*19. K. L. Narayana,”Physical orthonormalization criteria for gravitation

      spinor wave functions, chirality projections and hyper-number √1”,

       66^th Sess. of Ind. Sci. Cong. Hyderabad, Mathematics Section, 1978.

PHYSICAL ORTHOGONALIZATION CRITERIA OF GRAVITATION SPINOR WAVE FUNCTIONS, chirality projections AND HYPERNUMBER √1.

K. L. Narayana,

Shivaji University, Kolhapur 416004.

Prof. Dr. Kotcherlakota Lakshmi Narayana,

Permanent address: 17-11-10, Narasimha Ashram, Official Colony,

 Maharanipeta. P.O, Visakhapatnam -530002

ABSTRACT

The spinor wave function forms have been developed in detail within the framework of a spin 5/2 and Bosonic Statistical formulation given by the author earlier for the Gravitons. A method of orthonormalization procedure is suggested which employs the physical restraint of parity bound motion of the Gravitons. The expressions for the Chirality operators of the form (̂ ̂σ . ⃗p ) adopting the spin 5/2 matrices of angular representation have been given explicitly. Also the sums of positive and negative spinor wave functions ∑ u_r  ̅u_r;  ∑ v _r   ̅v _r are reported, which are useful to project out, in any given specific physical process of interactions, positive and negative freuency parts of the wave function satisfying the equation (β_μ ∂_μ   -  κ₂)Ψ =0,  κ₂ being the mass of the Graviton.

        The relation of the parity bound motion of Graviton with the HYPERNUMBER √1 is discussed.

*19. K. L. Narayana,”Physical orthonormalization criteria for gravitation

      spinor wave functions, chirality projections and hyper-number √1”,

       66^th Sess. of Ind. Sci. Cong. Hyderabad, Mathematics Section, 1978.

PHYSICAL ORTHOGONALIZATION CRITERIA OF GRAVITATION SPINOR WAVE FUNCTIONS, chirality projections AND HYPERNUMBER √1.

K. L. Narayana,

Shivaji University, Kolhapur 416004.

Permanent address: 17-11-10, Narasimha Ashram, Official Colony,

 Maharanipeta. P.O, Visakhapatnam -530002

SECTION – I

INTRODUCTION

The new formulation of General Theory of Relativity equations given by Narayana (1978a) for the Graviton field equations in empty space are of the form ( β_μ ∂_μ   -  κ₂) Ψ =0,  where Ψ is a 60 component spinor wave function, which involve both the field variables and its derivatives. These equations describe the spin 5/2 quanta and are exhibited in a solution of central force field problem given by Narayana (1978b) to have an explicit angular momentum representation.

        We expect that this formulation also yields projection operators, the eigenvalues of which specify the Chirality and the Helicities of Graviton. However, at the same time we might have also to expect existence of truly neutral Gravitons, with the charge conjugate states, the Graviton and Anti-Graviton yielding equivalent description of Gravitational field, since these quanta obey Bose-Einstein statistics.

        Subtle complexities of physical interpretation arise in the buildup of such a theory, by virtue that the higher spin matrices and associated Chirality operators etc. are not simple mathematical quantities for ease of handling them such as for the case of a spin ½ particles.

        The aim of this paper is to report results of the preliminary study made, in the new formulation regarding the physical normalization criteria of spinors its significance through the hyper number √1 and as well the details and certain properties of the projection operators.

                                

SECTION- II

A)  SPINOR SOLUTIONS

Instead of precise formulation, as it is the practice of field theory, particular forms of field equations and their solutions are highly useful and desirable. In the present method these solutions are to a certain extent dictated already by the particular form of the β-matrix representation that is necessarily adopted in the spin 5/2 Graviton theory formulation. The representation of the β’s form is, of course, by virtue that the field theory should indeed correspond with the General Theory of Relativity field equations of Gravity. Again, the chosen form of β-matrices is the same as those conventionally used in the angular momentum representation theory (Rose 1957).

At first we examine a set of specific solutions of the particle equations of motion from the new Lagrangian as they are worth investigating for deeper physical insight into the problem of negative and positive frequency and parity bound motion of the spin 5/2 Gravitational Quanta.

It is apparent that the equation (β_μ ∂_μ   -  κ₂) Ψ =0,  can be satisfied by a wave function while includes, either the periodic terms e ^{±i p.x} where p.x  = p_λ  x_λ . Adopting, the solutions of this equations, therefore as form,

  Ψ  = Ψ^u   e ^{+ i p.x}

or                  Ψ  = Ψ^v   e ^{- i p.x} ………….(1)

where Ψ^u   and    Ψ^v    are the spinors to be determined, we obtain the solution of Graviton motion with +p.x term referring to states of particles with positive energy +E and a positive momentum + ⃗p.

        The need of negative frequency solution arises in our problem due to the requirement of spinor character of the Graviton, however, as explicitly shown in an earlier paper, introduces no complexity of physical interpretation, despite they are spinors and obet Bose statistics.

        Customary as it is to solve the equations of this type, we set the solutions so as to result in appropriate form, in the limit of vanishing momentum, to describe specifically the spin 5/2 quanta, with the following specific spinor forms,

[1 0 0 0 0 0]’, [0 -1 0 0 0 0]’, [0 0 1 0 0 0]’,

[0 0 0 -1 0 0]’, [0 0 0 0 1 0]’, [0 0 0 0 0 1]’

where ‘ specifies the column matrices for positive frequency part. A similar set of spinor solutions may be adopted for the negative frequency part. For this purpose we have adopted the so-called large and small components of the Graviton Spinors indicated by the subscript letters L and S.

Listed below are the equations (2), (3), (4) and (5) of the present work. We then get the equations,

 Fig.1

Fig.2

           

If one chooses a frame of reference with the Z-axis parallel to the physical momentum, then ⃗p = (0, 0, p), the matrix A reduces to the form σ _zp. Under this choice of the system of co-ordinates the spin 5/2 coupled equations of motion reduce to a simple form. More interesting but is the possibility that,

σ_p U _r = (-1)^{r -1} U _r where σ_p   = (̂   ̂σ . ⃗p )/| ⃗p | provided,

of course, we take that

A  =     ̂σ . ⃗p  =  σ_z  p

Fig.3

analogous to the case of a spin ½ particle. The criteria being that (8) is transformed or “physically orthonormalized” to have the form (9). The details of this normalization procedure would be given in the next subsection, but to continue the discussion on the operator σ_p we neatly note that it is indeed (with the matrix A of the form given in equation (9) the Chirality projection operator for the spin 5/2 system. This projection operator is valid only in the special frame of reference chosen.

        Thus σ_p U_r = (-1)^{r -1} U_r  for r= 1, 2, 3, 4, 5, 6 is the Chirality operator, projecting out the positive frequency spinors.

C). NORMALIZATION OF SPINOR WAVE FUNCTIONS

        To give the generalized definition of Chirality operator needs precise understanding of the physical normalization criteria suggested casually in the previous section. Orthogonalized matrix A would now read as

Fig.4

                    

               The orthogonalization criteria require that expression of the type p₊ p₊   or  p_- p_- to vanish : these imply p_x²  = p _y² and p_{x .} p_y =0. The first of equations though asserts a necessity of axial symmetry of Graviton motion the second equation has a far reaching physical significance. This would be further discussed in a subsequent section. For conclusion of this sub section we state that individual columns and rows of the matrix

or the matrix

is required. However, only three distinct normalization factors need to be adopted. These are, with

Using these, the complete orthonormalized spinors would be of the form, as given below:

D) PROJECTION OPERATORS

                     The positive and negative frequency parts may be expected to be projected out, we give below the

∑_r u_r  ̅u_r    and    ∑ _r v _r   ̅v _r

Operators, for constructing the so-called projection operators

                                        Λ ⁺   and Λ ^- .

SECTION - III

PARITY BOUND MOTION OF GRAVITON AND HYPER NUMBER √1 :

                     Space inversion is an important transformation of paramount physical significance. The parity has been in built in the quantum electrodynamics, and both the strong and weak interaction symmetries through the operators 1+ γ₅ and 1- γ₅ for spin ½ particles.

Details of it may be found in any standard text book (Muirhead 1964). Attention may also be drawn to an article by Corinaldesi 1958, on particles and their symmetries. The comparability of it with other quantities has been recently discussed by Grober 1975.

                     Charles Muses 1977, specifically points out the relationship between the operators γ₅ and the hyper-number √1. He asserts that in mathematical reality the projection operators, spin and helicity are in fact “Hypernumbers”.

                     We note that, the present theory of author of adopting the physical orthonormalization criteria requires that p_{x .} p_y =0. The significance of this is that p_x  and  p_y may be of  the form (1+ε)  and (1- ε) where a product (1+ε).(1-ε)=0 with ε itself being neither +1 nor -1 but ε² =1. The Epsilon hyper-number has quaternion set known as the meta-quaternion set. The square root of Epsilon is given by ± ½ (1+ ε - i + i₀)    where i ε = ε i = i₀ and 1 is the ordinary number; i = √-1.

                     Intimate relationship of ε = √1 ≠   ± 1 with parity operation indicates therefore that physical orthonormalization criteria adopted in this paper does indeed leads to a “parity bound motion” of the spin 5/2 particle.

                     Further work on this, implication of this motion specifically for the Chirality, Helicity and Spin will be the subject of forthcoming papers by the author. TABLE - 1 and TABLE – II are listed below.

Table II

ACKNOWLEDGMENT

                              The author is indebted to late Prof.  K.  R.  Rao,  D.Sc. (Madras)  D.Sc.(London)  of  Andhra University,  Waltair  for  his  interest  and guidance  in  promoting my  research  endeavor.

REFERENCES

        1.Narayana. K. L et al , Nuovo Cimento Series 11,

      Vol.33A, p.641-648, 1976.

        2. Narayana K. L, ”Spin 5/2 Graviton-like particles and their

            relativistic motion in  a central force field”,

            66^th Sess. of Ind. Sci. Cong. Physics Section, 1978a;

             Invited talk presented at GRG 8^th National Symposium at

             Bhavanagar, 1978b

3. Rose M.E. “Elementary theory of Angular Momentum”,

    John Wiley & Sons. Inc, 1958.

4.Davydov. A.S, “Quantum Mechanics”, Pergmon Press,

    London,1956.

5. Muirhead H, “Physics of Elementary Particles”,

    Pergamon Press, London, 1965.

6. Corinaldesi “Nuclear Physics”,  Amsterdam, 1958.

7. Grober D, “Parity, Baryon number and Supersymmetry”,

    Preprint, Tujbingen 1975.

8.Charles Muses, Il Nuovo Cimento, Vol.33A, p.532-640, 1976.