Octonion Representation of the Superstring Theory

Octonion Representation of the Superstring Theory

B. C. Chanyal (1), P. S. Bisht(2) and O. P. S. Negi(3)

Department of Physics, Kumaun University

S. S. J. Campus, Almora -263601 (U.K.) India

Abstract

In this paper, we have made an attempt to discuss the role of octonions in superstring theory (i.e. a theory of everything to describe the unification of all four types of forces namely gravitational, electromagnetic, weak and strong) where, we have described the octonion representation of the super-string (SS) formulation as the combination of four complex (C) spaces namely associated with the gravitational (G-space), electromagnetic (EM-space), weak (W-space) and strong (S-space) interactions. We have discussed the octonionic differen- tial operator, octonionic valued potential wave equation, octonionic field equation and other various quantum equations of superstring formulation in simpler, compact and consistent manner. Consequently, the general- ized Dirac-Maxwells equations are studied with the preview of superstring theory by means of octonions.

Key words: octonion, superstring theory, generalized Dirac-Maxwells equations, wave equation.

1. Introduction

   Physics is a natural science that involves [1] the study of matter and its motion through space and time, along with related concepts such as energy and force. The search for unity and simplicity has been the theme of physics ever since Newton first showed that celestial and terrestrial mechanics could be unified . The 20th century has been a time for tremendous format and change in our understanding new phenomena or adding new features to the existing theories [2]. Particle physics [1, 2] is the branch of physics which deals with the study of matter, energy, space and time. Its objectives are to identify the most simple objects out of which all matter is composed and to understand the forces which cause them to interact and combine to make more complex things. Now a days particle physics is popularly known as high energy physics [1, 2, 3], which is the theory of basic structure of matter and its forces. The physics of elementary particles is currently described in terms of very successful theory called standard model [1, 4]. It describes all known elementary particles and their interactions except gravitational interactions. The standard model accommodates the quarks and the leptons, which are constituents of matter, the vector particles that mediate the strong and electroweak forces and Higgs

   Corresponding Author

   bosons, which is expected to account for the masses of particles. The standard model (SM) also describes [5] the unified picture of strong and electro-weak interactions within the framework of a SU (3) × SU (2) × U (1) non-Abelian gauge theory.

   Physics beyond the Standard Model [6, 7] refers to the theoretical developments needed to explain the deficiencies of the Standard Model, such as the origin of mass, the strong CP problem, neutrino oscillations, matterantimatter asymmetry, and the nature of dark matter and dark energy [6, 8]. Another problem lies within the mathematical framework of the Standard Model itself. The Standard Model is inconsistent with that of general relativity to the point that one or both theories break down in their descriptions under certain conditions (for example within known space-time singularities like the Big Bang and black hole event horizons).

   A Grand Unified Theory (GUT) [6, 7], is a model in particle physics in which at high energy, the three gauge interactions of the Standard Model which define the electromagnetic, weak, and strong interactions, are merged into one single interaction characterized by one larger gauge symmetry and thus one unified coupling constant. In contrast, the experimentally verified Standard Model of particle physics is based on three inde- pendent interactions, symmetries and coupling constants. The standard model has three gauge symmetries [9]; the colour SU (3), the weak isospin SU (2), and the hypercharge U (1) symmetry, corresponding to the three fundamental forces. Due to renormalization the coupling constants of each of these symmetries vary with the energy at which they are measured. Around 1016 GeV these couplings become approximately equal. This has led to speculation that above this energy the three gauge symmetries of the standard model are unified in one single gauge symmetry with a simple group gauge group, and just one coupling constant. Below this energy the symmetry is spontaneously broken to the standard model symmetries [10]. Unifying gravity with the other three interactions would provide a theory of everything (TOE), rather than a GUT. Nevertheless, GUTs are often seen as an intermediate step towards a TOE. The new particles predicted by models of grand unification cannot be observed directly at particle colliders because their masses are expected to be of the order of the so-called GUT scale, which is predicted to be just a few orders of magnitude below the Planck scale and thus far beyond the reach of currently foreseen collision experiments. Instead, effects of grand unification might be detected through indirect observations such as proton decay, electric dipole moments of elementary particles, or the properties of neutrinos [11]. Some grand unified theories predict the existence of magnetic monopoles. Popular choices for the unifying group are the special unitary group in five dimensions SU(5) and the special orthogonal group in ten dimensions SO(10) [12]. String theory [13] is one such reinvention, and many theoretical physicists think that such theories are the next theoretical step toward a true Theory of Everything. Theories of quantum gravity such as loop quantum gravity and others are thought by some to be promising candidates to the mathematical unification of quantum field theory and general relativity, requiring less drastic changes to existing theories [14]. However recent work places stringent limits on the putative effects of quantum gravity on the speed of light, and disfavors some current models of quantum gravity [15].

   On the other hand, there has been a revival in the formulation of natural laws so that there exists [16] four-division algebras consisting the algebra of real numbers (R), complex numbers (C), quaternions (H) and Octonions (O). All four algebras are alternative with totally anti symmetric associators. Quaternions [17, 18] were very first example of hyper complex numbers have been widely used [19]-[25] to the various applications of mathematics and physics. Since octonions [26] share with complex numbers and quaternions, many attractive mathematical properties, one might except that they would be equally as useful as others. Octonion analysis has been widely discussed by Baez [27]. It has also played an important role in the context of various physical prob-

   lems [28, 29, 30, 31] of higher dimensional supersymmetry, super gravity and super strings etc. In recent years, it has also drawn interests of many [32, 33, 34, 35] towards the developments of wave equation and octonion form of Maxwells equations. We [36, 37, 38] have also studied generalized octonion electrodynamics, gener- alized split octonion electrodynamics and octonion quantum chromodynamics and obtained the corresponding field equations (Maxwells equations) and equation of motion in compact and simpler formulation. In this pa- per, we have made an attempt to discuss the role of octonions in superstring theory (i.e. a theory of everything to describe the unification of all four types of forces namely gravitational, electromagnetic, weak and strong) where, we hav described the octonion representation of the super-string (SS) formulation as the combination of four complex (C) spaces namely associated with the gravitational (G-space), electromagnetic (EM-space), weak (W-space) and strong (S-space) interactions. We have discussed the octonionic differential operator, octonionic valued potential wave equation, octonionic field equation and other various quantum equations of superstring formulation in simpler, compact and consistent manner.

2. Mathematical Preliminaries

   An octonions (Cayley numbers) is the family of hypercomplex numbers and have eight dimensions. Thus, octonion x O is expressed [39, 40] as

   x = e0x0 + e1x1 + e2x2 + e3x3 + e4x4 + e5x5 + e6x6 + e7x7

   7

   = e0x0 + eAxA (1)

   A=1

   where eA(A = 1, 2, …, 7) are imaginary octonion units and e0 is the multiplicative unit element. Set of octets (e0, e1, e2, e3, e4, e5, e6, e7) are known as the octonion basis elements and satisfy the following multiplication rules

   e0 = 1; e0eA = eAe0 = eA;

   eAeB = ABe0 + fABCeC. (A, B,C = 1, 2, ….., 7). (2)

   The structure constants fABC is completely antisymmetric and takes the value 1 for following combinations,

   fABC = +1; (ABC) = (123), (471), (257), (165), (624), (543),(736). (3)

   It is to be noted that the summation convention is used for repeated indices. Here the octonion algebra O is described over the algebra of real numbers having the vector space of dimension 8. Octonion algebra is non associative and multiplication rules for its basis elements given by equations (2) and (3) are then generalized in the following table:

   ·	e1	e2	e3	e4	e5	e6	e7
   e1	1	e3	e2	e7	e6	e5	e4
   e2	e3	1	e1	e6	e7	e4	e5
   e3	e2	e1	1	e5	e4	e7	e6
   e4	e7	e6	e5	1	e3	e2	e1
   e5	e6	e7	e4	e3	1	e1	e2
   e6	e5	e4	e7	e2	e1	1	e3
   e7	e4	e5	e6	e1	e2	e3	1

   Table 1: Octonion Multiplication table

   As such we may write the following relations among octonion basis elements

   Octonion conjugate is defined as

   [eA, eB] = 2 fABC eC ;

   {eA, eB} = 2 ABe0;

   eA( eB eC ) /= (eA eB ) eC. (4)

   x = e0x0 e1x1 e2x2 e3x3 e4x4 e5x5 e6x6 e7x7

   7

   = e0x0 eAxA (5)

   A=1

   where we have used the conjugates of basis elements as e0 = e0 and eA = eA. Hence an octonion can be decomposed in terms of its scalar (Sc(x)) and vector (Vec(x)) parts as

   Sc(x) = 1 ( x + x ); 2

   Vec x 1 x x 7 e x

   (6)

   ( ) = 2 (

   ) =

   A=1

   A A.

   Conjugates of product of two octonions and its own are described as

   (xy) = y x; (x) = x. (7)

   while the scalar product of two octonions is defined as

   (x, y) = 1 (xy + yx) = 1 (xy + yx) =

   7

   x y. (8)

   2 2

   =0

   The norm N(x) and inverse x1 (for a nonzero x) of an octonion are respectively defined as

   N(x) =

   xx = xx

   7

   0

   0

   =

   =0

   x2 .e ; x1

   x

   = N(x) =

   xx1

   = x1 x

   = 1.

   (9)

   The norm N(x) of an octonion x is zero if x = 0, and is always positive otherwise. It also satisfies the following

   property of normed algebra

   N(xy) = N(x) N(y) = N(y) N(x). (10)

   Equation (4) shows that octonions are not associative in nature and thus do not form the group in their usual form. Non – associativity of octonion algebra O is provided by the associator (x, y, z) = (xy)z x(yz) x, y, z O defined for any three octonions. If the associator is totally antisymmetric for exchanges of any three variables,

   i.e. (x, y, z) = (z, y, x) = (y, x, z) = (x, z, y), then the algebra is called alternative.

3. Homogeneous and Inhomogeneous Octonion Wave Equation

   In order to consider the properties of octonions [41, 42] and its eight dimensional connection, we may now write the octonion differential operator D [43, 44] as

   7

   { }

   { }

   D 0, 1, 2, 3, 4, 5, 6, 7 = eµ µ , (µ = 0, 1, 2, 3, ……, 7) (11)

   µ=0

   where µ are described as the components of a differential operator in an eight dimensional representation. Here we may consider the eight dimensional space as the combination of two (external and internal) four dimensional spaces. As such, a function of an octonion variable may be described as

   7

   { }

   { }

   F (X ) f0, f1, f2, f3, f4, f5, f6, f7 = eµ fµ (X ), (12)

   µ=0

   Here fµ are scalar functions. Since octonions are neither commutative nor associative, one has to be very careful to multiply the octonion either from left or from right in terms of regularity conditions [45]. As such, a function

   7

   { }

   { }

   F (X ) of an octonion variable X X0, X1, X2, X3, X4, X5, X6, X7 = eµ Xµ is left regular at X if and only if

   µ=0

   F (X ) satisfies the condition

   DF (X ) = 0. (13)

   Similarly, a function G(X ){g0, g1, g2, g3, g4, g5, g6, g7} is a right regular if and only if

   G(X )D = 0, (14)

   where G(X ) = g0 + g1e1 + g2e2 + ….. + g7e7. Then we get

   7

   { }

   { }

   DF = I I0, I1, I2, I3, I4, I5, I6, I7 = eµ Iµ , (15)

   µ=0

   where
   I0	= 0 f0 1 f1 2 f2 3 f3 4 f4 5 f5 6 f6 7 f7;
   I1	= 0 f1 + 1 f0 + 2 f3 3 f2 + 6 f5 5 f6 7 f4 + 4 f7;
   I2	= 0 f2 + 2 f0 + 3 f1 1 f3 + 4 f6 6 f4 7 f5 + 5 f7;
   I3	= 0 f3 + 3 f0 + 1 f2 2 f1 + 6 f7 7 f6 + 5 f4 4 f5;
   I4	= 0 f4 + 4 f0 + 3 f5 5 f3 2 f6 + 6 f2 1 f7 + 7 f1;
   I5	= 0 f5 + 5 f0 + 1 f6 6 f1 + 7 f2 2 f7 3 f4 + 4 f3;
   I6	= 0 f6 + 6 f0 1 f5 + 5 f1 + 2 f4 4 f2 3 f7 + 7 f3;
   I7	= 0 f7 + 7 f0 + 1 f4 4 f1 + 2 f5 5 f2 6 f3 + 3 f6.	(16)

   The regularity condition (13) may now be considered as a homogeneous octonion wave equation for octonion variables without sources. On the other hand, equation (15) is considered as the inhomogeneous wave equation as

   DF =I. (17)

   where I is also an octonion. Similarly, we may also write the homogeneous as well as inhomogeneous octonion wave equations on using the right regularity condition (14). We may now interpret these octonion wave equa- tions as the classical wave (field) equations of physical variables. Thus, one dimensional octonion representation is identical to eight dimensional spaces over the field of real numbers.

4. Octonion form of Generalized Dirac-Maxwells (GDM) Equations

   In order to consider the generalized electromagnetic fields of dyon, we may write the various quantum equations of dyon in octonion formulation. Thus the octonion valued potential, in eight dimensional formalism as the combinations of two four dimensional spaces, is defined as

   7

   { }

   { }

   V V0,V1,V2,V3,V4,V5,V6,V7 = eµVµ , (µ = 0, 1, 2, 3, ……, 7). (18)

   µ=0

   We may now identify the components of generalized potential of dyons as

   V0 , V1 Ax, V2 Ay, V3 Az,

   V4 iBx, V5 iBy, V6 iBz, V7 i (i =

   1) (19)

   where ( , Ax, Ay, Az) = ( , A ) {Aµ } and (, Bx, By, Bz) = (, B ) {Bµ } are respectively described as the components of electric Aµ and magnetic Bµ four potentials of dyons. In order to obtain the generalized field equations of dyons in four dimensional space time, we may identify differential operator () to be four dimensional, so that the differential operator (11) be written as

   D D {e1, e2, e3, e7} = e j j + e77, (20)

   t

   t

   xj

   xj

   where 7 = i (i = 1), j = ( j = 1, 2, 3) and other components may be taken vanishing as we are

   concerned with classical electrodynamics of dyons in four dimensional space-time world [36, 37]. Octonion conjugate of equation (20) may then be written as

   D D {e1, e2, e3, e7} = ejj e77, (21)

   Now operating D given by equation (21) to octonion potential V of equation (18) for the octonionic potential wave equations, we get

   7

   { }

   { }

   DV = F F0, F1, F2, F3, F4, F5, F6, F7 = eµ Fµ , (µ = 0, 1, 2, 3, ……, 7). (22)

   µ=0

   where F is an octonion reproduces the generalized electromagnetic fields of dyons, may be express as

   F1 = (1 + 2Az 3Ay i7Bx) I Hx; F2 = (2 + 3Ax 1Az i7By) I Hy;

   F3 = (3 + 1Ay 2Ax i7Bz) I Hz; F4 = i (1 2Bz + 3By i7Ax) I i Ex;

   F5 = i (2 3Bx + 1Bz i7Ay) I i Ey; F6 = i (3 1By + 2Bx i7Az) I i Ez;

   F0 = ( · A + i7 ) I 0; F7 = i ( · B + i7) I 0. (23)

   Here E (Ex, Ey, Ez) and H (Hx, Hy, Hz) are the components of electric and magnetic four potentials. The gen- eralized electric (E ) and magnetic (H ) fields of dyons in terms of components of electric and magnetic four potentials

   t

   t

   E = A × B ;

   t

   t

   H = B + × A . (24)

   Now applying the differential operator (20) to equation (22), we get

   7

   { }

   { }

   DF = J J0, J1, J2, J3, J4, J5, J6, J7 = eµ Jµ , (µ = 0, 1, 2, 3, ……, 7).; (25)

   µ=0

   We may now identify the components of generalized current of dyons as

   J0 , J1 jx, J2 jy, J3 jz,

   J4 ikx, J5 iky, J6 ikz, J7 i (i =

   1) (26)

   Here (, j ) = {jµ } and (, j ) = {kµ } are respectively the four currents associated with electric charge and magnetic monopole of dyons. Generalized Dirac-Maxwells equations represent one of the most elegant and concise ways to state the fundamentals of electricity and magnetism. Thus, equations (25) and (26) thus lead to

   following differential equations,

   ( · E ) = ;

   t

   t

   ( × E ) = H _k;

   t

   t

   ( × H ) = E + j ;

   ( · H ) = . (27)

   which are the generalized Dirac-Maxwells (GDM) equations of generalized fields of dyons. Like quaternion formulation of generalized electromagnetic fields of dyons. Octonion formulation is compact and simpler. Since e7 is coupling two quaternions into one octonion and also reverses its sign in its combination with quaternion units.

5. Octonion Superstring Formulation

In the early days, one of the grate attractions of superstring theory was that it appeared to be uniquely determined – we live, it seemed, in the only mathematically possible world. The superstring theory still holds out the best hope for ultimate unification of all four interactions ( i.e. gravitational, electromagnetic, weak and strong interactions), and it is probably the most promissing candidate for quantum gravity. Thus, the octonionic representation of the super-string (SS) formulation may be consider as the combination of four complex (C) spaces namely associated with the gravitational (G-space), electromagnetic (EM-space), weak (W-space) and strong (S-space) interactions [46, 47], i.e. unification of the four fundamental forces. So, we may write the octonionic superstring space as

OSS =C C C C

=(Og , Oem , Ow , Os) = ((e0, e1) , (e2, e3) , (e4, e5) , (e6, e7)) , (28)

where Og, Oem, Ow, Os are respectively known as gravitational, electromagnetic, weak and strong spaces in su- perstring theory related with the octonionic basis (e0, e1), (e2, e3), (e4, e5), (e6, e7). Thus, the octonionic physical quantity X OSS is expressed as

X =Xg + Xem + Xw + Xs

=(Xg0 e0 + Xg1 e1) + (Xem0 e2 + Xem1 e3) + (Xw0 e4 + Xw1 e5) + (Xs0 e6 + Xs1 e7). (29)

The octonionic differential operator in case of superstring formulation (i.e. unification of four differential oper- ator) may be written as

EBSS = Dg + Dem + Dw +Ds

=(g0 e0 + g1 e1) + (em0 e2 + em1 e3) + (w0 e4 + w1 e5) + (s0 e6 + s1 e7). (30)

Octonionic conjugate of equation (30) is described as

EESS =(g0 e0 g1 e1) (em0 e2 + em1 e3) (w0 e4 + w1 e5) (s0 e6 + s1 e7). (31) Thus, the octonionic superstring valued potential with the combination of four potentials may be expressed as

VSS = (Vg , Vem , Vw , Vs) =((V0,V1) , (V2,V3) , (V4,V5) , (V6,V7))

=((Vg0 ,Vg1 ) , (Vem0 ,Vem1 ) , (Vw0 ,Vw1 ) , (Vs0 ,Vs1 )) , (32)

which is further simplified to

VSS =(Vg0 e0 +Vg1 e1) + (Vem0 e2 +Vem1 e3) + (Vw0 e4 +Vw1 e5) + (Vs0 e6 +Vs1 e7). (33)

In order to obtain the octonionic potential wave equations of the superstring space, let us operate EESS given by equation (31) to octonion superstring valued potential VSS of equation (33) and we get

EESS VSS =e0{g0 Vg0 + g1 Vg1 + em0 Vem0 + em1 Vem1 + w0 Vw0 + w1 Vw1 + s0 Vs0 + s1 Vs1 }

+e1{g0 Vg1 g1 Vg0 em0 Vem1 + em1 Vem0 w0 Vs1 + w1 Vs0 s0 Vw1 + s1 Vw0 }

+e2{g0 Vem0 + g1 Vem1 em0 Vg0 em1 Vg1 w0 Vs0 w1 Vs1 + s0 Vw0 + s1 Vw1 }

+e3{g0 Vem1 g1 Vem0 + em0 Vg1 em1 Vg0 + w0 Vw1 w1 Vw0 s0 Vs1 + s1 Vs0 }

+e4{g0 Vw0 + g1 Vs1 + em0 Vs0 em1 Vw1 w0 Vg0 + w1 Vem1 s0 Vem0 s1 Vg1 }

+e5{g0 Vw1 g1 Vs0 + em0 Vs1 + em1 Vw0 w0 Vem1 w1 Vg0 + s0 Vg1 s1 Vem0 }

+e6{g0 Vs0 + g1 Vw1 em0 Vw0 + em1 Vs1 + w0 Vem0 w1 Vg1 s0 Vg0 s1 Vem1 }

+e7{g0 Vs1 g1 Vw0 em0 Vw1 em1 Vs0 + w0 Vg1 + w1 Vem0 + s0 Vem1 s1 Vg0 }, (34)

which provides the following octonionic analogous of superstring formulation as

EESS VSS = FSS = ((F0, F1), (F2, F3), (F4, F5), (F6, F7)); (35)

where FSS(F0, F1, F2, F3, F4, F5, F6, F7) is an octonion which reproduces the superstring field strengths. So, equa- tion (35) may further be expressed as

( )

( )

FSS = Fg + Fem + Fw + Fs = (Fg0 , Fg1 ) , (Fem0 , Fem1 ) , (Fw0 , Fw1 ) , (Fs0 , Fs1 )

=(Fg0 e0 + Fg1 e1) + (Fem0 e2 + Fem1 e3)

+ (Fw0 e4 + Fw1 e5) + (Fs0 e6 + Fs1 e7), (36)

where the first term (Fg = Fg0 , Fg1 ) is defined as the gravitational field strength in G-space, the second term (Fem = Fem0 , Fem1 ) is described as the electromagnetic field strength in EM-space, the third term (Fw = Fw0 , Fw1 ) provides the weak interaction field strength in W-space and the forth term (Fs = Fs0 , Fs1 ) is responsible for the strong field strength in S-space. Thus, the component of FSS{(Fg0 , Fg1 ), (Fem0 , Fem1 ), (Fw0 , Fw1 ), (Fs0 , Fs1 )} are

expressed as the following octonionic representation

Fg0 e0 = {g0Vg0 e0 g1Vg0 e1 em0Vg0 e2 em1Vg0 e3

w0Vg0 e4 w1Vg0 e5 s0Vg0 e6 s1Vg0 e7};

Fg1 e1 = {g1Vg1 e0 + g0Vg1 e1 em1Vg1 e2 + em0Vg1 e3

s1Vg1 e4 + s0Vg1 e5 w1Vg1 e6 + w0Vg1 e7};

em0 e2 = {em0Vem0 e0 + em1Vem0 e1 + g0Vem0 e2 g1Vem0 e3

s0Vem0 e4 s1Vem0 e5 + w0Vem0 e6 + w1Vem0 e7};

Fem1 e3 = {em1Vem1 e0 em0Vem1 e1 + g1Vem1 e2 + g0Vem1 e3

+ w1Vem1 e4 w0Vem1 e5 s1Vem1 e6 + s0Vem1 e7};

Fw0 e4 = {w0Vw0 e0 + s1Vw0 e1 + s0Vw0 e2 w1Vw0 e3

+ g0Vw0 e4 + em1Vw0 e5 em0Vw0 e6 g1Vw0 e7};

Fw1 e5 = {w1Vw1 e0 s0Vw1 e1 + s1Vw1 e2 + w0Vw1 e3

em1Vw1 e4 + g0Vw1 e5 + g1Vw1 e6 em0Vw1 e7};

Fs0 e6 = {s0Vs0 e0 + w1Vs0 e1 w0Vs0 e2 + s1Vs0 e3

+ em0Vs0 e4 g1Vs0 e5 + g0Vs0 e6 em1Vs0 e7};

Fs1 e7 = {s1Vs1 e0 w0Vs1 e1 w1Vs1 e2 s0Vs1 e3

+ g1Vs1 e4 + em0Vs1 e5 + em1Vs1 e6 + g0Vs1 e7}. (37)

So, in order to obtain the octonionic superstring field equations, we apply the differential operator (30) to equation (36) as

EBSS FSS =

e0{g0 Fg0 + g1 Fg1 + em0 Fem0 + em1 Fem1 + w0 Fw0 + w1 Fw1 + s0 Fs0 + s1 Fs1 }

+e1{g0 Fg1 + g1 Fg0 + em0 Fem1 em1 Fem0 + w0 Fs1 w1 Fs0 + s0 Fw1 s1 Fw0 }

+e2{g0 Fem0 g1 Fem1 + em0 Fg0 + em1 Fg1 + w0 Fs0 + w1 Fs1 s0 Fw0 s1 Fw1 }

+e3{g0 Fem1 + g1 Fem0 em0 Fg1 + em1 Fg0 w0 Fw1 + w1 Fw0 + s0 Fs1 s1 Fs0 }

+e4{g0 Fw0 g1 Fs1 em0 Fs0 + em1 Fw1 + w0 Fg0 w1 Fem1 + s0 Fem0 + s1 Fg1 }

+e5{g0 Fw1 + g1 Fs0 em0 Fs1 em1 Fw0 + w0 Fem1 + w1 Fg0 s0 Fg1 + s1 Fem0 }

+e6{g0 Fs0 g1 Fw1 + em0 Fw0 em1 Fs1 w0 Fem0 + w1 Fg1 + s0 Fg0 + s1 Fem1 }

+e7{g0 Fs1 + g1 Fw0 + em0 Fw1 + em1 Fs0 w0 Fg1 w1 Fem0 s0 Fem1 + s1 Fg0 }, (38)

which can further be reduced and be written in following compact notation in terms of an octonionic superstring representation as

EBSS FSS = JSS = ((J0, J1) , (J2, J3) , (J4, J5) , (J6, J7)) . (39)

Here JSS(J0, J1, J2, J3, J4, J5, J6, J7) is an octonionic superstring current source, which may be expressed in the

following matrix form,

e5

e6

e5

e6

e0 e1

J(gg) J(gg) J(gem) J(gem) J(gws) J(gws) J(gsw) J(gsw)

e2

J(emem) J(emem) J(emg) J(emg) J(emsw) J(emsw) J(emws) J(emws)

J(s

J(s

e3

= JSS.

J(ww) J(ws) J(ws) J(ww) J(wgem) J(wemg) J(wemg) J(wgem)

J(s

s) J(sw) J(ss) J(sw) J(semg) J(sgem) J(sgem) J(semg)

J(s

s) J(sw) J(ss) J(sw) J(semg) J(sgem) J(sgem) J(semg)

e4

e7

(40)

The generalized Dirac-Maxwells (GDM) is visualized in superstring formulation by means of octonions and the generalized current has been discussed in equation (39). The generalized superstring current source described the various terms where J(gg), J(emem), J(emg), J(gem) defines as the octonionic representation of gravitational

– gravitational, electromagnetic – electromagnetic, electromagnetic – gravitational, gravitational – electromag- netic current source and J(ww), J(ss), J(sw), J(ws) sectors of the octonionic current source respectively as- sociated the weak-weak, strong-strong, strong-weak, weak strong interactions. Consequently, the remaining terms of equation (40) denotes as the parts of octonionic current sources for the combination of different three interactions. Thus, the present formulation has been provides the superstring theory in the terms of octonionic representations.

References

1. S. Weinberg, The quantum theory of fields, Book, Cambridge University Press, vol 2 (1996).

2. D. J. Griffiths, Introduction to Elementary Particles, Book, John Wiley & Sons (1987).

3. M. E. Peskin and D. V. Schroeder, An introduction to quantum field theory, Book, Harper Collins, (1995).

4. S. F. Novaes, Standard Model: An Introduction, arXiv: hep-ph/0001283, (2000).

5. P. Langacker and A. K. Mann, The unification of electromagnetism with weak force, Physics Today, December , 22 (1989).

6. J. W. F. Valle, Physics Beyond the Standard Model, arXiv: hep-ph/9603307, (1996).

7. S. Raby, Grand Unified Theories, arXiv: hep-ph/0608183, (2006).

8. M. Dine, Origin of the matter-antimatter asymmetry, Rev. Mod. Phys., 76, 1 (2003).

9. N. Craigie, Theory and Detection of magnetic Monopoles in Gauge Theories, World Scientific, Singapur, (1986).

10. M. E. Peskin and D. V. Schroeder, An introduction to quantum field theory, Addison-Wesley, 786 (1995).

11. G. Ross, Grand Unified Theories, Book, Westview Press. (1984).

12. Buchmüller, "Neutrinos, Grand Unification and Leptogenesis", arXiv: hep-ph/0204288v2, (2002).

13. E. Witten, Stringtheory dynamics in various dimensions, Nuclear Physics B, 443, 85 (1995).

14. L. Smolin and R. Sundrum, Three Roads to Quantum Gravity, Basic Books, ISBN 0-465-07835-4 (2001).

15. A. A. Abdo, "A limit on the variation of the speed of light arising from quantum gravity effects", arXiv:0908.1832 (Fermi GBM/LAT Collaborations) (2009).

16. L. E. Dickson, On Quaternions and Their Generalization and the History of the Eight Square Theorem, Ann. Math., 20, 155 (1919).

17. W. R. Hamilton, Elements of quaternions, Chelsea Publications Co., New York, (1969).

18. P. G. Tait, An elementary Treatise on Quaternions, Oxford Univ. Press, New York, (1875).

19. D. Finklestein, J. M. Jauch, S. Schiminovich and D. Speiser, Principle of general quaternion covariance, J. Math. Phys., 4, 788 (1963).

20. S. L. Adler, Quaternion Quantum Mechanics and Quantum Fields, Oxford Univ. Press, New York, (1995).

21. B. S. Rajput, S. R. Kumar and O. P. S. Negi, Quaternionic formulation for dyons, Lett. Nuovo Cimento, 34, 180 (1982).

22. V. Majernik, Quaternionic formulation of the classical fields, Adv. Cliff. Alg., 9, 119 (1999)

23. V. V. Kravchenkov, Applied Quaternion Analysis, Helderman Verlag, Germany (2003).

24. Shalini Bisht, P. S. Bisht and O. P. S. Negi, Revisiting quaternion formulation and electromagnetism, Nuovo Cimento, B113, 1449 (1998).

25. K. Morita, Octonions, Quarks and QCD, Prog. Theor. Phys., 65, 787 (1981).

26. R. P. Graves, Life of Sir William Rowan Hamilton, 3 volumes, Arno Press, New York, (1975).

27. J. C. Baez, The Octonions, Bull. Amer. Math. Soc., 39, 145 (2001).

28. S. Catto, Exceptional Projective Geometries and Internal Symmetries, arXiv: hep-th/0302079 v1 (2003).

29. R. Foot and G. C. Joshi, Space-time symmetries of super string and Jordan algebras, Int. J. Theor. Phys., 28, 1449 (1989).

30. J. Lukierski and F. Toppan, Generalized space-time supersymmetries, division algebras and octonionic M-theory, Phys. Lett.,

    B539, 266 (2002).

31. J. Schray, Octonions and Supersymmetry, Ph.D. thesis, Department of Physics, Oregon State University, Corvallis, (1994).

32. K. Imaeda, Quaternionic formulation of tachyons, superluminal transformations and a complex space-time, Lett. Nuovo Cimento, 50, 271 (1979).

33. R. Penny, Octonions and the Dirac equation, Amer. J. Phys., 36, 871 (1968).

34. A. Gamba, Peculiarities of the Eight Dimensional Space, J. Math. Phys., 8, 775 (1967).

35. M. Gogberashvili, Octonionic electrodynamics, J. Phys. A: Math.Gen., 39, 7099 ( 2006).

36. B. C. Chanyal, P. S. Bisht and O. P. S. Negi, Generalized Octonion Electrodynamics, Int. J. Theor. Phys., 49, 137 (2010).

37. B. C. Chanyal, P. S. Bisht and O. P. S. Negi, Generalized Split-Octonion Electrodynamics, Int. J. Theor. Phys., 50, 1919 (2011).

38. B. C. Chanyal, P. S. Bisht, Tianjun Li and O. P. S. Negi, Octonion Quantum Chromodynamics, Int. J. Theor. Phys., 51,3410 (2012).

39. B. C. Chanyal, P. S. Bisht and O. P. S. Negi, Octonionic non-Abelian Gauge Theory, Int. J. Theor. Phys. 52, 3522 (2013).

40. G. M. Dixon, Division Algebras:: Octonions Quaternions Complex Numbers and the Algebraic Design of Physics, book, Springer (1994).

41. R. Penny, Octonions and isospin, Nuovo Cimento, B3 (1971), 95; Amer. J. Phys., 36, 871 (1968).

42. S. Okubo, Introduction to Octonion and Other Non-Associative Algebras in Physics, Cambridge University Press, Cambridge, (1995).

43. K. Morita, Octonions, Quarks and QCD, Prog. Theor. Phys., 65, 787 (1981).

44. P. S. Bisht, B. Pandey and O. P. S. Negi, Octonion wave Equation and Generalized fields of Dyons, Proc. National Symp. Mathematical Sciences, Nagpur, 12, 137 (2001).

45. K. Imaeda, H. Tachibaba and M. Imaeda, Octonions, superstrings and ten-dimensional spinors, Nuovo Cim., 100B, 53 (1987).

46. Z. Weng, Octonionic electromagnetic and gravitational interactions and dark matter, arXiv: physics.class-ph/0612102v8, (2009).

47. Z. Weng, Octonionic Strong and Weak Interactions and Their Quantum Equations, arXiv: physics/0702054, (2007).