GRAVITON-ANTIGRAVITON ANNIHILATION AND CREATION OF FERMI QUARK - F-MESON PAIR MULTIPLET

Ref: Indian Science Congress 68^th Session,Paper No.114, 

        Mathematics Section, Banaras Hindu University, 3-8^th January 1981.

GRAVITON-ANTI GRAVITON ANNIHILATION AND CREATION OF

 FERMI QUARK - F-MESON PAIR MULTIPLET

K. L. Narayana,

Shivaji University, Kolhapur – 416004.

Present Address: Prof. Dr. Kotcherlakota Lakshmi Narayana,

17-11-10, Narasimha Ashram, Official Colony, Maharanipet.P.O., Visakhapatnam-530002.

ABSTRACT

      Within the context of an unified Quasi U(2,10) supersymmetric Gravity theory the process Graviton + Anti-Graviton à Fermi Quark + Anti-Fermi Quark + f-meson + Anti–f-meson has been analyzed as a four body decay process and the life time (Half-life) has been obtained as τ _{f i} = C ^{- 2} . 10 ^{-  59} where the new coupling constant is shown to be as fundamental and the Newtonian coupling constant. Also is the suggested “connectivity” between the two space-time domains of interactions one of the elementary particles and the other of Graviton-like particles. The relationship between the mass ratios of f-meson and Graviton-like Spin 2 (and Spin 5/2) and the new coupling constant C, of the universe has been explicitly given adopting the COSMOD transformations (Il Nuovo Cimento, Vol.33A, p.641, 1976) and the PERANGON  transformations(Int. Symp. on Appl. Math. Tech. to Phys. Sciences, December 31, 1976). The decay cross-section expressions are reported.

Ref: Indian Science Congress 68^th Session, Paper No.114, Mathematics Section, Banaras Hindu University, Varanasi, 3-8^th January 1981.

GRAVITON-ANTIGRAVITON ANNIHILATION AND CREATION OF

 FERMI QUARK - f-MESON PAIR MULTIPLET

K. L. Narayana,

Shivaji University, Kolhapur – 416004.

Present Address: Prof. Dr. Kotcherlakota Lakshmi Narayana,

17-11-10, Narasimha Ashram, Official Colony, Maharanipet.P.O., Visakhapatnam-530002.

INTRODUCTION

       Dirac [1] form of gravitation equation, of motion has been earlier formulated by the author. The underlying group has been suggested as one of a direct product of Kemmer-Duffin ring structure with the Spin 5/2 angular momentum operators. This set of generators analogous with SU(2,2) ≃ O(4,2) isomorphic group have been utilized to construct a unified Lie Algebraic structure i.e. isomorphic with the sub-algebra of a quasi U(2,10) group. The formulation also leads to other kinds of Gravity possibilities of gravitation such as the non-compact Graviton spectrum existence of a Graviton vital space and as well that the Charm Graviton to be able to be classified as a singlet of unified description of Graviton-Lepton-Hadron super-symmetric structures [2].

        The fact that the Graviton has spin 5/2 leads to either possibility that it may obey Einstein-Bose statistics or Fermi-Dirac statistics. The Bosonic Graviton of half-integral spin 5/2 was commented upon by the author extensively [2] previously. The object of the present paper is to examine more closely the feasibility of its obeying a Fermi-Dirac statistics in the context of four body decay processes that would consequently become allowed. Obviously, such a formulation has the immediate physical consequence of existence of an Anti-Graviton. The Anti-Graviton (though the authors approach is distinct) concept is not new one and is already discussed within the frame work of super-symmetry models, that are being investigated in literature by Ferrara,[3], Wess and Zumino[4] and recently by Davis [5] and Narayana [6] certain features of the Graviton- Anti-Graviton processes have been considered by Zeldovich[7].

        For our investigation presented herein firstly it may be recognized that the Fermionic Graviton of Spin 5/2 would have a corresponding Anti-Graviton of opposite momentum. The Graviton and Anti-Graviton system would lead to a compound system of spin angular momentum total of five units. The particular possibility that this system can decay into a Quark - Anti-Quark and two f-mesons is one of a great interest, as it would lead to events that are verifiable experimentally. The author earlier model that the hadrons and as well the Leptons are the bound structures of a fundamental super-quark system of Quasi U(2,10) group certainly implies a decay process of the kind above mentioned. It is interesting that a number of features of this decay mode are amenable for an analysis without explicitly adopting either the covariant-perturbation theory or the dispersion technique for the evaluation of the cross-section. Analogous approaches of analysis were previously adopted in the early stages of formulation of theories of β-decay (Low Energy) and in recent times, in the studies of Vector Boson mediated electro-weak forces (High Energy).

        In section-1 the details of a four body decay of final states are given. In section-2 the expressions of the decay mode in the center-of-mass frame of reference have been detailed. Section-3 presents the results and conclusions drawn by the present study of the Graviton – Anti-Graviton annihilation process.

SECTION- 1: Decay to four particles final state:

         The four body reaction is written as,

a + b  à d + e + f + g

with the decay cross-section given by,

                     

Fig.1

where σ is the cross-section of the process from the initial to the final state of momenta P_i to P_f respectively, V the volume of spatial extension and M_fi the matrix amplitude of the process. From the later one can construct the scattering matrix

Fig.2

with M_fi =  τ _{f i} / N where N is the appropriate normalization factor. The transition probability per unit time ω = (1/(VT)) |R_fi|² with the standard relation of perturbation theory connecting R_fi  with the M_fi.

      

               Also we can write

Fig.3

SECTION – 2:  Center-of-mass frame of reference

In the center-of-mass system, we have,

Fig.4

The differential cross-section,

Fig.5

          An important step that has been adopted in deriving this, the expressions are valid in the limit of zero mass as well just as in β-decay Fermi-coupling constant derivation. But, the quarks and as well the f-mesons we note are massive. The mass consideration however, does arise since the Graviton and Anti-Graviton rest mass energies are of finite magnitude. This is a rather radical departure from the conventional understanding of gravitation and its propagation by the Graviton mediator. In our theory, being presented here, the Graviton and rest-mass and as well has the Anti-Graviton has a negative energy and negative momentum.

Fig.6

where m_f,  E_f are the mass and energy of the F-meson.

Fig.7

    Where C² is proportional to the product of strong interaction strength g ² / 4π  = 14, and that of weak interaction strength

(g ²_ω  m² _π /4π)  ≃  10^-14.

SECTION – 3: PERANGON TRANSFORMATIONS AND RESULTS

        It is necessary for practical calculations then to take the derived formula for the half-life (τ _{f i} )  of the decay process.

        Graviton + Anti-Graviton à Quark Pair + (Pair) F-meson 

   

        or

g  +   ̅g   à  ν    +   ̅ν   + f   +  ̅f

  to be given as follows,

C ² m_f⁷ /( 64x64 π⁵) *(Δ / m_f)⁶    ≃   1/ τ _{f i}

when Δ = 2 m_f it simplifies to

1/ τ _{f i}  =  C ² m_f⁷ / 64 π⁵

when m_f is about 1250 MeV we get,

1/ τ _{f i} =   C² . 10⁵⁹

as the order of magnitude.

                Thus we arrive at an incredulously low half-life of the decay

process i.e. about 10 ^{- 45} sec. What may be the physically reasonable life-time for the process that is envisaged in the present work? It can be said that it is many fold move quicker than any known strong interaction order of time periods! What is the optimum half life-time order of magnitude for the creation process following the annihilation of Graviton - Anti-Graviton pair?

PERANGON Transformation has been defined earlier which connect the domains of two types of spin 5/2 particles while discussing about the “Cosmod Transformations”. The author has stressed that the ratio of masses between the two different kinds of Spin 2 particles, should be taken as fundamental as the Newtonian coupling constant. In relevance to these investigations, the present work reveals that there, exists a new type of coupling constant which is actually to be taken as fundamental as the Newtonian coupling constant. There exists an intimate relationship of this new coupling constant with the mass ratios of the two domains of particles, and also with the half life-time of the decay process outlined above. The interaction coupling strengths and the mass ratios both play a significant role in the determination of the half life-times of the particle creation and annihilation processes.

What’s the new “connectivity” that exists between the two domains of interactions one that of particle would as observed to possess the Charm, Hyper-charge, Strangeness etc. quantum numbers and the other domain of Graviton-like particles, such as spin 5/2 and spin 2 particles. This topic would be elaborated in a forthcoming publication.

CONCLUSION

                    To conclude the present work we state that the value of τ_fi = 10^-45 sec is almost double the usual interaction life-time of the order of 10^-23 sec., for the strong interaction processes. Since the gravitational interaction are much faster relative to the strong interactions. The value of τ_fi = 10^-45 sec is not physically unreasonable.

ACKNOWLEDGEMENT

   The author is grateful to Late Prof. K. R. Rao, D. Sc. (Madras) D.Sc. (London) for his stimulating research endeavor at his laboratories in Andhra University, Waltair.

REFERENCES

1.  Dirac

2.  Bosonic quanta  Narayana

3.  Ferrar

4.  Wess and Zumino

5.  Davis

6.  Narayana

7.  Zeldovich

*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.

*10. K. L. Narayana, “On the Unification of Gravitation and the Quantum

       Theory”, Shivaji University, 1979 and invited talk presented at

       GRG 8^th National Symposium at Bhavnagar, 1978.

A NEW FIELD QUANTIZATION APPROACH TO GRAVITATION

AND BOSONIC GRAVITONS OF SPIN 5/2

By

Dr. K. L. Narayana,

Shivaji University, Kolhapur – 416004.

Present address: Prof. Dr. Kotcherlakota Lakshmia Narayana,

17-11-10, Narasimham Ashram, Offical Colony, Maharanipet.P.O.,

Visakhapatnam-530002

ABSTRACT

A new formulation of Einstein’s field equations of gravitation is given adopting the field Lagrangian as

Fig.1

Where ψ is 60-component spinor and β’s are 60x60 spinor matrices. The method differs from the Dirac formulation in the sense that derivative graviton fields are utilized to construct ψ.

        Β’s define a new algebra and give rise to current Graviton terms

Fig.2

where g is the characteristic constant of gravitation.

        An important outcome of the theory is that Graviton field variables obey Bose statistics but in spinor contents is of Fermionic character with a graviton spin 5/2. The theory is most useful to for the Meso-Baryon symmetry models sand method of a unified approach is suggested. The Hamiltonian, Linear Momentum and Density of the gravitational field expressions are reported.

Fig.3

where R₂ = -2β²₄ + I describes the anti-gravitons creation or destruction of which are found equivalent to destruction or creation of gravitons of negative momentum.  

*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.

*10. K. L. Narayana, “On the Unification of Gravitation and the Quantum

       Theory”, Shivaji University, 1979 and invited talk presented at

       GRG 8^th National Symposium at Bhavnagar, 1978.

A NEW FIELD QUANTIZATION APPROACH TO GRAVITATION

AND BOSONIC GRAVITONS OF SPIN 5/2

By

Dr. K. L. Narayana,

Shivaji University, Kolhapur – 416004.

Present address: Prof. Dr. Kotcherlakota Lakshmia Narayana,

17-11-10, Narasimham Ashram, Offical Colony, Maharanipet.P.O.,

Visakhapatnam-530002

INTRODUCTION

          Gravitation if interpreted in a field sense yields Gravitons by a second quantization procedure and they would indeed the most fundamental particles of the Nature. In a weak field approximation it can be easily be shown (Dirac-Pauli-Fierz and Gupta) the Graviton to have the spin 2. Many Papers on the nature of spin 2 theories of Gravitation have since then appeared in literature. Earliest are those by Fierz followed by Olof Brulin and Stig Hjalmars and Hjalmars. Lord has also discussed spin 2 field gravity in interaction with other fields and adopting the Riemann-Curvature tensor. Dirac-like form of general spin equations which yield spin 2 particles with limiting conditions of vanishing mass are discussed by Berg Korff and E. P. Wigner. Dirac equation for the description of a spin 2 particle motion has been given by Madhava Rao with prescription of associated algebra. Relevance of this formulation and correspondence if any with the Einstein’s empty space field equation has been the subject of investigation by Narayana. Mercier contends that however, spin 2 seems to be the unique property of this most fundamental particle to exhibit similarity of it with the particles of other fields, and in itself is only a consequence of artificial construction and hence argues field of gravitation is not of the same nature as other fields. Narayana et al. on the other hand conjectured ‘Cosmod Transformation’ which predict possible existence of (from elementary particles of strong interaction symmetry) spin 2 graviton-like particles. This conjecture however, does not explain anything regarding the nature of gravitation but only adds to the complication by predicting yet another type of Newtonian-like gravitation field, with peculiar properties of elements with particle symmetry nature and their characteristic quantum numbers. Apart from the spin 2 aspect of graviton, Mercier points out that gravitation is not an interaction like other interactions and further he asserts that if at all gravitons do exist then quantization would amount to quantization of time. In the studies of gravitation physics, Hiida and Yamaguchi point out, the basic equations of any massless tensor field with arbitrary integral spin and parity can be written down and by simplicity principle show that graviton to possess reasonably a spin 2 and should be a particle of even parity. Their equation have many interaction terms involving C, P and T violations.

        Wienberg clearly demonstrates how the 10 independent components of metric tensor, in a weak field approximation, are reduced to only two degrees of freedom by virtue of the harmonic co-ordinate system and the gauze invariant criteria for Einstein’s field equations. These degrees of freedom correspond to ± Helicities of spin 2 gravitational wave. Feynmann and Wienberg have pointed out the severe constraints on the S-matrix, equivalent to the gauze invariance requirements of General Relativity, and establish the Graviton to have a Spin 2. Desser and Duff have discussed these and reviewed reasons of why Spin 2 exchanged particle is responsible for gravitational force. It is interesting to note that the arguments by these authors are based on the notion of weak principle of equivalence, exclusion of exotic Lorentz invariant theories of Graviton which would require ghost behavior and any kind of non-Lorentz theories. Not unusual that physical properties, such as universality of coupling strength and effects of light bending, of the Graviton and its long range character are cited to support the view of a Spin 2 for Graviton.

        Again if Gravitons are indeed of Spin 2⁺ quantum state, Abdus Salam queries regarding the selection rules for mixing of Gravitons and other Spin 2 particles of other fields. Can therefore, Graviton manifest physically in symmetry model of elementary particle physics? Narayana et al suggested realizing Spin 2 particles from a model of fundamental Meso-Baryon symmetry.

Quantum-mechanical analog models Zel’dovich has come to the conclusion that additional Einstein terms are imperative in the General Einstein Field equations to describe the process of pair creation in vacuum under a gravitation field. One of the aspects of his investigation refers to a possible estimate of entropy of the universe and its isotropy which, are both connected for pair creation presumably at the point of initial singularity.

In the Thirring’s field theoretic approach of gravitation, Spin 2 quanta are accompanied by Spin 0 component. When the ratio of these two components is such as to yield only Spin 2 content then it agrees with the Einstein’s Field equations but more interesting is that Thirring’s theory incorporates “the Mach Principle”. More ambitious are the approaches of super-gauze structures such as those by Volkov and Sovokar which yield a Spin 2 mixture with a Spin 3/2. Other theories are those by Yang and Dewitt and Utiyama and Dewitt.

In the present work the author is interested to re-examine the question of Spin of the Graviton and its statistics, from the view point that gravitational quanta though obey Bose statistics may not be of even integral spin. The immediate possibility of a spin 5/2 has been investigated. Motivation is due to the philosophy of a unified approach of Gravity and quantum field theories of particle physics. If anything akin, peculiarity of statistics and its spin behavior, of the Graviton, is one aspect from where we expect a new understanding of gravitation. It would be of immense benefit, nevertheless, to link up similar fields with meso-baryon supermultiplet models that are being investigated at our school.

In the next section of this article presented are the details of the new formulation proposed. The third section gives a discussion and results of the present theory by the author.

SECTION – 2

MATHEMATICAL FORMULATION:

        We adopt the Lagrangian given by Narayana et al,

Fig.4 a and Fig.4b

as the supplementary condition. This allows us only 6 independent components hence the spin of Graviton is 5/2. Demonstration of this would be made through the field variables at a subsequent stage of this article.

                Accordingly the above equation for the 6 independent variables and their derivatives are recast into a simple equation of the form

Fig.5

Here ψ is a 60-component spinor matrix with the six field variables and corresponding to each the four derivatives field variables. The β’ are also 60x60 matrices. Such multi-component spinors with large spinor matrices and associated algebras are not unusual feature of field theory. As examples, the work of de Broglie, Petiau, Pereira, may be cited.

        The field Lagrangian of the gravitation with Graviton mass given by κ₂ is then

Fig.6

This leads to the following spinor operator relations:

Fig.7

With the suffixes representing positive spinor component, S and L respectively the large and small components of spinors Wk and W₄ being the derivative field variables of a component of the field variables a_ik. 

Here

Fig.8

With                          φ = ∂ γ \ ∂x _k .

Thus analogous to the Thirring model of gravitation the present theory also has the possible admixture of Spin 0 content. But the merit of the theory proposed now is that the scalar quantity in its spatial-temporal variations adds to the Spin 2 aspects as a part of the Graviton. The novelty being that field theory incorporates the background (scalar field of matter) automatically in the field equations.

The

Fig.9

Leads to also definitions of current quantities

Fig.10

where g is a characteristic quantity of the field gravitation.

        Perhaps more interesting is that the Hamiltonian comes out to be,

Fig.11

with negative momentum states and as for any Bose symmetry we adopt that the creation of Anti-Graviton, for example, equivalent to the destruction of Graviton of negative momentum. This result is very interesting since it leads to an understanding of time reversal at microscopic level of gravitation.

        The factor 2 occurring in front of the expressions for H, P and ρ has a resemblance with that which occurs even in the Gupta’s expressions for the quantization of gravitational field. In our approach it arises solely due to the involvement of derivative field variables in ψ. But explicitly our formulation devoid of spin 0 component to be admixed. Also the present theory does not need to adopt the criteria that the contracted field variables γik  must be γ.

          Formal features of a canonical quantization procedure of a field such as Noetherm theorem validity and the definitions of Hamiltonian and the momentum are same in our theory, since we adopt,

Fig.12

        The theory developed is being investigated to study the current interaction terms, current-algebra and its relation with a fundamental Meso-Baryon symmetry, of other spin 5/2 particles of elementary particle physics.

ACKNOWLEDGEMENT

                 The author is indebted to the authorities of GRG Association for an invited talk of the author delivered at 8^th Annual Symposium held at Bhavanagar in 1978. Details given as well by K. L. Narayana  at Shivaji University, 1979 in a talk entitled “On the unification of Gravitation and the  Quantum Theory”.

REFERENCES

Mercier: Proc. of International Symposium on Gravitation and

               Unified field Theory, Calcutta, p.1, 1975.

Dirac P. A. M,  Proc. Roy. Oc. Vol. A246, p.333, 1958,

                        Contemporary Physics Vol.1, Trieste Symposium,

                        p.539, IAEA, Vienna, 1969.

Fierz M and Pauli W, Proc. Roy.  Soc. A. Vol.173, p.211, 1939.

Gupta Suraj N, Proc. Phys. Soc. Vol.65A, p.608, 1952.

Narayana K. L. et al, Il Nupvo cimento Serie ii Vol.33A, p.641, 1976.

Hiida K, and Y .Yamaguchi, Prog.Theor. Phys. Suppl.

                                             Extra number p.261 1965.

Narayana K. L., J. Shivaji University, 1978.

Narayana K.L, High Energy Physics symposium, Vol.1,

                       Papers II-6 and II-9 Bhubhaneswar,

                       Orissa , November 1976.

Feynmann R. P, Valtech Lectures on Gravitation unpublished.

S. Wienberg, Phy.Rev. Letters, Vol.9, p.357, 1964.

                     Phys. Rev. Vol.135, B1049, 1964.

                     Phys. Rev. Vol.138, B988, 1965.

Desser S, Lectures on Particles and Field Theory,

                   Ed. By K.Fors et al, Prentice Hall Ltd, 1965.

                   Also Quantum Gravity, Ed. By C.I.Isham, R. Penrose and

                   D.W.Sciama, Clarendon Press, Oxford, p.111, 1975.

R. A. Berg, J. Math. Phys, Vol.6, p.24, 1965.

D. Korff, J. Math. Phys, Vol.5, p.869, 1964.

E. P. Wigner, Ann. Math. Vol.40, p.149, 1939.

S. Hjalmars, Arkiv Fysik Vol.1, p.41, 1949.

                   J.Math.Phys. Vol.2, p.662, 1961.

Olof Brulin and Stig Hjalmars, Arkiv. Fysik, Vol.14, p.49-60, 1959.

                                                Also ibid Vol.16, p.19, 1960,

                                                          Ibid Vol.18,p.209,1960.

                             J. Math. Phys. Vol.5, p.1368-1390,1964.

W. Thirring, Ann. Phys. Vol.16, p.96, 1961.