In Honor of Nobel Laureate Prof. M Stanley Whittingham
| SESSION: MathematicsTueAM-R9 | 6th Intl. Symp. on Sustainable Mathematics Applications |
| Tue. 28 Nov. 2023 / Room: Showroom | |
| Session Chairs: Ruggero Maria Santilli; Mike Mikalajunas; Session Monitor: TBA | |
We recall the experimental evidence according to which nuclei are composed by extended protons and neutrons in conditions of partial mutual penetration with ensuing interactions that are: linear, local and potential, thus variationally self-adjoint (SA) [1], as well as non-linear in the wave functions (as pioneered by W. Heisenberg), non-local because defined on volumes (as pioneered by L. de Broglie and D. Bohm) and non-derivable from a potential because of contact, thus zero range and variationally non-self-adjoint (NSA) type, as pioneered by R. M. Santilli in 1978 then at Harvard University under DOE support [1]. We then review the foundations of the time reversal invariant Lie-isotopic mathematics, also known as Santilli iso-mathematics, proposed in the volume [2] which is based on the preservation of the abstract axioms of 20th century applied mathematics and the use of their broadest possible realization, thus including the isotopy of [2]-[6]: 1) The quantum mechanical enveloping associative algebra of Hermitean operators ξ : {A, B, ...; AB = A × B, I} representing SA interactions via a Hamiltonian H(r, p) into the iso-associative enveloping algebra 
with iso-product 
, iso-unit 
and the Santillian 
for the representation of NSA interactions; 2) Lie’s theory into the Lie-Santilli iso-theory; 3) Numeric fields 
into Santilli iso-fields 
of iso-numbers 
; 4) Functional analysis into the iso-functional form; 5) Metric spaces 
over 
into iso-metric iso-spaces 
over 
and iso-metrics 
; 6) Newton-Leibnitz local differential calculus into a non-linear, non-local and NSA form with iso-differential 
; 7) Geometries on S over F into iso-geometries on isospaces 
over . Iso-mathematics characterizes the iso-mechanical branch of hadronic mechanics with Lie-isotopic generalization of Heisenberg equation for the time evolution of an observable 
in the infinitesimal and finite forms [2] [3]
where 
is the exponentiation in 
. It should be indicated that iso-mathematics and iso-mechanics can be constructed via the simple non-unitary transformation 
provided it is applied to the totality of conventional formalisms. We finally indicate that iso-mathematics and iso-mechanics have permitted the first and only known numerically exact and time invariant representation of experimental data for stable nuclei [7]-[8]. Tutoring lecture [9] may be a good introduction to iso-mathematics for physicists. A knowledge of this lecture is a necessary pre-requisite for the subsequent lecture on the broader, irreversible, Lie-admissible mathematics used for irreversible nuclear fusions.
References:
[1] R. M. Santilli, Foundation of Theoretical Mechanics, Springer-Verlag, Heidelberg, Germany, Vol. I (1978), www.santilli-foundation.org/docs/Santilli-209.pdf
[2] R. M. Santilli, Foundation of Theoretical Mechanics, Springer-Verlag, Heidelberg, Germany, Vol. II (1983), www.santilli-foundation.org/docs/santilli-69.pdf
[3] R. M. Santilli, Elements of Hadronic Mechanics, Ukraine Academy of Sciences, Kiev, Vol. I (1995), Mathematical Foundations, http://www.santilli-foundation.org/docs/Santilli-300.pdf
[4] C.-X. Jiang, Foundations of Santilli Isonumber Theory, International Academic Press (2001), http://www.i-b-r.org/docs/jiang.pdf
[5] R. M. F. Ganfornina and J. N. Valdes, Fundamentos de la Isotopia de Santilli, International Academic Press (2001), English translation http://www.i-b-r.org/docs/Aversa-translation.pdf
[6] S. Georgiev, Foundations of IsoDifferential Calculus, Volumes 1 to 6, Nova Publishers, New York (2014-2016) and Iso-Mathematics, Lambert Academic Publishing (2022).
[7] R. M. Santilli and G. Sobczyk, Representation of nuclear magnetic moments via a Clifford algebra formulation of Bohm’s hidden variables, Scientific Reports 12, 1-10 (2022), http://www.santilli-foundation.org/Santilli-Sobczyk.pdf
[8] R. M. Santilli, ”Reduction of Matter in the Universe to Protons and Electrons via the Lie-isotopic Branch of Hadronic Mechanics,” Progress in Physics, Vol. 19, 73-99 (2023), http://www.ptep-online.com/2023/PP-65-09.PDF
[9] R. M. Santilli, ”Tutoring lecture on iso-mathematics,” http://www.world-lecture-series.org/santilli-tutoring-i
| SESSION: MathematicsTueAM-R9 | 6th Intl. Symp. on Sustainable Mathematics Applications |
| Tue. 28 Nov. 2023 / Room: Showroom | |
| Session Chairs: Ruggero Maria Santilli; Mike Mikalajunas; Session Monitor: TBA | |
Studies in the axiomatic formulation of the irreversibility of nuclear fusions with the most general possible SA and NSA forces were initiated in the 1967 Ph. D. Thesis [1] [2] via the generalization of Lie algebras into Albert’s Lie-admissible and Jordan-admissible algebras [3] and the formulation of the first known deformations of Lie algebras with product (Eq. (8) of [1])
which, twenty years later, were followed by tens of thousands of papers on q-deformations. In 1978, after joining the Department of Mathematics of Harvard University under DOE support, Santilli achieved [4] the axiomatic formulation of irreversibility via bi-modules in which motions forward (backward) in time are represented with the ordering of all products to the right (ordering of all products to the left)
The Lie-admissible branch of hadronic mechanics [5] [6] (see also [7]-[17]) is then characterized by two inequivalent iso-mathematics, one per each direction of time, resulting in the Lie-admissible generalization of Heisenberg’s equation for an observable Q in the infinitesimal and finite forms in which SA interactions are represented by the Hamiltonian H and forward NSA interactions are represented by the Santillian S [1] [2]
where e> (e
References:
[1] R. M. Santilli, ”Embedding of Lie-algebras into Lie-admissible algebras,” Nuovo Cimento 51, 570-585 (1967), www.santilli-foundation.org/docs/Santilli-54.pdf
[2] R. M. Santilli, ”Dissipativity and Lie-admissible algebras,” Meccanica 1, 3-12 (1969).
[3] A. A. Albert, Trans. Amer. Math. Soc. 64, 552-585 (1948).
[4] R. M. Santilli, ”Initiation of the representation theory of Lie-admissible algebras of operators on bimodular Hilbert spaces,” Hadronic J. 3, 440-467 (1979), http://www.santilli-foundation.org/docs/santilli-1978-paper.pdf
[5] R. M. Santilli, Lie-Admissible Approach to the Hadronic Structure, International Academic Press, Vols. I and II (1978), http://www.santilli-foundation.org/docs/santilli-71.pdf, http://www.santilli-foundation.org/docs/santilli-72.pdf
[6] R. M. Santilli, Elements of Hadronic Mechanics, Volumes I, II, III, Ukraine Academy of Sciences (1995), http://www.i-b-r.org/Elements-Hadronic-Mechanics.htm
[7] H. C. Myung and R. M. Santilli, Editors, Proceedings of the Second Workshop on Lie-Admissible formulations, Parts A and B: http://www.santilli-foundation.org/docs/hj-2-6-1979.pdf, http://www.santilli-foundation.org/docs/hj-3-1-1979.pdf
[8] H. C. Myung and R. M. Santilli, Editors, Proceedings of the Third Workshop on Lie-Admissible Formulations, Parts A, B, C: http://www.santilli-foundation.org/docs/hj-4-2-1981.pdf, http://www.santilli-foundation.org/docs/hj-4-3-1981.pdf, http://www.santilli-foundation.org/docs/hj-4-4-1981.pdf
[9] T. Arenas, J. Fronteau and R. M. Santilli, Editors, Proceedings of the First International Conference on Nonpotential Interactions and their Lie-Admissible Treatment, Part A, B, C, D: http://www.santilli-foundation.org/docs/hj-5-2-1982.pdf, http://www.santilli-foundation.org/docs/hj-5-3-1982.pdf, http://www.santilli-foundation.org/docs/hj-8-4-1982.pdf, http://www.santilli-foundation.org/docs/hj-5-5-1982.pdf
[10] H. C. Myung and R. M. Santilli, Editors, Proceedings of the First and Second Workshops on Hadronic Mechanics, http://www.santilli-foundation.org/docs/hj-6-6-1983.pdf, http://www.santilli-foundation.org/docs/hj-7-5-1984.pdf, http://www.santilli-foundation.org/docs/hj-7-6-1984.pdf
[11] Tuladhar Bhadra Man, Editor, Proceedings of the third international conference on the Lie-admissible treatment of non-potential interactions, Kathmandu University, Nepal (2011), Vol. I and II: 2 http://www.santilli-foundation.org/docs/2011-nepal-conference-vol-1.pdf, http://www.santilli-foundation.org/docs/2011-nepal-conference-vol-2.pdf
[12] A. Schoeber, Editor, Irreversibility and Non-potentiality in Statistical Mechanics, Hadronic Press (1984), http://www.santilli-foundation.org/docs/Santilli-110.pdf
[13] S. Beghella-Bartoli and R. M. Santilli, Editors, Proceedings of the 2020 teleconference on the Einstein-Podolsky-Rosen argument that ’Quantum mechanics is not a complete theory,” Curran Associates, New York, NY (2021), www.proceedings.com/59404.html
[14] J. Fronteau, A. Tellez-Arenas and R. M. Santilli, ”Lie-admissible structure of statistical mechanics,” Hadronic Journal 3, 130-176 (1979), http://www.santilli-foundation.org/docs/arenas-fronteau-santilli-1981.pdf
[15] J. Dunning-Davies, J. Dunning-Davies, ”The Thermodynamics Associated with Santilli’s Hadronic Mechanics,” Progress in Physics 4, 24-26 (2006), http://www.santilli-foundation.org/docs/Dunning-Davies-Thermod.PDF
[16] A. A. Bhalekar, ”Santilli’s Lie-Admissible Mechanics. The Only Option Commensurate with Irreversibility and Nonequilibrium Thermodynamics,” AIP Conf. Proc.
1558, 702-722 (2013), http://www.santilli-foundation.org/docs/bhalekar-lie-admissible.pdf
[17] T. Vougiouklis, ”The Santilli theory ’invasion’ in hyperstructures,” Algebras, Groups and Geometries 28, 83-104 (2011), http://www.santilli-foundation.org/docs/santilli-invasion.pdf
[18] R. M. Santilli, ”Lie-admissible invariant representation of irreversibility for matter and antimatter at the classical and operator levels,” Nuovo Cimento B 121, 443-485 (2006), http://www.santilli-foundation.org/docs//Lie-admiss-NCB-I.pdf
[19] R. M. Santilli, ”Apparent Resolution of the Coulomb Barrier for Nuclear Fusions Via the Irreversible Lie-admissible Branch of Hadronic Mechanics,” Progress in Physics, 18, 138-163 (2022), www.ptep-online.com/2022/PP-64-09.PDF
[20] R. M. Santilli, ”Tutoring on Lie-admissible mathematics,” Parts I, II, III: http://www.world-lecture-series.org/santilli-tutoring-iv-part-1, http://www.world-lecture-series.org/santilli-tutoring-iv-part-2, http://www.world-lecture-series.org/santilli-tutoring-iv-part-3