5 January 2017: A NEW FORMULATION OF A THIRD ORDER AND A FIFTH ORDER GROUP EQUATIONS TO DESCRIBE PARTICLES

        trusciencetrutechnology@blogspot.com                           Volume 2017, Issue No.1d,  Dt:5 January 2017, Time:14h  to  18h PM 

A NEW FORMULATION OF A THIRD ORDER AND A FIFTH ORDER GROUP EQUATIONS TO DESCRIBE PARTICLES 
                                                       by

               Professor Dr. Kotcherlakota Lakshmi Narayana,

M.Sc. (Math. Phys) M.Sc. (Nucl. Phys) Ph. D. (AU), Int. Edu & Res. (Sweden 1964-65) 
DOB:23-07-1940. [Retd. Prof of Physics; Shivaji University, Kolhapur - 416004]
Residential Address: 17- 11-10, Narasimha Ashram, Official Colony,

      Maharanipeta. P. O., Visakhapatnam-530002, A.P. Mobile: 09491902867.

kotcherlakota_l_n@hotmail.com;  lakshminarayana.kotcherlakota@gmail.com

MATHEMATICS SESSION PAPER 

Dr. Biresh Guha Street, Kolkata-700017.

ISC 2017, Sri Venkateswara University, Tirupati, January 3 to 8,

A NEW FORMULATION OF A THIRD ORDER AND A FIFTH ORDER GROUP EQUATIONS TO DESCRIBE PARTICLES

Professor Dr. Kotcherlakota Lakshmi Narayana,

(Retd. Professor.  SU, Kolhapur-416004)

17-11-10, Narasimha Ashram, Official Colony,

Maharanipeta. P. O., Visakhapatnam - 530002, AP.

Mobile No.9491902867

Dt. 23 July 2016 at 10 h 15 m 45 s, AM.

ABSTRACT

             The new equations of third order and fifth order both to describe the particles are enunciated. It involves in third order, all the momenta p1, p2, p3 of the particle and has a mass m term. It has the form m^3+m^2*a2+m*a1+a0=0 with the coefficients a2, a1 and a0 being elaborate functions of the momenta. This happens to be a subgroup {I, W, J, X} of a bigger Group G (S₃). [Given earlier by squdder@du.edu]. The equation formulated by me, is given as -m*I + p1*W + p2*J + p3*X = 0, where m is the mass of a three dimensional space and involving the subgroup elements as operators. The extensive analysis, reveals the present equation with complex form and yields a real solution as well. The cubic mass term is very distinct and, unlike the Dirac equation, reveals new mathematics. Further, I designate U=p1; V=p2; Q=p3; B=p0; A=E; as the other possibility of the description for the massive particle M in a 5D space-time group. We get -M*I + a*A +b*B + u*U + v*V + q*Q = 0. Here M becomes the mass of a five dimensional space-time particle. The addition of an energy expression E and the momentum p0, enlarges the present model. Details are explicitly be given. The present work describes the third order description of the particle with mass m and, as well the fifth order description of particle with mass M explicitly. Graphs given presented for both type of particles.

  

INTRODUCTION

                 Both the group third order and fifth order are described explicitly. The idea being that the involvement of energy E, and momenta p0 (that describes the time variable) are involved in the fifth order equation. The third order exclusively involves only the momenta p1, p2, p3. The study provided a contrasting picture of the odd order equations unlike the Dirac equation that involves only the fourth order equation, to describe the electrons and their antiparticles the positrons. The question of negative solutions do not occur in the present formulation. The antiparticles need to be described in a separate format. The details would be given later.

GENERAL GROUP

                                            Group G(S3) subgroups are (I,A,B); (I,C,D); (I,E,F); (I,G,H); (I,W,J,X); (I,A,B,U,V,Q); (A,B,C,D,E,F,G,H,I,J,K,L); (I,K);(I,L); (I,J);(I,M); (I,U); (I,V)  about 13 groups individually with M and N:(0 0 1;1 1 1; 0 -1 0) as generators of the groups.(See the article published by S. Gudder on ‘Discrete Quantum Gravity and Quantum Field Theory”  8 March 2016; 1603.03471vI [Math-ph].

MY FINDINGS

THIRD ORDER GROUP

3 order Equation is

                               -m*I + p1*W + p2*J + p3*X = 0; with matrices specified, 

-m*[1 0 0; 0 1 0; 0 0 1] + p1*[0 -1 0; 1 1 1;-1 0 0] + p2*[-1 -1 -1; 0 0 1; 0 1 0] + p3*[0 0 -1; -1 0 0; 1 1 1] = 0; p1, p2 and p3 as the momenta.

The third order equation is, m as the mass,

m^3 + m^2* (p2-p3-p1) + m*(p3^2+p1^2-p2^2-p2*p1-p1*p3-2*p1*p2*p3) +

(-p3^2-p2^2+p1*p3^2-p2*p3^2-p2^2*p3+2*p1p2*p3)=0;

Pictorial Presentation

As an example, I present here the picture of the expression

                     plot3d(1+(p2-p3)+(p3^2-p2^2-2*p2*p3)+(-p3^3-p2^3-p2*p3^2- p2^2*p3),p2=-3..3,p3=-3..3); 

                 Other examples would be given at 2017 Indian Science Congress Session to be held at Tirupati during 3 – 8 January 2017.

                 plot3d(1+(p2-p3)+(p3^2-p2^2-2*p2*p3)+(-p3^3-p2^3-p2*p3^2-                      

                     p2^2*p3),p2=-3..3,p3=-3..3);                              

FIFTH Order GROUP

                                                                     We consider E as the energy of the particle.

          Again considering a group G (S3) of six terms {I, A, B, U, V, Q} we may formulate including E=A.

                       A = [0 0 1; 1 0 0; 0 1 0]; B = [0 1 0; 0 0 1; 1 0 0]; U = [0 -1 0; -1 0 0; 0 0 -1]; V = [-1 0 0; 0 0 -1; 0 -1 0]; Q = [0 0 -1; 0 -1 0; -1 0 0];

Writing, 5 order equation as, with mass M,

-M*I + a*A +b*B + u*U + v*V + q*Q = 0; or with momenta p1, p2, p3, p0 and energy E specified, U=p1; V=p2; Q=p3; B=p0; A=E;

I get, -M*I + a*E + b*p0 +u*p1 + v*p2 + q*p3 =0 is another equation of higher order.

         I designate U=p1; V=p2; Q=p3; B=p0; A=E; as the other possibility of the description for the massive particle M.

The equation describing it is found from the group specified as,

M^3 +M^2 * (v + u + q) –M*(a*q +b*q-q*u-v*q + b*u-3*b*a-b*v-u^2 –q^2) + (2*v*u*a +q*a*v + b*u*a  + a*b*q + u*b*v –v*q*u + v*b*a+ u* b* q + q* u*a + b*v*q  +b*v^2 + u^3 +a^3 +q^3 + b^3 – u*v^2 – b*u^2 –b^2 * q –b^2*v –u^2*a – u*b^2 –a^2 *v – u*a^ - q*a^2- a*q^2 –b*q^2=0;

Pictorial Presentation

   As an example, I present here the picture of the expression

plot3d(2*p3*E*p2+1.5*p2*p3+0.5*E^2*p2-2*E*p3^2-3.0*E*p3-p3*E^2+E*p2^2+2.25*p2+2.25*p3+1.5*p3^2+2.0*E^3+p3^3+4.5*E^2+3.375,p2=-3..3,p3=-3..3);p0:=E;p1:=0;M:=1.5;E:=1.5;

   

                       plot3d(2*p3*E*p2+1.5*p2*p3+0.5*E^2*p2-2*E*p3^2-3.0*E*p3-            p3*E^2+E*p2^2+2.25*p2+2.25*p3+1.5*p3^2+2.0*E^3+p3^3+4.5*E^2+3.375,

            p2=-3..3,p3=-3..3);p0:=E;p1:=0;M:=1.5;E:=1.5;                                         

        Other examples would be given at 2017 Indian Science Congress Session to be held at Tirupati Sri Venkateswara University, during 3 – 7 January 2017.

        Many diagrams and details of the models enunciated would be presented at the Tirupati ISC 2017 meeting to be held during 3-7 Jan 2017 at TIRUPATI Sri Venkateswara University. 

PRESENTATION AT THE SESSION OF MATHEMATICS DEPARTMENT

IN TIRUPATHI Sri Venkateswara University on 5 Jan 2017.

ANOTHER PAPER GIVEN AT ISC 2017 TIRUPATI ON  6 Jan 2017:

    Dr. Biresh Guha Street, Kolkata-700017. ISC2017, SRI VENKATESWARA UNIVERSITY, TIRUPATI, January 3 to 7, Physics Session Paper "Water in Universe with Gravitons and its double lone-pair electrons at an apex distance of 1.4 Angstroms" Professor Dr. Kotcherlakota Lakshmi Narayana, (Retd. Prof.of Physics, SU, Kolhapur-416004),17-11-10, Narasimha Ashram, Official Colony, Maharanipeta. P. O., Visakhapatnam - 530002, AP. Mobile No.9491902867 and No. +919491362867 Dt. 16 July 2016 at 10 h 15 m 45 s, AM. ABSTRACT:Double pair electrons share in the middle the Graviton, as per my model considerations. This positioning of the individual gravitons helps to distribute the entire galaxy spread, water distribution. Several water molecules along with their associated gravitons and the lone-pair electrons may assemble to augment the entire spread of the expanse of the water distribution in the galaxy, each holding a graviton. This is totally a new look of the galaxy spread, with water molecules and the four electron embedded gravitons.  Key Words: Graviton, lone-pairs, Water, Universe, apex. 

ACKNOWLEDGMENT            

                    I am indebted to Late Prof. K. R. Rao, D.Sc. (Madras) D.Sc. (London) for his stimulation, on new thoughtful research and path breaking findings.