*18. K. L. Narayana,”Spin 5/2 Graviton-like particles and their relativistic motion in

      a central force field”,  66^th Sess. of Ind. Sci. Cong. Physics Section, Paper No.    ,1978.

Spin 5/2 Graviton-like particles and their

Relativistic motion in a central force field

K. L. Narayana,

Shivaji University, Kolhapur – 416004.

Present address: 17-11-10, Narasimham Ashram, Official Colony, Mharanipeta.P.O, Viskhapatnam-53002

ABSTRACT

                The relativistic motion of a spin 5/2 particle has been solved in an angular momentum representation solving explicitly the spin-content of the radial wave equations. The formalism of spin 5/2 quanta obeying Bose statistics developed by the author has been adopted with the basic equation of motion as ( E  -  V ) β₄ ψ = (i  ̅β  . ̅p   –  κ₂) ψ where κ₂ is the mass of Graviton. The spectrum of s_5/2,  p_3/2,  p_7/2,  d_1/2, d_9/2……….. etc. has been suggested, as well the relativistic motion of the relativistic graviton-like particle governed by projection operators, such as ( ⃗σ   .  ⃗L ) χ_ρμ  = 5 . |ρ| χ_ρμ  and ⃗σ (σ_i  . x_i) . ⃗L,  a dyadic relativistic motion. The radial equations solved in spin content using the appropriate Clebsch-Gordon and Racah coefficients are reported. Importance of these for Astrophysical experimental spectral observations has been stated.

*18. K. L. Narayana,”Spin 5/2 Graviton-like particles and their relativistic motion in

      a central force field”,  66^th Sess. of Ind. Sci. Cong. Physics Section, Paper No.    ,1978.

Spin 5/2 Graviton-like particles and their

Relativistic motion in a central force field

K. L. Narayana,

Shivaji University, Kolhapur – 416004.

Present address: 17-11-10, Narasimham Ashram, Official Colony, Mharanipeta.P.O, Viskhapatnam-53002

INTRODUCTION

                Number of theories, adopting relativistic wave equations for describing both particles and anti-particles, are in recent time being evolved to describe particular phenomenon.  Jose’. V. Pereira has put forward for instance the “theory of fusion” first suggested by de Broglie, to develop a general method of dealing with anti-particles. Such theories are of paramount importance to understand the fundamental differences between Bosons and Fermions.

        The author has also suggested a formalism of the general relativity gravitational equations, in terms of a Dirac-like equation to describe the spin 5/2 quanta of the field. Peculiarity of his theory being that it incorporates the possible behavior of Fermionic quanta of spin 5/2 with a statistical Bosonic character.(Narayana et al.)

        In the present paper are the details of the central field solutions of such a particle. Purpose is to formulate explicitly the wave function of a spin 5/2 particle emphasizing that such wave functions are indeed the correct one in an angular momentum representation.

        The utility of the solutions obviously lies in the fact that spin of a particle carries no energy in itself and in a sense becomes only through its coupling with orbital motion. This view point, though purely relativistic in its content, leads to results verifiable or useful to predict spectral features by Astrophysical experimental observation of Galaxies moving with velocities comparable with the speed of light.

        Coupling of spin with orbital motion has two apparent aspects of interest. It leads to find out the conserved total angular momentum and secondly to estimate the energy levels of a quantum gravitational atom.

SECTION – II: METHOD OF CALCULATIONS

(a).         Non-relativistic description.

We introduce a wave function ψ = R (r) χ_ρμ , where

χ_{ρμ =} ∑_m  C(l 5/2 j; m, μ-m) Y_lm   ψ_{5/2, μ – m} ;

with  μ – m = ± 5/2, ±3/2, ±1/2

with the l and j angular momenta of the particle of Spin 5/2 both given by the quantity ρ, as

j=| |ρ| - 5/2 |; l=ρ for ρ > 0 and

j  = ||ρ| + 5/2 |;  l = -ρ-1  for ρ < 0.

Here ρ = -1,1,-2,2 ….etc for  s_5/2, p_3/2, p_7/2, d_1/2, d_9/2 etc. 

The  ψ_{5/2,, ms} are the intrinsic spin functions described by respectively for ms= 5/2, 3/2,1/2,-1/2,-3/2,-5/2.

From these definitions it clearly follows that

⃗S.⃗L χ_ρμ  =  5 |ρ| χ_ρμ 

and the Schordinger’s equation of such a particle would then be,

Fig.1

(b). RELATIVISTIC DESCRIPTION

        We adopt now the equation suggested by Narayana for the spin 5/2 particle.

Hψ = Eψ = β₄ (i  ̅β  . ̅p   –  κ₂ +V(r)) ψ

or

((i  ̅β  . ̅p   –  κ₂ ) ψ =  (H- E) β₄ ψ

for a stationary state of total energy E.

 The conventional usage of ⃗p  = - i ⃗  \bigtriangledown  potential energy V(r) and other symbols of relativistic wave mechanics would adopted in what follows. Heaviside units

 ћ = c=1 would be used.

The equations now would become,

Fig.2

the σ’s according to the definition

Fig.3

and the σ’s are the usual spin 5/2 intrinsic spin matrices.

Since

Fig.4a and Fig.4b

                Here the dashed quantities are the derivatives with respect to r. The set of radial equations, which exhibit the relativistic structure of spin 5/2 particle, are highly complicated to solve since they involve number of components. For simplicity sake and to arrive at the subtlety of the structure of the radial wave behavior we adopt that 

f_W1 = f_w2  = f_w3 and g_W1 = g_w2 = g_w3

in their radial dependence. 

Fig.5a and Fig.5b

                As regards solving in spin-aspect of these relativistic radial equations, for the spin 2 particles in a central force field, we note the following,

Fig. 6

Since (5/2 || σ || 5/2) =√35

And

C(l’1l;0 0)= √(l’+1)/(2l’+1) for l= l’+1

= √l’/(2l’+1)  for l= l’-1.

We get

Fig.7

Obviously σ _r = being of odd-parity, we expect l’≠ l, so that ρ’=-ρ or l’=l±1. Choosing l’=0, we have W(1,5/2,0,5/2;j,1) which exists for j=5/2 and equals -1/√18.

                Similarly for l’=1, we have w(0,5/2,1,5/2;j,1)=-1/√18. In either case, we get σ_r χ_ρμ  = -( 1/ √18) . √6. √35 χ _{- ρμ}   = - √(35/3).

Thus    σ_r χ_ρμ   = - N χ _{- ρμ}   where N is a numerical constant.

            The evaluation of the operators ⃗σ (σ_j  . x_j) . ⃗L. χ_ρμ is also possible for instance if we adopt the relations such as,

σ_y ψ _{5/2, ms} = (-1) ^ms-5/2  ψ _{5/2, ms-1} .

Conclusions

      This paper illustrates how the basic equations of central force field of a spin 5/2 particle may be set up and solved for their radial behavior and fine structure spectrum. Form of the potential V(r), not known for spin 5/2 is necessarily for numerical predictions and solutions of that type would be the subject of our forth coming paper. Essential conclusions of this paper are however, (1). The relativistic motion of a spin 5/2 particle, such as a Graviton in a galaxy moving at relativistic speed, is a sort of superposition of different kind of motions described by the spin-orbit coupling j= l+5/2 to j=l-5/2. (2). Unlike the spin ½ Dirac particles, the spin 5/2 motion involving the “spin dyadic motion” arising out of the operators such as ⃗σ (σ_j  . x_j) . ⃗L 

where j=1,2,3. (Arfken. G [4]).

ACKNOWLEDGMENT

                        

                  The author is deeply indebted to Late Prof. K. R. Rao, D.Sc. (Madras), D.Sc. (London), for his stimulating ideas and encouragement to publish my thoughts on Gravity etc. spin considerations, and for whom I am grateful.

REFERENCES

1. J. V. Pereira, Int. J. Theor. Physics, Vol.16, p.147, 1977.
2. De Broglie L, “Etude Critique des bases de l’interpretation de la Me’cenique ondulatoire”, Gauther-Villars, Paris, Elsevier, 1964.
3. P.A. M. Dirac, “Quantum Mechanics”, p.46, Clarendon Press, Oxford, 1935.
4. G. Arfken, “Mathematical Methods for Physicists”, p.136,Academic Press, 1970.
5. Narayana. K. L, Accompanying Paper, Ind. Sci. Cong. 1979.

*17. K. L. Narayana,”A new field quantization approach to Gravitation and

Bosonic gravitons of Spin 5/2”, Indian Science Congress, Hyderabad,

Physics Section,(Late Paper), 1979.