A new approach to Elementary Particle Physics and Super Symmetry of Quantum Mechanics

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Volume 2012, Issue No.3, Dt. March 31st, 2012
Time: 16h47m PM

A New Quantum approach for Elementary Particle Spectroscopy & Super- Symmetry

by

Professor Dr Kotcherlakota Lakshmi Narayana

{Retd. Prof. of Phys, SU, Kolhapur} 17-11-10, Narasimha Ashram, Official Colony, Maharanipeta P. O, Visakhapatnam -530002. Mobile No: 9542717723 & 9491902867

ABSTRACT

A new version of the Quantum Energy levels of a Hamiltonian formalism to describe new types of Elementary Particles and the Super Symmetry envisaged.

Color Maps given to illustrate the formalism developed. Klnex formalism now given by the present author that is characterized by two fundamental Masses bet1 and bet2. These two masses have been contrasted, with a set of two masses like the Proton, Rao quanta or just the Proton and Positron etc. The momentum component alpr and the bet1 & bet2 masses seem to be intricately involved. Is it the Rao quanta play a definite role in Elementary Particle Physics? The four sets of energy value diagrams presented herewith.

INTRODUCTION

It is necessary to develop new approaches for the Quantum Mechanics and any attempt to safe guard the possible set of several Elementary Particles that arise in the description of the finer aspects of the Particle spectroscopy or the Universe at large. I am happy that a newer formalism of the Super-symmetry and quantum Mechanics is envisaged herewith. Hamiltonian approach seems to be the easiest method of analysis. Color map is given below in a design then 0 and 1 would have different colors. See the references sited below about Super Symmetry [1 and 2]

Main approach

I define the set of matrices f1, f2, f3, f4, f5 and f6 that allows one to formulate a Hamiltonian formalism to describe new types of elementary particles and their spectroscopy. I define the set of matrices f1, f2, f3, f4, f5 and f6 that allows one to formulate a Hamiltonian formalism to describe new types of Elementary Particles and their Spectroscopy.

f1=[0 1 0 0 0 0;1 2 0 0 0 0;0 0 0 0 0 0;0 0 0 0 0 0;0 0 0 0 0 0;0 0 0 0 0 0]; tr1=2;


f2=[0 0 2 3 0 0;0 0 3 0 0 0;2 3 0 0 0 0;3 0 0 0 0 0;0 0 0 0 0 0;0 0 0 0 0 0]; tr2=0;


f3=[0 0 0 0 4 5;0 0 0 0 5 4;0 0 0 0 0 0;0 0 0 0 0 0;4 5 0 0 0 0;5 4 0 0 0 0]; tr3=0;


f4=[0 0 0 0 0 0;0 0 0 0 0 0;0 0 0 1 0 0;0 0 1 2 0 0;0 0 0 0 0 0;0 0 0 0 0 0]; tr4=2;


f5=[0 0 0 0 0 0;0 0 0 0 0 0;0 0 0 0 2 3;0 0 0 0 3 2;0 0 2 3 0 0;0 0 3 2 0 0]; tr5=0;


f6=[0 0 0 0 0 0;0 0 0 0 0 0;0 0 0 0 0 0;0 0 0 0 0 0;0 0 0 0 0 1;0 0 0 0 1 0]; tr6=0;

Below is the presentation of a matrix that consists of about 6x6=36 elements with a zero value for the ground state of a quantum system with different specifications.

Color map is given below in a design then 0 and 1 would have different colors.

Klnex formalism now given by the present author leads then to a Hamiltonian

H=alpx*f6+alpy*f5+alpz*f3+alpr*f2+ bet1*f1+bet2*f4

that is characterized by two fundamental masses bet1 and bet2 which (may be in the style of COSMOD TRANSFORMATION earlier defined, introduced and proposed by the present author Prof K. L. Narayana (Nuvo Cimento ) may be identified with a set of two particles like the Proton, Rao quanta or just the proton and positron etc.

Here f1, f2, f3,f4, f5 and f6 play the role of certain set of matrices of which, f1 and f4 have trace 2 and others are traceless.

Here

alpx, alpy, alpz and alpr

are similar but a distinctly different set of may be the momentum components of a particle of dual nature with masses bet1 and bet2. 

Thus the newly invented klnex formalism holds good and is valid within its own frame work. The f1 product with f2 matrix is just the transpose of the product f2 with f1. Similarly, f1 with f3 gives the transpose of f3 with f1. While the product of f1 with f4 leads to a null matrix. Similarly, f1 product with both f5 and f6 leads to null matrix. The product f2*f3 is transpose of f3*f2. The product f2*f4 is a null matrix. The product f2*f5 is transpose of f5*f2. The product f2*f6 gives the null matrix just like the product f3*f4. The product f3*f5 is transpose of f5*f3. The product f3*f6 is transpose of f6*f3.The product f4*f5 is transpose of f5*f4. The product f4*f6 is a null matrix.

f1*f2= [0 0 3 0 0 0; 0 0 8 3 0 0;


            0 0 0 0 0 0; 0 0 0 0 0 0;


            0 0 0 0 0 0; 0 0 0 0 0 0];



f1*f3= [0 0 0 0 5 4; 0 0 0 0 14 13;


             0 0 0 0 0 0; 0 0 0 0 0 0;


             0 0 0 0 0 0; 0 0 0 0 0 0];

                      f1*f4=f1*f5=f1*f6=[0];


f2*f3= [0 0 0 0 0 0; 0 0 0 0 0 0;


             0 0 0 0 23 2; 0 0 0 0 12 15;


             0 0 0 0 0 0; 0 0 0 0 0 0];

 f2*f4=[0];


f2*f5=[ 0 0 0 0 13 12; 0 0 0 0 6 9;


             0 0 0 0 0 0; 0 0 0 0 0 0;


             0 0 0 0 0 0; 0 0 0 0 0 0];
     f2*f6=[0];


       f3*f4=[0];


f3*f5= [0 0 23 22 0 0; 0 0 22 23 0 0;


             0 0 0 0 0 0; 0 0 0 0 0 0;


             0 0 0 0 0 0; 0 0 0 0 0 0];




f3*f6= [0 0 0 0 5 4; 0 0 0 0 4 5;


             0 0 0 0 0 0; 0 0 0 0 0 0;


              0 0 0 0 0 0; 0 0 0 0 0 0];


The choice


alpx=0; alpy=0; alpz=0;


projects H as


Hproj =


[0, bet1, 2*alpr, 3*alpr, 0, 0;


bet1, 2*bet1, 3*alpr, 0, 0, 0;


2*alpr, 3*alpr, 0, bet2, 0, 0;


3*alpr, 0, bet2, 2*bet2, 0, 0;


0, 0, 0, 0, 0, 0;


0, 0, 0, 0, 0, 0]

Which, is isomorphic with


{1 , -I, I, –I}


or {E, C4, C4^2, C4^3 }

and the later components constitute a group of symmetry of a square.

The momentum component alpr and the bet1 & bet2 masses seem to be intricately involved. Is it the Rao quanta play a definite role in Elementary Particle Physics?

To realize the role played by the alpr component adopt

                             alpr=1; bet1=1; bet2=1;

then we get simplified. Other values of the bet1 and the bet2 masses may be chosen to realize other types of alpr component magnitudes. Few novel choices are also permitted. For the illustration a matrix is displayed with the values alpr=1;bet1=1;bet2=1.

 

These elements may also be represented in a matrix form given below.

The figure also gives the details of the Hproj values in (X,Y)-plot.

Hproj has the eigenvalues and the eigenvectors as given below:

v =[ -0.6533 0.5000 0 0 -0.2706 0.5000;

        -0.2706 -0.5000 0 0 0.6533 0.5000;

          0.6533 0.5000 0 0 0.2706 0.5000;

          0.2706 -0.5000 0 0 -0.6533 0.5000;

          0 0 1.0000 0 0 0;

          0 0 0 1.0000 0 0];

And the diagonal energy values

d =[ -2.8284 0 0 0 0 0; 0 -2.0000 0 0 0 0;

         0 0 0 0 0 0;0 0 0 0 0 0;

         0 0 0 0 2.8284 0; 0 0 0 0 0 6.0];

 



Fig.1 WFs of energy value -2.8284




Fig.2 wave functioms of energy value -2.0





Fig.3 WFs of the energy value 6.0



Fig.4 WFs of the energy value 2.8284

Note all the four excited energy levels. Two are positive and two are negative. The duality behavior holds good. The forbidden gaps are quite different in these two cases. But, the existence of two more eigenvectors corresponding to the zero energy levels, project the 3rd and the 4th potentials respectively in a six dimensional klnex formalism space.

CONCLUSIONS

• New Quantum Mechanics is more general than that envisaged by a Super Symmetry Model of Quantum Mechanics. A book is available in I I Sc, Bangalore library, on Super Symmetry and Quantum Mechanics dated 2012.

• Because of the restrictions of conventional Super-symmetry model on the couplings, two Higgs doublets are required, in order to give masses to all the charged fermions and the spontaneous breaking gives Goldstone Bosons.

• Dated 26th October 2011 propriety right held

References:

1. K.L.Narayana et al., 21 Giugno, Il Nuvo Cimento, Serie 11, Vol.33A, p.641-640 1976 { claim news 26th Feb and 24th Feb 1977 Times of India, 26th Feb 1977 Indian Express, Patriot and Indian Express (Vijayawada)} The scientist envisaged that the new gravitational force is endowed with hypercharge and strangeness. It may be mediated by particles less massive than the Graviton.

2. Supersymmetric Quantum Mechanics by A. Gangopadhyaya et al, World Scientific 2010 (031122) IISc Ref No.530.12dc22 also see the book by M. Drees, Rohini M. Godbole and Probhior Roy entitled ”Theory and phenomenology of S-particles 4D N=1 Super Symmetry in HEP” 183964; 539.725. These are very interesting books on the new approaches in Quantum Mechanics.

ACKNOWLEDGEMENT

I am indebted to Late Professor K. R Rao D.Sc. (Madras) D.Sc. (London) for his inspiring guidance and helping me to formulate new thoughts in the subjects of human endeavor.