“Workshops on Santilli’s Iso-,Geno- and HyperMathematics 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
1
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 AntiGravity, 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.
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”.
5
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].
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]
9
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].
10
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].
11
Antiparticles have been discovered in the
negative- energy solutions of celebrated Dirac’s
Equation
[(p- eA/c) + im](x) = 0,

0 

,
o




I 0











i

,
{


}

2

,


i








0

I





4

2

2
,
2

2

[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.
12
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:







0

I 0


~
~
~
~
~
~








,


i

,
{

,

}

2


,






i











0
0
I



~
~
[
(
p

e

A
/
c
)

im
]
(
x
)

0
 
d
k 4
k
k


d
2

2

d
d
d
d
2

2

4
d
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].
14
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.
Mathematical and physical equivalance between
positive-definite quantities referred to positivedefinite units ‘I’, characterizing matter, and
negative-definite quantities referred to negative
definite units, charactrizes antimatter[1].
16
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.
17
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].
18
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.
19
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].
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].
21
This isodual mathematics which is developed by
Prof. R. Santilli is anti isomorphic to conventional
mathematics therefore it can deal classically with
antimatter.
22
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].
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]
24
(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 AB among generic
quantities A,B (such as numbers, vector fields,
matrices etc.) is mapped in to the form
ABA 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 of
A d B is the isodual product. The mapping above is
the isodual map.
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.
27
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 sum
ad +d bd = −(a + b)† = cd  Fd,
associative and distributive isodual product
ad ×d bd = ad × (Id)−1 × bd = cd  Fd,
28
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
29
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.
30
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].
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.
32
Isodual Functional Analysis
Santilli further describe the Isodual special
functions and transforms that can be
constructed from conventional functions by
applying isoduality.
33
Isodual trignometric and Hyper geometric
functions:
Isodual trignometric functions are defined as
Sind d = − Sin(− )
Cosdd = − Cos(−)
Satisfying Sind2dd + Cosd2d d = 1d = -1.
The isodual hyper geometric functions are
defined as Sinhdd = -Sinh(-)
Coshdd = -Cosh(-)
satisfy Coshd2dd –d Sinhd2dd = 1d = -1 [1].
34
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 + Add Ad/d2!d +… = -eA† [1].
35
Isodual Differential and Integral Calculus:
Prof. Santilli also constructed Isodual
differential and Integral calculus.
Isodual differential is defined as ddxk = Iddxk =
-dxk i.e. ddxk = -dxk.
corrosponding isodual partial derivatives are
defined as
∂d/d ∂d xk = -∂/∂xk and ∂d/d ∂d xk = -∂/∂xk [1].
36
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].
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:
38
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;
39
.
(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.
40
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.
41
(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.
42
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)idd gdij d (x-y)jd] Id Defined
over the isodual field Rd = Rd(nd,+d, d ) with same
isodual unit Id [1].
43
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].
44
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].
45
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 = {xd} = { xd Id} Md over Rd.
Isodual Mathematics in the field of Special
Relativity
The 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].
47
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].
48
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.
49
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].
50
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].
51
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.
52
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].
53
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].
54
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.
55
It should be noted that The basic
postulates of the isodual special
relativity are also a simple isodual image
of the conventional postulates[1].
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].
57
[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.
58
[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].
59
My sincere thanks
to all of you!
60