Elementary particle Symmetry and Group theory

Published: 07th July 2006
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Elementary particle Symmetry and Group theory

Article was published in Ganita Chandrika (ISSN 0973-3493), 6, 13 (2006)

Dr. N.V. Prasad, J. Subrahmanyam, Dr. A. Hanumaih and Dr. K.S. Ravi

Department of Physics and Electronics, Postgraduate Centre

P.B. Siddhartha College of Arts & Science, Vijayawada-10

The focus on elementary particles is increasing tremendously on account of its unstability nature. Owing to the fact, both fundamental and theoretical research groups are being concentrating in this direction. After invention of neutron particle by Chadwick, Heisenberg suggested that atomic nuclei consist of neutron (n) and protons (p). Besides proton and neutron, the intermediate particles, namely photons, may exist between neutrons and electrons (e) in the atomic system. If matter (consist of p, n and e) exist then certainly antimatter may also exist [consisting anti-electron (positron), anti-proton and anti-neutron]. After prediction of new particle omega (mass two hundred times of electron) by Yukawa during 1935, many new particles, whose lifetime less than microseconds, were identified. On account short lifetime, these particles cannot be treated as normal constituent of matter. They are characterized by mass, spin, electric charge and magnetic moment. In view of this, these particles are referred to as Fundamental, Strange, and Elementary particles. The word fundamental referees that the particles are the basic

building blocks of matter.

In brief, classification of elementary particle is divided into main groups, namely Bosons and Fermi-ions. The important difference between the two classes is only Fermiions obey law of conservation. Classification of elementary particles is shown in the figure 1.


Figure 1. Classification of identical particles

The types of particle-particle interaction and its details are summarized in the bellow


Type of elementary


Relative magnitude Associate particle Characteristic time


Strong interaction 1 Pions, kaons 10-23



10-3 Photon 10-20

Weak interaction 10-13 Intermediate bosons 10-10



1039 Graviton (mass less




The aforementioned classifications of elementary particles (by observing cosmic rays and cloud chamber experiments) have met with success in predicting new particles.

In this connection, an alternative advanced mathematics, named 'Group theory' was found to be precious tool to predict new particles, in the absence of electromagnetic

interactions. In other words, group theory is an alternative theoretical method from which one can predict the new particles without any experimental evidence. For simplicity, by using group theory, the important assumptions and conclusions are given bellow:

Considering the following assumptions:

1. Set of operators represents the symmetry constituents of the group. The appropriate group operation can transform from one state to another state in the same


2. If the system is invariant with respect to displacement, angular displacement, and time in space then linear momentum, angular momentum, and energy are conserved.

Conclusions arrived:

1. Simplest group U (1) contains transformation, which adds a phase factor to particle wave function. The invariance such transformation gives conservation of charge (Q),

baryon, lepton (L) and hyper-charge (twice the average charge in the number group).

2. Proton and neutrons are assumed to be identical as per as nuclear forces are concerned. But they are different during the electromagnetic interactions. Therefore

by considering the group theory, one can transform proton to neutron or neutron to proton in the absence of electromagnetic field. In other words the matrix interaction

of matter, consisting of proton and neutron states (!p>, !n>), and antimatter,

3. consisting of antiproton and anti neutron (!p>, !n>) state can be represented in

matrix notation as:

⎢p ⎢ (p- n- ) (pp- +nn-) /2 ⎢1 0 ⎢ + ⎢ (p p- - nn-)1/2 pn- ⎢

⎢n ⎢ ⎢0 1 ⎢ ⎢ np- (pp- - nn-)1/2⎢

Where the first and second term of right hand side represents of Meson and Pions.

3. SU (3) represents the three-basis state fundamental representation. Under 3 x 3 =9 operators, only eight operators are valid. Therefore this group has eight generators.

Gell-Mann refers the present symmetry eight operators to Buddha eight-fold path to nirvana (right action).

(i) If matter (consisting of p, n and λ particles) and antimatter consist (p-, n- and λ-) then the matter and antimatter interaction in terms of matrix notation is as follows:

⎢p ⎢ (p- n- λ-) ⎢1/3 (2pp- - nn- - λλ-) p-n- pλ- ⎢

⎢n ⎢ ⎢np 1/3 (2pp- - nn- - λλ-) λn- ⎢

⎢λ ⎢ ⎢λp- λn- 1/3 (2pp- - nn- - λλ-) ⎢

(ii) From the diagonal element, the neutral meson was predicted. Ιt follows the

relation: η = (pn+nn+2λλ)/ √6

(iii) Gell-mann and Neemanm predicted the meson, from the boson octet. The meson

was denoted by symbol ηo. The meson was identified at Q (charge) = 0, Y (hyper

charge) = 0 and T (spin component) = 0. The axis terms extraneza and carga refers to hypercharge and charge in Japanese language.


(iv) The formulated baryon octet array (B=1) by Gell-mann and Neemann matrix is:

⎢(Σo/√2)+(Λ/√6) Σ+ p ⎢

⎢Σ- (-Σo/√2)+ (Λ /√6) n ⎢

⎢Σ- Σo -2 Λ/√6Σ- ⎢

(v) (a) Gell-mann suggested the following average mass component relation:

3Mo 2 + M1 2 = 4 M1/2 2, where M0, M1 and M 1/2 are the masses of meson, Pion

and kaons.

(b) The average mass of four components of baron is:

2(Mn + MΣ) = 3MA+MΣ

In this connection, Okubo suggested spatial general formula as:


M (T,Y) = Mo {1+ay+b(T(T+1)-1/4 Y2), where a and b are constants.

From the equation, the mass of the omega was found to be 1676MeV at Y = -2 and the value is tallied with the measured value 1675 MeV. Thus, one can say that

group theory is an advanced tool to predict new particles without any experimental evidences.

About Murray Gell-Mann:

Murray Gell-Mann (1929-) was born in New York and entered Yale University at fifteen. After obtaining his Ph.D. from the Massachusetts Institute of Technology in 1951, he was at the institute for advanced study in Princeton and at the

University of Chicago before joining the faculty of the California institute of Technology. In 1953 Gell-Mann introduced strangeness number and its conservation in certain interactions to help understanding the properties of

elementary particles. In 1961 he formulated a method of classifying elementary particles that enabled him to predict the Ω- particle, which was latter discovered.

Two years latter Gell-Mann come-up with the idea of quarks, the ultimate entities from which particle subject to the strong interaction or composed. He received the Noble prize in Physics in 1969.

Philosophical Approach: Universe consists of macroscopic and microscopic quantities and these quantities are encapsulated with energy form. The interaction between particle/matter (one truth) and particle /antimatter (another truth) is through exchange mechanism called myth (maya). In other words one can understand 'truth' through 'maya'.6 It is known that matter consist of nuclei (truth) and electrons (another truth) rotating around the nuclei. The intermediate particles photons (considered

another truth) and the interaction mechanism is myth. In other words truth and myth are complimentary to each other. They cannot separate out. The same phenomenon is applicable to solar system. Sun is the truth, which gives energy to

our living planet earth. Earth is also another truth, which gives energy to living kind. Therefore to understand truth, we human beings certainly should adopt myth or good karma (right actions). This statement indeed quoted Both Buddha and Gell-Mann.


1. Nuclear Physucs, D.C Tayal, Himalaya Publications

2. Concepts of Modern Physics, A. Beiser, Tata Mc Graw Hill Pub.

Acknowledgements: Authors thank Siddhartha Academy of General and Technical Education for the constant encouragement.

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