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“Workshops on Santilli’s Iso-,Geno- and Hyper-Mathematics for Matter and Its Isodual for Antimatter, in

ICNAAM 2013, Greece,at Rhodes Palace Hotel, Rhodes”.

Introduction to Isodual Mathematics and Its Applications to Special Relativity

Dr. P.S. Muktibodh

Deptt. Of Mathematics

Hislop College, Nagpur

India

In the beginning…..

It gives me a great pleasure, and it is my honor, to be present in this “Workshops on Santilli’s Iso-,Geno- and Hyper-Mathematics for Matter and Its Isodual for Antimatter, in ICNAAM 2013.

Title of my talk is an ‘Introduction to Isodual Mathematics and Its Applications to Special Relativity’. Earlier, I was working on the nature of singularities in case of gravitational collapse of star using Einstein’s theory of general Relativity.

I was also interested in the problem of Cosmic Censorship Hypothesis and its precise formulation which is still an open problem.

When I came in contact with Prof. Ruggero M. Santilli’s epoch making work through monographs and the book entitled ‘Isodual theory of Anti-matter with applications to Anti- Gravity, Grand Unification and Cosmology’, Springer 2006, It took me no time to get involved in the path breaking work of Prof. R.M. Santilli.

I am happy and delighted to study in depth the works of Prof. R. M. Santilli and to extensively propagate his work in the community of academicians.

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Acknowledgments:

I first express, from the bottom of my heart, my sincere thanks to R.M.Santilli Foundation for giving me this opportunity and full financial support to attend this “Workshop on Santilli’s Iso-,Geno- and Hyper-Mathematics for Matter and Its Isodual for Antimatter, in ICNAAM 2013 at Greece, Rhodes”.

I express my deepest regards and whole hearted thanks to Prof. Ruggero M. Santilli, whose epoch making discovery of isodual mathematics has revolutionarised the whole understanding of scientific knowledge. His personal interest and suggestions for improvement of my paper are extremly valuable for me.

I, further, wish to express my sincere thanks to Prof. Richard Anderson and Prof. Christian Corda for their interaction and valuable suggestions during the preparition of my paper on isodual mathematics. I also express my sincere thanks to Prof. Anil Bhalekar, Nagpur who introduced me to this wonderful world of isodual mathematics. His invaluable help in every respect is very precious for me.

This presentation entitled, ‘Introduction to Isodual mathematics and its applications to Special Relativity’ is totally based on the monumental work of Prof. Rugerro M. Santilli. I claim no originality or addition or new interpretation of his work. I have referred mostly to Prof. R. M. Santilli’s work presented in ‘ Isodual Theory of Antimatter with applications to Anti-Gravity, Grand Unification and Cosmology, Springer 2006, ISBN 1-4020-4517-4 [1].

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Discovery of Isodual Mathematics

Prof. Ruggero M. Santilli, the founder of epoch making discovery of isodual mathematics has stated that “A protracted lack of solution of physical problems is generally due to the use of insufficient or inadequate mathematics” He further wrote that “There cannot exist a really new mathematics without new numbers”[1a]

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Prof.R.M.Santilli, first presented isodual mathematics for the identification of all possible equivalence classes of the group of rotation without any reference to antimatter. The fundamental numbers of the new mathematics today are well known as Santilli’s isodual numbers[2,3].

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Discovery of isodual mathematics is also the result of Prof. R. Santilli’s indepth study of Dirac’s celebrated equations and particularily of the fact that the two dimensional unit of the anti particle component of Dirac’s equation is indeed negative definite[1a].

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Antiparticles have been discovered in the negative- energy solutions of celebrated Dirac’s Equation[(p- eA/c) + im](x) = 0,

[1a]. This equation was interepreted as representing “one spin ½ particle” the electron.The negative energies for the related antiparticle the positron, was violating the causality and other physical laws.

,0

o

iI

Ii ,2}{,

0

0,

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Dirac-Santilli isodual equation Inconsistency of the 20th century intrepretation of

Dirac’s equation were first pointed out by Prof.R. Santilli[1b]. He reformulated the Dirac’s equations with the help of isodual formulations. The ‘‘identical’’ Dirac- Santilli isodual equation now stands as:

d

ddd

d

d

k

dk

k iI

Ii

ximcAep

4

22

224 ~~,2}~,~{,0

0~,0

0~

0)(~])/(~[

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Prof. R. Santilli first presented treatment of antimatter via the isodual mathematics, establishing the equivalance at the operator level of isoduality and charge conjugation[4].

The charge conjugation is charactrized by the following transformation of wave function (x)C(x) = c†(x), where |c| = 1 and symbol ‘†’denotes Hermitean conjugation, This being manifestly equivalent to the isodual transform (x)d(xd) = -†(-xt), where ‘t’ denotes transpose[1]. The equivalence between these two transforms is because charge conjugation maps space time into itself, while isodulity maps spacetime into its isodual.

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Mathematical and physical equivalance between positive-definite quantities referred to positive- definite units ‘I’, characterizing matter, and negative-definite quantities referred to negative definite units, charactrizes antimatter[1].

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It is known that Astronomical bodies made up of antimatter are nutral. Thus General Relativity and its Reiemanian geometry cannot provide any difference between matter and antimatter stars with null total charge. Thus there is a need of suitable new theory of antimatter.

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Implications of the isodual theory of antimatter resulted in the prediction of antigravity for antimatter in the gravitational field of matter, or vice-versa while resolving all known objections against antigravity.[5].

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Gravitational Field of Antimatter cannot be studied without classical representation of antimatter. It was found that only reversing the sign of the charge of the particle does not serve the purpose[1]. Thus the gravitational field of antimatter galaxies cannot be studied by simply using Riemenian geometry with only change of sign of the charge.

Needed mathematics to deal classically with antimatter should be compatable with charge conjugation at the quantum level. Prof. Santilli introduced an anti Hermitean conjugation called isoduality. The central condition that has to be applied to all physical quantities to all their units and to all their operators can be written as Q(t,r,v,---)Qd(td,rd,vd,---) = -Q†(-t†,-r†,-v†---) where Q is the generic quantity depending on time ‘t’, co-ordinate ‘r’, velocity ‘v’etc[1].

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Convential charge conjugation and Santilli’s isoduality differs in fact that former solely applies at the quantum level while latter applies at the classical Newtonian level as well as at all subsequent levels of study, including quantum level in which charge conjugation and isoduality are equivalent[1].

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This isodual mathematics which is developed by Prof. R. Santilli is anti isomorphic to conventional mathematics therefore it can deal classically with antimatter.

Mathematical and physical equivalance between positive-definite quantities referred to positive- definite units I, characterizing matter, and negative-definite quantities referred to negative definite units, charactrizes antimatter[1].

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Fundamental assumption of isodual mathematics:

(1) The Most fundamental assumption of isodual mathematics is of negative basic unit ‘-1’, as the correct left and right unit at all levels. The resulting mathematics is anti asomorphic to that which deals with matter. This is applicable at all levels of study and resulting in being equivalent to charge conjugation after quantization.[1]

(2) The central idea of isodual mathematics and related theory of antimatter for the case of point like anti particle is the lifting of conventional trival unit I = +I for matter in to the negative definite unit. Id= -1 for all levels of treatment. I >0 Id = -I < 0.[1]

(3) Isodual product:Also the conventional product AB among generic quantities A,B (such as numbers, vector fields, matrices etc.) is mapped in to the form

ABA d B = A (Id-1) B=A (-I) B under which Id is the correct left and right unit of the theory. Id dA= A d Id = A, for all elements ofA d B is the isodual product. The mapping above is the isodual map.

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Units, Numbers and Fields of Isodual

Mathematics We begin with recalling basics of Santilli’s isomathematics[1]. Let the symbol†denotes Hermitean conjugation, Hence, for real numbers ‘n’ we have n† = n, for complex numbers ‘a’ we have a† = ac

and for qauternions ‘q’ we have q† = qtc.

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 Let F = F(a,+,×) be a field (of characteristic zero), Isodual number is defined as ad = −a†, additive isodual unit 0d = 0, multiplicative isodual unit Id = −I†, ad = −a†, ad Fd with associative and commutative isodual sumad +d bd = −(a + b)† = cd Fd, associative and distributive isodual productad ×d bd = ad × (Id)−1 × bd = cd Fd,

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Additive Isodual unit is 0d = 0,i.e. ad +d 0d = 0d +d ad = ad,

Multiplicative Isodual unit is Id = −I†,

i.e. ad ×d Id = Id ×d ad = ad, & ad, bd Fd

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thus, Fd = Fd(ad,+d,×d ) is a Field called as Santilli’s isodual fields [1]. Isodual fields have a negative-definite norm, called as isodual norm.

| ad|d = |a†| Id = - (aa†)1/2 < 0.

For isodual real numbers |nd|d = -|n| < 0.

Also when a quantity and its isodual are same it is called as isoself dual. Ex. Id = i. Isoselfduality is an important concept as it has important application in the field of Cosmology with referance to equal distribution of matter and antimatter in the universe[1].  

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If F is a field, then its image Fd under the isodual map is also a field. It can be seen that the field F and its isodual images Fd are anti-isomorphic. This is the basis of isodual mathematics which is required to understand antimatter. Santilli’s isodual mathematics is the perfect mathematics to deal with the problem of antimatter. Further we observe that isodual mathematics is the dual of conventional mathematics.

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Isodual Functional AnalysisSantilli further describe the Isodual special functions and transforms that can be constructed from conventional functions by applying isoduality.

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Isodual trignometric and Hyper geometric functions:Isodual trignometric functions are defined as Sind d = − Sin(− ) Cosdd = − Cos(−) Satisfying Sind2dd + Cosd2d d = 1d = -1. The isodual hyper geometric functions are defined as Sinhdd = -Sinh(-) Coshdd = -Cosh(-)

satisfy Coshd2dd –d Sinhd2dd = 1d = -1 [1]. 

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Isodual logarthemic and exponential functions Isodual Logarthemic and exponential functions are simillarly defined as: logd nd = -log(-n) and ed = Id + Ad/d 1!d + Add Ad/d2!d +… = -eA† [1].

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  Isodual Differential and Integral Calculus:

Prof. Santilli also constructed Isodual differential and Integral calculus. Isodual differential is defined as ddxk = Iddxk = -dxk i.e. ddxk = -dxk.

corrosponding isodual partial derivatives are defined as

∂d/d ∂d xk = -∂/∂xk and ∂d/d ∂d xk = -∂/∂xk [1].

It can be noted that conventional differential are isoself dual i.e.

(dxk)d = ddxkd = dxk ,

but derivatives are not self dual

i.e.(∂f/∂xk)d = -∂d fd/d∂dxkd [1].

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Consistency of Special Relativity Prof.R Santilli has very clearly shown the boundries of applicability of Special Relativity and its inapplicability( not violation).Following points of Consistency can be noted:

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1) We know that the formulation of Special Relativity is based on Minkowskian space time M(x, , R), where local space time coordinates are x = x= ( rk, t), = 1,2,3,0 & k=1,2,3, with c0=1.

The line element with = Diag.(1,1,1,-1), is ( x - y)2 = (x - y) (x - y); and I = Diag.(1,1,1,1,), over the field of real numbers R;

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.(2) Further, all laws of special relativity, are

invarient under the fundamental Poincare symmetry.

(3) Special Relativity admits basic unit and numerical prediction that are invarient in time.

Therefore, special relativity can be confidently

applied to experimental measurement. Thus special relativity is indeed applicable to classical and quantum representation of electromagnetic waves and point- particles when moving in vacuum.

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Limitations of Special Relativity:We note the limitations of Special Relativity:

(1) Special Relativity is unable to represent Gravitation due to lack of curvature in Minkowskian space time.

(2) The inability of Special Relativity to represent both particles and anti particles as both of them in the physical reality are extended and generally are non spherical and also deformable.

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(3) The inability of Special Relativity to represent classical representation of motion of extended particles and/or anti particles within physical media.

(4) The inability of special relativity to represent the vast experimental evidence that speed of light is not a universal constant but depends on the characteristics of the medium in which it propagates and can be smaller or bigger than the speed of light in Vacuum.

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Isodual Eucledian Geometry :The Isodual spaces are the spaces Sd = Sd (xd, gd, Rd) with isodual coordinates xd =-xt(t stands for transposed) and the isodual metric gd(xd,--) = -gt(-xt,--) = -g(-xt,--) and the corrosponding isodual interval is

(x –y )d2d =[(x-y)idd gdij d (x-y)jd] Id Defined over

the isodual field Rd = Rd(nd,+d, d ) with same isodual unit Id [1].

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The isodual Eucledian geometry is the geometry of isodual space Ed over Rd.We note that the Norm on Rd is negative- definite, the isodual distance among two points on isodual line is also negative definite and is given by Dd = D Id = -D where D is the conventional distance[1].

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Isodual Minkowskian Geometry

Let M(x, , R) be the conventional Minkowskian space time with local coordinates

x = ( rk, t ) = (x), k =1,2,3 and =1,2,3,4.

The metric = Diag.(1,1,1,-1) on Reals R=R(n,+,). The Minkowskian –Santilli isodual space time first introduced in Ref (7) and studied in details(8) [1].

with isodual metric: d = - = diag.( -1,-1,-1,1), and isodual unit: Id = Diag. (-1,-1,-1,-1). The Minkowski – Santilli isodual geometry is the geometry of isodual spaces: Md =( xd, d, Rd) where,

xd = {xd} = { xd Id} Md over Rd.

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Isodual Mathematics in the field of Special RelativityThe isodual numbers have been induced by Santilli for the generalization of the conventional space-time, algebras, geometries and mechanics which are the images of the conventional structures under an anti homomorphic conjugation called isoduality[1].

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It is shown that isoduality of different space-time, is equivalent to charge conjugation in our own space-time, which leads to charactrization of anti particle via isodual numbers, spaces, algebras, geometries and mechanics. This leads to isodual universe which is geometrically separate from our universe with characteristics of negative energy < 0 and also evolving backward in time < 0[1].

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It is well known that Special and General Relativity do not distinguish between matter and antimatter therefore entire antimatter content cannot be treated by Special or General Relativity.

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On Constancy of Speed of Light:

Universal constancy of speed of light, the basic postulate of special relativity does not stand as it is experimentally found that speed of light varies from medium to medium. The speed of light C is a local quantity dependent on the characteristics in which the propagation occurs, with speed C = c in vacuum, speeds C << c within physical media of low density and speeds C >> c within media of very high density[1].

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The variable character of the speed of light then seals the lack of universal nature of Special relativity. Further, Special Relativity is afflicted by the historical inability to represent irreversible processes [1].

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Isodual special relativity developed by Prof. Santillie deals with the classical relativistic treatment of point like anti-particles.

We know that special relativity is constructed on the fundamental 4-dimensional unit of Minkowskian space.

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Where I = Diag.(1,1,1,1) representing the diamensional less units of space and time.

Isodual Special Relativity is charactrized by the map

I = Diag.({1,1,1},1) > 0 I = Diag.( {-1,-1,-1},-1) < 0.

Antimatter relativity is based on negative units of space and time[1].

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This implies the reconstruction of the entire mathematics of the special relativity with respect to common isodual unit Id, Isodual field R with isodual nos. n = n I, the isodual Minkowskian space time Md( xd, d , Rd ) with isodual coordinates xd = xId, isodual metric d = - and basic invarient on R,( x - y ) ( x- y)IR[1].

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The application of map: t td , x xd , E Ed

Generates new notions called as isodual time, isodual space and isodual energy[1]. We note that while conventional time, space and energy are measured with respect to positive units then isodual images are computed with respect to negative units. This helps in resolving existing inconsistencies for motion backward in time, negative energies and anti gravity.

It should be noted that The basic postulates of the isodual special relativity are also a simple isodual image of the conventional postulates[1].

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References[1] Ruggero Maria Santilli, Isodual theory of Anti-matter with applications to Anti- Gravity, Grand Unification and Cosmology, Springer 2006, ISBN 1-4020-4517-4 ]

[1a] R.M. Santilli, Isonumber and genonumbers of dimension 1,2,4,8 their isodual and pscudoduals, and hidden numbers of dimension 3,5,6,7 Algebras, Groups and Geometries, Vol10,pp 273-321(1993). loc.cot.

[1b] R.M. Santilli, Isodual theory of Antimatter with Application toAntigravity, Grand Unification and the Space-time Machine, Springer, New York, (2001)

[2] D. McMahon (2008), Quantum Field Theory,McGraw Hill (USA), ISBN 978-0-07-154382-8].

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[3] O.Adriani et al, Nature 458, 607-609 (2 April 2009) | doi:10.1038/nature07942; Received 28 October 2008; Accepted 6 February 2009]

[4] David Tenenbaum, Daid, One step closer. University of Wiscosconsin- Madison News, Dec26, 2012.  

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[5] Ruggero Maria Santilli, Hadronic Mathematics, Mechanics And Chemistry, Vol II , International Academic Press, 2007].

[6] Ruggero Maria Santilli, Hadronic Mathematics, Mechanics And Chemistry, Vol I , International Academic Press, 2008].

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My sincere thanks to all of you!