## CHAPTER 3: |

Duncan | Medford | Liberty | Jamaica Plain | Cymbalta Reseda | Cymbalta San Benito | Cymbalta orsica | Saint Clairsville | Mobridge |

- coupon for cymbalta medication
- what will generic cymbalta cost
- cost of cymbalta in canada
- cost of generic cymbalta without insurance
- coupons for cymbalta medication
- cheapest price cymbalta 60 mg
- coupon for cymbalta with insurance
- cymbalta generic best price

Cymbalta 180 Pills 20mg $369 - $2.05 Per pill

Cymbalta 180 Pills 40mg $569 - $3.16 Per pill

Cymbalta 60 Pills 30mg $179 - $2.98 Per pill

Cymbalta 90 Pills 30mg $259 - $2.88 Per pill

Cymbalta 90 Pills 30mg $259 - $2.88 Per pill

Cymbalta price 60 mg zyban for depression ireland generic dapoxetine - priligy cost of generic cymbalta without insurance. Can you buy cialis in canada over the counter is cialis over the counter in canada cymbalta cost of over the counter cialis in canada. Buy antabuse online canada brand cialis online us pharmacy cymbalta cost in canada cialis price us pharmacy buy antabuse canada generic cialis au cymbalta vs effexor cost. Can i buy cialis over the counter in canada cymbalta price at cvs can cialis be purchased over the counter in canada can you buy cialis over the counter in canada. Cymbalta cost uk cymbalta price in canada cialis us pharmacy cymbalta discount prices is cialis sold over the counter in canada cialis generico consegna 24 ore. Can cialis be bought over the counter in canada is there real generic cialis generic cialis aurochem. Cialis generico al miglior prezzo generic cialis with dapoxetine cialis online from us pharmacy cialis generico consegna in 24 ore. Cymbalta cost per month zyban in ireland over the counter cialis canada is there a real generic cialis cialis australia generic cialis in canada over the counter. Zyban price ireland cialis us online pharmacy cymbalta price at costco pharmacy cymbalta prescription prices is cialis an over the counter drug in canada. Cymbalta cost united healthcare price of cymbalta generic cialis generico prezzo migliore how much does cymbalta cost in australia cymbalta price 2014 is zyban available in ireland. Cialis generico uso diario Tadalafil generic usa order antabuse canada priligy generico dapoxetine cost of zyban in ireland. Cymbalta cost comparison generic cialis us pharmacy cymbalta price rite aid best price for cymbalta 60 mg cialis generico aurochem can you purchase cialis over the counter in canada. Can you buy cialis over the counter in ontario price of cymbalta at cvs online pharmacy cialis united states. Generic tadalafil dapoxetine cymbalta dosage cost cymbalta price cvs over the counter generic cialis generic medications similar to cymbalta. Where to buy cialis over the counter in canada coupons for cymbalta 20 mg is cialis available over the counter in canada. Cialis generico uk price of cymbalta without insurance cialis generico prezzo piu basso cialis from us pharmacy. Online cialis us pharmacy cialis over the counter canada where to buy antabuse in canada generic levitra with dapoxetine. Cialis over the counter in canada Cymbalta 180 Pills 40mg $569 - $3.16 Per pill generic cialis w dapoxetine how much does generic cymbalta cost at walmart. Cymbalta 60 mg price canada cialis pharmacy usa cialis uk generic american made generic cialis generic viagra with dapoxetine generic viagra canada online pharmacy. Order cymbalta from canada generic levitra mit dapoxetine generic viagra w dapoxetine generic priligy dapoxetine 60 mg cymbalta price cvs. Real generic cialis where to buy antabuse canada retail price cymbalta 60 mg cheapest us pharmacy for cialis

- Cymbalta in Penticton
- Cymbalta in Leonora
- Cymbalta in South carolina

Cialis gÃ¼nstig kaufen deutschland cialis 5mg rezeptfrei kaufen in deutschland can i buy aciphex over the counter generic equivalent of flovent. Generic cialis for sale online flovent inhaler generic cialis 5mg rezeptfrei in deutschland kaufen haldol generic drug. Cymbalta and effexor same drug generic cialis online us cymbalta generic cost coupon for cymbalta prescription generic viagra cialis online. Buy aciphex online cheap how much does cymbalta cost 2012 doxazosin pill identification doxazosin pill picture what does cymbalta cost without insurance. Buy cialis generic online generic cialis online cheap aciphex buy online buy aciphex cheap buy generic cialis canada online cymbalta drug uses flovent inhaler generic price. Haldol decanoate generic name what is the generic name of the antipsychotic medication haldol where to buy generic aciphex. Haldol generic generic cialis viagra levitra online cialis rezeptpflichtig deutschland Generic viagra sale online cost of cymbalta in ireland. Cymbalta clonazepam drug interactions cymbalta drug manufacturer cymbalta interactions with other drugs cost of cymbalta australia generic cialis online uk. Can you buy aciphex over the counter cymbalta xanax drug interactions cymbalta cost in australia drug interactions between cymbalta and phentermine. Doxazosin pill shape cost of cymbalta prescription buy aciphex 20mg flovent hfa 110 mcg generic cialis generika in deutschland rezeptfrei kaufen. Cialis generika kaufen in deutschland flovent diskus generic name Amoxicillin potassium clavulanate cost how much does cymbalta cost with insurance. Cymbalta cost ireland flovent inhaler generic name cost of cymbalta brand how much does cymbalta cost in ireland

Cymbalta Beech Grove | Cymbalta Mount Holly | Clinton | Incline Village | Forest Hill |

Mount Carmel | Mentor On The Lake | Beachwood | Caldwell | Kingwood |

Savage | Morganfield | VohenstrauÃŸ | Skillman | Marsberg |

cost of cymbalta in mexico

online canadian pharmacy discount code

generic viagra canada pharmacy

largest online pharmacy in canada

online courses for pharmacy technician in canada

generic cost of cymbalta

Cymbalta 20 mg dose prilosec otc 84 count diflucan tablets over the counter diflucan over the counter mexico. Generic cymbalta dosage weight loss pills and cymbalta buy prilosec otc online prilosec otc usage is it ok to take prilosec otc while pregnant. Order cymbalta 60 mg online what is dosage of prilosec otc prilosec otc and klonopin best online pharmacy in canada. Prilosec otc and gastritis cymbalta generic dose is there an over the counter medicine like diflucan prilosec otc 14 day treatment. Prilosec otc gerd cymbalta dosage 180 mg prilosec otc 20 mg tablet diflucan over the counter nz buy cheap dapoxetine online. Prilosec otc good for gerd prilosec otc and kidney stones prilosec otc and ulcer prilosec liquid otc prilosec otc vs generic omeprazole reviews on prilosec otc. Cymbalta order online canada cymbalta dosage 15 mg prilosec otc take every day cymbalta dosage compared to lexapro. Prilosec otc dosage forms prilosec otc rite aid antabuse purchase online prilosec otc and breastfeeding prilosec otc headaches. Prilosec otc cost cvs Drugstore bronzer uk prilosec 40 mg otc price difference between prescription prilosec and prilosec otc prilosec otc target. Cymbalta price per pill cialis buy online europe cymbalta 30 mg dosage Dapoxetine 60mg uk cymbalta dosage 30 mg order cymbalta online cymbalta generic dosage. Cymbalta 20 mg dosage buy viagra generic online cymbalta 30 mg dose dapoxetine online purchase uk prilosec otc for ulcer treatment. Buy cialis soft tabs online purchase antabuse online prilosec otc zantac 150 cymbalta 60 mg dosage cymbalta and weight loss pills cymbalta dosage neuropathic pain. Prilosec otc dosage vs prescription cymbalta online order Nombre generico del prozac prilosec otc and constipation prilosec otc dosage cost of cymbalta per pill. Order generic cymbalta online contraindications of prilosec otc buy dapoxetine priligy online can you get oral diflucan over the counter.

The main references on hadronic mechanics are the following: the analytic foundations were treated in the two monographs of 1978 and 1982 hereon referred to as

**"Foundations of Theoretical Mechanics, I: The Inverse Problem in Newtonian Mechanics"
R. M. Santilli, ** Springer-Verlag (1978)

**"Foundations of Theoretical Mechanics, II: Birkhoffian Generalization of hamiltonian Mechanics"
R. M. Santilli,** Springer-Verlag (1982);

The first comprehensive axiomatically consistent treatment of hadronic mechanics can be found in the two monographs hereon referred to for brevity **1995 EHM Volumes I and II**

**"Elements of Hadronic Mechanics", Vol. I: "Mathematical Foundations"
R. M. Santilli**, Ukraine Academy of Sciences (1995),

**"Elements of Hadronic Mechanics"
Vol. II: "Theoretical Foundations"
R. M. Santilli,** Ukraine Academy of Sciences (1995)

A recent Lie-admissible formulation of hadronic mechanics can be found in the memoir published by the Italian Physical Society

**
Lie-admissible invariant representation of irreversibility for matter and antimatter at the classical and operator level
Ruggero Maria Santilli**
Nuovo Cimento B Vol. 121, p. 443-595 (2006)

and the most recent presentation is available in the five volumes hereon referred to as **2008 HMMC Volumes I, II, III, IV, V**

** Hadronic Mathematics, Mechanics and Chemistry, Volumes I, II, III, IV and V:
R. M. Santilli,** International Academic Press (2008)

**
Iso-, geno-, hyper-mechanics for matter, their isoduals for antimatter, and their novel applications to physics, chemistry and biology
R. M. Santilli,**
Journal of Dynamical Systems and Geometric Theories, Vol. 2, pages 121-194 (2003)

**
"Santilli's Lie-Isotopic Generalization of Galilei and Einstein Relativities"
A. K. Aringazin, A. Jannussis, F. Lopez, M. Nishioka and B. Veljanosky,**
Kostakaris Publishers, Athens, Greece (1991)

**
"Santilli's Isotopies of Contemporary Algebras
Geometries and Relativities"
J. V. Kadeisvili,**
Ukraine Academy of Sciences
Second edition (1997)

**
"Mathematical Foundation of the Lie-Santilli Theory"
D. S. Sourlas and G. T. Tsagas,**
Ukraine Academy of Sciences (1993)

**Prizes and nominations**

Santilli has received large financial rewards from the new industrial applications of hadronic mechanics in physics, chemistry and biology. He has been listed by the Estonia Academy of Sciences among the most illustrious applied mathematicians of all times because of his discovery of the Lie-admissible covering of all of 20th century mathematics that encompasses all possible mathematics with an algebra (Chapter 2) and, consequently, all possible physical and chemic;l theories with an algebra in the brackets of their time evolution (Chapters 3-9), the listing of Santilli name being done with the quotation his 1967 initiation paper on Lie-admissible algebras jointly with the names of Gauss, Hamilton, Lie, Jordan, Wigner, and others very famous mathematicians (the only name of Italian origin appearing in the list). A motivation has been that

* .... several other mathematicians have discovered individual mathematical structures, for instance, Hamilton discovered the quaternions, Jordan discovered his algebras, and Lie discovered his theory, but no other mathematician in history discovered, as Prof. Santilli did, structural generalizations of the totality of mathematics in sequential series [isotopic, genotopic, hyperstructural and isodual].*

Additionally, a lecture room at a research center in Australia has been called "Santilli Hall."Besides various gold medals for scientific merits, Santilli has received in January 2009 the prestigious prize of the Mediterranean Foundation, previously granted to Price Albert of Monaco , France President Nicolas Sarkozy, Juan Carlos King of Spain, international architect Renzo Piano, and other famous people. Finally, Santilli has received hundreds of nominations for the Nobel prize in physiscs because of the construction of hadronic mechanics and more recently also for the Nobel prize in Chemistry. For details, one may visit the web site

**
Prof. Santilli's prizes and nominations**

**Acknowledgments**

Jointly with the completion in 1997 of the formal construction of hadronic mechanics and its primary experimental verifications as well as applications in the 1997 paper

**
Relativistic hadronic mechanics: nonunitary, axiom-preserving
completion of relativistic quantum mechanics
R. M. Santilli,**
Found. Phys. Vol. 27, 625-729 (1997)

Santilli released a rather vast acknowledgment to all institutions, journals and colleagues who helped the, or were exposed to the construction of hadronic mechanics. The Foundation has retrieved the preprint and provides below the original version of the Acknowledgments since they had to be reduced in the published version by editorial request.

*1996 SANTILLI ACKNOWLEDGMENTS
FOR THE CONSTRUCTION OF HADRONIC MECHANICS, *

released in the preprint

Relativistic hadronic mechanics: nonunitary, axiom-preserving completion of relativistic quantum mechanics

R. M. Santilli, IBR preprint TH-06-25 (1996)

For the final version, download the published version

*It is a pleasant duty to express my sincere appreciation to th referees of Foundations of Physics for a very accurate control of the manuscript and for simply invaluable critical suggestions.*

It is also a duty to express my appreciation to a number of institutions, journals and colleagues for hospitality and invaluable help during the laborious studies in the construction of hadronic mechanics and its verification conducted during the past three decades.

First, I would like to thank the following Institutions:

The University of Naples, Italy, where I conducted my undergraduate studies in physics for an unforgettable human and scientific experience. I want to remember and thank in particular my mathematics teacher Renato Caccioppoli for propagating to be his passion for mathematics that set the direction of the rest of my scientific life.

The Department of Physics of the University of Torino, Italy, where I put the foundations of hadronic mechanics\ in the late 1960's as part of my Ph. D. thesis;

The Avogadro Institute in Torino, Italy, that gave me a chair in nuclear physics when quite young, with various students still remembering and tracing me down to this day;

The Center for Theoretical Physics of the University of Miami, Coral Gables, Florida, where I had a very enjoyable stay during the academic year 1967-1978;

The Department of Physics of Boston University where I taught, from prep courses to post Ph. D. Seminar courses in mathematics and physics from, 1968 to 19074;

The Center for Theoretical Physics of the Massachusetts Institute of Technology, where most background technical preparation was conducted in the mid 1970's, such as the papers on the existence and construction of a Lagrangian in field theory, the paper on the identification of gravitational and electromagnetic interactions, the preliminary versions of monographs published by Springer Verlag, and other studies;

The Department of Mathematics of Harvard University, were the main papers proposing the construction of hadronic mechanics and numerous other works were written in the late 1970's and early 1980's under support from the U. S. Department of Energy;

The Joint Institute for Nuclear Research, Dubna, Russia, for summer hospitality in recent years, where several papers were written, such as the crucial paper on isonumbers, genonumbers and their isoduals, the paper on the synthesis of the neutron first appeared as JINR Communication Number E4-93-352, and other papers,

The Institute for High Energy Physics, Protvino-Sherpukov, Russia, also for summer hospitality in recent years, where the most innovative studies in gravitation were initiated,

The International Center for Theoretical Physics in Trieste, Italy, for a short visit in 1992;

CERN, Geneva, Switzerland, also for a short stay in 1992;

The Institute for Basic Research on Harvard Grounds from 1982 to 1989 and then in Palm Harbor Florida from 1989 to present where the main research on hadronic mechanics has been conducted and continued to this day;

and numerous other Institutions for shorter stays.

I would like to express my appreciation for recent hospitality I received for presentations on various aspects of hadronic mechansics at the following meetings (up to 1996):
Three Workshops on Lie-admissible Formulation, Harvard University, 1978-1981;

International Conference on nonpotential interactions and their Lie-admissible treatment, University of orleans, France, 1982;

Nine Workshops on Hadronic mechanics from 1981 to present held at various institutions in the Boston, Area (USA), Belgrad (Yugoslavia), Patras (Greece), Como (Italy), London (England), Beijing (China), and other locations;

International Workshop on Symmetry Methods in Physics, J.I.N.R., Dubna, Russia, July 1993;

Third International Wigner Symposium, Oxford University, Oxford, England,September 1993;

International Conference

XVI-th [1993], XVII-th (1994) and XIX-th (1996) International Workshop on High Energy Physics and Field Theory, I. H. E. P., Protvino-Sherpukov, Russia, September 1993;

International Conference on the Frontiers of Fundamental Physics, Olympia, Greece, September 1993;

VI-th Seminar on High Temperature Superconductivity, J.I.N.R., Dubna, Russia, September 1993;

Seventh Marcel Grossmann Meeting on General Relativity and Cosmology, Stanford University, Stanford, CA, U.S.A., July 1994;

1996 Sanibel Symposium, St. Augustine, Florida, March 1995 and February 1996;

First Meeting for the Saudi Association for Mathematical Sciences, Riyadh, Saudi Arabia, May 1994;

International Conference on Selected Topics in Nuclear Structure, J.I.N.R., Dubna, Russia, July 1994;

International Workshop on Differential Geometry and Lie Algebras, Thessaloniki, Greece, December 1994;

HyMag Symposium, National High Magnetic Field Laboratory, Tallahassee, Florida, December 1995;

International Workshop on new Frontiers in Gravitation, Istituto per la Ricerca di Base, Castle Prince Pignatelli, Monteroduni, Italy, August 1995;

National Conference on Geometry and Topology, Iasi, Rumania, September 1995
International Symposium for New Energy, Boulder Colorado, April 1996;

International Workshop on the Gravity of Antimatter and Anti-Hydrogen Atom Spectroscopy, Sepino, Italy, May, 1996;

Workshop on Differential geometry, Palermo, Italy, June 1996;

International Workshop on Polarized Neutrons, J.I.N.R., Dubna, Russia, June 1996.

Special thanks are also due for the recent opportunity of delivering lectures or short seminar courses on the various aspects of hadronic mechanics at:
Moscow State University,Moscow, Russia, August 1993;

Estonia Academy of Sciences, Tartu, August 1993;

Theoretical Division, J.I.N.R., Dubna, Russia, September 1993; August 1994; August 1995; August 1996;

Ukraine Academy of Sciences,Kiev, September 1993;

Institute for Nuclear Physics, Alma Ata, Kazakhstan, October 1993;

Institute for High Energy Physics, Protvino, Russia, June 1993, June 1994, June 1995;

Theoretical Division, C.E.R.N, Geneva Switzerland, December 1994;

Department of Mathematics, Aristotle University, Thessaloniki, Greece;

Department of Mathematics, King Saud University, Riyadh, Saudi Arabia;

Demokritus Institute, Athens, Greece, December 1994;

Institute of Nuclear Physics. Democritos University of Thrace Xanthi, Greece, December 1994;

Institute for Theoretical Physics, Wien, Austria, December 1994;

Department of Mathematics, University of Constanta,Romania, September 1995;

Research Center COMSERC, Howard University, Washington, D.C., U.S.A. April, 1995;

Department if Mathematics, Howard University, Washington, D. C., U.S.A., April 1995;

The International Center for Theoretical Physics (ICTP), Trieste, Italy, 1992;

Department of Nuclear Physics, University of Messina, Italy, June 1996;

Department of Mathematics, University of Palermo, Italy, June 1996;

Academia Sinica, Beijing, China, supper 1995;

The Italian national Laboratories in Frascati, Italy, 1977;

The Center for Theoretical Physics of the Massachusetts Institute of Technology, 1976

The Lyman Laboratory of Physics, Cambridge, MA, 1978, delivering a seminar course on the integrability conditions for the existence of a Lagrangian in Newtonian mechanics and field theory;

The University of illinois in bloomington, 1968;

Russia Academy of Sciences,Moscow, June 1996;

and other institutions in various countries. I have no word to express my sincere appreciation and gratitude to all colleagues at the above meetings or institutions for invaluable critical comments.

Additional thanks for the critical reading of parts of this paper are due to: M. Anastasiei, Yu. Arestov, A. K. Aringazin, A. K. T. Assis, M. Barone, Yu. Barishev, J. Ellis, T. Gill, J. V. Kadeisvili, A. U. Klimyk, A. Jannussis, N. Makhaldiani, R. Miron, M. Mijatovic, D. Rapoport-Campodonico, D. Schuch, G. T. Tsagas, N. Tsagas, C. Udriste, T. Vougiouklis, H. E. Wilhelm, and others.

Finally, this paper has been made possible by rather crucial publications appeared in the following Journals, here acknowledge with sincere appreciation:

Foundations of Physics, for publishing: this memoir, the first after the achievement of axiomatic maturity in relativistic hadronic mechanics; the 1981 article on the apparent impossibility for quarks to be elementary at a time of widespread belief to the contrary; and several related articles in classical and operators studies not quoted for brevity;

Physical Review A, for publishing the important article by Schuch on the need for nonunitary treatment of nonlinear operator systems;

Physical Review D, for publishing the 1981 article on the need to verify the validity of Pauli's principle under nonconservative conditions due to external strong interactions;the 1978 article {3c} on the isotopies of electroweak interactions with a breaking of the gauge invariance; and other papers;

Hyperfine Interactions, for publishing the paper on the prediction of a novel light emitted by antimatter;

Nuovo Cimento, for the publication of: the 1967 article on the first Lie-admissibity in the physical literature; the 1983 article on the first isotopies of Minkowski spaces, the Lorentz symmetry and the special relativity the 1983 article {4f} on the first operator realization of isosymmetries via a lifting of WignerÂs theorem; the 1982 article on the first Lie-admissible time-irreversible formulation of open strong interactions; the article on the first isotopies of SU(3), article, the scattering theory, and several other seminal papers;

The (MIT) Annals of Physics, for the publication of the 1976 articles on the integrability conditions for the existence and computation of a Lagrangian in field theory, the 1982 article on the crucial identification of the gravitational and electromagnetic fields from the primary electromagnetic origin of mass (that subsequently rendered unavoidable the prediction of antigravity), and others;

Journal of Physics G, for publishing the 1981 articles on the rather crucial isominkowskian representation of the behavior of the meanlives of K-o with energy, and other papers;

Physica, for publishing the 1985 article on the possibility of regaining convergent perturbative series for strong interactions, and others;

Physics Essays, for publishing the 1992 article on classical realizations of Santilli's isogalilean relativity, and the article on the representation of the difference between cosmological redshifts of physically connected quasars and galaxies via Santilli's isospecial relativity, and others;

Communications in Theoretical Physics, for publishing a number of crucial articles, such as the first article on the isotopic quantization of gravity, the first article on the isoquark theory, the first article on the isodual representation of antimatter, the first article on the paradox of quantum mechanics at the limit of gravitational singularities, and several others;

Annales de la Fondation Louis de Broglie, for publishing the crucial articles on the limitations of current generalized theories, and others;

Revista Tecnica, for the publication of articles on the isotopies of Newtonian, analytic and quantum mechanics;

Journal of Moscow Physical Society, for the publication of the comprehensive 1993 article on the isotopies and isodualities of the Poincare~ symmetry, including the universal symmetry of all possible Riemannian and Finslerian line elements, which is the single most important paper of these studies from which all results can be uniquely derived;

J.I.N.R. Rapid Communications for the publication of the crucial 1993 article on the isotopies of SU(2)- spin with the isopauli's matrices and the reconstruction of the exact isospin symmetry in nuclear physics;

International Journal of Quantum Chemistry, for the publication of the crucial 1981 article on the application and experimental verification of hadronic mechanics to superconductivity, with the first attractive force among two identical electrons in singlet couplings at mutual distances smaller than their coherent length;

Chinese Journal of Systems Engineering and Electronics, for the publication of the crucial 1995 article on the isotopies of the spinorial covering of the Poincare~ symmetry and of DiracÂs equations, with application to the synthesis of the neutron from protons and electrons only, and other articles;

Mathematical Methods in Applied Sciences, for the publication of the recent comprehensive study {5g} by Kadeisvili on the Lie-Santilli isotheory and related methods;

Rendiconti Circolo Matematico di Palermo for the publication of an entire 1996 issue of of their Supplemento entirely dedicated to the new mathematics underlying hadronic mechanics;

Acta Applicxandae Mathematica for the publication in 1995 of th crucial application of hadronic mechanics to Bell's inequality, the isotopies of the SU(2) spin symmetry and all that;

The Indian mathematical Society, for the publication of numerous seminal papers in pure and applied mathematics at the foundation of hadronic mechanics.

,p>
and other Journals.

Particular thanks are additionally due to all past and present Editors of the Hadronic Journal and Algebras, Groups and Geometries for their continued encouragement, support and control of various publications quoted in this paper.

Additional thanks are due to the participants, editors and and publishers of the Proceedings of some eighteen international workshops and conferences held in the field of hadronic mechanics in the USA, Europe, and China resulted in a total of over thirty volumes, which are too numerous to mention here individually.

I must also express my utmost gratitude to G. F. Weiss, S. Smith and P. Fleming, staff of our Institute for basic Research in Palm Harbor, Florida, and numerous other members and visitors through the years, for simply invaluable help, assistance and control in the preparation of this manuscript.

It is also my pleasant duty to thank several colleagues for their invaluable contributions in the construction of the hadronic mechanics, particularly during the early years of its study, including: S. Okubo, H. C. Myung, R. Mignani, F. cardone, A. K. Aringazin, A. Kalnay, A. O. E. Animalu, D. Schuch, T. L. Gill, Gr. Tsagas, D. S. Sourlas, J. V. Kadeisvili, E. B. Lin, M. Nishioka, A. Jannussis, G. Eder, J. Fronteau, M. Gasperini, D. Brodimas, P. Caldirola, M. Mijatovic, Y. Prigogine, K. Popper, B. Veljanoski, A. Tellez-Arenas, and others.

I cannot close these Acknowledgments without expressing my appreciation to the American, British, Italian, Swedish, French, German, Russian, Chinese physical and other societies for their role in the construction of hadronic mechanics, On my side, I would lille to indicate that. when facing truly fundamental structural advances of pre-existing knowledge as it is the case here, the "burden of proof" on their validity belongs to the author(s) and definitely not to the societies, since their historical as role is that of exercising caution for the very protection of science. On the other side, scientific societies are suggested to exercise tolerance when attacked for insufficient scientific democracy at the time when the battle for new scientific vistas reaches its climax.

I cannot close these Acknowledgments without expressing my deepest appreciation to the United States of America for being so generous to me and my family, by permitting me to realize my scientific dreams on hadronic mechanics as well as my personal dreams in the American way of life, sports cars and boats, generosity that has caused in me an unbounded allegiance.

It is a truism to say that. without my conduction of research in the U.S.A., hadronic mechanics would not have been completed and established because, even though its main lines had been conceived in Italy, the realization of the above indicated "burden of proof" required "experiumental verifications and novel industrial applications relevant to society" that would have been of difficult realization elsewhere because they must be achieved nowadays outside academia whenever dealing with basic advances over pre-established doctrines, as well known to insiders.

On my part, I considered myself a "special immigrant" because: I came here: from a rich Italian family, my father Ermanno Santilli being an Italian Medical Doctor and my grandfather Ruggero Santilli being an Italian industrialist; after achieving in Europe the highest possible education in mathematics physics and chemistry; and while being the recipient of a chair in nuclear physics at the Avogadro Institute in Torino.

The construction and proof of hadronic mechanics were possible "by" (rather than "in") the U.S.A. amidst incredible, well known and documented academic obstructions (at time reaching true levels of hysteria against the surpassing of beloved old doctrines), because of: the inspired values of the U. S. Constitution, the best throughout history I ever read; the crucial democracy of its Institutions; and its unique multitude of overlapping social, governmental and industrial structures offering people a variety of ways to realize their dreams, but only following fierce determination, relentless commitment and true values.

Most special thanks are finally due to my wife Carla for her grace, class, patience and support in enduring predictable obstructions in the conception, completion and proof of hadronic mechanics.

Needless to say, I am solely responsible for the content of this paper owing to the numerous changes occurred during the preparation of the final version.

** 3.11C. Interior and exterior dynamical systems**

As santilli recalls, physical systems were classified by Lagrange, Hamilton, Jacobi and other founders of mechanics into:

1) *Exterior dynamical systems,* consisting of a finite number of point-like particles moving in vacuum (conceived as empty space) without collisions. Note that the lack of collisions is sufficient to admit an effective point-like approximation of particles and, vice versa, the assumption of a point-like structure implies the tacit assumption of lack of collisions since dimensionless points cannot collide. Typical classical examples are given by the Solar system or a spaceship in orbit around Earth in vacuum since in both cases the actual size and shape of the constituents (the planets or the spaceship) do not affect the dynamical evolution, and said constituents can be well approximated as massive points. Typical particle counterparts are given by the atomic structure, particles in accelerators, crystals and other systems admitting a good approximation of the constituents as being dimensionless. Note also that *all exterior systems are purely Lagrangian or Hamiltonian,* in the sense that the knowledge of only one quantity, a Lagrangian or a Hamiltonian, is sufficient to characterize the entire dynamics.

2) *Interior dynamical systems,* consisting of a finite number of constituents moving within a physical medium, in which case point-like abstraction are no longer valid, since the actual size and shape of the constituents has direct implications in the dynamical evolution. Typical classical examples are given by the structure of a planet such as Jupiter or a spaceship during re-entry in our atmosphere. Typical particle examples are given by the structure of the Sun or, along similar lines, the structure of nuclei and hadrons since, in all these cases, motion of one constituent occurs within the medium characterized by the wavepacket of other surrounding constituents. Note that *interior systems are non-Lagrangian and non-Hamiltonian,* in the sense that a given Lagrangian or Hamiltonian is insufficient to characterize the dynamics due to the need for a second quantity characterizing the contact interactions represented with external terms in the analytic equations (1.2).

As reviewed in Section 3.9, the above classification was eliminated in the 20th century by organized interests on Einsteinian doctrines via the abstraction of all particles as being point-like, consequential elimination of the contact non-Lagrangian or non-Hamiltonian interactions, and consequential elimination of interior dynamical systems.

As indicated in Section 1.1, the first and perhaps most fundamental scientific contribution by Santilli has been to prove via Theorem 1.1 that the above abstraction was a figment of academic imagination. In any case, the inconsistency of most of the 20th century particle physics can be unmasked by noting that both elastic and inelastic scattering events are impossible for dimensionless particles by conception, again, because dimensionless particles cannot influence the trajectories of other dimensionless particles except for Coulomb interactions. Alternatively, the experimental evidence of deflection of trajectories in scattering processes from a purely Coulomb behavior is evidence on the existence of non-Lagrangian and non-Hamiltonian interactions precisely according to Theorem 1.1.

It is evident that Santilli's studies, including those on hadronic mechanics, specifically refer to *interior* dynamical systems that will be the sole system considered hereon. As we shall see, the second quantity needed for the representation of size, shape and dynamics of interior systems will be given by the isounit. Hence, * special relativity and quantum mechanics are hereon assumed as being exactly valid for exterior dynamical systems, and Santilli's isorelativity and hadronic mechanics are hereon assumed as being exactly valid for interior dynamical systems with unique and unambiguous interconnecting limits characterized by the isounit alone.*

For references in the above classification, including an accurate historical analysis, we refer the serious scholar to the 1995 FTM Volumes I and II. An instructive reading in the topic of this section is also that of Santilli's ICTP paper

**
Inequivalence of exterior and interior dynamical problems
R. M. Santilli,**
ICTP preprint # IC/91/258 (1991)

published in "Santilli's 1991 Papers at the ICTP", International Academic Press (1992)

**3.11D. Closed and open dynamical systems**

Lagrange, Hamilton, Jacobi and other founders of mechanics introduced the following additional classification of dynamical systems:

A) *Closed dynamical systems,* given by systems that can be well approximated as being isolated from the rest of the universe, thus verifying the ten conservation laws of total quantities characterized by the Galilei or the Poincare' symmetry (the conservation of the total energy, linear momentum, angular momentum and the uniform motion of the center of mass). This is typically the case for both exterior and interior systems, whether at the classical or operator levels, when isolated from the rest of the universe.

B) *Open dynamical systems,* given by system in interaction with an external component under which at least one of the ten Galilei's or Poincare' conservation laws is not verified due to exchanges of physical quantities between the system considered and the external component. Needless to say, when the external component is included, the open system is completed into a closed form.

Again, for the intent to adapt nature to Einsteinian and quantum theories, another widespread belief of the 20th century physics has been that "closed systems can solely admit conservative-potential forces" or, equivalently, that internal, contact, nonpotential interactions do not verify all ten Galilean or Poincare' conservation laws and, consequently, the contact-nonpotential forces "do not exist in particle physics".

The above belief has caused an alteration of physical research of historical proportions because the belief is at the foundation of some of the most equivocal assumptions of the 20th century physics, such as the belief that Einstein's special relativity and quantum mechanics are exactly valid for the structure of hadrons, nuclei and stars. The political argument (political because without a serious scientific basis) is that said systems verify the ten total conservation laws when isolated from the rest of the universe. Hence, the argument says, Einsteinian doctrines and quantum mechanics hold for their interior.

Santilli has disproved this additional academic belief with his notions of:

I) * Closed non-Hamiltonian system,* or, more technically, *closed variationally nonselfadjoint systems* (see Section 2.9), given by systems verifying the ten Galilean or Poincare's conservation laws, thus being closed, yet they admit internal forces that are Hamiltonian as well as non-Hamiltonian or, more technically, variationally selfadjoint (SA) and nonselfadjoint (NSA).

II) *Open non-Hamiltonian systems,* or *open variationally nonselfadjoint systems,* given by systems that do not verify at least some of the ten Galilean or Poincare' conservation laws due to non-Hamiltonian, or nonselfadjoint interactions with an external system. It is evident that these systems are *irreversible over time.*

In fact, Santilli proved in the 1982 FTM Volume II, page 235, that a Newtonian system of two or more particles with potential/selfadjoint and nonpotential/nonselfadjoint forces

verifies all ten conventional total conservation laws when the nonselfadjoint forces verify the following simple algebraic conditions

where ∗ and ∧ denote scalar and vector products, respectively.

The operator counterpart of closed non-hamiltonian system is easily provided by Santilli's Lie-isotopic theory (Section 2.7), in general, and the Galilei-Santilli or Lorentz-Poincare'-Santilli isosymmetry, because: the ten conventional generators, representing the ten total conserved quantities are preserved identically by the isotopic symmetries; the selfadjoint forces are represented by the Hamiltonian; and the nonpotential forces are represented by the isounit I^{*}(t, r, p, ...) = 1/T(t, r, p, ...), as we shall see. The totally symmetric character of the Lie-isotopic product [Q, H]^{*} = QTH - HTQ assures total conservation laws.

Nevertheless, *closed non-Hamiltonian systems admit internal exchanges of all physical quantities, that is, we have internal exchanges not only of the energy, but also of mass, charge, angular momentum, spin, etc.* without any conflict with total conservation laws since we merely have internal exchanges that compensate each other in their sum due to the isolated character of the system. As we shall see in the next chapters, this feature alone of hadronic mechanics has far reaching implications and applications mostly beyond our imagination at this writing.

The case of open non-Hamiltonian systems is the second fundamental class of systems studied by hadronic mechanics and includes all energy releasing processes. These systems require Santilli's Lie-admissible theory (Section 2.8), since the lack of totally antisymmetric character of the brackets (Q, H)^{*} = QRH - HSQ in the time evolution law (3.110) assures the description of *time rate of variations* of physical quantities of which conventional conservation laws are a particular case, in the same way as Santilli isoalgebras are a particular case of Santilli's Lie-admissible algebras.

The classical notion of closed non-Hamiltonian systems was introduced in the 1982 FTM Volume II, with the operator counterpart presented in various papers (see EHM and HMMC). An instructive reading is also that of the ICTO paper

**
Closed systems with non-Hamiltonian internal forces
R. M. Santilli,**
ICTP [preprint # IC/91/259 (1991)

published in "Santilli's 1991 Papers at the ICTP", International Academic Press (1992)

**3.11E. Newton-Santilli isoequations**

>From Theorem 1.1, the central problem addressed by Santilli was the achievement of a mathematically and physically consistent, classical and operator formulation of non-Hamiltonian (or variationally nonselfadjoint) forces, whose correct quantization had escaped all attempts during the 20th century. Santilli knew that such an objective cannot be achieved without an action principle, since the latter is crucial for a consistent map from classical to operator forms.

But, Newtonian systems with nonpotential forces F^{NSA}(t, r, v, ...) do not admit any action principle (when formulated with conventional mathematics). Thus, Santilli searched for an identical reformulation of Newton's equation (3.127) capable of admitting a covering action principle suitable for consistent maps to operator forms. It is at this point where the dimension of Santilli's scientific edifice can be appraised, since it encompasses a variety of discoveries in various branches of mathematics, physics and chemistry, all part of one single monolithic structure that will indeed resist the test of time due to its axiomatic consistency, beauty, experimental verification and industrial applications.

Santilli struggled for decades to reformulate Newton's equations into a form admitting a covering variational principle without success, until he discovered the iso-, geno- and hyper-differential calculus in the mid 1995, that allowed him to achieved a series of structural generalization of Newton equations since Newton's "Principia" of 1687, the first known to the Foundation (evidence of dissident views is solicited for presentation in this section). The broader equations are today known as **Newton-Santilli iso-, geno-, hyper- and isodual equations.** Regrettably, we can solely indicate here the Newton-Santilli isoequations and refer the scholar to the literature available in free download.

Let S_{tot}(t, r, p) = E(t, x, I_{t}) x E(r, x, I_{r}) x E(v, x, I_{v}) be the Kronecker product of the representation spaces for the Newton equations with time t, coordinates r and velocity v, conventional associative multiplication axb = ab, and units I_{t} = 1, I_{r} = I_{p} = Diag. (1, 1, 1). Santilli introduces the following isotopies of the Newtonian representation space with related isocoordinates, isoproducts and isounits (Section 2)

in the isotime, isocoordinates and isovelocities

with real-valued, positive-definite isounits

Then, the

namely,

for which Eqs. (3.134) become for the simpler one-dimensional case with n

with simple solution for v constant

from which endless examples can be derived.

To understand the advance over Newton's original conception, the serious scholar should note that the conventional Newton equations can only represent *point-like particles* due to the background local-differential topology and geometry, while the Santilli's covering equations represent *particles with their actual extended shape under the most general possible potential and nonpotential interactions,* due to the background novel isotopology.

Additionally, Santilli has provided the **genotopic, hyperstructural and isodual coverings of Newton's equations** for irreversible and multivalued matter systems and antimatter systems, respectively, that we cannot possibly review here.

Hence, to select the appropriate covering of Newtonian mechanics, one should identify whether the considered classical equations deal with: A) matter or antimatter; B) Closed or open systems; and C) Single-valued or multi-valued systems. Then, one should select the appropriate covering mechanics. Mathematically inclined scholars should know that Santilli has provided one single abstract formulation encompassing all possible * eight* different equations, including the conventional, iso-, geno-, hyper-systems and their isoduals, although such a unified treatment is not recommended for physical applications because excessively abstract.

Santilli's coverings of Newton's equations and mechanics can be studied in the 1996 RCMP memoir, and in EHM Volumes I and II.

**3.11F. Hamilton-Santilli isomechanics**

The embedding of the external terms in Lagrange's and Hamilton's equations in the generalized units, and the consequential regaining of a variationally selfadjoint formulation on isospaces over isofields, have far reaching implications. To begin, the true Hamilton's equations (1.2) are identically rewritten in the form known as **Hamilton-Santilli isoequations,**

namely,

Recall that Hamilton's equations with external terms do not characterize *any* algebra with the brackets of the time evolution, let alone violate *all* Lie algebras (Section 1.1). Via Eqs.(3.139), Santilli restores an algebra in the brackets of the time evolution with external terms, and this algebra results to be a Lie isoalgebra as a covering of the algebra for the truncated analytic equations. In fact, Eqs. (3.139) characterize the time evolution of a physical quantity Q(t)

whose brackets coincide with the conventional Poisson brackets at the abstract level.

Among an infinite number of *algebraic* solutions, a simple one is given by

for which

The first important consequence is that the Hamilton-Santilli isomechanics admits indeed an action principle. In fact, under the preceding simple realization Eqs. (3.139) can be derived from the

where one should note that the isoproduct for the space component is different than that for the time component.

The **Hamilton-Jacobi-Santilli isoequations** on isospaces over isofields expressed in terms of isocoordinates are given by

For open irreversible single-valued or multi-valued or antimatter systems we have the

Note from Section 3.11D that *the Hamilton-Santilli isomechanics is solely applicable to closed non-Hamiltonian systems,* trivially, because the antisymmetric character of the brackets of the time evolution imply the conservation of the Hamiltonian and other physical quantities.

Again, to select the appropriate covering mechanics, one should identify whether the considered system deals with: A) matter or antimatter; B) Closed or open systems; C) Single-valued or multi-valued systems. The selection of the appropriate mechanics is then consequential.

The topic of this section can be best studied in the 1996 RCMP memoir, or in EHM Volumes I and II.

**3.11G. Animalu-Santilli isoquantization**

The conventional naive quantization maps the Hamiltonian action into an expression depending on Planck's constant

thus setting the foundations for "quantized orbits" characterized by h/2π.

The map of the Hamilton-Santilli isoaction into an operator form was first identified by A. O. E. Animalu and R. M. Santilli at the XII Workshop on Hadronic Mechanics of 1990, it is today called the **Animalu-Santilli isoquantization, **and can be written

where one should note that I

under the subsidiary condition (verified naturally by all isounits used in hadronic mechanics)

Expressions (3.150), (3151) constitute the conceptual foundations of hadronic mechanics. Recall that, by central assumption, quantum mechanics is valid for the exterior problem of point particles in vacuum, while hadronic mechanics is assumed valid for the interior problem of extended particles moving within a medium composed by other particles, as expected for the constituents of hadrons, nuclei and stars, of course, according to different degrees of mutual penetrations.

Consequently, *map (3.150) represents the fundamental assumption of hadronic mechanics according to which Planck's constant becomes a locally varying operator representing the impossibility to have quantized orbits for an extended particle immersed within a hyperdense medium as it is the case, for instance, for an electron in the core of a star, under the condition (3.151) of recovering conventionally quantized orbits when motion returns to be in vacuum.*

Hence, the serious scholar accustomed to the usually quantized orbits for the structure of atoms should not expect the same quantized orbits in the interior of hadrons, nuclei or in the core of stars to avoid evident contradictions. More specifically, when a hadronic constituent is subjected to an excited orbit, that orbit is expected to be in vacuum, rather than in the interior of hadrons, thus belonging to quantum rather than hadronic mechanics. As we shall see in Section 4, this aspect is very insidious and confuses the problem of *classification* of hadrons generally searched via a *spectrum* of *quantum* states, with the *structure* of one individual hadron for which only one orbit is possible at mutual distances smaller than the size of the wavepackets of particles.

For references and a detailed presentation, the serious scholar is suggested to study EHM Volume II and HMMC Volume III. The original contribution by Animalu and Santilli is available from the pdf file

**title to be added
A. O. E. Animalu and R. M. Santilli,** in "Hadronic Mechanics and Nonpotential
Interactions," M. Mijatovic, Editor, Nova Science, New York, pp. 19--26 (l990).

**3.11H. Hilbert-Santilli isospaces**

The isotopic branch of hadronic mechanics is formulated on *Hilbert-Santilli isospaces* Η^{*} that are the image of conventional Hilbert spaces Η over a conventional field F under nonunitary transformations (see Section 3.xx below), with *isostates* |ψ^{*}), *isoinner product* defined on an isofield F^{*}

or

and related

A fundamental property is that, if an operator Q is Hermitean on Η over F, then it is *iso-Hermitean,* namely, it verifies the condition of Hermiticity on Η^{*} over F^{*},

Consequently,

Note that I^{*} is indeed the correct right and left unit of the isotopic branch of hadronic mechanics because it verifies the identities

with isoexpectation value

For details, extention to geno-, hyper- and isodual cases, and historical notes we refer the interested scholar to the 1995 EHM Volumes I and II.

**3.11I. Schroedinger-Santilli isoequations**

As indicated earlier, the first lifting of Schroedinger's equations was done by Santilli in 1979, and reinspected in various works. The final version was reached by Santilli in the 1996 RCMP memoir as part of the discovery of the differential calculus. The desired equations can be expressed via the image of the Hamilton-Jacobi-Santilli isoequations (3.145)-(3.147) under map (3.149). For the simple case of a constant isounit, or an isounit averaged to constant, the isoequation can be written

where all coordinates and their derivatives are isotopic (even if not indicated due to limitations of the hmtl language).

Via elementary calculations, the above equations can be written in the final form known as **Schroedinger-Santilli isoequations**

where: one should note the

The study of open irreversible single or multi valued matter systems and their antimatter counterparts requires the use of **Schroedinger-Santilli geno-, hyper- and isodual equations,** respectively, we cannot possibly review here.

Serious scholars are suggested to study EHM Volumes I and II and HMMC Volume III.

**3.11J. Heisenberg-Santilli isoequations**

The isotopies of Heisenberg's equations were discovered by Santilli in the 1978 original memoirs, their final version was also reached in the 1996 RCMP memoir jointly with the discovery of the isodifferential calculus, are today called **Heisenberg-Santilli isoequations,** and can be written for the time evolution of an iso-Hermitean operator Q(t) in the finite form (with simplifications of inessential isoproducts and the simple assumption I^{*}_{t} = 1)

with infinitesimal form easily derivable from the preceding expression (where we ignore again for simplicity the isotopy of time)

and

For details, we suggest study EHM Volumes I and II and HMMC Volume III.

**3.11K. Dirac-Myung-Santilli isodelta function and elimination of quantum divergencies**

One of the main limitations of quantum mechanics has been the emergence of divergencies, such as the divergent character of the perturbation theory for strong interactions, divergencies in Feynman's diagrams, and others. One of the main contributions of hadronic mechanics is the elimination of quantum divergencies *ab initio,* thus permitting, for the first time in scientific history, *convergent perturbative expansions for strong interactions.*

As it is well known, the origin of the divergencies in quantum mechanics rests with the point-like abstraction of particles, which abstraction is technically represented by the Dirac delta function δ(r - r_{o}) that is divergent at r = r_{o}. However, the image of the Dirac delta function in hadronic mechanics, today known as **Dirac-Myung-Santilli isodelta function** from a paper of said originators of 1982, is given by

where, as one can see,

Additionally, for any given divergent or weekly convergent series Q(w) = I + w (Q H - H Q)/1! + ... → ∞, I = 1 there always exists an isounit I^{*} = 1/T whose value (or average value) is much bigger than w (the isotopic element is much smaller than w) under which the above series becomes strongly convergent, namely, it verifies the expression where N is a finite positive number

The isodelta function was presented for the first time in the paper

**Foundation of the hadronic generalization of atomic mechanics, II: modular-isotopic Hilbert space formulation of strong interactions
H. C. Myung and R. M. Santilli,**
Hadronic Journal Vol. 5, pages 1277-1366 (1982).

The name of *Dirac-Myung-Santilli delta function* was introduced by M. Nishioka in the following paper of 1984

**Extension of the Dirac-Myung-Santilli delta function to field theory
M. Nishioka,** Lettere Nuovo Cimento Vol. 39, pages 369-372 (1984).

See also by the same author

**Realizations of hadronic mechanics
M. Nishioka,**
Hadronic J. Vol. 7, 1636-1679 (1984)

The above pioneering studies established the absence of quantum divergencies in hadronic mechanics and were followed by several studies reviewed in EHM Vol. II, including the convergence of isoperturbation expansions. The most recent contribution in the new scattering theory of hadronic mechanics (that will be reviewed in Chapter 5) is that of vthe paper.

**Nonunitary-isoscattering theory, I: Basic formalism without difergencies for low energy reversible scattering
A. O. E. Animalu and R. M. Santilli, **for the Procveedings of the 2008 Yard Conference, submoitted for publication.

**3.11L. Lie-admissible genotopic and hyperstructural branches of hadronic mechanics**

The starting point for the geno- and hyper-coverings of isomechanics is, again, Newton's equation, this time for the embedding of irreversibility in the mathematical foundations of the dynamics, via the genotopic lifting of the basic unit of the Euclidean space and related associative product among two generic quantities G_{k}, k = 1, 2, into two inequivalent formulations, one to the right and a complementary one to the left (see Section 2.8), where, again, the symbols f and b denote forward and backward dynamics, respectively,

with interconnection crucial for consistent time reversal images

in which case the right and left genounits are indeed the correct units for both products.

The next step is the selection of one direction in time, generally assumed to be the forward, and represent it with Santilli genomathematics to the right, that is, with genonumbers to the right, genospaces to the right, genogeometries to the right, etc. To avoid catastrophic inconsistencies often not noted by non-experts in the field, the above selection requires the religious restriction of *all* multiplication and other operations to the right.

Under the above foundations, we have the **Newton-Santilli genoequations to the right**

that, as one can see, is indeed irreversible because it is inequivalent to its time reversal image. Similarly, we have: the

related

action on a geno-Hilbert space to the right, and the

with corresponding genotopies of all remaining aspects of the isotopic branch of hadronic mechanics.

The **hyperstructural branch to the right** (primarily used for biological structures but also for multi-dimensional universes in physics) is essentially given by the above genotopic branch in which the genounits are assumed to be multi-valued, that is, to have a finite ordered set of values

with all multi-valued hyperstructures following from the above basic assumption on the fundamental unit.

A serious study of the above geno- and hyper-mechanics can only be achieved with a serious study of Santilli's 1996 RCMP memoir, the 1995 EHM Volumes I and II and the 2008 HMMC Volume III.

**3.11M. Isodual branches of hadronic mechanics**

Hadronic mechanics admits four different isodual branches for the representation of antimatter in conditions of increasing complexity according to the following classification:

1) **isodual quantum mechanics,** for the description of point-like abstractions of antiparticles in exterior dynamical conditions in vacuum (presented in Section 3.10);

2) **Isodual isomechanics,** for the description of closed non-Hamiltonian systems of extended antiparticles;

3) **Isodual genomechanics,** for the description of open systems of extended antiparticles; and

4) **Isodual hypermechanics,** for the description of multi-valued universes of antimatter..

All the above isodual mechanics can be constructed from the corresponding mechanics for matter via the application of the **isodual map**

to the

For a serious knowledge we suggest again the study of Santilli's 1996 RCMP memoir, the 1995 EHM Volumes I and II and the 2008 HMMC Volume III.

**3.11N. Two-body hadronic system**

A typical two-body quantum mechanical system is given by the hydrogen atom in which the two constituents are well approximated as being point-like since the mutual distance is much bigger than the size of the wavepacket of the constituents. In this case, the system is entirely represented with a Hamiltonian of the type

In the corresponding case of two body hadronic systems, the constituents are at mutual distances equal or smaller than 1 fm = 10

Suppose that the two particles have the shape of spheroid ellipsoids with semiaxes n_{ak}^{2}, a = 1, 2, k = 1, 2, 3. Clearly, the representation of these shapes is beyond any capability of a Hamiltonian, but shapes can be easily represented via Santilli's isounit.

Suppose that the above two extended particles with wavefunctions ψ_{1} and ψ_{2} are in conditions of partial mutual penetration (Figure 1.3), as it is the case for electrons in valence bonds, hadronic constituents, nuclear constituents and other structures. These physical conditions evidently cause nonlocal interactions extended over the volume of mutual overlapping that can be represented with volume integral ∫ ψ_{1}^{†}(r) ψ_{2}(r) dr^{3}.

Clearly, this mutual penetration cannot be represented with a quantum Hamiltonian for numerous reasons, beginning with a granting of potential energy to contact nonpotential effects, let alone the violation of the background local-differential topology. However, the same interactions can be readily represented with Santilli's isounit because the underlying topology is indeed nonlocal-integral.

By combining these and other aspects, we can see that the considered two-body hadronic system can be characterized by the Schroedinger-Santilli isoequation (3.162), or the Heisenberg-Santilli isoequation (3.166), with the same Hamiltonian H as in Eq. (3.182), plus the isotopic element T given by

x exp[ − F(t, r, p, E, μ, ψ ψ

where the exponent in general and the F function in particular, originate at the Newtonian level as in Eq. (3.138) and represent nonpotential interactions whose explicit form depends on the case at hand (see the applications in Chapters 4 and 5). Note that isotopic element (3.183) verifies the condition for strong isoconvergence of divergent quantum series, Eq. (3.169).

A most important feature of the above isotopic element is that, for mutual distances much bigger than 1 fm, the volume integral is null and the shapes become spherical due to absence of nonlocal interactions, thus verifying the basic condition (3.151), i.e.,

namely,

As a result, *hadronic mechanics has been built to provide a "completion" of quantum mechanics solely applicable at short distances essentially along the historical argument by Einstein, Podolsky and Rosen (see below for more comments). As we shall see in the next chapters, two body hadronic bound states with Hamiltonian (3.182) and isotopic element (3.183), when applicable, provide exact numerical representations in various fields that are impossible with quantum mechanics.*

**3.11O. Simple construction of hadronic mechanics**

It is important for readers to know that all mathematical and physical methods of hadronic mechanics can be constructed via the simple nonunitary transform of quantum models. This construction was first identified by Santilli in the 1978 original memoirs, studied extensively by various authors and will be heavily used in the subsequent outline of experimental verifications and applications of hadronic mechanics.

**Construction of isomodels**. The starting point is the identification of the nonunitary transform with the basic isounit of the model. For the case of two-body hadronic particles, the isounit is the inverse of the isotopic element (3.183), therefore yielding the identification

Diag. (n

x exp [ F(t, r, p, E, μ, ψ ψ

Once Santilli's isounit has been identified on groups of physical requirements (see the Chapters 4 and 5 for numerous realizations), the lifting of a quantum model into the hadronic form is simply achieved via the application of the above nonunitary transform to the *totality* of the mathematics and physics of the considered quantum model, without exceptions to avoid catastrophic inconsistencies.

In this way, we have: the very simple lifting of: the unit I of quantum mechanics into the isounit,

the lifting of numbers n into isonumbers

the lifting of conventional associative product nm between two numbers n and m into the isoproduct

the lifting of Hilbert states | ψ ) into Hilbert-Santilli isostates | ψ^{*} )

the lifting of the conventional Hilbert product into the * inner isoproduct* over the isofield of isocomplex isonumbers

the lifting of the conventional Schroedinger equation into the Schroedinger-Santilli isoequation

U [ H | ψ ) ] U

H

U [ E | ψ

where one should note the *change in the numerical value of the eigenvalue,* E → E' called *isorenormalization.* In fact, E is the eigenvalue of H, while E' is the eigenvalue of the different operator HT, thus implying that E ≠ E'. Clearly, *the isorenormalization of the energy is a fundamental feature of hadronic mechanics for numerous applications.*

**Construction of geno- and hyper-models**. Genomodels are constructed via *two* different nonunitary transforms,

and the following identification of the forward and backward genounit

The entire forward and backward genotopic branch of hadronic mechanics can then be constructed by applying the above nonunitary transforms to the totality of the quantum formalism. A similar procedure holds for the construction of the forward and backward hyperstructural branches of hadronic mechanics..

**3.11P. Invariance of hadronic mechanics**

As indicated earlier, the physical consistency of quantum mechanics is due to the invariance over time of: the basic units of measurements, the observability of operators and the preservation of the same numerical predictions under the same conditions at different times. Hadronic mechanics does indeed verify these central conditions of physical consistency, although at a covering level.

This feature can be simply seen as follows. Recall that the time evolution of hadronic mechanics is nonunitary when defined on a conventional Hilbert space defined over a conventional field of complex numbers. It is easy to see that, under these assumptions, hadronic mechanics is *not* invariant over time. In fact, following the identification of the isounit with a nonunitary transform, Eq. (3.186), a repeated application of the same transform does not leave invariant the isounit,

But, as stressed before, hadronic mechanics must be elaborated with its own mathematics to prevent inconsistencies. Hence, nonunitary transforms must be reformulated in the following

It is then easy to see that *isounitary transformations preserve Santilli's isounit, thus preserving over time the basic units of measurements and the actual shape of particles,*

It is also easy to prove that * isounitary transforms preserve Hermiticity, thus preserving the observability of operators,*

Finally, it is easy to see that *isounitary transforms predict the same numerical values under the same conditions at different times* because of the verification of the following condition at the isounitary level

W

in which one should note the

The invariance of Lie-admissible branch of hadronic mechanics, when formulated on Hilbert-Santilli genospaces over genofields, follows the same lines. This invariance was first studied in the following 1997 paper

**
Invariant Lie-admissible formulation of quantum deformations
R. M. Santilli,**
Found. Phys. Vol. 27, 1159- 1177 (1997)

**3.11Q. Relativistic hadronic mechanics**

**Foreword**

Relativistic hadronic mechanics is, of course, the most important branch of the mew discipline for experimental verifications (chapter 6), theoretical predictions (Chapters 7. 8) and industrial applications (Chapters 4, 5, 9). It comprises the isotopic, genotopic and hyperstructural liftings of conventional relativistic quantum mechanics for matter in non-Hamiltonian reversible, irreversible and multi-valued conditions, respectively, and their isoduals for antimatter in corresponding conditions.

Evidently, we cannot possibly review such a vast structure and are regrettably forces to provide the main lines solely for the isotopic branch, hereon referred to as *isorelativistic hadronic mechanics.*. The following paper presents relativistic isomechanics in a final invariant form

**
Relativistic hadronic mechanics: nonunitary, axiom-preserving
completion of relativistic quantum mechanics
R. M. Santilli,**
Found. Phys. Vol. 27, 625-729 (1997)

The most comprehensive presentation of the field remains Santilli's 1995 monograph

**"Elements of Hadronic Mechanics"
Vol. II: "Theoretical Foundations"
R. M. Santilli,** Ukraine Academy of Sciences (1995)

The primary scope of isorelativistic hadronic mechanics is to provide a quantitative representation of the *mutations of "particles" into "isoparticles", namely, the alteration of the "intrinsic" as well as kinematic characteristics of particles in the transition from motion in empty space to motion within a hadronic medium,* while recovering relativistic quantum mechanics uniquely and identically when the particles return to move in vacuum or, equivalently, when particles are at sufficient mutual distances to allow their point-like abstraction.

Recall that particles can be defiend as unitary irreducible representations of the Lorentz-Poincare' symmetry, while isoparticles can be defined as isounitary irreducble representations of the covering Lorentz-Poincare'-Santilli isosymmetry studied in Section 3.10 for the conventional case and in this section for the covering isospinorial form.

The mutation (also called isonormalization) of the rest energy of particles is an unavoidable consequence of all nontrivial isotopies of the Lorentz-Poincare' symmetry. However, the mutation of spin, charge and other intrinsic characteristics depends on the energy or, equivalently, the density of the hadronic medium considered.

This setting led Santilli to identify two main main cases, the first in which isoparticles maintain the conventional values of spin, charge and other characteristics, and the second in which these characteristic too are mutated.

We can now clarify the *title* of the memoir proposing the construction of hadronic ,mechanics,

**Need of subjecting to an experimental verification the validity within a hadron of Einstein special relativity and Pauli exclusion principle
R. M. Santilli,** Hadronic J. Vol. 1, 574-901 (1978).

In essence, a particle with spin 1/2 preserves its spin under *external electromagnetic interactions,* as well known, in which case Pauli's principle is evidently verified. However, Santilli argued that *particles may experience a mutation of their spin under external strong interactions,* such as for nucleons passing very near nuclei considered as fixed and external, in which case an experimental verification of Pauli's principle and, consequentl,y of special relativity, is necessary.

The aspect that does not appear to have sufficiently propagated in the physics community, thus leading to misinterpretation or vacuous judgments, is that *spin mutations are "internal" effects within hadronic matter that, as such, are not visible from the outside.* Alternatively, Santilli argues that if a hadron has the conventional spin 1/2, this does not necessarily imply that its constituents have conventional spin because there could be internal mutations such to compensate each other resulting in the total spin 1/2, in a way similar to the mutual compensation of internal nonconservative forces resulting in total conservation laws (Section 3.11D). Hence, the :external" character of strong interactions is crucial to avoid vacuous claims of "experimental verification" of Pauli's exclusion principle.

Some 30 years following Santilli's call in 1978 to test Pauli's principle, a number of meetings have been recently organized in the subject (without consulting Santilli or quoting his 1978 origination). We assume the serious scholar is aware of the fact that any deviations from Pauli's principle is impossible when data are elaborated via quantum mechanics, since no spin mutation is the possible. Similarly, the serious scholar is assumed to know that hadronic mechanics is the only known axiomatically consistent mechanics predicting deviations from Pauli's principle under the indicated external strong interactions (the verification of Pauli's principle in heavy atoms causing deep wave overlappings of the wavepacksts of peripheral electrons with consequentioal nonlocalm nonunitary and nonquantum effects, can be done in a similar way by considering one peripheral electron while the rest of the system is assumed as external).>br>

**Isolinearization of second order isoinvariants.**

Nonrelativistic hadronic mechanics outlined in the preceding sections is characterized by the Galilei-Santilli isosymmetry not presented in these lines for brevity, but treated in detail in the monographs

**"Isotopic Generalization of Galilei and Einstein Relativities",
Volume I: "Mathematical Foundations"
R. M. Santilli,**
Hadronic Press (1991)

**"Isotopies of Galilei and Einstein Relativities"
Vol. II: "Classical Foundations"
R. M. Santilli,**
Hadronic Press (1991)

Isorelativistic hadronic mechanics is then characterized by the Lorentz-Poincare'-Santilli isosymmetry of Section 3.10 defined on an iso-Minkowskian space M^{*}(r^{*}, m^{*}, R^{*}) under the interpretation of the generators as Hermitean operators on a Hilbert-Santilli isospace over the isofield R^{*} with isounit I^{*} = 1/T > 0 andrealization of the 4-dimensional isolinear (meaning linear on isospaces over isofields) momentum operator

with isostates |e

The second order Casimir-Santilli isoinvariant (3.81) then yields the following **Klein-Gordon-Santilli isorelativistic equation** here written in its projection in our spacetime for simplicity

or equivalently

where: the isometric (namely a matrix with isonumbers as elements) has been simplified to the form M

The "isolinearization" of the above second order isoequation has been studied extensively by Santilli, (see EHM Volume II) resulting in the **Dirac-Santilli isoequation** that we write in the simplified form also projected in our spacetime

where ∂

showing the appearance of the fundamental isometric directly in the structure of the isoequation. We assume the reader has acquired at least a minimal knowledge of preceding sections to understand that

**Pauli-Santilli isomatrices**

To identify the structure of the Dirac-Santilli isoequation, we must first review the isotopies of SU(2)-spin with particular reference to the isotopies of its fundamental representation via Pauli's matrices, first studies by Santilli in various works, such as

**Isotopic lifting of SU(2)-symmetry with
application to nuclear physics,
R. M. Santilli,**
JINR rapid Comm. Vol. 6. 24-38 (1993)

**Isorepresentation of the Lie-isotopic SU(2) algebra with application to nuclear physics and local realism,
R. M. Santilli,**
Acta Applicandae Mathematicae Vol. 50, 177-190 (1998)

and reviewed extensively in EHM-II Chapter 6. As indicated above, we have to distinguish the following two cases:

*CASE I: Pauli-Santilli isomatrices without spin mutation*

This case is characterized by the so-called *regular isounitary isorepresentations* of the Lie-Santilli isosymmetry SU^{*}(2). This case can be easily constructed via a nonunitary transformation of the conventional Pauli matrices.

Let σ_{k}, k = 1, 2, 3, be the conventional Pauli matrices defined on a two-dimensional, complex valued, Euclidean space E(r, δ, R) with trivial metric δ = Diag. (1, 1, 1). Consider the Euclid-Santilli isospace E^{*}(r^{*}, δ^{*}, R^{*}) on a Hilbert-Santilli isospace with isostates |s^{*}> and isometric

where s

Then, the

and verify the following isocommutation relations and isoeigenvalues expressions

The preservation of the conventional eigenvalues for spin 1/2 is evident, a feature that Santilli proved to extend to all spins (see EHM-II).

Prior to venturing vacuous judgments of triviality, serious readers should be aware that the above Pauli-Santilli isomatrices provide an explicit and concrete realization of hidden variables for

by consequently voiding Bell's inequality of final character, since no longer valid under Santilli isotopies. For technical details, one should study the seminal paper

**Isorepresentation of the Lie-isotopic SU(2) algebra with application to nuclear physics and local realism,
R. M. Santilli,**
Acta Applicandae Mathematicae Vol. 50, 177-190 (1998)

*CASE II: Pauli-Santilli isomatrices with spin mutation*

This case is characterized by the *irregular isorepresentations* of the Lie-Santilli SU^{*}(2). the latter cannot any longer be derived via a trivial nonunitary transform of the Lie case and constitute an intrinsic new feature of the Lie-Santilli isotheory without any correspondence with the conventional theory, although the latter always remains a particular case.

Among various cases identified by Santilli (see above quoted papers and EHM-II), an example of *irregular Pauli-Santilli isomatrices* is given by

The mutation of spin is then evident, as desired by Santilli and as needed by his physical and industrial applications (see next chapters).

Note that the irregular case can indeed be derived via a nonunitary transformation of the Lie case, but *six dimensional* (while that of the regular case was two dimensional, according to

that ensures the Lie-Santilli character of the isoalgebra.

**Dirac-Santilli isoequation**

Recall that *the conventional Dirac equation represents an electron under the "external" electromagnetic field of the proton* as well known, since a consistent extension of Dirac's equation to the two-body system constituted by the H-atom has not been achieved to this day. In this case, all conventional intrinsic characteristics of particles are preserved and, therefore, there are no mutations. In this case, we have ordinary "particles" characterized by the Lorentz-Poincare' symmetry (3.75) with generators (3.76) and commutation rules (3.77)-(3.79).

By comparison, *the Dirac-Santilli isoequation represents an isoelectron under "external" electromagnetic and contact nonpotential interactions,* as necessary for the synthesis of the neutron from protons and electrons occurring in stars and studied in Chapter 7, since this case the wavepackets of the proton and electron are in conditions of mutual penetration, thus causing additional non-Hamiltonian interactions and related isorenormalizations.

Since the electron in vacuum has spin 1/2, the symmetry needed for the characterization of the isoelectron is given by the isotopy of the spinorial covering of the Lorentz-Poincare' symmetry, first studies by Santilli during his visit at the JINR in Dubna, Russia, Communication number E4-93-252 (1993), published in the 1995 paper

**Recent theoretical and experimental evidence on the apparent
synthesis of neutrons from protons and electrons,
R. M. Santilli,**
Chinese J. System Engineering and Electronics Vol. 6, 177-199 (1995)

and today known as *Santilli isospinorial covering of the Lorentz-Poincare' symmetry,* that we write

,p>

with generators

and the same commutation rules as in Eqs. (3.77)-(3.79).

By comparing isosymmetries (3.219) and (3.75), it is evident that SL^{*}(2.c) is the isospinorial covering of SO^{*}(3.1), T^{*}(4) continues to represent isotranslations as in eqs. (3.88), and T^{*}(1) continues to represent isotopic transforms as in Eq. (3.90).

Recall that, contrary to popular beliefs, Santilli has discovered a fundamental 11-th symmetry of the conventional Minkowskian spacetime used for grand unification, operator gravity and other important advances. Consequently, the Lorentz-Poincare' symmetry P(3.1), its isotopic covering P^{*}(3.1) and its isospinorial covering Π^{*}(3.1) are all *eleven dimensional.*

The characterization of isosymmetry (3.219) requires two isospaces and related isounits, one for the mutation of spacetime (st) with *spacetime isounit* I_{st}^{*} and one for the mutation of the two-dimensional complex unitary spin space with*spin isounit* I_{spin. From the positive-definiteness of these isounits, we assume the following diagonal realization (and leave very intriguing off-diagonal realizations to interested reader, see EHM-II)}

As for the Pauli-Santilli isomatrices, we have the following two cases:

*CASE I: Dirac-Santilli isoequation without spin mutation*

Let |e> be the eigenstates of the conventional Dirac equation on the conventional Hilbert space over the field of complex numbers for the representation of an *electron, and consider the following nonunitary transforms* Let

The isostate on the iso-Hilbert space over the isofield of complex numbers representing the

The simplest possible version of the

where the γ

It is easy to prove that isogenerators (3.220) realized via isogammas (3.225) verify all isocommutators (3.77)-(3.79) and the interested reader is encouraged to verify.
Note that, in this case, no isotopy for the spin is needed because automatically provided by the assumed spacetime isotopy, resulting in a new realization of the regular Pauli-Santilli isomatrices, as the reader is suggested to verify. In any case, the spin isotopy can indeed be added, but has to preserve the spin 1/2 by assumption of the case considered, thus being inessential.

*CASE II: Dirac-Santilli isoequation with spin mutation*

This is the most important case for the synthesis of the neutron from a proton and an electron inside a stars studied in Chapter 5, because the latter synthesis requires a mutation of spin.

In this case, we have the irregular realization of Eqs. (3.203), first identified by santilli in the above quoted paper of 1993-1995, today known as *irregular Dirac-Santilli isoequation,* that can be written:

In this case, , the

with isoeigenvalues

Note that the above particular realization of the isogroup SO

From generators (3.201), the *isotopic formulation of the spin* of the isoelectron is given by

with isoeigenvalues

illustrating the spin mutation

Note that the isocommutation rules of Π^{*} are the same as those of P^{*}(3.1), Eqs. (3.77)-(3.79), as the reader is encouraged to verify and that, despite the indicated differences,Π(3.1) is isomorphic to the conventional spinorial symmetry Π(3.1). in particular, the above isotopic SU(2)-spin remains isomorphic to SU(2), of course, at the abstract, realization-free level..

Additional mutations characterized by the Dirac-Santilli isoequation are those of the magnetic moment μ and electric dipole moment d, whose derivation has been worked out by Santilli in the above quoted 1993-1995 paper via a simple isotopy of the conventional derivation, resulting in the isolaws valid for the case of an axial symmetry along the third axis

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

It is an instructive exercise for the interested reader to verify that the above realization of the above irregular Dirac-Santilli isoequation *cannot* be constructed via a nonunitary transform of the conventional Dirac equation as for the regular case, but requires special maps.

**3.11R. Direct universality and uniqueness of hadronic mechanics**

The following properties are important for an understanding of the verifications and applications of hadronic mechanics:

1) Hadronic mechanics has been proved to be "directly universal," namely, admitting as particular cases all possible generalizations of quantum mechanics with brackets of the time evolution characterizing an algebra as defined in mathematics (universality), directly in the frame of the experimenter, thus avoiding any coordinate transformation (direct universality). This property is a consequence of the fact that Santilli's Lie-admissible algebras (Section 2.8) are the most general possible algebras admitting as particular cases all possible algebras as conventionally understood in mathematics.

2) All possible true generalizations of quantum mechanics, namely, those outside its classes of unitary equivalence but preserving an algebra in the brackets of the time evolution, are particular cases of hadronic mechanics.

3) Any modification of hadronic mechanics for the intent of claiming novelty, such as the formulation of basic laws via conventional mathematics, verifies the Theorems of Catastrophic Inconsistencies of Nonunitary Theories.

Note that the above direct universality applies not only for nonrelativistic but also for relativistic hadronic mechanics.

Yet another aspect studied in detail by Santilli for years is whether the structure of hadronic mechanics is unique or there exist *inequivalent* nonunitary generalizations of quantum mechanics that are equally invariant over time. The result of this study is that hadronic mechanics is indeed the sole mechanics verifying the conditions indicated (nonunitary time invariant structure).

As an example, in his original proposal to build hadronic mechanics, Santilli classified all possible modifications of the associative product AB of two matrices A, B via the use of a fixed matrix with the same dimension,

and concluded that the only acceptable isotopy is the form ATB, because the alternative forms TAB (ABT) violate the right (left) distributive and scalar laws, thus preventing the use of an algebra in the enveloping operator algebras with consequential catastrophic inconsistencies. A reason for the uniqueness is that the only possible representation of contact non-Hamiltonian interactions verifying the condition of time invariance is that via Santilli isounit. Invariance then follows since the unit is the basic invariant of all theories. Nonequivalent generalizations of quantum mechanics must then use a representation of non-Hamiltonian effects other than that via the isounit, by activating the Theorems of Catastrophic Inconsistency of Nonunitary Theories.

**3.11S. EPR completion of quantum mechanics, hidden variables and all that**

Santilli has repeatedly presented hadronic mechanics as a form of "completion" of quantum mechanics in honor of Einstein, Podolsky and Rosen who expressed historical doubts on the completeness of quantum theories. In fact, *hadronic mechanics provides an explicit and concrete realization of hidden variables* λ that are realized via the isotopic operator T according to the isoassociative eigenvalue equations

The hidden character emerges from the fact that, at the abstract, realization-free level, there is no distinction between the conventional associative action of the Hamiltonian on a Hilbert state and its isoassociative covering. In fact, at the abstract level one can write the modular action in the abstract right-associative form "H |ψ

More generally, **all branches of hadronic mechanics preserve the abstract axioms of quantum mechanics and merely provide broader realizations of the same axioms.**

Santilli has also studied the nonunitary covering of Bell's inequalities and shown that, contrary to the quantum case, they do admit indeed a classical counterpart, thus altering the entire field of local realism.

**
Isorepresentation of the Lie-isotopic SU(2) algebra with
application to nuclear physics and local realism,
R. M. Santilli,.**
Acta Applicandae Mathematicae Vol. 50, 177-190 (1998)

**3.11T. Operator isogravity**

As indicated in Chapter 1, one of the biggest scientific imbalances of the 20th century physics has been the absence of a consistent quantum formulation of gravity, since the quantization of the Riemannian representation is afflicted by a litany of inconsistencies. In particular, the noncanonical character of the classical formulation requires, for consistency, a nonunitary operator counterpart, thus activating the Theorems of Catastrophic Inconsistencies of Nonunitary Theories.

Santilli studied for decades the problem of a consistent operator form of gravity without any publication. He finally presented his solution at the 1994 M. Grossmann Meeting on Gravitation held at Stanford Linear Accelerator Center

**
Isotopic quantization of gravity and its universal isopoincare' symmetry
R. M. Santilli,**
in the Proceedings of "The Seventh Marcel Grossmann Meeting", R. T. Jantzen, G. M. Keiser and R. Ruffini, Editors, World Scientific Publishers pages 500-505(1994).

See also EHM Volumes I and II and the paper

**
Quantum isogravity
R. M. Santilli,**
Communication in Theor. Phys. Vol. 2, pages 1-14 (1995)

Santilli's argument is essentially the following. The impossibility of achieving a consistent operator form of gravity is due to *curvature,* since the latter requires a noncanonical classical structure with consequential nonunitary operator formulation and related catastrophic inconsistencies.

Hence, Santilli formulated his **isogravitational theory** indicated in Section 3.10H in which Riemannian line elements are identically reformulated in the Minkowski-Santilli isospace via the decomposition of the metric g(r) = T_{gr}(r)m, Eq. (3.100), where m is the Minkowski metric, and T_{gr} is the gravitational isotopic element. The formulation of the isometric m^{*} = T_{gr}(r)m with respect to the isounit as the * inverse* of the gravitational isotopic element, I^{*}_{gr} = 1/T_{gr}, eliminates curvature, thus restoring unitary on the Hilbert-Santilli isospace over isofields with isounits I^{*}_{gr}.

This discovery was made possible by the unification of the Minkowskian and Riemannian geometries into the Minkowski-Santilli isogeometry presented in detail in EHM Volume I, as well as in the memoir

**Isominkowskian geometry for the gravitational treatment of matter and its isodual for antimatter,
R. M. Santilli,**
Intern. J. Modern Phys. D Vol. 7, 351-407 (1998)

Following the above advances, the achievement of a consistent operator formulation of gravity was elementary. In fact, **relativistic hadronic mechanics includes gravity without any modification of its structure via the mere interpretation of its isotopic element as being that of gravitational nature.** Again, the procedure merely requires the factorization of the Minkowski metric m from any given Riemannian metric m^{*}(r) = T_{gr}(r)m, such as for the Schwartzschild's metric, and the use of relativistic hadronic equations. As an illustration, the procedure yields the Dirac-Santilli isoequation (3.203), for which the anticommutation of the isogamma matrices yields precisely the Schwartzschild's metric, Eq. (3.204).

**3.11U. Iso-grand-unification**

There is no doubt that one of Santilli's biggest scientific contributions has been the achievement of the first axiomatically consistent grand unification of electroweak and gravitational interactions without pre-existing comparisons for consistency, mathematical beauty and physical content, to the Foundation's best knowledge (the indication of equally consistent grand unification is encouraged for comparative listing in this section). Here are summary comments released by Santilli:

*The achievement of a consistent grand unification has been, by far, the most complex research problem I ever confronted due to the vastity and diversification of the required knowledge. Also, the more I worked at a solution, the bigger the problems with consequential widening of the field. Without any expectation that colleagues would agree, my conclusions following decades of work at the problem are the following:*

1) **Antimatter.** I had to reject all preceding attempts at a grand unification, including that by Einstein, because of unsurmontable inconsistencies caused by antimatter. In fact, electroweak theories beautifully represent matter and antimatter, while a Riemannian gravitation does not, as nowadays well known. Only after achieving the isodual mathematics and related isodual theory of antimatter I was finally able to resolve these inconsistencies with a judicious decomposition of electroweak theories into advanced solutions and their isoduals with a corresponding gravitational and isodual counterpart allowing full democracy between matter and antimatter at all levels.

2) **Curvature.** After years of failed attempts along orthodox lines, I had to admit to myself that the representation of gravity via a curved spacetime renders any grand unification simply impossible. This was due to a litany of inconsistencies originating from attempting the combination of a theories structurally flat in spacetime, such as electroweak theories, and a gravitational theory that is structurally curved in spacetime. In particular, any reformulation of electroweak theories on a curved manifold to achieve geometric compatibility with gravitation, lead to unsurmontable catastrophes, such as the loss of physical meaning of electroweak theories at the operator level. These inconsistencies were determinant for my decision to cross the scientific "Rubicon" and abandon curvature for a covering theory of gravitation without curvature. That generated the birth of isogravitation.

3) **Covariance.** A third litany of inconsistencies originated from the fact that electroweak theories are beautifully structured by gauge and spacetime symmetries, while gravitation had none. The use of the customary "covariance" adopted by gravitational studies throughout the 20th century caused additional catastrophic inconsistencies, such as the lack of physical meaning of electroweak theories due to the general impossibility to predict the same numerical values under the same conditions at different times. The resolution of this third class of inconsistencies required the laborious construction of the Lie-isotopic theory that, in turn, permitted the construction of the Lorentz-Poincare'-Santilli universal isosymmetry of isogravitation.

The combination of all my studies, including the various new mathematics, the isodual theory of antimatter, the Lie-isotopic theory and relativistic hadronic mechanics, then finally lead to the iso-grand-unification with an axiomatically consistent inclusion of mutually compatible electroweak and gravitational theories for matter and antimatter.

The final solution I proposed is so elementary to be deceptive, because I essentially introduced gravitation where nobody looked for, in the unit of electroweak theories. However, by looking in retrospect, I can say that the virtual entirety of my research was ultimately aimed at the achievement of an axiomatically consistent grand unification. The diversification and novelty of the research illustrates the complexity of the problem of grand unification beyond the level of biased academic views.

In fact, following decades of research, Santilli finally released his iso-grand-unification at the VIII Marcel Grossmann Meeting on Gravitation held in Jerusalem, Israel, in 1996, as well as in related papers provided below,

**
Unification of gravitation and electroweak interactions
R. M. Santilli,**
in the proceedings of the "Eight Marcel Grossmann meeting", Israel 1997, T. Piran and R. Ruffini, Editors, World Scientific, pages 473-475 (1999)

**
Isotopic grand unification with the inclusion of gravity
R. M. Santilli,**
Found. Phys. Letters Vol. 10. 307-327 (1997)