.1. 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, which can be written ´ = Diag:(1; 1; 1;¡c2 o ). Then, the transition from co to c = co=n in the metric can only be achieved via a noncanonical or nonunitary transform ´ = Diag:(1; 1; 1;¡c2 o ) ! ˆ´ = Diag:(1; 1; 1;¡co=n2) = U £ ´ £ Uy; (3:68a) U £ Uy = Diag:(1; 1; 1; 1=n2) 6= I: (3:68b) 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-Poincar´e-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 done in Section 3.13. Isorelativity was first proposed by R. M. Santilli in Ref. (26) of 1983 via the first invariant formulation of 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 Poincar´e 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.). 35 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 meanlifes 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. The central notion of isorelativity is the lifting of the basic unit of the Minkowski space and of the Poincar´e symmetry, I = Diag:(1; 1; 1; 1), into a 4 £ 4-dimensional, nowhere singular and positive-definite matrix ˆI = ˆI4£4 with an unrestricted functional dependence on local spacetime coordinates x, speeds v, frequency !, wavefunction Ã, its derivative @Ã, etc., I = Diag:(1; 1; 1) ! ˆI(x; v; !; Ã; @Ã; :::) = 1= ˆ T(x; v; !; Ã; @Ã; :::) > 0: (3:69) Isorelativity can then be constructed via the method of Section 3.10, namely, by assuming that the basic noncanonical or nonunitary transform coincides with the above isounit (where the diagonalization is permitted by its Hermiticity) U £ Uy = ˆI = Diag:(g11; g22; g33; g44); g¹¹ = g¹¹(x; v; !; Ã; @Ã; :::) > 0; ¹ = 1; 2; 3; 4; (3:70) and then subjecting the totality of quantities and their operation of special relativity to the above transform. Let M(x; ´;R) be the Minkowski space with local coordinates x = (x¹), metric ´ = Diag:(1; 1; 1;¡1) and invariant x2 = (x¹ £ ´¹º £ xº £I 2 R. The fundamental space of isorelativity is the Minkowski isospace (26,15) and related topology (14), ˆM (ˆx; ˆ´; ˆR) characterized by the dual lifting of the basic unit (and related field) and the inverse lifting of the metric as per rules (3.35) I = Diag:(1; 1; 1; 1) ! U £ I £ Uy = ˆI = 1= ˆ T; (3:71a) ´ = Diag:(1; 1; 1;¡1) £ I ! (Uy¡1 £ ´ £ U¡1) £ ˆI = ˆ´ = = ˆ T £ ´ = Diag:(g11; g22; g33;¡g44) £ ˆI; (3:71b) with consequential isotopy of the basic invariant x2 = (x¹ £ ´¹º £ xº £I 2 R ! 36 ! U £ x2 £ Uy = ˆxˆ2 = (ˆx¹ ˆ£ ˆm¹º £ xº £I 2 R; (3:72) whose projection in conventional spacetime can be written ˆxˆ2 = [x¹ £ ˆ´¹º(x; v; !; Ã; @Ã; :::) £ xº ] £ ˆI; (3:73) The nontriviality of the above lifting is illustrated by the fact that Minkowski-Santilli isospaces include as particular spaces all possible spacetimes, such as the Riemannian, Finslerian, non-Desarguesian and any other space with the signature (+;+;+;¡). Moreover, the iso-Minkowskian metric ˆ´ depends explicitly on the local coordinates. Therefore, the Minkowski-Santilli isogeometry requires for its formulation the isotopy of all tools of the Riemannian geometry, such as the iso-Christoffel symbols, isocovariant derivative, etc. (see for brevity Ref. (15)). Despite that, one should keep in mind that, in view of the positive-definiteness property (34.79), the Minkowski-Santilli isogeometry coincides at the abstract level with the conventional Minkowski geometry, thus having a null isocurvature (because of the basic mechanism of deforming the metric ´ by the amount ˆ T(x; :::) while deforming the basic unit of the inverse amount ˆI = 1= ˆ T). It should be also noted that, following the publication in 1983 of Ref. (26), numerous papers on ”deformed Minkowski spaces” have appeared in the physical and mathematical literature (generally without a quotation of their origination in Ref. (29)). These ”deformations” are formulated via conventional mathematics and, consequently, they all suffer of the catastrophic inconsistencies of Theorem 3.1. By comparison, isospaces are formulated via isomathematics and, therefore, they resolve the inconsistencies of Theorem 3.1, as shown in Section 3.9. This illustrates the necessity of lifting the basic unit and related field jointly with any noncanonical lifting of canonical metrics. Let P(3:1) be the conventional Poincar´e symmetry with the well known ten generators J¹º; P¹ and related commutation rules. The second basic tool of isorelativity is the Poincar´e-Santilli isosymmetry ˆ P(3:1) which can be constructed via the isotheory of Section 3.5, resulting in the isocommutation rules (26,29) [J¹ºˆ;J®¯] = i £ (ˆ´º® £ J¯¹ ¡ ˆ´º® £ J¯º ¡ ˆ´nu¯ £ J®¹ + ˆ´¹¯ £ J®º); (3:74a) [J¹ºˆ;P®] = i £ (ˆ´¹® £ Pº ¡ ˆ´º® £ P¹); [P¹ˆ;P º] = 0; (3:74b) . where we have followed the general rule of the Lie-Santilli isotheory according to which isotopies leave observables unchanged (since Hermiticity coincides with iso-Hermiticity) and merely change the operations among them. Isorelativistic kinematics is then based on the following two iso-invariants: Pˆ2 = P¹ ˆ£P¹ = P¹ £ ´¹º £ Pº = Pk £ gkk £ Pk ¡ p4 £ g44 £ P4; (3:75a) Wˆ2 = W¹ ˆ£W¹;W¹ = ˆ²¹®¯½ ˆ£J®¯ ˆ£P½: (3:75b) Since ˆI > 0, it is easy to prove Lemma 3.3, namely, that the Poincar´e-Santilli isosymmetry is isomorphic to the conventional symmetry. It then follows that the isotopies 37 increase dramatically the arena of applicability of the Poincar´e symmetry, from the sole Minkowskian spacetime to all infinitely possible spacetimes. To understand the physical, chemical and biological applications outline in this paper, the reader should be aware that all ”particles” considered hereon are assumed to be ”isoparticles”, that is, irreducible isorepresentation of the Poincar´e-Santilli isosymmetry, namely, particles are assumed to be extended, generally nonspherical and deformable under Hamiltonian and non-Hamiltonian interactions. Since any interaction imply a renormalization of physical characteristics, it is evident that the transition form particles to isoparticles, that is. from motion in vacuum to motion within physical media, implies an alteration (called isorenormalization) of all the intrinsic characteristics, such as rest energy, magnetic moment, charge, etc. As we shall see in Section 3.14, such isorenormalization have permitted the first exact numerical representation of nuclear magnetic moments which had resulted to be impossible for quantum mechanics despite about 75 years of attempts. The explicit form of the Poincar´e-Santilli isotransforms leaving invariant line element (3.73) are given by: (1) The isorotations ˆO(3) : ˆx0 = ˆ<(ˆµ) ˆ£ˆx; ˆµ = µ £ ˆIµ 2 ˆRµ (11) which, for isorotations in the (1, 2) isoplane, are given by x10 = x1 £ cos[µ £ (g11 £ g22)1=2] ¡ x2 £ g22 £ g¡1 11 £ sin[µ £ (g11 £ g22)1=2]; (3:76a) x20 = x1 £ g11 £ g¡1 22 £ sin[µ £ (g11 £ g22)1=2] + x2 £ cos[µ £ (g11 £ g22)1=2]: (3:76b) For the general expression in three dimensions interested reader can inspect Ref. (55) for brevity. Note that, since ˆO (3) is isomorphic to O(3), Ref. (11) proved that, contrary to a popular belief throughout the 20-th century, the rotational symmetry remains exact for all possible signature-preserving (+,+,+) deformations of the sphere, of course, when treated with the appropriate mathematics. The above reconstruction of the exact rotational symmetry can be geometrically visualized by the fact that all possible signature-preserving deformations of the sphere are perfect spheres in isospace called isosphere. This is due to the fact that ellipsoidical deformations of the semiaxes 1k ! 1=n2 k are compensated on isospaces over isofields by the inverse deformation of the related unit 1k ! n2 k. Therefore, by recalling structure (3.35) of the isoinvariant, on iso-Euclidean space we have the perfect isosphere ˆrˆ2 = ˆr2 1 + ˆr2 2 + ˆr2 3 with exact ˆO(3) symmetry while its projection on the conventional Euclidean space is the ellipsoid r2 1=n21 + r2 2=n22 + r2 3=n23 with broken O(3) symmetry. (2) The Lorentz-Santilli isotransforms ˆO(3:1) : ˆx0 = ˆ¤(ˆv; :::) ˆ£ ˆx; ˆv = v £ ˆIv 2 ˆRv (26,29) which, for the case of isorotation in the (3; 4) isoplane, can be written x10 = x1; (3:77a) x20 = x2; (3:77b) x30 = x3 £ cosh[v £ (g33 £ g44)1=2]¡ 38 ¡x4 £ g44 £ (g33 £ g44)¡1=2 £ sinh[v £ (g33 £ g44)1=2] = = ˆ° £ (x3 ¡ ¯ £ x4); (3:77c) x40 = ¡x3 £ g33 £ (g33 £ g44)¡1=2 £ sinh[v(g33 £ g44)1=2]+ +x4 £ cosh[v £ (g33 £ g44)1=2] = = ˆ° £ (x4 ¡ ˆ ¯ £ x3); (3:77d) ˆ ¯ = vk £ gkk £ vk co £ g44 £ co 1 (1 ¡ ˆ ¯2)1=2
 * (3:46)
 * (3:47)
 * n22
 * n23
 * ˆ° =
 * (3:77e)

For the general expression interested readers can inspect Ref. (55). Ref. (26) proved that, contrary to another popular belief throughout the 20-th century, the Lorentz symmetry remains exact for all possible signature preserving (+,+,+,1) deformations of the Minkowski space, of course, when treated with the appropriate mathematics. The above exact reconstruction of the Lorentz symmetry can be geometrically visualized by noting that the light cone x23 ¡ c2 o £ t2 = 0 can only be formulated in vacuum while within physical media we have the generic hyperboloid r2 3 ¡ c2 o £ t2=n2(!; :::) = 0. However, it is an instructive exercise for interested readers to prove that the isolight cone (that is, the light cone on isospace over isofields) is the perfect cone ˆr2 3 ¡ c2 o £ ˆt = 0 with the exact symmetry ˆO(3:1) while its projection on conventional space is given by r2 3 ¡ c2 o £ t2=n2(!; :::) = 0 with broken Lorentz symmetry. (3) The isotranslations ˆ T (4) : ˆx0 = ˆ T (ˆa; :::)ˆ] £ x = ˆx + ˆ A(ˆa; x; :::); ˆa = a £ ˆIa 2 ˆRa which can be written x¹0 = x¹ + A¹(a; :::); (3:78a) A¹ = a¹(g¹¹ + a® £ [g;i¹ˆ;P®]=1! + :::); (3:78b) where there is no summation on the ¹ indices. Note that the isotranslations are highly nonlinear (thus non-inertial) in conventional spacetime although they are isolinear (thus inertial) in isospace. This illustrates the reason why conventional notion of relativity are solely applicable in spacetime, thus illustrating the reason of the name ”isorelativity.” (4) The novel isotopic invariance ˆI : ˆx0 = ˆ wˆ£ ˆx = w £ ˆx; ˆI0 = w £ ˆI, where w is a constant (29), ˆI ! ˆI0 = ˆ wˆ£ ˆI = w £ ˆI = 1= ˆ T0; (3:79a) ˆxˆ2 = (x¹ £ ˆ´¹º £ xº ) £ ˆI ´ ˆx0ˆ2 = [x¹ £ (w¡1 £ ˆ´¹º) £ xº ] £ (w £ ˆI); (3:79b) Therefore, the Poincar´e-Santilli isosymmetry can be written ˆ P(3:1) = ˆO (3:1) ˆ£ ˆ T (4) ˆ£ ˆI (3:80) thus having eleven (rather than ten) dimensions with parameters µk; vk; a¹;w; k = 1; 2; 3; ¹ = 1; 2; 3; 4; the 11-th dimension being characterized by invariant (3.78). Note that, contrary 39 to popular beliefs, the conventional Poincar´e symmetry is also eleven dimensional since invariance (3.78) also holds for conventional spacetime. The simplest possible realization of the above formalism for isorelativistic kinematics can be outlined as follows (see Section 3.13 for the isogravitational realization). The first application of isorelativity is that of providing an invariant description of locally varying speeds of light propagating within physical media. For this purpose a realization of isorelativity requires the knowledge of the density of the medium in which motion occurs. The simplest possible realization of the fourth component of the isometric is then given by the function g44 = n24 (x; !; :::) normalized to the value n4 = 1 for the vacuum (note that the density of the medium in which motion occur cannot be described by special relativity). Representation (3.68) then follows with invariance under ˆ P(3:1). In this case the quantities nk; k = 1; 2; 3; represent the inhomogeneity and anisotropy of the medium considered. For instance, if the medium is homogeneous and isotropic (such as water), all metric elements coincide, in which case ˆI = Diag:(g11; g22; g33; g44) = n24 £ Diag:(1; 1; 1; 1); (3:81a) ˆxˆ2 = x2 n24 £ n24 £ I ´ x2; (3:81b): thus confirming that isotopies are hidden in Minkowskian axioms, and this may be a reason why they have nog been discovered until recently. 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 with semiaxes n21 . Note that them minimum number of constituents of a closed non- Hamiltonian system is two. In this case we have shapes represented with n®k; ® = 1; 2; ; :::; n. Applications finally require the identification of the nonlocal interactions, e.g., whether occurring on an extended surface or volume. As an illustration, two spinning particles denoted 1 and 2 in condition of deep mutual penetration and overlapping of their wavepackets (as it is the case for valence bonds), can be described by the following Hamiltonian and total; isounit total isounit H = p1 £ p1 2 £ m1 + p2 £ p2 2 £ m2 + V (r); (3:82a) ˆITot = Diag:(n2 11; n2 12; n2 13; n2 14) £ Diag:(n2 21; n2 22; n2 23; n2 24)£ £eN£( ˆ Ã1=Ã1+ ˆ Ã2=Ã2)£R ˆ Ã1"(r)y£ ˆ Ã2#(r)£dr3 where N is a constant. Note the nonlinearity in the wavefunctions, the nonlocal-integral character and the lack of representation of all the above features via a Hamiltonian. From the above examples interested readers can then represent any other closed non- Hamiltonian systems. 40 The third important part of isorelativity is given by the following isotopies of conventional relativistic axioms which, for the case of motion along the third axis, can be written (29): ISOAXIOM I. The projection in our spacetime of the maximal causal invariant speed is given by: VMax = co £ g1=2 44 g1=2 33 = co n3 n4 = c n3
 * n22
 * n23
 * (3:82b)
 * (3:83)

This isoaxioms 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 g44 = g33 = 1=n2, where n is the index of refraction. In this case the maximal causal speed for a massive particle is co as experimentally established, e.g., for electrons, while the local speed of electromagnetic waves is c = co=n., as also experimentally established. Note that such a resolution requires 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 generally believed. At any rate, physical media are generally opaque to light but generally not to particles. Therefore, the assumption as the maximal causal speed as that of light which cannot propagate within the medium considered would be evidently vacuous. It is an instructive exercise for the interested readers to prove that the maximal causal speed of particles on isominkowski space over an isofield remains co. ISOAXIOM II. The projection in our spacetime of the isorelativistic addition of speeds within physical media is given by: vTot = v1 + v2 1 + v1£g33£v2 co£g44£co = v1 + v2 1 + v1£n24 £v2 co£n23 £co (3:84) We have again the correct occurrence that the sum of two maximal causal speeds in water, Vmax = co£(n3=n4), yields the maximal causal speed in water, as the reader is encouraged to verify. Note that the such a result is impossible for special relativity. Note also that the isorelativistic sum of two speeds of lights in water, c = co=n, 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 to and contraction of length `o and the variation of mass mo with speed are given by: t = ˆ° £ to; ` = ˆ°¡1 £ `o;m = ˆ° £ mo: (3:85) 41 Note that in water these values coincide with the relativistic one as it should be since particles such as the electrons have in water the maximal causal speed c0. Note again the necessity of avoiding the interpretation of the local speed of light as the maximal local causal speed. Note 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 90o angle of aberration): ! = ˆ° £ !o: (3:86) This isorelativistic axioms 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 exit 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). ISOAXIOM V. The projection in our spacetime of the isorelativistic law of equivalence of mass and energy is given by: E = m £ c2 o £ g44 = m £ c2 o n24
 * (3:87)

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 g44 À 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 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). The isodual isorelativity for the characterization of antimatter can be easily constructed via the isodual map of Section 2, and its explicit study is left to the interested reader for brevity. 3.12: Isorelativistic Hadronic Mechanics and its isodual. The isorelativistic extension of nonrelativistic hadronic mechanics is readily permitted by the Poincar´e-Santilli isosymmetry. In fact, iso-invariant (3.75a) implies the following iso-Gordon equation on ˆH over ˆ C ˆp¹ ˆ £j ˆ Ã >= ¡ˆi ˆ£ ˆ@¹j ˆ Ã >= ¡i £ ˆIº ¹ £ @ºj ˆ Ã >; (3:88a) (ˆp¹ ˆ£ˆp¹ + ˆm2 o ˆ£ ˆc4) ˆ £j ˆ Ã >= (ˆ´®¯ £ @® £ @¯ + m2 o £ c4) £ j ˆ Ã >= 0: (3:88b) 42 The linearization of the above second-order isoinvariant into the iso-Dirac equation has been studied in detail in Ref. (29) as well as by several other authors (although generally without the use of isomathematics, thus losing the invariance). By recalling the correct structure (2.34) of Dirac’s equation as the Kronecker product of a spin 1/2 massive particle and its antiparticle, the iso-Dirac’s equation is formulated on the total isoselfadjoint isospace and related isosymmetry ˆM Tot = [ ˆM orb(ˆx; ˆ´; ˆR) £ ˆ Sspin(2)] £ [ ˆM dorb(ˆxd; ˆ´d; ˆRd) £ ˆ Sdspin(2)] = ˆM dT ot; (3:89a) ˆ STot = ˆ P(3:1) £ ˆ Pd(3:1) = ˆ SdT ot; (3:89b) and can be written (29) [ˆ°¹ ˆ£(ˆp¹ ¡ ˆe ˆ£ ˆ A¹) +ˆi ˆ£ ˆm] ˆ £jÁ(x) >= 0; (3:90a) ˆ°¹ = g¹¹ £ °¹ £ ˆI (3:90b) where the °’s are the conventional Dirac matrices. Note the appearance of the isometric elements directly in the structure of the gamma matrices and their presence also when the equation is projected in the conventional spacetime. A realization via the iso-Dirac equation of the Poincar´e-Santilli isosymmetry with isocommutators (3.74) is given by (29) J¹º = (Sk;Lk4); P¹; (3:91a) Sk = (ˆ²kij ˆ£ˆ°i ˆ£ˆ°j)=2;Lk4 = ˆ°k ˆ£ ˆ°4=2; P¹ = ˆp¹ (3:91b) The notion of ”isoparticle” can be best illustrated with the above realization because it implies that,in the transition from motion in vacuum (as particles have been solely detected and studied until now) to motion within physical media, particles generally experience the alteration, called ”mutation,” of all intrinsic characteristics, as illustrated by the following isoeigenvalues which are implied by isocommutation rules (3.74), ˆ Sˆ2 ˆ £j ˆ Ã >= g11 £ g22 + g22 £ g33 + g33 £ g11 4 £ j ˆ Ã >; (3:92a) ˆ S3 ˆ £j ˆ Ã >= (g11 £ g22)1=2 2 £ j ˆ Ã > : (3:92b) The mutation of spin then implies a necessary mutation of the intrinsic magnetic moment which is given by (29) ˜¹ = ( g33 g44 )1=2 £ ¹; (3:93) where ¹ is the conventional magnetic moment for the same particle when in vacuum. The mutation of the rest energy and of the remaining characteristics has been identified before via the isoaxioms. The construction of the isodual isorelativistic hadronic mechanics is left to the interested reader by keeping in mind that the iso-Dirac equation is isoselfdual as the conventional equation. 43 To properly understand the above results, one should keep in mind that the mutation of the intrinsic characteristics of particles is solely referred to the constituents of a hadronic bound state under condition of mutual penetration of their wavepackets (such as one hadronic constituent) under the condition of recovering conventional characteristics for the hadronic bound state as a whole (the hadron considered), much along the original Newtonian subsidiary constrains on non-Hamiltonian forces, Eqs. (3.42b)-(3.42d). The reader should also keep in mind that, at this kinematical level prior to the introduction of gravity, g44 = 1=n24 represent the density of the medium in which motion occurs, normalized to the value g44 = 1; n4 = 1 for the vacuum. Also, the inhomogeneity of the medium is represented by the functional dependence of its density, e.g., from the radial distance r and other variables, g44 = g44(r; :::). The anisotropy of the medium is represented by g33, e.g., for the case of the spheroidal ellipsoid for which g11 = g22 6= g33. Finally, isotopic invariance (3.79) implies the capability of rescaling the radius of a sphere. Therefore, for the case of the perfect sphere we can always have g11 = g22 = g33 = g44 in which case the magnetic moment is not mutated. These results recover conventional classical knowledge according to which the alteration of the shape of a charged and spinning body implies the necessary alteration of its magnetic moment. It should be also stressed that the above mutations violate the unitary condition when formulated on a conventional Hilbert spaces, with consequential catastrophic inconsistencies. As an illustration, the violation of causality and probability law has been established for all eigenvalues of the angular momentum M different than the quantum spectrum M2 £ jÃ >= `(` + 1) £ jÃ >; ` = 0; 1; 2; 3; :::. As a matter of fact, these inconsistencies are the very reason why the mutations of internal characteristics of particles for bound states at short distances could not be admitted within the framework of quantum mechanics. By comparison, hadronic mechanics has been constructed precisely to recover unitarity on iso-Hilbert spaces over isofields, thus permitting an invariant description of internal mutations of the characteristics of the constituents of hadronic bound states, while recovering conventional features for states as a whose. As we shall indicate at the end of this section, far from being mathematical curiosities, the above mutations imply basically new structure models of hadrons, nuclei and stars, with consequential, new clean energies and fuels. These new advances were prohibited by quantum mechanics precisely because of the preservation of the intrinsic characteristics of the constituents in the transition from bound states at large mutual distance, for which no mutation is possible, to the bound state of the same constituents in condition of mutual penetration, in which case mutations have to be admitted in order to avoid the replacement of a scientific process with unsubstantiated personal beliefs one way or the other. The best illustration of the iso-Dirac equation is, therefore, that for which it was constructed (30), to describe the transition for the electron in the hydrogen atom, to the same electron when compressed in the hyperdense medium in the interior of the proton, namely, to achieve a quantitative and invariant representation of the synthesis of the neutron according to Rutherford as a ”hydrogen atom compressed in the core of a star.” 44 If special relativity, relativistic quantum mechanics and the conventional Dirac equation are assumed to be exactly valid also for the motion of the electron within the hyperdense medium in the interior of the proton, the neutron cannot be a bound state of a proton and an electron at short distances, thus mandating the assumption of undetectable constituents which cannot be produced free, as well known. One of the most important results of hadronic mechanics has been the proof at the nonrelativistic (214) and relativistic level (30) that a hadronic bound state of an isoprotons and an isoelectron represents all characteristics of the neutron, including its rest energy, spin, charge, parity, charge radius, anomalous magnetic moment and spontaneous decay. The societal implications of the above alternative are such to require the surpassing of traditional academic interests on pre-established doctrines. In fact, no new energy is conceivably possible under the assumption of the exact validity within a hadron of the Minkowski geometry, the special relativity and relativistic quantum mechanics, with consequential hadronic constituents which cannot be produced free. On the contrary, the assumption that isorelativity within the hyperdense medium inside hadrons implies that the hadronic constituents are indeed produced free in the spontaneous decay and, therefore, they can indeed be stimulated to decay, thus implying basically new energies (58). 3.13: Isogravitation, iso-grand-unification and isocosmology. 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 which 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 reason why special relativity in vacuum has a majestic axiomatic consistence is its invariance under the poincar´e symmetry. Recent studies have shown that the formulation of gravitation on a curved space or, equivalently, rathe formulation of gravitation based on as ”covariance,” is necessarily noncanonical at the classical level and nonunitary at the operator level, thus suffering of all catastrophic inconsistencies of Theorem 3.1 (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 to be 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 Rieman- 45 nian space and its formulation instead on an iso-Minkowskian space (15) via the following basic steps: I) Factorization of any given Riemannian metric g(x) into a nowhere singular and positive-definite 4 £ 4 matrix ˆ T(x) times the Minkowski metric ´, g(x) = ˆ Tgrav(x) £ ´; (3:94) II) Assumption of the inverse of ˆ Tgrav as the fundamental unit of the theory, ˆIgrav(x) = 1= ˆ Tgrav(x); (3:95) III) Submission of the totality of the Minkowski space and relative symmetries to the noncanonical/nonunitary transform U(x) £ Iy(x) = ˆIgrav: (3:96) The above procedure yields the isominkowskian spaces and related geometry ˆM (ˆx; ˆ´; ˆR), (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 Poincar´e-Santilli isosymmetry ˆ P(3:1) for the gravity of matter with isounit ˆIgrav(x) with the isodual isosymmetry ˆ Pd(3:1) for the gravity of antimatter, and the isodual symmetry ˆ P(3:1) £ ˆ Pd(3:1) for the gravity of matter-antimatter systems; 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 3.1; iv) An axiomatically consistent operator version of gravity always existed and merely creeped in un-noticed through the 20-th century because gravity is embedded where nobody looked for, in the unit of relativistic quantum mechanics, and it is given by isorelativistic hadronic mechanics as in Eqs. (3.88) and (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(x) referred to the unit I) and its replacement with isoflatness, namely, the verification of the axioms of flatness in isospace, while preserving conventional curvature in its projection on conventional spacetime (or, equivalently, curvature characterized by the g(x) = ˆ Tgrav(x) £ ´ referred to the isounit ˆIgrav(x) in which case curvature becomes null due to the interrelation ˆIgrav(x) = 1= ˆ Tgrav(x)) (15). 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 ˆ P(3:1) symmetry by mere visual 46 verification that the generators of the conventional and isotopic Poincar´e symmetry are the same, since they represent conserved quantities in the absence and in the preserve of gravity); 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 ˆIgrav(x) ! I); and others. 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 ˆ T = ˆ Tgrav which is as axiomatically consistent as the conventional relativistic quantum mechanics because the two formulations coincide, by construct, at the abstract, realization-free level. As an illustration, whenever ˆ Tgrav = Diag:(g11; g22; g33; g44), the iso-Dirac equation (3.90) provides a direct rtepresentation of the conventional electromnagnetic interactions experienced by an electron, represented by the fevot potential A¹, plus gravitational interactions represented by the isogamma matrices. 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 Poincar´e invariance of electroweak interactions; isogravity can be formulated on the same flat isospace of electroweak theories; and isogravity admits positive and negative energies in the same way as it occurs for electroweak theories. An axiomatically consistent iso-grand-unification then follows (32,35). Note that the above grand-unification requires the prior geometric unification of the special and general relativities, which 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. In particular, black holes are now characterized by the zeros of the isounit (55) ˆIgrav(x) = 0: (3:97) 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 ˆ Tgrav(x) of g(x) = ˆ Tgrav(x) £ ´, since ´ 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 gravitation, etc. These additional problematic aspects are also resolved by isospecial relativity due to the unrestricted character of the functional dependence of the isometric which 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 universal invariance ˆ P(3:1)£ ˆ Pd(3:1) in which there is no need for the ”missing mass” (as indicated earlier), time and the speed of light become 47 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 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). 3.14: Experimental verifications and scientific applications. Nowadays, isotopies in general, and the isotopic branch of hadronic mechanics in particular, have clear experimental verifications in classical physics, particle physics, nuclear physics, chemistry, superconductivity, biology and cosmology, among which we quote the following representative verifications: ? The first and only known optimization of the shape of extended objects moving within resistive media via the optimal control theory (14) following the first achievement of the universality of the isoaction (3.50) for all possible resistive forces; ? The axiomatically correct formulation of special relativity in terms of the proper time by T. Gill and his associates (202)-(206); ? The first identification of the connection between Lie-admissibility and supersymmetries by Adler (211); ? The proof by Aringazin (192,197) of the ”universality” of Isoaxiom III, namely, its capability of admitting as particualr cases all available anomalous time dilations via different expensions in terms of different quantities and with different truncatiuons; ? The exact representation of the anomalous behavior of the meanlives of unstable particles with speed by Cardone et al (110,11) that to Isoaxioms III of isorelativity; ? The exact representation of the experimental data on the Bose-Einstein correlation by Santilli (112) and Cardone and Mignani (113) under the exact iso-Poincar´e symmetry; ? 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 achievement of an exact confinement of quarks by Kalnay (216) and Kalnay and Santilli (2.17) thanks to incoherence between the external and internal Hilbert spaces; ? The proof by Jannussis and Mignani (186) of the convergence of isotopic perturbative series when conventionally divergent based on the property for all isotopic elements used in actual models ˆ T ¿ 1, thus implying that perturbative expansions which are divergent when formulated with the conventional associative product A £ B become convergent when re-expressed in terms of the isoassociative product Aˆ£B = A £ ˆ T £ B; ? The initiation by Mignani (182) of a nonpotential-nonunitary scattering theory reformulated by Santilli (55) as isounitary on iso-Hilbert spaces over isofields, thus recovering causality and propability laws for the first known description ofscattering among extended particles with consequential contact-nonpotential inetractions; ? The first and only known exact and invariant representation by Santilli (114,115) of nuclear magnetic moments and other nuclear characteristics thanks to interior mutation of type (3.93), which representation has escaped quantum mechanics for about one century; 48 ? The first and only known model by Animalu (170) and Animalu and Santilli (116) of the Cooper pair in superconductivity with an attractive force between the two identical electrons in excellent agreement with experimental data; ? The exact representation via isorelativity by Mignani (118) of the large difference in cosmological redshifts between quasars and galaxies when physically connected; ? The exact representation by Santilli (117) of the internal blue- and red-shift of quasar’s cosmological redshift; ? The elimination of the need for a missing mass in the universe by Santilli (34) thanks to isoaxiom V. Additional important applications of isotopies have been studied by by A. O. E. Animalu, A. K. Aringazin, R. Aslaner, C. Borgi, F. Cardone, J. Dunning-Davies, F. Eder, J. Ellis, J. Fronteau, M. Gasperini, T. L. Gill, J. V. Kadeisvili, A. Kalnay, N. Kamiya, S. Keles, C. N. Ktorides, M. G. Kucherenko, D. B. Lin, C.-X. Jiang, A. Jannussis, R. Mignani, M. R. Molaei, N. E. Mavromatos, H. C. Myung, M. O. Nishioka, D. V. Nanopoulos, S. Okubo, D. L. Rapoport, D. L. Schuch, D. S. Sourlas, A. Tellez-Arenas, Gr. Tsagas N. F. Tsagas, E. Trell, R. Trostel, S. Vacaru,H. E. Wilhelm, W. Zachary, and others. These studies are too numerous to be effectively reviewed in this memoir. Above all, hadronic mechanics achieved the main objective for which it was built: the first exact and invariant representation from unadulterated first axiomatic principles of all experimental data of the hydrogen, water and other molecules by R. M. Santilli and D. D. Shillady (125,126) (see also the comprehensive treatment in monograph (59)). The representation was achieved via the use of nonrelativistic hadronic mechanics based on the simple isounit (3.16) in which, as one can see, there are no free parameters for ad hoc fits of experimental data, but only a quantitative description of wave-overlappings, with isorelativistic extension characterized by isounit (3.82). It should be noted that, whether in valence coupling or not, electrons repel each other. Also, the total electric or magnetic forced between neutral atoms are identically null, while exchange, van der Waals and other forces of current use in chemistry are basically insufficient to represent the strength of molecular bonds (59). Studies (125,126) achieved the first and only known strongly attractive force between pairs of identical electrons in singlet coupling at short distance, and proved to originate from nonlocal, nonlinear and nonpotential interactions due to deep overlappings of electron’s wavepackets in singlet coupling (where the word ”strongly” it is not evidently referred to strong interactions). The birth of this new, nonpotential, strongly attractive force between particles in conditions of mutual penetration then implies new structure models of hadrons, nuclei and stars (see below). As it is well known, the exact representation of molecular data had escaped about one century of attempts via conventional chemistry, because the missing 2% originated precisely from the nonlocal-integral, nonlinear and nonpotential interactions due to deep overlapping of the wavepackets of the valence electrons (Figure 2) which are beyond any descriptive capacity of quantum mechanics. As it is also well known, improved representations of molecular data have required the ”screening of the Coulomb potential,” which screening cannot be qualified as char- 49 acterizing a ”quantum” theory since the quantum of energy only exists for the pure Coulomb potential. In any case, screened Coulomb potentials are nonunitary images of the Coulomb law, thus being particular cases of the nonunitary/isounitary structure of hadronic mechanics and chemistry (59). The achievement of a deeper understanding of molecular bonds has far reaching scientific implications. In fact, it confirms that nonlocal, nonlinear and non potential interactions exist in all interior problems at large, such as the structure of hadrons, nuclei and stars, and imply basically new structure models in which the constituents are isoparticles (irreducible representation of the Poincar´e-Santilli isosymmetry), rather than conventional particles in vacuum. The original proposal to build hadronic mechanics (38) of 1978 included the proof that all characteristics of the ¼o meson can be represented in an exact and invariant way via a bound state of one iso-electron ˆe¡ and its antiparticle ˆe+ under conditions of mutual penetration within 10¡13cm, ¼o = (ˆe¡; ˆe+)HM; (3:98) the ¼§ meson can be represented via a bound state of three isoelectrons, ¼§ = (ˆe¡; ˆe§; ˆe+)HM; (3:99) and the remaining mesons can be similarly identified as hadronic bound states of massive isoparticles produced free in the spontaneous decays with the lowest mode. Following the prior achievement of the isotopies of the SU(2) spin (28), Ref. (214) of 1990 achieved for the first time the exact and invariant representation of all characteristic of the neutron as a nonrelativistic hadronic bound state of one isoproton and one isoelectron according to Rutherford’s original conception of the neutron, n = (ˆp+; ˆe¡)HM; (3:100) while the relativistic extension was reached in Ref. (30), jointly with the first isotopies of Dirac’s equation. Subsequently, it was easy to see that all unstable baryons can be considered as hadronic bound states of massive isoparticles, again those generally in the spontaneous decays with the lowest modes. Compatibility of the above new structure models of hadrons with ordinary massive constituents and SU(3)-color theories was achieved via the assumption that quarks are composite, a view first expressed by Santilli (225) in 1981), and the use of hypermathematics (see Section 5) with different units for different hadrons (31). This approach essentially yields the hyperrealization ˆ SU(3) in which composite hyperquarks are characterized by the multivalued isounit with isotopic element ˆ T = ( ˆ Tu; ˆ Td; ˆ Ts), resulting in hypermultiplets of mesons, baryons, etc. The compatibility of this hypermodel with conventional theories is established by the isomorphism between conventional SU(3) and the hyper- ˆ SU(3), the latter merely being a broader realization of the axioms of the former. The significance of this hypermodel is illustrated by the fact that all perturbative series which are divergent for SU(3) are turned into convergent forms because ˆ Tu; ˆ Td; ˆ Ts ¿ 1 under which, as indicated earlier, all 50 divergent perturbative series expressed in terms of the conventional product A£B become convergent when re-expressed in terms of the hytperproduct A £ ˆ T £ B. Compatibility with the structure model of hadrons with ordinary massive constituents is evident from the fact that quarks result to be composed of ordinary massive isoparticles. It should be recalled that none of the above hadronic models are possible for quantum mechanics, e.g., because the representation of the rest energies of hadrons would require ”positive binding energies”(since, unlike similar occurrences in nuclear physics, the rest energy of bound states (3.98)-(3.100) is much bigger than the sum of the rest energies of the constituents. Such ”positive binding energies are prohibited by quantum mechanics because Schroedinger’s equation become inconsistent. These and other objections were resolved by the covering hadronic mechanics due to the isorenormalizations (also called mutations) of the rest energies and other features of the constituents caused by nonlocal, nonlinear and nonpotential interactions. Predictably, the reduction of the neutron to a bound state of an isoproton and an isoelectron permitted a new structure model of nuclei as hadronic bound states of isoprotons and isoelectron (113,114), with the conventional quantum models based on protons and neutron remaining valid in first approximation. The new isonuclear model permitted the first known understanding of the reason why the deuteron ground state has spin 1 since it is a three-body system for hadronic mechanics, D = (ˆp+; ˆe¡; ˆp+)HM; (3:101) thus admitting 1 as the lowest possible angular momentum, while the ground state of the deuteron quantum mechanics, D = (p+; no)QM should have spin zero since it is a two body system. The isonuclear model also permitted the interpretation of other features that had remained unexplained in nuclear physics for about one century such as why the correlation among nucleons is restricted to pairs only. In particular, the old process of keep adding potentials to the nuclear force without ever achieving an exact representation of nuclear data has been truncated by hadronic mechanics, due to a component of the nuclear force which is nonlocal, nonlinear and nonpotential originating from the mutual penetration of the charge distribution of nucleons established by nuclear data (such as the ratio between nuclear volumes and the sum of the volume of the nucleon constituents). The reduction of neutron stars and other astrophysical bodies to isoprotons and isoelectrons is then consequential. 3.15: Industrial applications to new clean energies and fuels. In closing, it should be indicated that the studies on isotopies have long passed the level of pure scientific relevance, because they now have direct industrial applications fort new clean energies and fuels so much needed by our contemporary society. As an illustration at the particle level, the synthesis of the neutron from one proton and one electron according to Rutherford, Eq. (3.100), has been experimentally confirmed by C. Borghi et al. (123) to occur also at low energies, although under a number of conditions studied in monograph (58), and additional tests are under way. 51 Once Rutherford’s original conception of the neutron is rendered acceptable by hadronic mechanics, the electron becomes a physical constituent of the neutron (although in a mutated state). In this case, hadronic mechanics predicts the capability of stimulating the decay of the neutron via photons with suitable resonating frequencies and other means, thus implying the first known form of ”hadronic energy” (58) (that is, energy originating in the structure of hadrons, rather than in their nuclear aggregates), which has already been preliminarily confirmed via an experiment conducted by N. Tsagas et al (124) (see monograph (58) for scientific aspects and the web site www.betavoltaic.com for industrial profiles). As an illustration at the nuclear level, hadronic mechanics predicts a basically new process for controlled nuclear syntheses which is dramatically different than both the ”hot” and the ”cold” fusions, and which is currently also under industrial development, which condition prohibits its disclosure in this memoir. As an illustration at the molecular level, the deeper understanding of the structure of molecules has permitted the discovery and experimental verifications in Ref.s (27) (see also the studies by Aringazin and his associates in Refs. (128-1130) and monograph (59)) of the new chemical species of magnecules consisting of clusters of individual atoms, dimers and molecules under a new bond originating from the electric and magnetic polarization of the orbitals of atomic electrons. In turn, the new species of magnecules has permitted the industrial synthesis of new fuels without hydrocarbon structure, whose combustion exhaust resolves the environmental problems of fossil fuels by surpassing current exhaust requirement by the U. S. Environmental Protection Agency without catalytic converter or other exhaust purification processes (see monograph (59) for scientific profiles and the web site www.magnegas.com for industrial aspects). 4. CONSTRUCTION OF GENOMECHANICS FROM IRREVERSIBLE PROCESSES 4.1: The scientific unbalance caused by irreversibility. As it is well known, physical, chemical or biological systems are called irreversible when their images under time reversal, t ! ¡t, are prohibited by causality and other laws, as it is the case for nuclear transmutations, chemical reactions and organisms growth. Systems are called reversible when their time reversal images are as causal as the original ones, as it is the case for planetary and atomic structures when considered isolated from the rest of the universe (see reprint volume (81) on irreversibility and vast literature quoted therein). Yet another large scientific unbalance of the 20-th century has been the treatment of irreversible systems via mathematical and physical formulations of reversible systems which are themselves reversible, resulting in serious limitations in virtually all branches of science. The problem was compounded by the fact that all used formulations were essentially of Hamiltonian type, with the awareness that all known Hamiltonians are reversible (since all known potential interactions are reversible). 52 This third scientific unbalance was dismissed by academicians with vested interests in reversible theories with unsubstantiated statements, such as ”irreversibility is a macroscopic occurrence which disappears when all bodies are reduced to their elementary constituents.” The underlying belief is that mathematical and physical theories which are so effective for the study of one electron in a reversible orbit around a proton are tacitly assumed to be equally effective for the study of the same electron when in irreversible motion in the core of a star with the local nonconservation of energy, angular momentum, etc. These academic beliefs have been disproved by the following: THEOREM 4.1 (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. The occurrence established by the above theorems dismiss all nonscientific conversations which have occurred on irreversibility in the 20-th century, and identify the real scientific needs, the construction of formulations which are structurally irreversible, that is, irreversible for all known reversible Hamiltonians, and are applicable at all levels of study, from Newtonian mechanics to second quantization. 4.2: The forgotten legacy of Newton, Lagrange and Hamilton. It should be indicated that the above scientific unbalance existed only in the 20-th century becauseNewton’s equations (1) are generally irreversible since, as recalled in the preceding section, Newton’s force F(t; x; v) can be decomposed into the sum of variationally selfadjoint and nonselfadjoint components (48,51) m® £ ak® = FSA k® + FNSA k® ; (4:1a) FSA = ¡@V=@x; FNSA 6= ¡@V=@x; k = 1; 2; 3; ® = 1; 2; :::; n: (4:1b) It is evident that, since all known FSA are reversible, in Newtonian mechanics irreversibility originates in the contact nonpotential forces FNSA. 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(x; v) and the Hamiltonian H(x; p), representing FSA, plus external terms representing precisely the contact nonpotential forces FNSA, d dt @L(x; v) @v ¡ @L(x; v) @x = FNSA(t; x; v); (4:2a) dx dt ¡ @H(x; p) @p = 0; dp dt + @H(x; p) @x = FNSA(t; x; p); (4:2b) with time evolution for an observable A(x; p) in phase space over R characterized by the brackets dA dt = (A;H; FNSA) = 53 Figure 3: An illustration via sea shells growth of the third scientific unbalance of the 20-th century, the lack of a structurally irreversible mathematics (that is, a mathematics whose basic axioms are not invariant under time reversal) for quantitative representations of irreversible processes. The unbalance is due to the fact that all formulations used until now are of Hamiltonian type, while all known Hamiltonians and their background mathematics are reversible, thus implying the study of irreversible systems via fully reversible formulations. = ( @A @xk ® £ @H @pk® ¡ @A @pk® £ @H @xk ® ) + @A @pk® £ FNSA k® : (4:3) Since all known Lagrangians and Hamiltonians are reversible in time, according to the teaching of Lagrange and Hamilton, irreversibility is characterized, again, by the external terms representing contact zero-range interactions among extended particles. At the beginning of the 20-th century, Lagrange’s and Hamilton’s external terms were truncated, resulting in analytic equations d dt @L(x; v) @v ¡ @L(x; v) @x = 0; (4:4a) dx dt ¡ @H(x; p) @p = 0; dp dt + @H(x; p) @x = 0; (4:4b) with time evolution characterized by the familiar Lie brackets dA dt = [A;H] = = @A @xk ® £ @H @pk® ¡ @A @pk® £ @H @xk ® which are fully reversible. 54 The above occurrence was due to the successes of the truncated analytic equations for the representation of planetary and atomic structures, resulting in their use for virtually all scientific inquiries of the 20-th century. In turn, the assumption of the truncated analytic equations as the ultimate formulation of science implied the scientific unbalance under consideration here because planetary and atomic structures are fully reversible, thus lacking sufficient generalities for all of nature. 4.3: Catastrophic inconsistencies of formulations with external terms. More recent studies (23,38) have shown that the true Lagrange’s and Hamilton’s equations (those with external terms) cannot be used in applications due to a number of insufficiencies, such as: (1) the lack of invariant numerical predictions in accordance with Theorem 3.1 (due to their evident noncanonical character); (2) the lack of characterization of any algebra by the brackets of the time evolution, let alone the loss of all Lie algebras, because brackets (A;H; FNSA) of Eqs. (4.3) violate the right distributive and scalar laws as necessary to characterize an algebra commonly understood in contemporary mathematics since they are triple systems); (3) the lack of a topology suitable to represent contact nonpotential interactions among extended particles since the topology of conventional Hamiltonian formulation is strictly local-differential, thus solely characterizing point particles; and other limitations. The only resolution of these problematic aspects known to this author was the construction of the novel structurally irreversible mathematics indicated earlier. Stated in different terms, the manifestly inconsistent reduction of irreversible macroscopic systems to elementary particles in reversible conditions was due, again, to insufficiencies of the used mathematics. It should be noted that the isomathematics of the preceding section is also reversible in time because the isounit is Hermitean, thus lacking the mathematical characterization of time reversal, and confirming the need of constructing of a broader mathematics specifically suited to represent irreversibility. 4.4: Initial versions of irreversible mathematics. The achievement of a structurally irreversible mathematics resulted to be a long scientific journey due to the need of achieving invariance under irreversible conditions. The first studies can be traced back to Ref. (8) of 1967 which presented the first known parametric deformation of Lie algebras with product (A;B) = p £ A £ B ¡ q £ B £ A = = v £ (A £ B ¡ B £ A) + w £ (A £ B + B £ A) = = v £ [A;B] + w £ fA;Bg; (4:6) where p, q, and p § q are non-null parameters, v = p + q;w = q ¡ p, and A, B are Hermitean matrices. 55 The studies continued with the first known presentation in Ref. (38) of the operator deformations of Lie algebra with product (Aˆ;B) = A £ P £ B ¡ B £ Q £ A = = (A £ T £ B ¡ B £ T £ A) + (A £W £ B + B £W £ A) = = [Aˆ;B] + fAˆ;Bg; T +W = P;W ¡ T = Q; (4:7) where P, Q and P § Q are nowhere singular matrices. On historical grounds, the above deformations were introduced in Refs. (8,38) as realizations of Albert’s Lie-admissible and Jordan-admissible products (7), namely, products whose antisymmetric and symmetric parts are Lie and Jordan, respectively, (Aˆ;B) ¡ (Bˆ;B) = 2 £ [Aˆ;B] = Lie; (4:8a) (Aˆ;B) + (Bˆ;A) = 2 £ (Aˆ;B) = Jordan: (4:8b) Note, however, that the Lie and Jordan algebras attached to brackets (Aˆ;B) are not conventional because of their broader isotopic nature [4]. The transition from the parameter to the operator deformations of Lie algebras was mandatory because all time evolution which can be characterized by the former brackets are nonunitary. Therefore, the reader can easily verify that the application of a nonunitary transform to the parametric deformations leads to the operator ones, i £ dA=dt = (A;B);A(t) = U £ A(0) £ Uy;U £ Uy 6= I; (4:9a) U £ (A;B) £ Uy = (A0ˆ;B0)0; (4:9b) A0 = U £ A £ Uy;B0 = U £ B £ Uy; (4:9c) T = v £ (U £ Uy)¡1;W = w £ (U £ Uy)¡1: (4:9d) Operator deformations (4.7) (rather than the parametric deformations (4.6)) are promising for the representation of irreversibility because they are no longer totally anti-symmetric (as it is the case for Lie brackets) and, therefore, they can indeed represent nonconservation as needed in irreversible processes. Moreover, operator deformations (4.7) are universal in the sense of admitting as particular cases all infinitely possible algebras as currently known in mathematics (those characterized by a bilinear product), including Lie, Jordan, Kac-Moody, supersymmetric and all other possible algebras. Finally, the joint Lie- and Jordan-admissibility is preserved by any additional nonunitary transforms, Z £ (Aˆ;B) £ Zy = A0 £ P0 £ B0 ¡ B0 £ Q0 £ A0 = (A0ˆ; ˆB)0; (4:10a) Z £ Zy 6= I;A0 = Z £ A £ Zy;B0 = Z £ B £ Zy (4:10b) P0 = Zy¡1 £ P £ Z¡1;Q0 = Zy¡1 £ Q £ Z¡1; (4:10b) thus confirming that brackets (Aˆ;B) characterize the most general possible algebras. 56 Nevertheless, all parametric and operator deformations are afflicted by the catastrophic mathematical and physical inconsistencies of Theorem 3.1 because of the lack of invariance of the deformation parameters P ! P0 6= P;Q ! Q0 6= Q (or, equivalently, of the product). During the last two decades of the 20-th century, the physical and mathematical literature saw an explosion of contributions in Lie deformations which continues to this day, although generally without a quotation of their origination in Refs. (8,23,38), without a quotation of their Lie- and Jordan-admissible content (7), and, above all, without a quotation of the rather vast literature on their catastrophic inconsistencies (see, Refs. (171-175) and memoir (46) and literature quoted therein). By contrast, by the mid 1980’s this author had abandoned the study of Lie deformations according to their original formulations (8,23,38) because of said catastrophic inconsistencies. 4.5: Elements of genomathematics. A breakthrough occurred with the discovery, apparently done for the first time by R. M. Santilli in Ref. (12) of 1993, that the axioms of a field also hold when the ordinary product of numbers a £ b is ordered to the right, a > b, or, separately, ordered to the left, a < b. In turn, such an order permitted the construction of two generalized units, called genounits to the right and to the left I = Diag:(1; 1; :::; 1) ! ˆI>(t; x; v; Ã; @xÃ; :::) = 1= ˆ T>(t; x; v; Ã; @xÃ; :::) > 0; (4:11a) I = Diag:(1; 1; :::; 1) !<> ˆI(t; x; v; Ã; @xÃ; :::) = 1=< ˆ T(t; x; v; Ã; @xÃ; :::); (4:11b) ˆI> = (< ˆI)y; (4:11c) with corresponding ordered genoproducts to the right and to the left A £ B ! A > B = A £ ˆ T> £ B; (4:12a) A £ B !! A < B = A £< T £ B; (4:12b) A > B = (B < A)y; (4:12c) under which the left and rights character of the genounits is preserved, I £ A = A £ I = A ! ˆI> > A = A > ˆI> = A; (4:13a) I £ A = A £ I = A !< ˆI < A = A << ˆI = A; (4:13b): for all (Hermitean elements A, B, of the considered set. Examples of genounits and genoproducts will be provided shortly. In this way, the ordering ”>” can describe motion forward in time while the ordering ”<” can describe motion backward in time, with interconnecting Hermitean (or transposed) conjugation. This approach permitted the embedding of irreversibility in the most fundamental quantities, the basic units and operations, thus assuring ab initio the construction of a structurally irreversible mathematics, today known as genomathematics, as summarized below. 57 DEFINITION 4.1: Let F = F(a;+;£) be a field as per Definition 2.1. The forward genofields (first introduced in Ref. (12) of 1993) are rings ˆ F> = ˆ F>(ˆa>; ˆ+>;>) with forward genonumbers ˆa> = a £ ˆI>; (4:14) associative, distributive and commutative forward genosum ˆa> ˆ+>ˆb> = (a + b) £ ˆI> = ˆc> 2 ˆ F>; (4:15) associative and distributive forward genoproduct ˆa> > ˆb> = ˆa > £ˆ T> £ˆb> = ˆc> 2 ˆ F; (4:16) additive forward genounit ˆ0 > = 0; ˆa> ˆ+>ˆ0> = ˆ0> ˆ+>ˆa> = ˆa> 2 ˆ F>; (4:17) and multiplicative forward genounit ˆI> = 1= ˆ T>; ˆa> > ˆI> = ˆI> > ˆa> = ˆa> 2 ˆ F>; 8ˆa>; ˆb> 2 ˆ F>; (4:18) where ˆI> is a complex-valued non-Hermitean, or real-value non-symmetric, everywhere invertible quantity generally outside F. The backward genofields < ˆ F(<ˆa;< ˆ+;<), their elements, units and their operations are given by the Hermitean conjugate (or transposed) of the corresponding quantities and their operations in ˆ F>(ˆa>; ˆ+>; ˆ£>), e.g., < ˆI = (ˆI>)y; etc: (4:19) LEMMA 4.1: Forward and backward genofields are fields with characteristic zero (namely, they verify all axioms of said fields). In Sect. 2 we pointed out that the conventional product ”2 multiplied by 3” is not necessarily equal to 6 because, depending on the assumed unit and related product, it can be ¡6. In Section 3 we pointed out that the same product ”2 multiplied by 3” is not necessarily equal to +6 or ¡6, because it can also be equal to an arbitrary number, or a matrix or an an integrodifferential operator. In this section we point out that ”2 multiplied by 3” can be ordered to the right or to the left, and yield different numerical results for different orderings, ”2 > 3 6= 2 < 3, all this by continuing to verify the axioms of a field per each order (12). Once the forward and backward fields have been identified, the various branches of genomathematics can be constructed via simple compatibility arguments, resulting in the genofunctional analysis, genodifferential calculus, etc (14,54,55). We have in this way the genodifferentials and genoderivatives ˆ d>x = ˆ T> x £ dx; ˆ@> ˆ@>x = ˆI> x £ @ @x 58 Particularly intriguing are the genogeometries (loc. cit.) because they admit nonsymmetric metrics, such as the genoriemannian metrics g>(x) = ˆ T>(x) £ ´; (4:21) where ´ is the Minkowski metric and ˆ T>(x) is a real-values, nowhere singular, 4 £ 4 nonsymmetric matrix, while bypassing known inconsistencies since they are referred to the nonsymmetric genounit ˆI> = 1= ˆ T>: (4:22) In this way, genogeometries are structurally irreversible and actually represent irreversibility with their most central geometric notion, the metric. 4.6: Lie-Santilli genotheory and its isodual. Particularly important for this note is the lifting of Lie’s theory permitted by genomathematics, first identified by R. M. Santilli in Ref. (23) of 1978, and today knows as the Lie-Santilli genotheory [7,8], which is characterized by: (1) The forward and backward universal enveloping genoassociative algebra ˆ»>;< ˆ», with infinite-dimensional basis characterizing thePoincar´e-Birkhoff-Witt-Santilli genotheorem ˆ»> : ˆI; ˆXi; ˆXi > ˆXj ; ˆXi > ˆXj > ˆXk; :::; i · j · k; (4:23a) < ˆ» : ˆI; ˆXi; ˆXi < ˆXj ; ˆXi < ˆXj < ˆXk; :::; i · j · k; (4:23b) where the ”hat” on the generators denotes their formulation on genospaces over genofields and their Hermiticity implies that ˆX > =< ˆX = ˆX ; (2) The Lie-Santilli genoalgebras characterized by the universal, jointly Lie-and Jordanadmissible brackets (4.7), < ˆL> : ( ˆXiˆ; ˆXj) = ˆXi < ˆXj ¡ ˆXj > ˆXi = ˆ Ck ij ˆ£ ˆX k; (4:24) although now formulated in an invariant form (see below); (3) The Lie-Santilli genotransformation groups < ˆG> : ˆ A( ˆ w) = (ˆeˆi ˆ£ ˆX ˆ£ ˆ w > ) > ˆ A(ˆ0) < (<ˆe¡ˆi ˆ£ ˆ wˆ£ ˆX ) = = (ei£ ˆX £ˆ T>£w) £ A(0) £ (e¡i£w£< ˆ T£ ˆX ); (4:25) where ˆ w> 2 ˆR> are the genoparameters; the genorepresentation theory, etc. The mathematical implications of the Lie-Santilli genotheory are significant because of the admission as particular cases of all possible algebras, as well as because, when computed on the genobimodule < ˆ»£ˆ»> Lie-admissible algebras verify all Lie axioms, while deviations from Lie algebras emerge only in their projection on the bimodule <» £ »> of the conventional Lie theory. This is due to the fact that the computation of the left action A < B = A £< ˆ T £ B on < ˆ» (that is, with respect to the genounit < ˆI = 1=< ˆ T) yields the save value as the computation of the conventional product A£B on <» (that is, with 59 respect to the trivial unit I), and the same occurs for the value of A > B on ˆ»>. In this way, thanks to genomathematics, Lie algebras acquire a towering significance in view of the possibility of reducing all known algebras to primitive Lie axioms. The physical implications of the Lie-Santilli genotheory ar equally significant. In fact, Noether’s theorem on the reduction of conservation laws to primitive Lie symmetries can be generalized to the reduction of, this time, nonconservation laws to primitive Lie- Santilli genosymmetries. As a matter of fact, this reduction was the very first motivation that suggested the construction of the genotheory in memoir (23) (see also monographs (49,50)). The reader can then foresee similar liftings of all remaining physical aspects treated via Lie algebras. The construction of the isodual Lie-Santilli;i genotheory is an instructive exercise for readers interested in learning the new methods. The physical theories characterized by genomathematics can be summarized as follows. 4.7: Geno-Newtonian Mechanics and its isodual. Recall that, for the case of isotopies, the basic Newtonian systems are given by those admitting nonconservative internal forces restricted by certain constraints which verify total conservation laws (closed non- Hamiltonian systems). For the case of the genotopies under consideration here, the basic Newtonian systems are the conventional nonconservative systems (4.1) without subsidiary constraints (open non-Hamiltonian systems). In this case irreversibility is characterized by nonselfadjoint forces, as indicated earlier. The forward geno-Newtonian mechanics and its isodual is a generalization of Newtonian mechanics for the description of motion forward in time of the latter systems via a structurally irreversible mathematics. The new mechanics is characterized by (14): the forward genotime ˆt> = t £ ˆI> t with (nowhere singular and non-Hermitean) forward time genounit ˆI> t = 1= ˆ T> t 6= ˆI>y t, related forward time genospace ˆ S> t over the forward time genofield ˆR> t ; the forward genocoordinates ˆx> = x £ ˆI> x with (nowhere singular non-Hermitean) forward coordinate genounit ˆI> x = 1= ˆ T> x 6= ˆI>y x with forward coordinate genospace ˆ S> x and related forward coordinate genofield ˆR> x ; and the forward genospeeds ˆv> = ˆ d>ˆx>= ˆ d>ˆt> with (nowhere singular and non-Hermitean) forward speed genounit ˆI> v = 1= ˆ T> v 6= ˆI>y v with related forward speed genospace ˆ S> x and forward speed genofield ˆR > v. Note that, to verify the condition of non-Hermiticity, the time genounits should be at least complex valued, and the same then occurs for the other genounits. The representation space is then given by the Kronecker product ˆ S> Tot = ˆ S> t £ ˆ S> x £ ˆ S> v ; (4:26) defined over the genofield ˆR > tot = ˆR> t £ ˆR> x £ ˆR> v ; (4:27) with total genounit ˆI> tot = ˆI> t £ hatI> x £ ˆI> v : (4:28) The basic equations are given by the forward geno-Newton equations, also known as Newton-Santilli genoequations, first proposed in memoir (14) via the genodifferential 60 calculus, also known as forward Newton-Santilli genoequations [8-11] ˆm> >® ˆ d>ˆv> k® ˆ d>ˆt> = ¡ ˆ@> ˆ V > ˆ@>ˆx>k ®
 * (4:5)
 * etc: (4:20)
 * (4:29)

The backward geno-Newton equations is characterized by backward genounits can be obtained via transpose conjugation of the forward formulation. As one can see, the representation of Newton’s equations is done in a way similar to the isotopic case, the main difference being that the basic unit is now no longer symmetric. Note that in Newton’s equations the nonpotential forces are part of the applied force F, while in the geno-Newton equations nonpotential forces are represented by the forward genounits, or, equivalently, by the forward genodifferential calculus, in a way essentially similar to the case of isotopies. The main difference is that isounits are Hermitean, thus implying the equivalence of forward and backward motions, while genounits are non- Hermitean, thus implying irreversibility. Note also that the topology underlying Newton’s equations is the conventional, Euclidean, local-differential topology which, as such, can only represent point particles. By contrast, the topology underlying the geno-Newton equations is the Santilli-Sourlas- Tsagas genotopology (14,139) for the representation of extended, nonspherical and deformable particles via forward genounits, e.g., of the diagonal type ˆI> = Diag:(n21 4) £ ¡>(t; x; v; :::); (4:30) where n2 k; k = 1; 2; 3 represents the semiaxes of an ellipsoid, n24 represents the density of the medium in which motion occurs (with more general nondiagonal realizations here omitted for simplicity), and ¡> represents contact interactions occurring for the motion forward in time. The construction of the isodual image of the above geno-Newtonian mechanics is instructive to understand the difference between isoduality and motion backward in time. 4.8: Geno-Hamiltonian mechanics and its isodual. The most effective setting to introduce real-valued and non-Hermitean (thus non-symmetric) genounits is in the 6ndimensional forward genocotangent bundle (geno-phase-space) with local genocoordinates and their conjugate ˆa>¹ = a½ £ ˆI>¹ 1½ ; (ˆa>¹) = Ã ˆx>k ® ˆp> k® !; ˆR> ¹ = R½ £ ˆI>½ 2¹ ; ( ˆR> ¹ ) = (ˆpk®; ˆ0); (4:31a); ˆI> 1 = 1= ˆ T> 1 = (ˆI> 2 )T = (1= ˆ T> 2 )T ; k = 1; 2; 3; ® = 1; 2; :::; n; ¹; ½ = 1; 2; :::6n; (4:31a) where the superscript T stands for transposed, with nowhere singular, real-valued and non-symmetric genometric and related invariant ˆ±> = ±6n£6n £ ˆ T> 1 6n£6n; (4:32a) ˆa>¹ > ˆR> ¹ = ˆa>½ £ ˆ T>¯ 1½ £ ˆR > ¯ = a½ £ ˆI>¯ 2½ £ R¯: (4:32b) 61 In this case we have the following genoactionprinciple (14) ˆ±> ˆ A> = ˆ±> ˆ Z > [ ˆR> ¹ >a ˆ d>ˆa> ¡ ˆH > >t ˆ d>ˆt>] = = ±Z [R¹ £ ˆ T>¹ 1º (t; x; p; :::) £ d(a¯ £ ˆI>º 1¯ ) ¡ H £ dt] = 0; (4:33) where the second expression is the projection on conventional spaces over conventional fields and we have assumed for simplicity that the time genounit is 1. It is easy to prove that the above genoprinciple characterizes the following forward geno-Hamilton equations, also called forward Hamilton-Santilli genoequations (originally proposed in Ref. (23) of 1978 with conventional mathematics and in ref. (14) of 1996 with genomathematics; see also Refs. (28,51,52,55)) ˆ!¹º ˆ£ ˆ dˆaº ˆ dˆt ¡ ˆ@ ˆH(ˆa) ˆ@ˆa¹ = = Ã 0 ¡1 1 0 !£ Ã dx=dt dp=dt !¡ Ã 1 K 0 1 !£ Ã @H=@x @H=@p ! = 0; (4:34a) ˆ! = ( ˆ@Rº ˆ@ˆa¹ ¡ ˆ@ ˆR¹ ˆ@ˆaº ) £ ˆI = Ã 0 ¡1 1 0 !£ ˆI; (4:34b) K = FNSA=(@H=@p): (4:34c) The time evolution of a quantity ˆ A>(ˆa>) on the forward geno-phase-space can be written in terms of the following brackets d ˆ A> dt = ( ˆ A>; ˆH >) = ˆ@> ˆ A> ˆ@>ˆa>¹ ˆ£ ˆ!¹º ˆ£ ˆ@> ˆH > ˆ@ˆa>º = = @ ˆ A> @ˆa>¹ £ S ¹º £ artoa; ˆH > @ˆa>º = = ( @ ˆ A> @ˆx>k ® £ @ ˆH> @ ˆp> k® ¡ @ ˆ A> @ ˆp> k® £ @ ˆH > @ˆx>k ® ) + @ ˆ A> @ ˆp> k® £ Kk k £ @ ˆH > @ ˆp> k®
 * n22
 * n23
 * n2
 * (4:35a)

S>¹º = !¹½ £ ˆI2º ½ ; !¹º = (jj!®¯jj¡1)¹º; (4:35b) where !¹º is the conventional Lie tensor and, consequently, S¹º is Lie-admissible in the sense of Albert (7). As one can see, the important consequence of genomathematics and its genodifferential calculus is that of turning the triple system (A;H; FNSA) of Eqs. (4.3) in the bilinear form (Aˆ;B) of brackets (4.35a),m thus regaining the existence of a consistent algebra in the brackets of the time evolution, for which central purpose genomathematics was built (since the multiplicative factors represented by K are fixed for each given system). The invariance of such a formulation will be proved shortly. 62 It is easy to verify that the above identical reformulation of Hamilton’s historical time evolution (4.3) correctly recovers the time rate of variations of physical quantities in general, and that of the energy in particular, dA dt = [ ˆ A>; ˆH >] + @ ˆ A> @ ˆp> k® £ FNSA k® : (4:36a) dH dt = [ ˆH >; ˆH >] + @ ˆH > @ ˆp> k® £ FNSA k® = vk ® £ FNSA k® : (4:36b) It is easy to show that genoaction principle (4.33) characterizes the following Hamilton- Jacobi-Santilli genoequations ˆ@>A> ˆ@>ˆt> + ˆH > = 0; (4:37a) ( ˆ@>A> ˆ@>ˆa>¹ ) = ( ˆ@>A> ˆ@>x>k ® ˆ@>A> ˆ@>p> k ® ) = ( ˆR> ¹ ) = (ˆp> k®; ˆ0); (4:37b) which confirm the property (crucial for genoquantization as shown below) that the genoaction is indeed independent of the linear momentum. Note the direct universality of Eqs. (4.33) for the representation of all infinitely possible Newton equations (4.1) (universality) directly in the fixed frame of the experimenter (direct universality). Note also that, at the abstract, realization-free level, Geno-Hamilton equations (4.34) coincide with Hamilton’s equations without external terms, yet represent those with external terms. The latter are reformulated via genomathematics as the only known way to achieve invariance while admitting a consistent algebra in the brackets of the time evolution (38). Therefore, genohamilton equations (4.34) are indeed irreversible for all possible reversible Hamiltonians, as desired. The origin of irreversibility rests in the contact nonpotential forces according to Lagrange’s and Hamilton’s teaching. Note finally that the extension of Eqs. (4.9) to include nontrivial genotimes implies a major broadening of the theory we cannot review for brevity (14,55). The above geno-Hamiltonian mechanics requires, for completness, three additional formulations, the backward geno-Hamiltonian mechanics for the description of matter moving backward in time, and the isoduals of both the forward and backward mechanics for the description of antimatter. The construction of these additional mechanics is lefty to the interested reader. 4.9: Genotopic Branch of Hadronic Mechanics and its isodual. A simple genotopy of the naive or symplectic quantization applied to Eqs. (4.37) yields the genotopic branch of hadronic mechanics defined on the forward genotopic Hilbert space ˆH> with forward genoinner product < ˆ Ãj > j ˆ Ã > £ˆI> 2 ˆ C>. The resulting genotopy of quantum mechanics is characterized by the forward geno-Schroedinger equations (first formulated in Refs. (42,179) via conventional mathematics and in Ref. (14) via genomathematics) ˆi > > ˆ@> ˆ@>ˆt> j ˆ Ã> >= ˆH > > j ˆ Ã> >= 63 = ˆH (ˆx; ˆv) £ ˆ T>(ˆt>; ˆx>; ˆp>; ˆ Ã>; ˆ@> ˆ Ã>::::) £ j ˆ Ã> >= E> > jÃ> >; (4:38a) ˆp> k ˆ £j ˆ Ã> >= ¡ˆi> > ˆ@> k j ˆ Ã> >= ¡i £ ˆI>i k £ @ij ˆ Ã> >; ˆI> > j ˆ Ã> >= j ˆ Ã> >; (4:38b) with conjugate backward equations obtained via Hermitean conjugation. Note the crucial independence of isoaction ˆ A> in principle (4.33) from the linear momentum, as expressed by the Hamilton-Jacobi-Santilli genoequations (4.37). In fact, such independence assures that genoquantization yields a genowavefunction solely dependent on time and coordinates, ˆ Ã> = ˆ Ã>(t; x). Other geno-Hamiltonian mechanics do not verify such a condition, thus implying genowavefunctions with an explicit dependence also on linear momenta, ˆ Ã> = ˆ Ã>(t; x; p) which violate the abstract identity of quantum and hadronic mechanics andwhose treatment in any case is beyond our operator knowledge at this writing. The complementary geno-Heisenberg equations are given by in their finite and infinitesimal forms (first formulated in Ref. (38) via conventional mathematics and in Ref. (14) via genomathematics) ˆ A(ˆt) = (ˆeˆi ˆ£ ˆH ˆ£ ˆt > ) > ˆ A(ˆ0) < (<ˆe¡ˆi ˆ£ ˆt ˆ£ ˆH ) = = (ei£ ˆH£ˆ T>£t) £ A(0) £ (e¡i£t£< ˆ T£ ˆH ); (4:39a) ˆi ˆ£ ˆ d ˆ A ˆ dˆt = ( ˆ Aˆ; ˆH ) = ˆ A < ˆH ¡ ˆH > ˆ A = = ˆ A £< ˆ T(ˆt; ˆx; ˆp; ˆ Ã; ::::) £ ˆH ¡ ˆH £ ˆ T>(ˆt; ˆx; ˆp; ˆ Ã; ::::) £ ˆ A; (4:39b) (ˆxiˆ;ˆpj) = i £ ±i j £ ˆI>; (ˆxiˆ;ˆxj) = (ˆpiˆ;ˆpj) = 0; (4:39c) where time has no arrow, since Heisenberg’s equations are computed at a fixed time. The genoexpectation values of an observable for the forward motion ˆ A> are then given by < ˆ Ãj > ˆ A> > j ˆ Ã > < ˆ Ãj > j ˆ Ã > £ ˆI> 2 ˆ C> (4:40) In particular, the genoexpectation values of the genounit recover the conventional Planck’s unit, < ˆ Ãj > ˆI> > j ˆ Ã > < ˆ Ãj > j ˆ Ã > = I; (4:41) thus confirming that the genotopies are ”hidden” in the abstract axioms of quantum mechanics much along the celebrated Einstein-Podolsky-Rosen argument. Note that geno-Hermiticity coincides with conventional Hermiticity. As a result, all quantities which are observable for quantum mechanics remain observable for the above formulation. However, unlike quantum mechanics, physical quantities are generally nonconserved, as it must be the case for the energy, ˆi > > ˆ d> ˆH > ˆ d>ˆt> = ˆH £ (< ˆ T ¡ ˆ T>) £ ˆH : (4:42): 64 Therefore, the genotopic branch of hadronic mechanics is the only known formulation permitting nonconserved quantities to be Hermitean, thus being observability. In fact, since they are Hamiltonian, other formulation attempt to represent nonconservation, e.g., by adding an ”imaginary potential” to the Hamiltonian. In this case the Hamiltonian is non- Hermitean and, consequently, the nonconservation of the energy cannot be an observable. besides, said ”nonconservative models” with non-Hermitean Hamiltonians are nonunitary and are formulated on conventional spaces over conventional fields, thus suffering all the catastrophic inconsistencies of Theorem 3.1. For additional aspects of genomechanics interested readers may consult Refs. [2,5]. 4.10: Invariance of genotheories. Recall that a fundamental Axiomatic feature of quantum mechanics is the invariance under its time evolution of numerical predictsions and physical law, which invariance is due to the unitary structure of the theory. However, quantum mechanics is reversible and can only represent conventional conservation laws. As indicated earlier, the representation of irreversibility and nonconservation require the use of nonunitary time evolutions which, however, are afflicted by the catastrophic inconsistencies of Theorem 3.1. The only resolution of such a basic impasse known to this author has been the achievement of invariance under nonunitarity and irreversibility via the use of genomathematics, provided that such genomathematics is applied to the totality of the formalism to avoid evident inconsistencies caused by mixing different mathematics for the selected physical problem. Such an invariance was first achieved by R. M. Santilli in Ref. (44) of 1997 can be illustrated by reformulating any given nonunitary transform in the genounitary form U = ˆU £ ˆ T>1=2;W = ˆW £ ˆ T>1=2; (4:43a) U £Wy = ˆU > ˆW y = ˆW y > ˆU = ˆI> = 1= ˆ T>; (4:43b) and then showing that genounits, genoproducts, genoexponentiations, etc., are indeed invariant under the above genounitary transform in exactly the same way as conventional unit, products, exponentiations, etc. are invariant under unitary transforms, ˆI> ! ˆ I>0 = ˆU > ˆI> > ˆW y = ˆI>; (4:44a) ˆ A > ˆB ! ˆU > (A > B) > ˆW y = = ( ˆU £ ˆ T> £ A £ T> £ ˆW y) £ ( ˆ T> £Wy)¡1 £ ˆ T>£ £( ˆU £ ˆ T>)¡1 £ ( ˆU £ T> £ ˆ A £ T> £ ˆW >) = = ˆ A0 £ ( ˆU £ ˆ T> £ ˆW y)¡1 £ ˆB = ˆ A0 £ ˆ T> £ B0 = ˆ A0 > ˆB 0; etc: (4:44b) from which all remaining invariances follow, thus resolving the catastrophic inconsistencies of Theorem 3.1. Note the numerical invariance of the genounit ˆI> ! ˆ I>0 = ˆI>, of the genotopic element ˆ T> ! ˆ T>0 = ˆ T>, and of the genoproduct >!>0=> which are necessary to have invariant numerical predictions. 65 4.11: Simple construction of genotheories. As it was the case for the isotopies, a simple method has been identified in Ref. (44) for the construction of genotheories from any given conventional, classical or quantum formulation. It consists in identifying such genounits as the product of two different nonunitary transforms U £ Uy 6= 1;W £Wy 6= 1;U £Wy = ˆI>;W £ Uy =< ˆI = (ˆI>)y; (4:45) and subjecting the totality of quantities and their operations of conventional models to said dual transforms, I ! ˆI> = U £ I £Wy; I !< ˆI = W £ I £ Uy; (4:46a) a ! ˆa> = U £ a £Wy = a £ ˆI>; a !< ˆa = W £ a £ Uy =< ˆI £ a; (4:46b) a£b ! ˆa> > ˆb> = U£(a£b)£W> = (U£a£Wy)£(U£Wy)¡1£(U£b£Wy); (4:46c) @=@x ! ˆ@>= ˆ@>ˆx> = U £ (@=@x) £Wy = ˆI> £ (@=@x); (4:46d) < Ãj £ jÃ >!<< Ãj > jÃ> >= U £ (< Ãj £ jÃ >) £Wy; (4:46e) H £ jÃ >! ˆH > > jÃ> >= (U £ H £Wy) £ (U £Wy)¡1 £ (U £ Ã >); etc: (4:46f) As a result, any given conventional, classical or quantum model can be easily lifted into the above genotopic form. Note that the above construction implies that all conventional physical quantities acquire a well defined direction of time. For instance, the correct genotopic formulation of energy, linear momentum, etc., is given by ˆH > = U £ H £Wy; ˆp> = U £ p £W>; etc:; (4:47) and this explains the reason for having represented in this section energy, momentum and other quantities with their arrow of time >. Such an arrow can indeed be omitted for notational simplicity, but only after the understanding of its existence. 4.12: Genorelativity. Another important implication of genomathematics is the construction of yet another lifting of special relativity, this time intended for the invariant characterization of irreversible classical and quantum processes, today known as genorelativity. This new relativity is characterized by the genotopies of: the background Euclidean topology (14); the Minkowski space (15); the Poincar´e symmetry (29); the physical laws; etc. The geno-Galilean case is treated in monographs (52,53) which appeared prior to the advent of the genodifferential calculus(14). The relativistic case is oulined in Rdf. (29). Regrettably, we cannot review genorelativity in details to avoid a prohibitive length. For the limited scope of this presentation it is sufficient to indicate that genospecial relativity can be constructed either from the isospecial relativity indicated in the preceding section via the lifting of the isounits into nonsymmetric forms, or by subjecting conventional special relativity to the dual noncanonical or nonunitary transform,ms as in Eqs. (4.45) and (4.46). 66 4.13: Experimental verifications and applications. The experimental validity of genotheories in classical mechanics is established by the direct representation of all nonconservative and irreversible Newtonian systems by Geno-Hamilton equations via a simple algebraic calculation of the term K in Eqs. (4.34c). The experimental validity of genotheories in particle physics is established by the fact that all dissipative nuclear models represented via imaginary potentials in the Hamiltonian and other nonunitary theories can be identically reformulated in terms of the genotopic branch of hadronic mechanics, while preserving the representation of experimental data identically. Above all, genomathematics and its related formulations have indeed achieve the objective for which they were built, namely, an invariant representation of irreversibility at all levels, from Newton to quantum mechanics. Such an objective can be achieved via the following main rules: (i) Identify the classical origin of irreversibility in the contact nonpotential forces among extended particles, much along the historical teaching of Newton (1), Lagrange (2) and Hamilton (3); (ii) Represent said nonpotential forces via real-valued, nowhere singular, non-symmetric genounits and construct a mathematics which is structurally irreversible for all reversible Hamiltonians in the sense indicated earlier; (iii) Achieve identical reformulation (4,34) of Hamilton’s equations with external terms with a consistent algebra in the brackets of the time evolution of Lie-admissible type according to Albert (7); (iv) Complement the latter mechanics with the underlying genosymplectic geometry, permitting the mapping of the classical formulations into operator formulations preserving said Lie-admissible character; and (v) Identify the origin of irreversibility in the most elementary layers of nature, such as elementary particles in their irreversible motion in the interior of stars. Note that a requirement for the above rules is the nonconservation of the energy and other physical quantities, which is readily verified by the geno-Hamilton equations (4.34) due to the lack of anti-symmetric character of brackets (A;B) of the time evolution. An important application of genomechanics has been done by J. Ellis et al. (122) who have shown that its Lie-admissibility provides an axiomatically consistent, direct representation of irrebersibility in interior quasares structures. In closing, it is hoped that Lagrange’s and Hamilton’s legacy of representing irreversibility with the external terms in their analytic equations is seriously considered because it implies covering theories with momentous advances in mathematics and all quantitative sciences. 67 5. CONSTRUCTION OF HYPERMECHANICS FROM BIOLOGICAL SYSTEMS 5.1: The scientific unbalance caused by biological systems. In this author’s opinion, by far the biggest scientific unbalance of the 20-th century has been the treatment of biological systems (herein denoting DNA, cells, organisms, etc.) via the mathematics developed for inanimate matter. The unbalance is due to the fact that conventional mathematics and related formulations, such as quantum mechanics, are inapplicable for the treatment of biological systems for various reasons. To begin, biological events, such as the growth of an organism, are irreversible. Therefore, any treatment of biological systems via conventional reversible mathematics and related physical formulations cannot pass the test of time. Quantum mechanics is ideally suited for the treatment of the structure of the Hydrogen atom or of crystals. These systems are represented by quantum mechanics as being ageless. Recall also that quantum mechanics is unable to treat deformations because of incompatibilities with basic formulations, such as the group of rotations. Therefore, the rigorous application of quantum mechanics to biological structures implies that all organisms from cells to humans are perfectly rigid and eternal. 5.2: The multivalued complexity of biological systems. Even after achieving the invariant formulation of irreversibility outlined in the preceding section, it was easy to see that the underlying genomathematics remains insufficient for in depth studies of biological systems. Recent studies conducted by C. R. Illert (56) have pointed out that the shape of sea shells can certainly be represented via conventional mathematics, such as the Euclidean geometry. However, the latter is inapplicable for a representation of the growth in time of sea shells. Computer simulations have shown that the imposition to sea shell growth of conventional geometric axioms (e.g., those of the Euclidean or Riemannian geometries) implies the lack of proper growth, as expected because said geometries are strictly reversible, while the growth of sea shells is strictly irreversible. The same studies by C. R. Illert (loc. cit.) have indicated the need of a mathematics which is not only structurally irreversible, but also multi-dimensional. As an example, C. R. Illert achieved a satisfactory representation of sea shells via the doubling of the Euclidean reference axes, namely, a geometry which appears to be six-dimensional. A basic problem in accepting such a view is the lack of compatibility with our sensory perception. In fact, when holding sea shells in our hands, we can fully perceive their shape as well as their growth with our three Eustachian tubes. In particular, our senses are sufficiently sensitive to perceive deviations from the Euclidean space, as well as for the possible presence of curvature. These occurrences pose a rather challenging problem, the achievement of a representation of the complexity of biological systems via the most general possible mathematics which is: (1) is structurally irreversible (as in the preceding section); (2) admits the deformation theory; (3) is invariant under the time evolution; (4) is multi-dimensional; and, last but not least, (5) is compatible with our sensory perception. 68 Figure 4: An illustration of the fourth scientific unbalance of the 20-th century, the representation of biological structures (including cells, organisms and their functions) via a mathematics which is not only Hamiltonian and reversible, but also single valued, while said biological systems require a mathematics which is not only nonhamiltonian and irreversible, but also multi-valued, as illustrated by the growth of a sea shell which, besides not being representable with a Hamiltonian and being irreversible, requires the doubling of all Euclidean reference axes. A search in the mathematical literature revealed that a mathematics verifying all the above five requirements did not exist and had to be constructed from the main features of biological systems. As an example, hyperstructures in their current formulations (see Ref. (96) lack a well defined left and right generalized unit even under their weak equalities, they are not structurally irreversible, and lack invariance, thus not being suitable for applications in biology. 5.3: Elements of hypermathematics. After numerous trials and errors, a yet broader mathematics verifying the above five conditions was identified by R. M. Santilli in Ref. (14) (see also Refs. (13,47) monograph (57)); it is today known under the name of hypermathematics; and it is characterized by the following hyperunits here expressed for the lifting of the Euclidean unit I = Diag:(1; 1; 1) ! ˆI>(t; x; v; Ã; @xÃ; :::) = Diag:(ˆI> 1 ; ˆI> 2 ; ˆI> 3 ) = = Diag:[(ˆI> 11; ˆI> 12; :::; ˆI> 1m); (ˆI> 21; ˆI> 22; :::; ˆI> 2m); (ˆI> 31; ˆI> 32; :::; ˆI> 3m)] = 1= ˆ T>; I = Diag:(1; 1; :::; 1) !< ˆI(t; x; v; Ã; @xÃ; :::) = Diag:(< ˆI1;< ˆI2;< ˆI3) = = Diag:[(< ˆI11;< ˆI12; :::;< ˆI1m); (< ˆI21;< ˆI22; :::;< ˆI2m);< (ˆI31;< ˆI32; :::;< ˆI3m)] = 1=< ˆ T; ˆI> = (< ˆI)y; (5:1) 69 with corresponding ordered hyperproducts to the right and to the left as in Eqs. (4.3) and identifies as in Eqs. (4.4) expressed in terms of ordinary equalities and operations. A further broadening is then permitted by weak operations. As one can see, the above mathematics is not 3m-dimensional, but rather it is 3- dimensional and m-multi-valued. Such a feature permits the increase of the reference axes, e.g., for m = 2 we have the six axes used by C. R. Illert (loc cit.), while achieving compatibility with our sensory perception because at the abstract, realization-free level the hypermathematics characterized by hyperunit (5.1) is indeed 3-dimensional. The various branches of hypermathematics (hypernumbers, hyperspaces, hyperalgebras, etc.) can be constructed via mere compatibility arguments with hyperunit (5.1) (see monograph (57) for brevity. The isodual hypermathematics can be constructed via the use of map (2.4). A main difference of hypermathematics with the preceding formulations is that in the latter the product of two numbers is indeed generalized but single-valued, e.g., 2 > 3 = 346:765. By comparison, in hypermathematics the product of two numbers yields, by conception, a set of values, e.g., 2 > 3 = (12:678; 341:329; 891:556; :::). Such a feature appears to be necessary for the representation of biological systems because the association of two atoms in a DNA (mathematically representable with the hypermultiplication) can yeld a large variety of different organs. This feature serves to indicate that the biological world has a complexity simply beyond our imagination at this time, and that the study of biological problems, such as understanding the DNA code via numbers dating back to biblical times, is manifestly insufficient. An important application of hyperformulations has already been indicated in Section 3.14, the achievement of compatibility between structure models of hadrons with ordinayr massive particles as constituents in mutated conditions, and the SU(3)-color models of classification, which compatibility is permitted by the assumption that quarks are composite, thus admitting an hyperstructure with mutivalued hyperunits. 5.4: The complexity of hypertime and hyperrelativity for biological systems. The reader should be aware that the complexity of biological structures requires the use of hypermathematics as well as its isodual, e.g., for quantitative interpretations of bifurcations. In fact, a quantitative interpretation of bifurcations, e.g., in sea shells, requires four different hypertimes and their isoduals, as indicated below. In turn, this is sufficient to illustrate the departure from conventional notions of a relativity suitable for quantitative studies on biological systems, known as hyperrelativity and its isodual (57). In fact, such new relativity requires the most general notion of numbers, those with a multi-valued hyperunits characterized by an ordered, yet unlimited number of non-Hermitean elements, with consequential most general possible geometries and mechanics, plus their isoduals. This results in an ordered, yet unlimited variety of spaces and their isoduals all coexisting in our three-dimensional Euclidean space, plus corresponding, equally co-existing varieties of time. There is little doubt that such features imply dramatic departures from the simplicity, thus insufficiency, of special relativity. An illustration of the complexity of hyperformulations and corresponding hyperrelativ- 70 ity is given by the four different notions of hypertime which are needed for the description of complex biological processes, such as bifurcations in seashells, all in a coexisting form and each having a multi-valued character: motion forward in future time ˆt>; motion backward in past time <ˆt; motion forward from past time <ˆtd; and motion backward from future time ˆt>d. The necessity of these four directions in time also illustrates the need of the isodual map. 5.5: Eric Trell’s hyperbiological structures. A new conception of biological systems, which constitute a truly fundamental advance over rather simple prior conceptions, has been recently proposed by Erik Trell (see Ref. (164) and contributions quoted therein). It is based on representative blocks which appear in our space to be next to each other, thus forming a cell or an organism, while having in reality hypercorrelations, thus having the structure of hypernumbers, hypermathematics and hyperrelativity, with consequential descriptive capacities immensely beyond those of pre-existing, generally single-valued and reversible biological models. Regrettably, we cannot review Trell’s new hyperbiological model to avoid an excessive length, and refer interested readers to the original literature (164). SUMMARY Mathematics Isodual mathematics Isomathematics Isodual isomathematics Genomathematics Isodual genomathematics Hypermathematics Isodual hypermathematics ??? Isodual ??? Figure 5: A schematic view of the main advances outlined in this memoir, from which all applications uniquely follow, a progressive generalization of ”mathematics” (here referred to the conventional formulation with left and right multiplicative unit defined over a field of characteristic zero) for the description of matter in conditions of increasing complexity and methodological needs, and their anti-isomorphic isoduals for corresponding progressive descriptions of antimatter. The last line illustrates the lack of final character of mathematics and, therefore, of scientific theories. 5.6: The lack of final character of all scientific theories. Main works of art, such as Michelangelo’s Piet´a, remain unsurpassed with the passing of time. By comparison, scientific theories, including mathematics, have a limited value in time, because, no matter how advanced and valid a given formulation may appear to be, the discovery of its structural generalizations is only a matter of time. Such a fate also holds for all generalized formulations outline in this memoir. In fact, the further broadening of hyperrelativity via 71 the so-called weak operations is already within mathematical reach (13), the only missing elements being future discoveries sufficient to motivate their use. Acknowledgments The author would like to thank several colleagues for invaluable and inspiring, constructive and critical remarks, including Professors E. Trell, S. Johansen, P. A. Bjorkum, L. Horwitz, T. L. Gill, A. K. Aringazin, M. C. Duffy, K. P. Shum, F. Winterberg, and several others. 72 References [1] HISTORICAL REFERENCES: (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 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 15, 395 (1998). (5) A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 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. 64, 552 (1948). [2] BASIC MATHEMATICAL PAPERS: (8) R. M. Santilli, Nuovo Cimento 51, 570 (1967). (9) R. M. Santilli, Suppl. Nuovo Cimento 6, 1225 (l968). (10) R. M. Santilli, Hadronic J. 3, 440 (l979). (11) R. M. Santilli, Hadronic J. 8, 25 and 36 (1985). (12) R. M. Santilli Algebras, Groups and Geometries 10, 273 (1993). (13) R. M. Santilli and T. Vougiouklis, contributed paper in New Frontiers in Hyperstructures, T., Vougiouklis, Editor, Hadronic Press, p. 1 (1996). (14) R. M. Santilli, Rendiconti Circolo Matematico di Palermo, Supplemento 42, 7 (1996). (15) R. M. Santilli, Intern. J. Modern Phys. D 7, 351 (1998). [3] ISODUAL FORMULATIONS: (16) R. M. Santilli, Comm. Theor. Phys. 3, 153 (1993). (17) R. M. Santilli, Hadronic J. 17, 257 (1994). (18) R. M. Santilli, Hadronic J. 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 New Frontiers of Hadronic Mechanics, T.L.Gill, ed., Hadronic Press (1996). (21) R. M. Santilli, Hyperfine Interactions, 109, 63 (1997). 73 (22) R. M. Santilli, Intern. J. Modern Phys. A 14, 2205 (1999). [4] ISOTOPIC FORMULATIONS: (23) R. M. Santilli: Hadronic J. 1, 224 (1978). (24) R. M. Santilli, Phys. Rev. D 20, 555 (1979). (25) C. Myung and R. M. Santilli, Hadronic J. 5, 1277 (1982). (26) R. M. Santilli, Novo Cimento Lett. 37, 545 (1983). (27) R. M. Santilli, Hadronic J. 8, 25 and 36 (1985). (28) R. M. Santilli, JINR Rapid. Comm. 6, 24 (1993). (29) R. M. Santilli, J.Moscow Phys.Soc. 3, 255 (1993). (30) R. M. Santilli, Chinese J.Syst.Ing. & Electr.6, 177 (1996). (31) R. M. Santilli, Found. Phys. 27, 635 (1997). (32) R. M. Santilli, Found. Phys. Letters 10, 307 (1997). (33) R. M. Santilli, Acta Appl. Math. 50, 177 (1998). . (34) R. M. Santilli, contributed paper to the 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 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 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. 1,574 and 1267 (1978). (39) R. M. Santilli, Hadronic J. 2, 1460 (l979) and 3, 914 (l980). (40) R. M. Santilli, Hadronic J. 4, 1166 (l981). (41) R. M. Santilli, Hadronic J. 5, 264 (l982). (42) H. C. Myung and R. M. Santilli, Hadronic J. 5, 1367 (l982). (43) R. M. Santilli, Hadronic J. Suppl. 1, 662 (l985). (44) R. M. Santilli, Found. Phys. 27, 1159 (1997). (45) R. M. Santilli, Modern Phys. Letters 13, 327 (1998). (46) R. M. Santilli, Intern. J. Modern Phys. A 14, 3157 (1999). [6] HYPERSTRUCTURAL FORMULATIONS: (47) R. M. Santilli, Algebras, Groups and Geometries 15, 473 (1998). 74 [7] MONOGRAPHS: (48) R. M. Santilli, Foundations of Theoretical Mechanics, Vol. I, Springer–Verlag, Heidelberg–New York (1978). (49) R. M. Santilli, Lie-admissible Approach to the Hadronic Structure, Vol.I, Hadronic Press, Palm Harbor, Florida (1978). (50) R. M. Santilli, Lie-admissible Approach to the Hadronic Structure, Vol. II, Hadronic Press, Palm Harbor, Florida (1981). (51) R. M. Santilli, Foundations of Theoretical Mechanics, Vol. II, Springer–Verlag, Heidelberg–New York (1983). (52) R. M. Santilli, Isotopic Generalizations of Galilei and Einstein Relativities, Vol. I, Hadronic Press, Palm Harbor, Florida (1991). (53) R. M. Santilli, Isotopic Generalizations of Galilei and Einstein Relativities, Vol. II, Hadronic Press, Palm Harbor, Florida (1991). (54) R. M. Santilli, Elements of Hadronic Mechanics, Vol I, Ukraine Academy of Sciences, Kiev, Second Edition (1995). (55) R. M. Santilli, Elements of Hadronic Mechanics, Vol II, Ukraine Academy of Sciences, Kiev, Second Edition (1995). (56) C. R. Illert and R. M. Santilli, Foundations of Theoretical Conchology, Hadronic Press, Palm Harbor, Florida (1995). (57) R. M. Santilli Isotopic, Genotopic and Hyperstructural Methods in Theoretical Biology, Ukraine Academy of Sciences, Kiev (1996). (58) R. M. Santilli, 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, Foundations of Hadronic Chemistry with Applications to New Clean Energies and Fuels, Kluwer Academic Publishers, Boston-Dordrecht-London (2001). (60) R. M. Santilli, Isodual Theory of Antimatter, with Applications to Cosmology, Antigravity and Spacetime Machines, In preparation. (61) R. M. Santilli, Elements of Hadronic Mechanics, Vol. III, in preparation. (62) H. C. Myung, Lie Algebras and Flexible Lie-Admissible Algebras, Hadronic Press (1982). (63) A. K. Aringazin, A. Jannussis, D.F. Lopez, M. Nishioka, and B. Veljanoski, Santilli’s Lie-isotopic Generalization of Galilei’s and Einstein’s Relativities, Kostarakis Publishers, Athens (1991). (64) D. S. Sourlas and G. T. Tsagas, Mathematical Foundations of the Lie-Santilli Theory, Ukraine Academy of Sciences, Kiev (1993). 75 (65) J. Lˆohmus, E. Paal and L. Sorgsepp, Nonassociative Algebras in Physics, Hadronic Press, Palm Harbor, FL, USA (1994). (66) J. V. Kadeisvili, 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, 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, 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, Applications of Lie-Admissible Algebras in Physics, Volume I, Hadronic Press (1978). (70) H. C. Myung and S. Okubo, Editors, Applications of Lie-Admissible Algebras in Physics, Vol. II, Hadronic Press (1978). (71) H. C. Myung and R. M. Santilli, Editor, 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, 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, Proceedings of the Third Workshop on Lie-Admissible Formulations,Part A, Hadronic J. Vol. 4, issue no. 2, pp. 183-607 (l9881) (73). (74) H. C. Myung and R. M. Santilli, Editor, 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, 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, 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, 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, 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). 76 (79) J. Fronteau, A. Tellez-Arenas and R. M. Santilli, Editor, 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, Proceedings of the First Workshop on Hadronic Mechanics, Hadronic J. Vol. 6, issue no. 6, pp. 1400-1989 (l983). (81) A. Shoeber, Editor, Irreversibility and Nonpotentiality in Statistical Mechanics, Hadronic Press (1984). (82) H. C. Myung, Editor, Mathematical Studies in Lie-Admissible Algebras, Volume I, Hadronic Press (1984). (83) H. C. Myung, Editor, Mathematical Studies in Lie-Admissible Algebras, Volume II, Hadronic Press (1984). (84) H. C. Myung and R. M. Santilli, Editor, Applications of Lie-Admissible Algebras in Physics, Vol. III, Hadronic Press (1984). (85) J. Fronteau, R. Mignani and H. C. Myung, Editors, 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, 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, Tomber’s Bibliography and Index in Nonassociative Algebras, Hadronic Press, Palm Harbor, FL (1984). (88) H. C. Myung, Editor, 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, Fourth Workshop on Hadronic Mechanics and Nonpotential Interactions, Nova Science, New York (l990). (90) H. M. Srivastava and Th. M. Rassias, Editors, Analysis Geometry and Groups: A Riemann Legacy Volume, Hadronic Press (1993). (91) F. Selleri, Editor, Fundamental Questions in Quantum Physics and Relativity, Hadronic Press (1993). (92) J. V. Kadeisvili, Editor, The Mathematical Legacy of Hanno Rund, Hadronic Press (1994). (93) M. Barone and F. Selleri Editors, Frontiers of Fundamental Physics, Plenum, New York, (1994). (94) M. Barone and F. Selleri, Editors, Advances in Fundamental Physics, Hadronic Press (1995). 77 (95) Gr. Tsagas, Editor, New Frontiers in Algebras, Groups and Geometries , Hadronic Press (1996). (96) T. Vougiouklis, Editor, New Frontiers in Hyperstructures, Hadronic Press, (1996). (97) T. L. Gill, Editor, New Frontiers in Hadronic Mechanics, Hadronic Press (1996). (98) T. L. Gill, Editor, New Frontiers in Relativities, Hadronic Press (1996). (99) T. L. Gill, Editor, New Frontiers in Physics,, Volume I, Hadronic Press (1996). (100) T. L. Gill, Editor, New Frontiers in Physics,, Volume II, Hadronic Press (1996). (101) C. A. Dreismann, Editor, New Frontiers in Theoretical Biology, Hadronic Press (1996). (102) G. A., Sardanashvily, Editor,New Frontiers in Gravitation, Hadronic Press (1996). (103) M. Holzscheiter, Editor, Proceedings of the International Workshop on Antimatter Gravity, Sepino, Molise, Italy, May 1996, Hyperfine Interactions, Vol. 109 (1997). (104) T. Gill, K. Liu and E. Trell, Editors, 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, 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, Fundamental Open Problems in Science at the end of the Millennium, Volume III, Hadronic Press (1999). (107) V. V. Dvoeglazov, Editor Photon: Old Problems in Light of New Ideas, Nova Science (2000). (108) M. C. Duffy and M. Wegener, Editors, Recent Advances in Relativity Theory Vol. I, Hadronic Press (2000). (109) M. C. Duffy and M. Wegener, Editors, 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. 18, L61 (1992). (111) F. Cardone, R. Mignani and R. M. Santilli, J. Phys. G: Nucl. Part. Phys. 18, L141 (1992). (112) R. M. Santilli, Hadronic J. 15, Part I: 1-50 and Part II: 77-134 (l992). (113) F. Cardone and R. Mignani, JETP 88, 435 (1995). (114) R. M. Santilli, Intern. J. of Phys. 4, 1 (1998). (115) R. M. Santilli Communications in Math. and Theor. Phys. 2, 1 (1999). 78 (116) A. O. E. Animalu and R. M. Santilli, Intern. J. Quantum Chem. 26,175 (1995). (117) R. M. Santilli, contributed paper to Frontiers of Fundamental Physics, M. Barone and F. Selleri, Editors Plenum, New York, pp 41-58 (1994). (118) R. Mignani, Physics Essays 5, 531 (1992). (119) R. M. Santilli, Comm. Theor. Phys. 4, 123 (1995). (120) Yu. Arestov, V. Solovianov and R. M. Santilli, Found. Phys. Letters 11, 483 (1998). (121) R. M. Santilli, contributed paper in the 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. Nanopoulos in 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. 56, 147 (1993) (123). (124) N. F. Tsagas, A. Mystakidis, G. Bakos, and L. Seftelis, Hadronic J. 19, 87 (1996). (125) R. M. Santilli and D., D. Shillady, Intern. J. Hydrogen Energy 24, 943 (1999). (126) R. M. Santilli and D., D. Shillady, Intern. J. Hydrogen Energy 25, 173 (2000). (127) R. M. Santilli, Hadronic J. 21, pages 789-894 (1998). (128) M. G. Kucherenko and A. K. Aringazin, Hadronic J. 21, 895 (1998). (129) M. G. Kucherenko and A. K. Aringazin, Hadronic Journal 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. 5, 1564 (1982). (132) J. V. Kadeisvili, Algebras, Groups and Geometries 9, 283 and 319 (1992). (133) J. V. Kadeisvili, N. Kamiya, and R. M. Santilli, Hadronic J. 16, 168 (1993). (134) J. V. Kadeisvili, Algebras, Groups and Geometries 9, 283 (1992). (135) J. V. Kadeisvili, Algebras, Groups and Geometries 9, 319 (1992). (136) J. V. Kadeisvili, contributed paper in the 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. 19 1349 (1996). (138) J. V. Kadeisvili, Algebras, Groups and Geometries, 15, 497 (1998). 79 (139) G. T. Tsagas and D. S. Sourlas, Algebras, Groups and Geometries 12, 1 (1995). (140) G. T. Tsagas and D. S. Sourlas, Algebras, Groups and geometries 12, 67 (1995). (141) G. T. Tsagas, Algebras, Groups and geometries 13, 129 (1996). (142) G. T. Tsagas, Algebras, Groups and geometries 13, 149 (1996). (143) E. Trell, Isotopic Proof and Reproof of Fermat’s Last Theorem Verifying Beal’s Conjecture. Algebras Groups and Geometries 15, 299-318 (1998). (144) A. K. Aringazin and D. A. Kirukhin, Algebras, Groups and Geometries 12, 255 (1995). (145) A. K. Aringazin, A. Jannussis, D.F. Lopez, M. Nishioka, and B. Veljanoski, Algebras, Groups and Geometries 7, 211 (1990). (146) A. K. Aringazin, A. Jannussis, D. F. Lopez, M. Nishioka, and B. Veljanoski, Algebras, Groups and Geometries 8, 77 (1991). (147) D. L. Rapoport, Algebras, Groups and Geometries, 8, 1 (1991). (148) D. L. Rapoport, contributed paper in theProceedings of the Fifth International Workshop on Hadronic Mechanics, H.C. Myung, Edfitor, Nova Science Publisher (1990). (149) D. L. Rapoport, Algebras, Groups and Geometries 8, 1 (1991). (150) C.-X. Jiang, Algebras, Groups and Geometries 15, 509 (1998). (151) D. B. Lin, Hadronic J. 11, 81 (1988). (152) R. Aslaner and S. Keles, Algebras, Groups and Geometries 14, 211 (1997). (153) R. Aslaner and S. Keles,. Algebras, Groups and Geometries 15, 545 (1998). (154) M. R. Molaei, Algebras, Groups and Geometries 115, 563 (1998) (154). (155) S. Vacaru, Algebras, Groups and Geometries 14, 225 (1997) (155). (156) N. Kamiya and R. M. Santilli, Algebras, Groups and Geometries 13, 283 (1996). (157) S. Vacaru, Algebras, Groups and Geometries 14, 211 (1997). (158) Y. Ylamed, Hadronic J. 5, 1734 (1982). (159) R. Trostel, Hadronic J. 5, 1893 (1982). [11] PHYSICS PAPERS: (160) J. P. Mills, jr, Hadronic J. 19, 1 (1996). (161) J. Dunning-Davies, Foundations of Physics Letters, 12, 593 (1999). (162) E. Trell, Hadronic Journal Supplement 12, 217 (1998). (163) E. Trell, Algebras Groups and Geometries 15, 447-471 (1998). (164) E. Trell, ”Tessellation of Diophantine Equation Block Universe,” contributed paper to Physical Interpretations of Relativity Theory, 6-9 September 2002, Imperial College, London. British Society for the Philosophy of Science, in print, (2002). 80 (165) J. Fronteau, R. M. Santilli and A. Tellez-Arenas, Hadronic J. 3, 130 (l979). (166) A. O. E. Animalu, Hadronic J.7, 19664 (1982). (167) A. O. E. Animalu, Hadronic J. 9, 61 (1986). (168) A. O. E. Animalu, Hadronic J. 10, 321 (1988). (169) A. O. E. Animalu, Hadronic J. 16, 411 (1993). (170) A. O. E. Animalu, Hadronic J. 17, 349 (1994). (171) S. Okubo, Hadronic J. 5, 1667 (1982). (172) D. F. Lopez, in Symmetry Methods in Physics, A.N.Sissakian, G.S.Pogosyan and X.I.Vinitsky, Editors ( JINR, Dubna, Russia (1994) . (173) D. F. Lopez, Hadronic J. 16, 429 (1993). (174) A. Jannussis and D. Skaltsas,Ann. Fond. L.de Broglie 18,137 (1993). (175) A. Jannussis, R. Mignani and R. M. Santilli, Ann.Fonnd. L.de Broglie 18, 371 (1993). (176) A. O. Animalu and R. M. Santilli, contributed paper in Hadronic Mechanics and Nonpotential Interactions M. Mijatovic, Editor, Nova Science, New York, pp. 19-26 (l990). (177) M. Gasperini, Hadronic J. 6, 935 (1983). (178) M. Gasperini, Hadronic J. 6, 1462 (1983). (179) R. Mignani, Hadronic J. 5, 1120 (1982). (180) R. Mignani, Lett. Nuovo Cimento 39, 413 (1984). (181) A. Jannussis, Hadronic J. Suppl. 1, 576 (1985). (182) A. Jannussis and R. Mignani, Physica A 152, 469 (1988). (183) A. Jannussis and I. Tsohantis, Hadronic J. 11, 1 (1988). (184) A. Jannussis, M. Miatovic and B. Veljanosky, Physics Essays 4, (1991). (185) A. Jannussis, D. Brodimas and R. Mignani, J. Phys. A: Math. Gen. 24, L775 (1991). (186) A. Jannussis and R. Mignani, Physica A 187, 575 (1992). (187) A. Jannussis, R. Mignani and D. Skaltsas, Physics A 187, 575 (1992). (188) A. Jannussis et al, Nuovo Cimento B103, 17 and 537 (1989). (189) A. Jannussis et al., Nuovo Cimento B104, 33 and 53 (1989). (190) A. Jannussis et al. Nuovo Cimento B108 57 (l993). (191) A. Jannussis et al., Phys. Lett. A 132, 324 (1988). (192) A. K. Aringazin, Hadronic J. 12, 71 (1989). (193) A. K. Aringazin, Hadronic J. 13, 183 (1990). 81 (194) A. K. Aringazin, Hadronic J. 13, 263 (1990). (195) A. K. Aringazin, Hadronic J. 14, 531 (1991). (196) A. K. Aringazin, Hadronic J. 16, 195 (1993). (197) A. K. Aringazin and K. M. Aringazin,Invited paper, in the Proceedings of the Intern. Conference ’Frontiers of Fundamental Physics’, Plenum Press, (1993). (198) A. K. Aringazin, K. M. Aringazin, S. Baskoutas, G. Brodimas, A. Jannussis, and K. Vlachos, contribuuted paper in the Proceedings of the Intern. Conference ’Frontiers of Fundamental Physics’, Plenum Press, (1993). (199) A. K. Aringazin, D. A. Kirukhin, and R. M. Santilli, Hadronic J. 18, 245 (1995). (200) A. K. Aringazin, D. A.Kirukhin, and R. M. Santilli, Hadronic J. 18, 257 (1995). (201) T. L. Gill, Hadronic J. 9, 77 (1986). (202) T. L. Gill and J. Lindesay, Int. J. Theor, Phys 32, 2087 (1993). (203) T. L. Gill, W. W. Zachary, M. F. Mahmood and J. Lindesay, Hadronic J. 16, 28 (1994). (204) T. L. Gill, W. W. Zachary and J. Lindesay, Foundations of Physics Letters 10, 547 (1997). (205) T.L. Gill, W.W. Zachary and J. Lindesay, Int. J. Theo Phys 37, 22637 (1998). (206) T. L. Gill, W. W. Zachary and J. Lindesay, Foundations of Physics, 31, 1299 (2001). (207) D. L. Schuch, K.-M. Chung and H. Hartmann, J. Math. Phys. 24, 1652 (1983). (208) D. L. Schuch and K.-M. Chung, Intern. J. Quantum Chem. 19, 1561 (1986). (209) D. Schich 23, Phys. Rev. A 23, 59, (1989). (210) D. Schuch, contributed paper in New Frontiers in Theoretical Physics, Vol. I, p 113, T. Gill Editor, Hadronic Press, Palm Harbor (1996); D. L. Schuch, Hadronic J. 19, 505 (1996). (211) S. L. Adler, Phys.Rev. 17, 3212 (1978). (212) Cl. George, F. Henin, F. Mayene and I. Prigogine, Hadronic J. 1, 520 (1978). (213) C. N. Ktorides, H. C. Myung and R. M. Santilli, Phys. Rev. D 22, 892 (1980). (214) R. M. Santilli, Hadronic J. 13, 513 (1990). (215) R. M. Santilli, Hadronic J. 17, 311 (1994). (216) A. J. Kalnay, Hadronic J. 6, 1 (1983). (217) A. J. Kalnay and R. M. Santilli, Hadronic J. 6, 1873 (1983). (218) M. Nishioka, Nuovo Cimento A 82, 351 (1984). (219) G. Eder, Hadronic J. 4, (1981) and 5, 750 (1982). 82 (220) H. E. Wilhelm, contributed paper in Space, Time and Motion: Theory and Experiments, H., E. Wilhelm and K. Liu, editors,Chinese J. Syst Eng. Electr., 6, issue 4 (1995). (221) C. A. Chatzidimitriou-Dreismann, T. Abdul-Redah, R.M.F. Streffer, J.Mayers, Phys. Rev. Lett. 79, 2839 (1997). (222) C. A. Chatzidimitriou-Dreismann, T. Abdul-Redah, B. Kolaric J. Am. Chem. Soc. 123, 11945 (2001). (223) C. A. Chatzidimitriou-Dreismann, T. Abdul-Redah, R. M. F. Streffer, J. Mayers, J. Chem. Phys. 116, 1511 (2002). (see also www.ISIS.rl.ac.uk/molecularspectroscopy/EVS). (224) R. M. Santilli, Hadronic J. Suppl. 1, 662 (l985) (225) R. M. Santilli, Found. Phys. 11, 383 (l981). 83

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http://home1.gte.net/ibr/ir00018.htm

THE NEW ISO-, GENO-, AND HYPER-MATHEMATICS OF HADRONIC MECHANICS

Original content uploaded February 15, 1997. Revisions uploaded on February 22 and March 29, April 4, and June 15, 1997 thanks to numerous critical comments by various visitors, which are acknowledged with gratitude. Additional critical comments should be sent to ibr@gte.net and will be appreciated.

Important note. This page may be upgraded at any time. To inspect the latest upgrade, visitors should reload this page each time it is inspected.

This section lists open research problems in pure and applied mathematics. All interested mathematicians in all countries, including graduate students, are welcome to participate in the research.

Following the introductory section, individual open problems are presented via:

a) a brief summary of the topic. b) a statement of the open problem(s) suggested for study. c) the motivation for the proposed research. d) the suggested IBR member(s) and/or Editor(s) for technical assistance. e) representative references.

Papers resulting from the proposed research will be listed at the end of each section. We assume the visitor of this site is aware of the inability at this time to have technical symbols and formulae in the www. Therefore, the symbols used in the presentation below have been rendered as simple as possible and they do not correspond to the symbols generally used in the technical literature.

OPEN RESEARCH PROBLEMS IN MATHEMATICS Prepared by J. V. KADEISVILI, IBR

CONTENTS

I. INTRODUCTION II. OPEN RESEARCH PROBLEMS IN NUMBER THEORY III. OPEN RESEARCH PROBLEMS IN GEOMETRIES IV. OPEN RESEARCH PROBLEMS IN FUNCTIONAL ANALYSIS V. OPEN RESEARCH PROBLEMS IN LIE-SANTILLI THEORY VI. OPEN RESEARCH PROBLEMS IN TOPOLOGY VII. MISCELLANEOUS OPEN RESEARCH PROBLEMS

I. INTRODUCTION Studies initiated in the late 1970's under the support of the Department of Energy at Department of Mathematics of Harvard University by the theoretical physicists Ruggero Maria Santilli have indicated that current mathematical knowledge is generally dependent on the assumption of the simplest conceivable unit e which assumes either the numerical value e = +1, or the n-dimensional unit form e = diag. (1, 1, . . ., 1).

Systematic studies were then initiated for the reformulation of contemporary mathematical structures with respect to a generalized unit E of the same dimension of the original unit e (e.g., an nxn matrix) but with an arbitrary functional dependence on a local chart r, its derivatives with respect to an independent variable (e.g., time t) v = dr/dt, a = dv/dt, and and any other needed variable,

(1) e -> E = E(t, e, v, a, . . . ), under the conditions of being everywhere invertible and admitting e as a particular case. Jointly, the conventional associative product axb among generic quantities a, b (e.g., numbers, vector fields, operators, etc.) is lifted into the form

(2) axb -> a*b = axTxb, T = 1/E, where T is fixed, and axT, Txb are the original associative products, in which case the quantity E = 1/T is indeed the correct left and right unit of the new theory, E*a = a*E = a for all elements a of the original set. To achieve consistency, the dual liftings (3) e -> E(t, r, v, a, ... ), ab -> a*b = axTxb, E = 1/T, must be applied to the totality of the original mathematical structure. By conception and construction, the new formulations are locally isomorphic to the original ones for all positive-definite generalized units E > 0. As a result, maps (3) do not yield "new mathematical axioms", but only "new realizations" of existing mathematical axioms and, for this reason, they were called "isotopic" in the Greek meaning of being "axiom-preserving".

When E is no longer Hermitean (e.g., it is nowhere singular and real-valued but non-symmetric), then we have the general loss of the original axioms in favor of more general axioms (see below for examples) and, for this reason maps (3) were called "genotopic" from the Greek meaning of being "axiom inducing". In this case we have two different units  = 1/S, generally interconnected by the conjugation )Ý with corresponding ordered products to the left and to the right,

(4a)  = 1/S, a>b = AxSxb, R = SÝ.

Additional classes of mathematical structures occur when the generalized units are multivalued, or subjected to anti-isomorphic conjugation (see below).

The new lines of mathematical inquiries emerged from these studies imply novel formulations of: number theory, functional analysis, differential geometries, Lie’s theory, topology, etc. For example, ordinary numbers and angles, conventional and special functions and transforms, differential calculus, metric spaces, enveloping algebras, Lie algebras, Lie groups, representation theory, etc., must be all reformulated under isotopies for the generalized product a*b = axTxb in such a way to admit E(t, r,v, a, ... ) = 1/T as the new left and right unit, and a more general setting occurs under genotopies.

To illustrate the nontriviality of these .liftings it is sufficient here to recall that Lie's theory with familiar product [A, B] = AxB -- BxA (where A, B are vector fields on a cotangent bundle or Hermitean operators on a Hilbert space, and AxB, BxA are conventional associative products), is linear, local-differential and potential-Hamiltonian, thus possessing clear limitations in its applications.

The isotopies and genotopies of Lie's theory, called Lie-Santilli isotheory and genotheory, respectively, include the corresponding liftings of universal enveloping algebras, Lie algebras, Lie groups, transformation and representation theories, etc. and are based on the following corresponding generalized products first proposed by Santilli in 1978

(5a) [A, B]* = A*B - B*A = AxT(t, r, ...)xB - BxT[t, r,...)xA, T = TÝ, (5b) (A, B) = AA = AxR(t, r, ...)xB - BxS(t, r, ...)xA, R = SÝ,

where product (5a) is "axiom preserving" in the sense of preserving the original Lie axioms, while product (5b) is "axiom-inducing" in the sense of violating Lie's axioms in favor of the more general axioms of Albert's Lie-admissible algebras (a generally nonassociative algebra U with abstract elements a, b, c, and product ab is said to be Lie-admissible when the attached antisymmetric algebra U_, which is the same vector space as U equipped with the product [a, b]U = ab - ba, is Lie). As expected, the theories with products (5a) and (5b) have been proved to provide an effective characterization of nonlinear, nonlocal and nonhamiltonian systems of increasing complexity (the former applying for stable-reversible condiions, and the latter for open-irreversible conditions, see the next Web Page 19). Their consistent treatment requires corresponding new mathematics, called iso- and genlo-mathematics, respectively. For instance, it would be evidently inconsistent to define an algebra with generalized unit E(t, r, ...) = EÝ over a conventional field of numbers with trivial unit e = +1, and the same happens for functional analysis, differential calculus, geometries, etc.

The studies initiated by Santilli were continued by numerous scholars including Gr. Tsagas, D. S. Sourlas, J. V. Kadeisvili, H. C. Myung, S. Okubo, S. I. Vacaru, B. Lin, D. Rapoport-Campodonico, R. Ohemke, A. K. Aringazin, M. Nishioka, G. M. Benkart, A. Kirukhin, J. Lohmus, J. M. Osborn, E. Paal, L. Sorgsepp, N. Kamiya, P. Nowosad, D. Juriev, C. Morosi, L. Pizzocchero,R.Aslander, S. Keles,and other scholars. A comprehensive list of contributions in related fields up to 1984 can be find in Tomber's Bibliography in Nonassociative Algebras, C. Baltzer et al., Editors, Hadronic Press, 1984. A bibliography on more recent contributions can be found in the monograph by J. Lohmus, A. Paal and L. Sorgsepp, Nonassociative Algebras in Physics, 1994, Hadronic Press (see Advanced Titles in Mathematics in this Web Site).

The mathematical nontriviality of the above studies is also illustrated by the fact that, at a deeper analysis, isotopies and genotopies imply the existence of SEVEN DIFFERENT LIFTINGS of current mathematical structures with a unit, each of which possess significant subclasses, as per the following outline:

1) ISODUAL MATHEMATICS. It is characterized by the so-called isodual map, first introduced by Santilli in 1985 (see [I-1] for a recent account), given the lifting of a generic quantity a (a number, vector-field, operator, etc.) into its anti-Hermitean form

(6) a -> isod(a) = - aÝ,

which must also be applied, for consistency, to the totality of the original structure. This implies that the isodualities of conventional mathematics, called isodual mathematics, have a "negative-definite unit" and related new product, according to the liftings (7) e = 1 -> E = - 1, axb -> a*b = (-bÝ)x(-1)x(-aÝ) = - bÝxaÝ. The above maps permitted the identification of new numbers with negative unit -1 (see Problem 1 below). In turn, the identification of new numbers permitted the identification of new spaces, algebras, geometries, etc. Note that in this first lifting the unit remains the number 1 and only changes its sign. Since the norm of isodual numbers is negative-definite, isodual mathematics has resulted to be useful for a novel representation of antimatter (see Page 19).

The visitor should be aware that contemporary mathematics appears to be inapplicable for a physically consistent representation of antimatter at the CLASSICAL level, with corresponding predictable shortcomings at the particle level. In fact,we only have today one type of quantization, e.g., the symplectic quantization. As a result, the operator image of contemporary mathematical treatments of antimatter does not yield the needed charge conjugate state. At any rate, the map from matter to antimatter must be anti-automorphic (or, more generally,anti-isomorphic), as it is the case for charge conjugation in second quantization.

The only known map verifying these conditions at all levels of treatment is Santilli's isodual map (6). This yields a novel classical representation of antimatter with a corresponding novel isodual quantization which does indeed yield the correct charge conjugate state of particles (see Web Page 19). Thus, the isodual mathematics resolves the historical lack of equivalence in the treatment between matter and antimatter according to which the former is treated at all levels, from classical mechanics to quantum, field theories, while the latter was treated only at the level of second quantization.

To understand the implications, the visitor should keep in mind that contemporary mathematics does not appear to be applicable for an effective treatment of antimatter, thus requiring its reconstruction in an anti-isomorphic form.

2, 3) ISOTOPIC MATHEMATICS AND ITS ISODUAL. The isotopic mathematics or isomathematics for short [I-1] is today referred to formulations for which the generalized unit E, called isounit, has a nontrivial functional dependence and it is Hermitian, E = E(t, r, v, ...) = EÝ. An important case is the Lie-Santilli isotheory with basic isoproduct (5a).

The isodual isotopic mathematics is the image of the isomathematics under maps (6) and therefore has the unit isod(E) = -EÝ = -E.

These structures have been classified by the theoretical physicist J. V. Kadeisvili in 1991 into:

CLASS I, when E is positive-definite (isotopies), CLASS II when E is negative definite (isodualities), CLASS III, given by the union of Classes I and II, CLASS IV, including the preceding classes plus null values of E (or singular values of T), and CLASS V, when E is arbitrary, e.g., a step-function or a distribution.

At this writing only Classes I, II and III have been preliminarily studied, while the remaining Classes IV and V are vastly unknown. 4, 5) GENOTOPIC MATHEMATICS AND ITS ISODUAL. The isotopies were proposed in 1978 as particular cases of the broader genotopies [I-1], which are characterized by two different generalized units  = 1/S for the genomultiplication to the left and to the right according to Eqs. (4). The resulting genotopic mathematics, or genomathematics for short, is given by a duplication of the isomathematics, one for ordered products to the left and the other to the right.

The isodual genomathematics is the isodual image of the preceding one, and it is characterized by the systematic application of map (6) to each of the left and right genomathematics.

In an evening seminar delivered at ICM94 Santilli proved that the genotopies can also be axiom-preserving and can therefore provide a still broader realization of known axioms. The proof was presented for product (5b) which, when considered on ordinary spaces and fields with the conventional unit e, is known to verify Albert’s axiom of Lie-admissibility. The same product was proved to verify the Lie axiom when each of the two terms AA is computed in the appropriate genoenvelope and genofield with the corresponding genounit.

6, 7) HYPERSTRUCTURAL MATHEMATICS AND ITS ISODUAL. At the IBR meeting on multivalued hyperstructures held at the Castle Prince Pignatelli in August 1995, the mathematician Thomas Vougiouklis and R. M. Santilli presented a new class of hyperstructures, those with well defined hyperunits characterized by hyperoperations. A subclass of the latter hyperstructures important for applications is that with hyperunits characterized by ordered sets of non-Hermitean elements,

(8a)  = {A>, B>, ... } = 1/S = {1/S1, 1/S2, ...},

with corresponding multivalued hypermultiplications ab = axSxb. The latter structures evidently permit a third layer of generalized formulations which are also axiom-preserving when treated with the appropriate hypermathematics. The isodual hypermathematics is the isodual image of the above hypermathematics and is therefore itself multivalued. Needless to say, the above studies are in their first infancy and so much remains to be done.

The material of this Web Page is organized following the guidelines of memoir [I-1] according to which there cannot be really new applications without really new mathematics, and there cannot be really new mathematics without new numbers. We shall therefore give utmost priority to the lifting of numbers according to the above indicated seven different classes. All remaining generalized formulations can be constructed from the novel base fields via mere compatibility arguments.

We shall then study the novel spaces and geometries which can be constructed over the new fields because geometries have the remarkable capability of reducing the ultimate meaning of both mathematical and physical structures to primitive, abstract, geometric axioms.

We shall then study: the generalized functional analysis which can be constructed on the preceding structures, beginning from Kadeisvili’s new notions of continuity; the all fundamental Lie-Santilli theory; the underlying novel Tsagas-Sourlas integro-differential topology; and other aspects.

Only primary references with large bibliography are provided per each section. Subsequent calls to references of preceding sections are indicated with [I-1], [II-1], etc.

REFERENCE OF SECT. 1: We recommend to study the following memoir and some of the large literature quoted therein

[I-1] R. M. Santilli, Nonlocal-integral isotopies of differential calculus, mechanics, and geometries, Rendiconti Circolo Matematico Palermo, Supplemento No. 42, pages 7-83, 1996.

See also

[I-2] J. V. Kadeisvili, An Introduction to the Lie-Santilli Isotopic theory, Mathematical Methods in Applied Sciences, Vol. 19, pages 1349-1395, 1996.

[I-3] J. V. Kadeisvili, "Santilli's Isotopies of Contemporary Algebras, Geometries and Relativities, Second Edition, Ukraine Academy of Sciences, Kiev, 1997 (First Edition 1992).

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II: OPEN RESEARCH PROBLEMS IN NUMBER THEORY

Prepared by the IBR staff. PROBLEM II.1: STUDIES ON THE ISODUAL NUMBER THEORY DEFINITION II.1: Let F = F(a,+,x) be a conventional field of (real R, complex C or quaternionic Q) numbers a with additive unit 0, multiplicative unit e = 1, sum a+b and product axb. Santilli’s isodual field [II-1] isodF = isodF(isoda, isod+,isodx) is a ring of elements isoda = -aÝ, called isodual numbers, with isodual sum (isoda)isod+(isodb) = (-aÝ-bÝ) and isodual multiplication (isoda)isodx(isodb) = (-aÝ)(-1)(-bÝ) = -(aÝ)(bÝ) under which the additive unit is isisod0 = 0 and the multiplicative unit is isode = -e = -1. LEMMA II-1 [II-1]: The isodual field is a field (i.e., it verifies all axioms of a field).

PROPOSITION II-1 [loc. cit.]: The map F -> isodF is anti-isomorphic.

PROPOSED RESEARCH II-1: Study the isodual number theory, including theorems on prime, factorization, etc.

SIGNIFICANCE: Isodual numbers have a negative norm, thus being useful to represent antimatter.

FOR TECHNICAL ASSISTANCE: Contact the IBR staff at ibr@gte.net

PROBLEMS II-2/II-3: STUDIED IN ISOFIELD THEORY AND ITS ISODUAL DEFINITION II-2: Let F = F(a,+,x) be a conventional field of (real R, complex C or quaternionic Q) numbers a with additive unit 0, multiplicative unit e = 1, sum a+b and product axb. Santilli’s isofields [II-1] F* = F*(a*,+*,x*) are rings of elements a* =axE called isonumbers, where a is an element of F, axE is the multiplication in F, and E = 1/T is a well behaved, everywhere invertible, Hermitean and positive-definite quantity (e.g., a matrix, vector field, operator, etc. of Kadeisvili’s Class I) generally outside the original field F, equipped with the isotopic sum a*+*b* = (a+b)xE and product a*x*b* = a*xTxb* = (axb)xE, with additive unit 0* = 0 and multiplicative unit E, called isounit. Santilli’s isodual isofields isodF*(isoda*,isod+*,isodx*) are the anti-Hermitean images of F* under the isodualities of the elements of F* and all its operations. LEMMA II-2 [II-2]: Isofields verify all axioms of a field (including closure under the combined associative and distributive laws). The lifting F -> F* is therefore an isotopy.

PROPOSITION II-2 (ref.[II-1], p. 284): When E is an element of the original field F (e.g., an ordinary real number for F = R), the isofield F*(a,+*,x*) is also a field (i.e., closure occurs for conventional numbers a without need to use the isonumbers a* = axE).

PROPOSED RESEARCH II-2: Formulate the real isonumber theory: 1) with a basic unit given by an arbitrary, positive, real number E = n > 0; and 2) under isoduality to a negative-definite unit isodE = -n < 0. These problems can be studied via the simplest possible class of Santilli isofields R*(a,+*,x*) and their isoduals isodR*(isoda,isod+*,isodx*) in which the elements a are not lifted, as per Proposition II.2 above. The study implies the re-inspection of all conventional properties of number theory in order to ascertain which one is dependent on the selected unit. As an example, it is known that the notion of prime depends on the selected unit [II-1] because, e.g., the number 4 becomes prime for the isounit E = 3.

SIGNIFICANCE: An important advance of memoir [II-1] is that the axioms of a field need not to be restricted to the simplest possible unit +1 dating back to biblical times, because they equally hold for arbitrary units. This basic property has far reaching implications. In mathematics the property implies the lifting of all structures defined on numbers; in physics the broadening of the unit implies basically novel applications (See the subsequent Web Page 19 on Open Research Problems in Physics); and in biology it implies a structural revision of current theories (see the subsequent Web Page 20 on Open Research Problems in Biology).

FOR TECHNICAL ASSISTANCE: Contact the IBR staff at ibr@gte.net

PROBLEM II-4/II-5: STUDIES ON GENONUMBERS THEORY AND ITS ISODUAL DEFINITION II-3: Let F = F(a,+,x) be a conventional field of (real, complex C or quaternionic Q) numbers a with additive unit 0, multiplicative unit e = 1, sum a+b and product axb. Santilli’s genofields to the right [II-1] F> = F>(a>,+>,x>) are rings of elements a> = axE> called genonumbers, where a is an element of F, axE> is the multiplication in F, and E> = 1/S is a well behaved, everywhere invertible and non-Hermitean quantity generally outside F, equipped with all operations ordered to the right, i.e., the ordered genosum to the right (a>)+>(b>) = (a+b)xE>, the ordered genoproduct to the right (a>)>(b>) = (a>)xSx(b>) = (axb)xE>, etc., conventional additive unit to the right 0> = 0 and generalized (left and right) unit E> for the multiplication to the right called genounit. Santilli’s genofields to the left <F(<a,<+,<) are rings with genonumbers <a = <Exa, all operations ordered to the left, such as genosum (b or b multiplying a from the right ab = ab = b>a and ab ‚ a)>(a>) = (a>)>(E>) = a> for all possible a>.

LEMMA II-3 [II-1]: Each individual genofield to the right F> or to the left  F> and F ->  and (a,+>,x>) and <F(a,<+,<x) is a field.

PROPOSED RESEARCH II-4/II-5: Formulate the number theory with 1) a basic unit given by a positive real number E> = n in which all operations are ordered to the right; 2) formulate the same theory under an ordering to the left with a different positive-definite genounit <E = b; 3) construct the isoduals of both theories. These problems can be studied via the simplest possible class of Santilli genofields F>(a,+>,x>) and <F(a,<+,<x) in which the elements are not lifted, as per Proposition II-3 above. The study implies the re-inspection of all conventional properties of the isonumber theory.

SIGNIFICANCE: Another significant advance of memoir [II-1] is that the axioms of a field, not only do not need the restriction to the unit +1, but the operations can be all restricted to be EITHER to the right OR to the left. This simple property has additional far reaching mathematical, physical and biological implications. In mathematics, it implies a dual lifting of all isotopic structures. In physics it implies an axiomatic representation of the irreversibility of the physical world via the most fundamental mathematical notion, the unit. In fact, operations ordered to the right can represent motion forward in time, while operations ordered to the left can represent motion backward in time.

Irreversibility is then reduced to the differences between E> and <E or, equivalently, between a>b and a<b. The inclusion of the isodualities implies the capability to represent all possible four different motions in time: motion forward to future time E>, motion backward to past time <E, motion forward from past time isod(<E), and motion backward from future time isod(E>). In theoretical biology, Santilli’s genonumbers are the foundation of the first known consistent mathematical representation of the irreversibility of biological structures. The addition of isoduality then permits the mathematical representation of certain bifurcations in biology whose treatment is simply beyond conventional mathematics.

SUGGESTED TECHNICAL ASSISTANCE: Contact the IBR staff at ibr@gte.net

PROBLEM II-6/II-7: STUDIES IN HYPERNUMBER THEORY AND ITS ISODUAL Repeat the studies of Problem II-4/II-5 but for the case when the genounits to the right and to the left and given by an ordered set of nonhermitean invertible elements equipped, first, with conventional and, then, with hyperoperations. For a definition of hyperfields with conventional operations see ref. [I-1] of Sect. I. For their broader definition with hyperoperations see ref. [II-2] below. SUGGESTED TECHNICAL ASSISTANCE: Consult

Prof. T. Vougiouklis Department of Mathematics Democritus University of Thrace GR-67100 Xanthi, Greece, fax +30-551-39348, or

Prof. M. Stefanescu Department of Mathematics Ovidius University, Bd. Mamaia 124 Costanta 8700, Romania,

REFERENCES OF SECT. II:

[II-1] R. M. Santilli, Isonumbers and genonumbers of dimension 1, 2, 4, 8, their isoduals and pseudoduals, and “hidden numbers” of dimension 3,5,6,7, Algebras, Groups and Geometries Vol. 10, pages 273-322, 1993 [II-2] T. Vougiouklis, Editor, New Frontiers in Hyperstructures, Hadronic Press, 1996.

RECENT PAPERS APPEARED IN THE RESEARCH PROPOSED IN SECT. II

ON SANTILLI'S ISOTOPIES OF THE THEORY OF REAL NUMBERS, COMPLEX NUMBERS, QUATERNIONS AND OCTONIONS N. Kamiya Department of Mathematics Shimane University Matsue 690, Japan "New Frontiers in Algebras, Groups and Geometries", Gr. Tsagas, Editor, Hadronic Press, Palm Harbor (1996), pp. 523-551.

A CHARACTERIZATION OF PSEUDOISOFIELDS N. Kamiya Department of Mathematics Shimane University Matsue 690, Japan and R. M. Santilli Institute for Basic Research P., O. Box 1577 Palm harbor, FL 34682, U.S.A. ibr@gte.net "New Frontiers in Algebras, Groups and Geometries", Gr. Tsagas, Editor, Hadronic Press, Palm Harbor (1996), pp. 559-570.

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III: OPEN PROBLEM IN GEOMETRIES

Prepared by the IBR staff. PROBLEM III.1: STUDIES IN ISODUAL GEOMETRIES DEFINITION III-1: Let S = S(r,g,R) be a conventional n-dimensional metric or pseudo-metric space with local chart r, nowhere singular, real-valued and symmetric metric g and invariant (rt)xgxr (where rt denotes transposed of r) over a conventional field R = R(a,+,x) of real numbers. Santilli’s isodual spaces isodS = isodS(isodr,isodg,isodR) is the vector space with local chart isodr = -r, isodual metric isodg = -g and isodual invariant (isodrt)x(isodg)x(isodr) = [(rt)xgxr](isode) on isodR. Isaodual geometries are the geometries on isodual spaces, thus based on negative-definite units. PROPOSITIONS III-1 [I-1]: Isodual spaces are anti-isomorphic to the original space.

PROPOSED RESEARCH III-1: Study the isodual Euclidean, isodual Minkowskian, isodual Riemannian, isodual symplectic and other isodual geometries, including the isodual calculus, the isodual sphere (i.e., the sphere with negative radius), the isodual light cone, etc. [I-1].

SIGNIFICANCE: Isodual geometries are fundamental for the recent isodual representation of antimatter, e.g., to characterize the shape of an antiparticle with negative units.

SUGGESTED TECHNICAL ASSISTANCE: Contact the IBR staff at ibr@gte.net

PROBLEMS III-2/III-3: STUDIES ON ISOGEOMETRIES AND THEIR ISODUALS

DEFINITION III-2: Let S = S(r,g,R) be a conventional n-dimensional metric or pseudo-metric space with local chart r, nowhere singular, real-valued and symmetric metric g and invariant (rt)xgxr (where rt denotes transposed) over a conventional real field R = R(a,+,x). Santilli’s isospaces [I-1] S*(r*,G*,R*) are vector spaces with local isocoordinates r* = rxE, isometric G* = g*xE = TxgxE and isoinvariant (r*t)*G*r* = [(rt)x(g*)xr]xE over an isofield R* = R*(a*,+*,x*) with a common isounit E = 1/T of Kadeisvili Class I. Santilli’s isogeometries [III-1] are the geometries of isospaces. The isodual isospaces and isodual isogeometries are the corresponding images under isoduality [I-1,III-1]. LEMMA III-2 [I-1,III-1]: Isospaces S*(x*,G*,F*) are locally isomorphic to the original space S(x,g,F). The lifting S -> S* is therefore an isotopy.

Proof. Each component of the metric g is lifted by the corresponding element of T, while the unit is lifted by the corresponding inverse amount E = 1/T, thus preserving the original geometric axioms.q.e.d.

PROPOSITION III-3 [III-2]: The axioms of the Euclidean geometry in n-dimension admit as particular cases all possible well behaved, real-valued, symmetric and positive-definite metrics of the same dimension.

REMARKS.To be consistently defined, Santilli’s isogeometries require the isotopies of the totality of the mathematical aspects of the original geometry, all formulated for a common isounit E with the same dimension of the isospace. This requires, not only the isotopies of fields and vector spaces, but also those of all other aspects.

PROBLEM III-2/III-3: Provide a mathematical formulation of Santilli’s isoeuclidean, isominkowskian, isoriemannian, isosymplectic and other isogeometries and their isoduals which have been only preliminarily studied for physicists in ref. [III-1]. SIGNIFICANCE: The mathematical and physical implications are significant indeed. Mathematically, the studies permit advances such as: the unification of all geometries of the same dimension into one single isotope; the admission under the Riemannian axioms of metric with arbitrary, nonlinear, integro-differential dependence in the velocities and other variables; the representation of nonhamiltonian vector-fields in the local chart of the observer (see the alternative to Darboux’s theorem, ref. [I-1], p. 63 motivated by the fact that, in view of their nonlinearity, Darboux’s transforms cannot be used in physics because the transformed frames cannot be realized in experiments and, in any case, they violate the axioms of Galilei’s and Einstein’s special relativity due to their highly noninertial character). Physically, the studies permit truly basic advances, such as the first quantitative research on the origin of the gravitational field, a geometric unification of the special and general relativity via the isominkowskian geometry in which the isometric is a conventional Riemannian metric, a novel operator formulation of gravity verifying conventional quantum axioms; and other advances (see Web Pages 19 and 20).

SUGGESTED TECHNICAL ASSISTANCE: Contact the IBR staff at ibr@gte.net, or

Prof. D. Sourlas Department of Mathematics University of Patras Gr-26100 patras, Greece Fax +30-61-991 980

Prof. Gr. Tsagas Department of Mathematics Aristotle University Thessaloniki 54006, Greece Fax +30-31-996 155

Prof. R. Miron Department of Mathematics “Al. I. Cuza” University 6600 Iasi, Romania rmiron@uaic.ro

PROBLEMS III-4/III-5: STUDIES ON GENOGEOMETRIES AND THEIR ISODUALS DEFINITION III-3: Let S = S(r,g,R) be a conventional n-dimensional metric or pseudo-metric space with local chart r, nowhere singular, real-valued and symmetric metric g and invariant (rt)xgxr over a conventional real field R = R(a,+,x). Santilli’s n-dimensional genospaces to the right [I-1] S>(r>,G>,R>) are vector spaces with local genocoordinates to the right r> = rxE>, genometric G> = (g>)x(E>) = SxgxE>, genoinvariant (r>t)>(G>)(r>) = [(rt)x(g>)xr]xE> over the genofield R> = R>(a>,+>,x>), common genounit to the right E> = 1/S given by an everywhere invertible, real-valued, non-symmetric nxn matrix, and all operations ordered to the right. Santilli’s genogeometries [III-1] are the geometries of genospaces. The isodual genospaces and isodual genogeometries are the corresponding images under isoduality [I-1,III-2]. Santilli’s n-dimensional genospaces to the left [I-1] <S(<r,<+,<R) are genospaces over genofields with all operations ordered to the left and a common nxn-dimensional genounit to the left <E = 1/R which is generally different than that to the right E> = 1/S. Genogeometries to the left and their isoduals are the geometries over the corresponding genospaces. LEMMA III-3: Genospaces to the right S> and, independently, those to the left <S, are locally isomorphic to the original spaces S. Proof. The original metric g is lifted in the form g -> Sxg, but the unit is lifted by the inverse amount I -> E = 1/S, thus preserving the original axioms. q.e.d.

PROPOSED RESEARCH: Provide a mathematical formulation of Santilli’s genoeuclidean, genominkowskian, genoriemannian, genosymplectic and other genogeometries to the left, their corresponding forms to the right and their isoduals which have been preliminarily studied in ref. [III-1] for physicists.

SIGNIFICANCE: Another important aspect of memoir [I-1] is that the Riemannian axioms do not necessarily need a symmetric metric because the metrics can also be nonsymmetric with structure g> = Sxg, S nonsymmetric, provided that the geometry is formulated on an isofield with isounit given by the INVERSE of the nonsymmetric part, E = 1/S, and the same occurs for the left case. This property has permitted the first quantitative studies on irreversibility of interior gravitational problems via the conventional Riemannian axioms, as it occurs in the physical reality, e.g., the irreversibility of the structure of Jupiter or of a collapsing star, for which purpose the genogeometries were constructed in the first place [III-1].

SUGGESTED TECHNICAL ASSISTANCE: Contact the IBR staff at ibr@gte.net

PROBLEMS III-6/III-7: STUDIES ON HYPERGEOMETRIES AND THEIR ISODUALS Extend the studies of the genogeometries to the right and to the left to the case when the corresponding genounits are given by an ordered set of invertible, real-valued and nonsymmetric nxn elements, first, with ordinary operations and then with hyperoperations. The existence of these geometries has been only indicated in ref. [I-1] without any detailed treatment.

SUGGESTED TECHNICAL ASSISTANCE: Consult the IBR staff at ibr@gte.net

REFERENCES OF SECT. III:

[III-1] R. M. Santilli, Elements of Hadronic Mechanics, Vol. I, Mathematical Foundations, Ukraine Academy of Sciences, Kiev, Second Edition, 1995.

RECENT PAPERS APPEARED IN THE RESEARCH PROPOSED OF SECT. III

ISOAFFINE CONNECTION AND SANTILLI'S ISORIEMANNIAN METRIC ON AN ISOMANIFOLD Gr. Tsagas Department of Mathematics Aristotle University Thessaloniki 54006, Greece Algebras, Groups and Geometries, Vol. 13, pages 149-169, 1996

STUDIES ON SANTILLI'S LOCALLY ANISOTROPIC AND INHOMOGENEOUS ISOGEOMETRIES, I: ISOBUNDLES AND GENERALIZED ISOFINSLER GRAVITY Sergiu I. Vacaru Institute of Applied Physics Academy of Sciencves of Moldova 5,. Academy Street CHISINAU 2028, REPOUBLIC OF MOPLDOVA Fax +3732-738149, E-address lises@cc.acad.md In press at Algebras Groupos and Geometries, Vol. 14, 1997

CARTAN'S STRUCTURE EQUATIONS ON SANTILLI-TASGAS-SOURLAS ISOMANIFOLDS Recept Aslander Inonu Universitesi Egitim Fakultesi Matematik Egitimi Bolumu 44100 Malatya, Turkey and Sadik Keles Inonu Universitesi Fen-Edebiyat Fakultesi Matematik Bolumi 44100 Malatya, Turkey In press at Algebras, Groups and Geometries, Vol. 14, 1997

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IV: OPEN RESEARCH PROBLEMS IN FUNCTIONAL ANALYSIS

Conventional and special functions and transforms and functional analysis at large are dependent on the assumed basic unit. As an example, a change of the two-dimensional unit of the Gauss plane implies a change in the very definition of angles and trigonometric functions, and the same happens for hyperbolic functions, Fourier, Laplace and other transforms, Dirac and other distributions, etc. To be operational, the seven classes of novel mathematical methods of the preceding sections require seven corresponding generalized forms of functional analysis, which are here recommended for study in a progressive way, beginning with the simplest possible case of isoduality.

Mathematical work done to date in these new topics has been rather limited. We here mention: Santilli has preliminarily studied the structure of the seven forms of differential calculus [I-1], isotrigonometric and isohyperbolic functions, the isofourier transforms and few other aspects {III-1]; Kadeisvili [IV-1,2] has studied basic definitions of isocontinuity and its isodual and reinspected some of the studies in the field; H. C. Myung and R. M. Santilli [IV-3] studied the isotopies of the Hilbert space, Dirac delta distributions and few other notions; A. K. Aringazin, D. A. Kirukhin and R. M. Santilli [IV-4] have studied the isotopies of Legendre, Jacobi and Bessel functions and their isoduals; M. Nishioka [IV-5] studied the Dirac-Myung-Santilli isodelta distribution (see [III-1] for a review up to 1995).

SUGGESTED TECHNICAL ASSISTANCE: Contact Prof. J. V. Kadeisvili at ibr@gte.net, or A. K. Aringazin and D. A. Kirukhin at aringazin@kargu.krg.kz

REFERENCE FOR SECT. IV:

[IV-1] J. V. Kadeisvili, Elements of functional isoanalysis, Algebras, Groups and Geometries vol. 9, pages 283-318, 1992. [IV-2] J. V. Kadeisvili, Elements of the Fourier-Santilli isotransforms, Algebras, Groups and Geometries Vol. 9, pages 319-242, 1992 [IV-3] H. C. Myung and R. M. Santilli, Modular-isotopic Hilbert space formulation of the exterior strong problem, Hadronic J. Vol. 5, pages 1277-1366, 1982. [IV-4] A. K. Aringazin, D. A. Kirukhin and R. M. Santilli, Isotopic generalization of Legendre, Jacobi and Bessel functions, Algebras, Groups and Geometries Vol. 12, pages 255-359, 1995. [IV-5] M. Nishioka, Extension of the Dirac-Myung-Santilli delta functions to field theory, Lett. Nuovo Cimento Vol. 39, pages 369-372, 1984.

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V. STUDIES ON THE LIE-SANTILLI ISO-, GENO- AND HYPER-THEORIES AND THEIR ISODUALS

Prepared by the IBR staff PROBLEM V-1: STUDIES ON THE ISODUAL LIE THEORY DEFINITION V-1: Let L be an n-dimensional Lie algebra with ordered Hermitean basis X = {A,B, ...} = XÝ, conventional commutator [A, B] = AxB - BxA (where AxB is conventionally associative) over a field F (of characteristics zero). A Lie-Santilli isodual algebra [I-1] isodL is the image of L under the isodual map (5), thus including isodual generators isodX = -XÝ = -X, isodual commutator isod[A, B] = (isodA)isodx(isodB) - (isodB)isodx(isodA) = - [A, B], etc., all defined on an iusodualF with negative-definite unit E = -Diag. (1, 1, ..., 1), and norm. LEMMA V-1 [I-1]: IsodL is anti-isomorphic to L.

PROPOSED RESEARCH V-1: Reformulate Lie's theory (enveloping associative algebras, Lie algebras, Lie groups, transformation and representation theories, etc.) for the Lie-Santilli isodual theory with an n-dimensional negative-definite unit E = - Diag(1, 1, ... 1).

SIGNIFICANCE: Isodual symmetries characterize antiparticles.

SUGGESTED TECHNICAL ASSISTANCE: Contact the IBR staff at ibr@gte.net

PROBLEM V-2/V-3: STUDIES ON LIE-SANTILLI ISOTHEORY AND ITS ISODUAL DEFINITION V-2: Let U(L) be the enveloping associative algebra of a Lie algebra L with infinite-dimensional basis (Poincare’-Birkhoff-Witt theorem) 1, X, (Xi)x(Xj )(i ¾ j), .. and conventional exponentiation expA = 1 + A/1! + AxA/2! + ...over a field F. Santilli’s universal enveloping isoassociative algebra U*(L) of L [V-1] of Class I is characterized by the isotopies of the Poincare’-Birkhoff-Witt theorem) with infinite-dimensional isobasis (9) E, X, (Xi)*(Xj) (i ¾ j), ...,

where the isounit E = 1/T is of Kaidesvilki Class I (positive-definite) has the same dimension of the representation at hand and A*B = AxTxB, with base isofield F* characterized by the same isounit E of U*(L). The isodual isoassociative envelope is the image of U* under isoduality. REMARKS. The conventional exponentiation is no longer applicable for U*(L) and must be replaced by the isoexponentiation

(10) isoexpA = E + A/1! + A*A/2! + ... = {exp(AxT)}xE = Ex{exp(TxA)}.

The symbol U*(L) rather than U*(L*) is used to indicate that the basis X of L is unchanged under isotopy and merely redefined in isospace,thus X* = X. This is the most significant case on physical grounds because generators of Lie symmetries represent quantities such as energy, linear momentum, angular momentum, etc. which, as such, cannot be changed. Only the operations defined in them can be changed. Since Lie’s theory leaves invariant its basic unit, the functional dependence of the isounit E is left unrestricted and, therefore, can depend on an independent variable t (say time), local chart r of the carrier space, its derivatives v = dr/dt, a = dv/dt and any other needed variable, E = E(t, r, v, a, ...) = 1/T. The nontriviality of the Lie-santilli isotheory can therefore be seen up-front because it implies the appearance of a nonlinear, integro-differential element T in the EXPONENT of the group structure, Eq. (10).

DEFINITION V-3. Let L be an n-dimensional Lie algebra as per Definition V-1. A Lie-Santilli isoalgebra [V-1] L* of Class I is the algebra homeomorphic to the antisymmetric algebra [U*(L)]- of U*(L). It can be defined as an isovector space with the same ordered basic X* = X of L equipped with the isoproduct

(11) [A, B]* = A*B - B*A = AxT(t, r, v, a, ...)xB - BxT(t, r, v, a, ...)xA

REMARKS. Lie-Santilli isoalgebras verify the conventional Lie axioms (anti-commutativity of the product and Jacobi identify), only formulated in isospace (that is, with respect to the isoassociative produce A*B) over the isofield F*. LEMMA V-3 [III-1]: Lie-Santilli isoalgebras are left and right isolinear, i.e., they verify the left linearity conditions on L* as an isovector space over F*,

(12a) {[(a*)*A* + (b*)*B*, C*]* = (a*)*[A*,B*]* + (b*)*[B*, C*]*

(12b) [(A*)*B*, C*] = (A*)*[B*, C*]* + [A*, C*]*(B*)

and corresponding right conditions. Isoalgebras L* are isolocal (in the sense of being everywhere local-differential except at the isounit E) and isocanonical (in the sense of admitting a canonical structure in isospace, see [III-1] for brevity). DEFINITION V-4: Let G be an n-dimensional connected Lie transformation group r’ = K(w)xr on a space S(r,F), where w are the parameters in F, verifying the usual conditions (differentiability of the map GxS -> S, invariance of the basic unit e = I, and linearity), as well as the conditions to be derived from the Lie algebra L via exponentiation

(13a) Q(w) = {exp(ixXxw)} x Q(0) x {exp(-ixwxX)}

(13b) i [Q(dw) - Q(0) ] / dw = QxX - XxQ = [ Q, X].

A connected Lie-Santilli isotransformation group [V-1] G* of Class I is the set of isotransforms (14) r*’ = Q*(w*)] * (r*) = [Q*[w*)] x T(t, r, v, a, ...) x r

on a Class I isospace S*(r*,F*), where now the isoparameters w* = wxE belong to F*, which verifies the usual conditions in their isotopic form (Kadeisvili’s isodifferentiability of the isomap (G*)*S* -> S*, invariance of the isounit E, and isolinearity), as well as the conditions to be derivable from the Lie-Santilli isoalgebra L* [V-1] (15a) Q*(w*) = {isoexp[i(X*)*(w*)]} * [Q*(0)] * {isoexp[-i(w*)*(X*)]} =

= { exp (i X x T x w) } x Q*(0) x { exp(-iwTX) },

(15b) i [Q*(dw*) - Q*(0) ] / dw* = (Q*) *(X*) - (X*) * (Q*) = [ Q*, X*]*

with isogroup laws [V-1] (16) [Q*(w*)]*[Q*(w*’)] = Q*(w* + w*’], [Q*(w*)]*[Q*(-w*)] = Q*(0*) = E

The isodual Lie-Santilli isogroups isodG* are the isodual image of G* under map (5). LEMMA V-3 [V-1]: Lie-Santilli isoenvelopes U*, isoalgebras L* and isogroup G* are locally isomorphic to the original structures U, L, and G, respectively for all possible positive-definite isounit E (not so otehrwise). The liftings

(17) U -> U*, L -> L* and G -> G*

are therefore isotopies. PROPOSED RESEARCH V-2: Conduct mathematical studies on the Lie-Santilli isotheory of Class I and its isodual with particular reference to: the isostructure theory; the isorepresentation theory; and related aspects.

REMARKS. At the abstract, realization-free level, isoenvelopes U*, isoalgebras L* and isogroups G* coincide with the conventional envelopes U, algebras L and groups G, respectively, by conception and construction for all positive-definite isounits E (not necessarily so otherwise). This illustrates the insistence by Santilli in indicating that the isotopies do not produce new mathematical structures, but only new realizations of existing abstract axioms.

As a result of the, the isorepresentation theory of U, L and G on isospaces over isofields is expected to coincide with the conventional representations of the original structures U, L and G on conventional spaces over conventional fields. The aspect of the isorepresentation theory which is important for applications is the PROJECTION of the isorepresentation on conventional spaces. Stated differently, Lie’s theory admits only one formulation, the conventional one. On the contrary, the covering Lie-Santilli isotheory admits two formulations, one in isospace over isofield and one given by its projection on conventional spaces over conventional fields.

The latter are important for applications, e.g., because the physical space-time is the conventional Minkowski space, while the isominkowski space is a mathematical construction. As a result, the isorepresentation theory of the Poincare’-Santilli isosymmetry [V-4] on isominkowski space over isofields is expected to coincide with that of the conventional symmetry on the conventional Minkowski space over the conventional field of real numbers. The mathematically and physically significant aspects are given by the PROJECTION of the isorepresentation on the conventional Minkowski space-time.

SIGNIFICANCE: The isotheory characterizes all infinitely possible, well behaved, arbitrarily nonlinear, nonlocal-integral and nonhamiltonian, classical and operator systems by reducing them to identical isolinear, isolocal and isocanonical forms in isospaces over isofields, thus permitting a significant simplification of notoriously complex structures.

PROPOSED RESEARCH V-3: Study the Lie-Santilli isotheories of Classes III (union of Class I with positive-definite and II with negative definite isounits E), Class IV (Class III plus null isounit E) and Class V arbitrary isounits E, including discontinuous realzioations). As a particular case unify all simple Lie algebras of the same dimension in Cartan's classification into one single isoalgebra of the same dimension of Class III, whose study has been initiated by Tsagas and Sourlas [V-4].

REMARK 1. In his original proposal on the isotopies of Lie's theory of 1978 (see the references inn [V-1]), Santilli proved the loss at the abstract level of all distinction between compact and noncompact Lie algebras of the same dimension provided that the isounit has an arbitrary positive- or negative-definite signature (Class III). This was illustrated via the algebra of the rotation group in three dimension O(3). When its conventional generators X1, X2, X3 (the components of the angular momentum) are equipped with the isounit E = Diag. (+1, +1., -1) and isoproduct (11) they characterize the noncompact O(2.1) algebra. The isoalgebra O*(3) with the fixed generators X1, X2, X3 equipped with isoproduct (11) and a isotopic element T of Class III therefore unifies all simple Lie algebras of dimension 3. This result has been proved to hold also for all orthogonal and unitary algebras, and it is expected to hold for all possible Lie algebras, including the exceptional ones.

REMARK 2. As indicated in the subsequent Web Page 19, the zeros of the isounit represent gravitational singularities. The study of the Lie-Santilli theory of Class IV is therefore important for applications. No study in on record at this writing in this field which requires the prior study of numbers, spaces, geometries, etc., whose units can be psoitive, negative as wel as null. No study is also on record on the isotoppies of Class V.

IMPORTANT NOTE. Visitors of this page should be aware that the treatment of the isoproduct [A, B]* = AxTxB - BxTxA on conventional spaces over conventional fields is not invariant under the group action and, as such, it has no known physical value. In fact, when realized on a Hilbert space over a conventional field, isogroups G* are characterized by nonunitary transforms WWÝ ‚ I. As a result, the base unit of a conventional treatment of the isoproduct [A, B]* is not left invariant by the isogroup, I -> I’ = WxIxWÝ ‚ I, and, consequently, the isoproduct itself is not invariant, [A, B]* -> Wx[A, B]*xWÝ = A’xT’xB’ - B’xT’xA’, where T’ = (WÝ to -1)xTx(W to -1) ‚ T. The loss of the traditional invariance of Lie’s theory then implies the lack of meaningful applications.

On the contrary, when treated via the isotopic mathematics, that is, formulated on isospaces over isofields, the isoproduct [A, B]* is fully invariant. For instance, by considering again the operator realization, the originally nonunitary structure of G* is turned into identical isounitary forms, i.e., we can write W = (W*)x(square root of T) for which WxWÝ = (W*)*(W*Ý) = (W*Ý)*(W*) = E, in which vase the base isounit E of the isofield is invariant, E -> E’ = (W*)*E*(W*Ý) = (W*)*(W*Ý) = E, and the isoproduct is consequently invariant, (W*)*[A, B]*x(W*Ý) = A’xTxB’ - B’xTxA’, where one should note that T is numerically preserved. The above occurrence illustrate the necessity of using Santilli’s isonumbers and isospaces for meaningful applications.

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PROBLEM V-4/V-5: STUDIES ON LIE-SANTILLI GENOTHEORY AND ITS ISODUAL HISTORICAL NOTE: Lie algebras are "nonassociative" in the sense that their product [A, B] = AxB - BxA is nonassociative. Yet their representations are reduced to those of their universal enveloping associative algebras U(L) with product AxB because of the homeomorphism between L and the attached antisymmetric algebra U(L)-. In 1948 the American mathematician A. A. Albert (Trans. Amer. Mat. Soc. Vol. 64, p. 552, 1948) introduced the notion of Lie-admissible algebras (presented in Sect. I and called the First Condition of Lie-admissibility). The formulation was done within the context of nonassociative algebras, in which context they have been studied by various mathematicians until recently. Also, Albert was primarily interested in the "Jordan content" of a given "nonassociative" algebra and, for this reason, he studied the product (a, b) = pxaxb + (1-p)xbxa, where p is a real parameter, which admits the commutative Jordan product for p = 1/2.

In 1967 Santilli (Nuovo Cimento Vol 51, p. 570, 1967) was the first physicist to study Lie-admissible algebras. He noted that Albert's definition did not admit Lie algebras in their classification and, for this reason, the algebras had limitations in physical applications. He therefore introduced a new notion of Lie-admissibility which is Albert's definition plus the condition of admitting Lie algebras in their classification (this is called the Second Condition of Lie-admissibility). In particular, Santilli studied the product (a,b) = pxaxb - qxbxa, where p, q, and p+or-q are non-null parameters, which does indeed admit the conventional Lie product as a (nondegenerate) particular case, and which constitutes the first formulation in scientific record of the "deformations" of Lie algebras of the contemporary physical literature (see the next Web Page 19).

In 1978 Santilli (Hadronic J. Vol. 1, p. 574, 1978) notes his Second Condition of Lie-admissibility was still insufficient for physical applications because Lie-admissibility implies "nonunitary" time evolutions under which the "parameters" p and q become "operators". He therefore introduced a more general definition of Lie-admissibility which is Albert's definition plus the conditions that the attached antisymmetric algebra is Lie-isotopic, rather than Lie, and the algebras admit conventional Lie algebras in their classification (this is the Third Condition of Lie-admissibility, also called Albert-Santilli Lie-admissibility, or General Lie-admissibility).

In this way, Santilli introced the product (A, B) = AxRxB - BxSxA, Eq.s (5b), where R, S, and R+or-S are fixed and nonull vector-fields, matrices, operators, etc. for which the attached antisymmetric algebra is the isotopic form [A,B] = (A, B) - (B, A) = AxTxB - BxTxA, T = R+S, while admitting of the conventional Lie product for R+S = 1. The product (A, B) is also the first on scientific records of the so-called "quantum groups" of the contemporary physical literature. The same product, being neither totally antisymetric nor totally symmetric, includes as particular cases supersymmetric and other generalizations of the Lie product (see the next Web Page 19).

In the same memoir of 1978, Santilli reduced the study of the Lie-admissible product (A, B) = AxRxB - BxSxA to its two isoassociative envelopes AxRxB and BxSxA, that is, he reduced the representation theory of the nonassociative product (A, B) to that of its two, right and left envelopes with "isoassociative" product AxRxB and BxSxA [V-5], in essentially the same way as the study of the Lie product [A, B] = AxB - BxA is reduced to that of the associative ones AxB and BxA.

The terms Santilli's Lie-admissible theory or genotheory are referred to the latter context, that is, to a dual left and right lifting of Lie's theory (enveloping associative lagebras, Lie algebras, Lie groups, representation theory, etc.).

The tool which permitted this formulation is that of a bi-representation (split-null extension) [V.5]. The main point is that bi-modular Lie-admissible structures are contained in the structure of CONVENTIONAL Lie’s groups. In fact, Eq.s (13) can be written [V-1]

(18a) Q(w) = {exp(ixXxw)} > Q(0) < {exp(-ixwxX)} (18b) i [Q(dw) - Q(0) ] / dw = W < X - X > W

where > means conventional modular-associative “action to the right” and < “action to the left”. The bi-modular character is trivial in Lie’s case because the action of a conventional Lie group from the left is minus the transpose action from the right. An important observation of Ref. [V-1] is that group structure (18) can also be written in a non-trivial bi-modular form characterized, first, by the isotopic modular actions to the right and to the left and, then, their differentiation into genotopic forms. To put it bluntly, a bimodular Lie-admissible structure is already contained in the conventional structure of Lie groups. It merely remained un-noticed until 1978. In fact, the modular associative product to the right can be realized via the right genoassociative algebra U> with product A>B = AxSxB and that to the left via the left genoassociative algebra <U with product A<B = AxRxB with corresponding genounits to the right and left E> = 1/S and <E = 1/R. Eq.s (18) then yield Santilli’s Lie admissible theory [V-1]

(19) Q(w) = {exp>(ixXxw)} > Q(0) < {exp<(-ixwxX)} = = {[exp(ixXxSxw)]xE>} x S x Q(0) x R x { (19 i [Q(dw) - Q(0) ] / dw = Q < X - X > Q =

= Q x R x X - X x S x Q = (Q, X)

In this way the representation theory of the Lie-admissible algebra with nonassociative product (A, B) is first reduced to a bi-representation theory on {<U, U>} and then shown to admit a Lie-admissible group structure in a way fully parallel to the conventional Lie case. In an evening seminar delivered at ICM94 Santilli completed his Lie-admissible theory by showing that the algebra with Product (A, B) = AxRxB - BxSxA does indeed verify the Lie axioms (antisymmetry and Jacobi law), provided that the terms A<B and B>A are represented in their respective genoenvelopes <U and U>over corresponding genofields <F and.

To understand better how the Lie-admissible product (A, B) = AxRxB - BxSxA, with R different than S, can be antisymmetric, recall that conventional Lie algebra admit one single realization, that on conventional spaces and fields (read: with respect to the trivial unit I = Diag.(1, 1, ...1)); the isoalgebras admit instead a dual realization, that on isospaces over isofield (read: with respect to the isounit E) as well as the projection on conventional spaces over conventional fields (read: with respect to the conventional unit I); for the genoalgebras we have essentially the a similar occurrence, namely, they can be computed on the right and left genospaces over right and left genofields (read: right and left genounits E> and <E), in which case the product (A, B) verifies the Lie axioms, or it can be computed in its projection in conventional spaces and fields (read: with respect to the conventional unit I), in which case the product (A, B) is manifestly non-Lie.

Equivalently, the Lie character of the product (A, B) = A<B - B>A on genospaces over genofields can be seen from the fact that the lifting of the associative envelope AxB -> A>B = AxSxB is compensated by an INVERSE lifting of the unit I -> E> = 1/S, thus preserving the original structure (i.e., U and U> are isomorphic), and the same occurs for the right product (i.e., U and <U are also isomorphic). Thus, at the abstract, realization-free level, the product (A, B) verifies the anti-commutative law and the Jacobi law

(20a) (A, B){<S,S>} = -(B, A){<S, S>} (20b) ((A, B), C){<S, S>} + ((B, C), A){<S, S>} + ((C, A), B){<S, S>} = 0.

PROPOSED RESEARCH V-4 : Study the Lie-admissible theory and its isodual with particular reference to: the genostructure theory, the transition from the genoalgebras to related genogroups; the representation theory; etc. SIGNIFICANCE: Lie algebras are the algebraic counterpart of conventional geometries; Lie-Santilli isoalgebras are the algebraic counterpart of the isogeometries; and, along similar lines, genoalgebras are the algebraic counterpart of the genogeometries. The conventional, modular representation theory of Lie algebras characterize particles in linear, local, canonical and reversible conditions; the isomodular representation theory of Lie-Santilli isoalgebras characterize particles in nonlinear, nonlocal and noncanonical but still reversible conditions; the bi-modular representation theory of the genoalgebras characterizes particles in nonlinear, nonlocal, noncanonical as well as irreversible conditions, such as a neutron in the core of a neutron star. The most advanced definition of “particle” in physics, admitting all other as particular case, including those characterized by string and supersymmetric theories, is a bi-isomodular representation of the Lie-admissible covering of the Poincare’ symmetry.

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PROBLEMS V-6/V-7: STUDIES ON THE LIE-SANTILLI HYPERTHEORY AND ITS ISODUAL A further advance of the recent memoir [I-1] is that the genounits E> and <E can be ordered sets of nonhermitean quantities under ordinary operations, in which case the Lie-admissible theory becomes multivalued yet still preserving the original Lie axioms at the abstract level. A form of hyper-Lie theory defined via hyperoperations was also introduced by Santilli and Vougiouklis in ref. [II-2]. PROPOSED RESEARCH V-5: Study the multivalued realization of the Lie-admissible theory, first, with ordinary operations, and then with hyperoperations.

SIGNIFICANCE: Besides the evident mathematical significance, multi-valued spaces have already seen their appearance in cosmology, and their need in biology is now established in view of the complexity of biological systems.

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REFERENCES FOR SECT. V: [V-1] R. M. Santilli, Foundations of Theoretical Mechanics, Vol. II, Springer-Verlag, 1983; [V-2] D. S. Sourlas and Gr. Tsagas, Mathematical Foundations of the Lie-Santilli Theory, Ukraine Academy of Sciences, Kiev, 1993. [V-3] J. V. Kadeisvili, An introduction to the Lie-Santilli isotheory, Rendiconti Circolo Matematico Palermo, Suppl. No. 42, pages 83-136, 1996. [V-4] R. M. Santilli, Nonlinear, nonlocal and noncanonical, axiom-preserving isotopies of the Poincare’ symmetry, J. Moscow Phys. Soc. Vol. 3, pages 255-297, 1993. [V-5] R. M. Santilli, Initiation of the representation theory of Lie-admissible algebras on a bimodular Hilbert space, Hadronic J. Vol. 3, pages 440-506, 1979.

RECENT PAPERS APPEARED IN THE RESEARCH PROPOSED OF SECT. V

STUDIES ON THE CLASSIFICATION OF LIE-SANTILLI ALGEBRAS Gr. Tsagas Department of Mathematics Aristotle University Thessaloniki 54006, Greece Algebras, Groups and Geometries Vol., 13, pp. 129-148, 1996

AN INTRODUCTION TO THE LIE-SANTILLI NONLINEAR, NONLOCAL AND NONCANONICAL ISOTOPIC THEORY J. V. kadeisvili Institute for Basic Research P. O. Box 1577 Palm Harbor, FL 34682 U.S.A. (br> ibr@gte.net Mathematical Methods in applied sciences Vol. 19, pp.1349-1395, 1996

REMARKS ON THE LIE-SANTILLI BRACKETS M. Nishioka Yamaguchi University Yamaguchi 753, Japan "New Frontiers in Algebras, Groups and Geometries", Gr. Tsagas, Editor, Hadronic Press, Palm Harbor (1996), pp.553-558.

COMMENTS ON A RECENT NOTE BY MOROSI AND PIZZOCCHERO ON ON SANTILLI'S ISOTOPIES OPF LIE'S THEORY J. V. kadeisvili Institute for Basic Research P. O. Box 1577 Palm Harbor, FL 34682 U.S.A. (br> ibr@gte.net Submitted for ppublication