On Spin-Gravity in a new Renormalization Theory of Electro-Weak interactions and Gravy-Changing (neutral) Currents existence

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Volume 2014, Issue No.2, February 13, 2014, Time: 3h41.P.M.

  103. K. L. Narayana,”On Spin-Gravity in a new Renormalization Theory of Electro-

        Weak interactions and Gravy-Changing (neutral) Currents existence”, 7th Jan, 

         Paper No.147, the 69th Sess. of Ind. Sci. Cong. Mathematics Section, 
         Manasagangotri, (Indian Express 7-1-1982), 1982.
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On Spin-Gravity in a new Renormalization Theory of Electro-Weak

 interactions and Gravy-Changing (neutral) Currents existence

K. L. Narayana

Shivaji University, Kolhapur – 416004 

Permanent address:

Professor Dr. Kotcherlakota Lakshmi Narayana 

 (Retd. Prof of Phys. SU, Kolhapur); 17-11-10, Narasimha Ashram,

 Official Colony, Maharanipeta. P.O,  Visakhapatnam – 530002.

Cell No.09491902867.

ABSTRACT

General Theory of Relativity of Einstein is shown to
contain a third rank fundamental tensor whose components possess  a doublet spinor structure. A scattering matrix orthogonality  theorem which states the conditions of orthogonality between usual S-matrix and a defined S-matrix has been established  for the spinor doublet structure “in” and “out” field operators.

The modified photon field due to presence of an iso-spin triplet
Vector Bosonic mesons is shown to lead to finitness of the
otherwise divergent loop-diagrams of electrodynamics. Next the
Electro-weak interaction theory of Salam-Wienberg has been
Perfected to get rid of the difficulties of High-Energy behavior 
Terms by adopting the above spinors, characterizing them with the  new Quantum number GRAVY of magnitude G=1. Thus the new renormalized theory of Electro-Weak interactions involves not only GRAVY- neutral currents but also GRAVY-changing currents along with an iso-spin doublet structure, which is a noteworthy feature. (K. L. Narayana,Sym.Theor. Studies. Shivaji University, Kolhapur July 12, 1981).

Finally a dyad spin-frame of GRAVY-QUARKS (q,   ̰q) has been Formulated with an attended spinor-calculus analogous to that of Newman-Penrose and Geroch et al. The diagram of spin-weight and Boost-weights of the two-parameter subgroup of Lorentz group are diagrammatically represented.

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103. K. L. Narayana,”On Spin-Gravity in a new Renormalization Theory of Electro- Weak interactions and Gravy-Changing (neutral) Currents existence”, 7th Jan, 
 Paper No.147, the 69th Sess. of Ind. Sci. Cong. Mathematics Section, Manasagangotri, (Indian Express 7-1-1982), 1982.
===================================================================

On Spin-Gravity in a new Renormalization Theory of Electro-Weak

 interactions and Gravy-Changing (neutral) Currents existence

K. L. Narayana

Shivaji University, Kolhapur – 416004 

Permanent address:

Professor Dr. Kotcherlakota Lakshmi Narayana 

 (Retd. Prof of Phys. SU, Kolhapur); 17-11-10, Narasimha Ashram,

 Official Colony, Maharanipeta. P.O,  Visakhapatnam – 530002.

Cell No.09491902867.

Introduction:

Attempts at constructing a finite theory of Quantum
Electrodynamics(or the recent chromodynamics etc.,) are not new. The difficulties in extending to handle the divergence for renormalization procedure to Hadronic system are also know. One or other way these attempts, in spite of their spectacular success, such as by Feynmann(1958), Schwinger (1957), Tomoaga(not traced), Dyson(1966), A’ Hooft  C. T, (197 ) and the recent super symmetry pseudo-Potential gauge theories etc., are handicapped in the sense that the way of handling the divergent expressions  remains ambiguous due to their arbitrary nature of mathematical tricks. Lee(1971) emphasized further that not only the first order but as well the second order electromagnetic processes are responsible for the finitness of the observed quantities such as fractional mass differences between Hadrons in the same iso-spin multiplet, and the radiative corrections to Gv / Gμ . The ratio of Fermi constant Gv in β-decay and that of μ-decay constant Gμ . He formulated a theory in which the S-matrix is taken to be strictly unitary but the Lagrangian is not Hermitian. He pointed out the existence of a very large class of Field Theories with non-hermitian Lagrangians which do satisfy both the relativistic invariance and the unitarity condition. The problem of renormalization and divergence continues to be the subject of
discussion and refinement of the statement of the associated
problems for the strong, Electro-Weak and as well for the
Gravitational interaction.  Hawking(1979) has discussed the
Last mentioned interaction in terms of its coupling constants
[ Narayana, K. L. (1976), and (1981)].

Confining our attention to Quarks, and the nature of a
Universal Electro-Weak strength, I note that in order to construct a finite theory free from divergence, at the lowest orders, there must be a fundamental change in the basic formulation of a quark-field theory. I expect that such a theory to have the natural aim of obtaining all the observables as finite even though the are actually related to the unrenormalized coupling constants and the unrenormalized masses.

  At an earlier occasion (Narayana K. L. (1981)}, I have 
Introduced such a theory a new spinor quark doublet
structure that is already and intimately related to the Fundamental Tensor H_ijk , of the General Theory of Relativity. This is a conjugate tensor to the dual Riemann-Curvature tensor.

·        R _ijkl [Lorentz (1962,1949); Einstein (1961)]

     Section 1 of this paper deals with the Spinors ( q ,     ̰q),
Scattering matrices, an jndefinite metric (Dirac 1942) η Pauli
(1943, 1949) and states and proofs the theorem of orthogonality
of the scattering matrices. Section 2 elaborates how the Vector
Bosonic mesons of an iso-spin triplet, makes it possible to
remove all infinities from the electromagnetic mass differences
between Hadrons, as well as those associated with the radiative
corrections to weak decays at a better order of convergence.
Section 3 presents modified Electro-Weak currents and how they lead to renormalization, with the help of Gravy Quarks
In a perfect analogy with the Newman-Penrose and Geroch formulations. It also presents the diagrams that describe the Spin-Weights and the Boost-Weights

Section 1:

The new theory proposed recognizes first that there exists
a fundamental third rank tensor H_ijk  which is the conjugate 
variable to the duel Riemann-Curvature tensor *R_ijkm. The full canonical Lagrangian

L ‘ = L ( *R ^ijkm , g _ik ) + H_ijk *R ^ijkm,_m 

                                     + p^ik _j ( Γ^j _ik  - {ik,j}) + [R _ik – F(Γ^j _ik)]

adopts the canonical field variables and their conjugate given
by matrix

(g _ik        Γ^j _ik        *R ^ijkm  )

(ρ  ^ik        p^ik _j         H_ijk  )

has demonstrated that variation with respect to *R ^ijkm  yields

H_ijk  =   H’_ijk  - Φ_j g _ik    +   Φ_i g _jk

The tensor Hijk is stated to be as fundamental as the metric
Tensor g _ij    and the Riemann-Curvature tensor  
R ^ijkm . 
               This tensor has not been exploited as yet, for its physical significance nor used in practical problems in field theories. (Lorentz 1962;1949). (Einstein 1936).

Utilizing the components of this tensor it has been shown
by the author earlier (Narayana, K. L., Sym. Theor. Studies, Shivaji University, Kolhapur, July 12th 1981) that the Dirac equation for a field and an adjoint field can be generated for apparently a Spin ½ system denoted by q and   ̰q. Thus the basic spinor equations. For description of quarks, according to the present theory, arise directly from gravitational field theory. This fact has a far reaching  physical significance, explsained in the subsequent  sections of the present paper. We represent in an abbreviated  notation these as  q_μ  and  _~q_μ  for their space-time components.
The physical S matrix is, by definition, the matrix

S = S_μ’μ =  ( qⁱⁿ⁺_μ’  η  q^out_μ)

and    _~S  =   _~S_μ’μ =  (  _~qⁱⁿ⁺_μ’  η    _~q^out_μ)

where it occupies only the upper left corner of the leading
diagonal matrices of the bigger matrix ψ.

Theorem:
 
S⁺S =1;      _~S⁺  _~S =1   and     _~S⁺ S = S⁺ _~S =0.

if   q⁺_r  η  q_r > 0   and      _~q⁺_r  η    _~q_r > 0   for all q_r and  _~q_r.

Proof:  
q^out_r (qⁱⁿ _r)   denote the Eigenstates. All the four

sets { q^out_r, q_c}, { qⁱⁿ_r, q_c},{ _~q^out_r,  _~q_c}, { _~qⁱⁿ_r,   _~q_c} 

are assumed to be  complete. Adopting the normalizations,

q⁺_r’  η  q_r = δ _r’r and  _~q⁺_r’  η   _~q_r = δ _r’r we have the relations

valid for the two sets { q^out_r, q_c} and { qⁱⁿ_r, q_c} equally

valid for the sets { _~q^out_r,  _~q_c} and { _~qⁱⁿ_r, _~q_c}.  

  Here H q_r ^in,out     = E_r q_r ^in,out  ;

           H q_c                 = E_c q_c   ;

           H  _~q_c                =  _~E_{c  ~}q_c  ;

           H   _~q_r ^in,out   =   _~E_{r   ~}q_r ^in,out  ;

where E_r ,  _~E_r  are real for all q_r , and E_c and _~E_c are not

real. From these it follows, that q⁺_r  η  q_r’ = 0 if E_r ≠ E_r’ 

etc. relations.

A set of transformations У which transform the complete

set of base vectors,

        { q^out_r, q_c} ⇔  { qⁱⁿ_r, q_c}

           ⇕                           ⇕

      {  _~q^out_r, _~q_c} ⇔  { _~qⁱⁿ_r, _~q_c}

        Fig.1a

aas schematically represented in the Fig.1a may be written down in the matrix form by utilizing the invariance property of η matrix under such set of transformations У.

У⁺ η У   = η

Explicitly,

У = 1/√2 [ S 0  _~S 0; 0 ▓ 0 0;  _~S  0 0 0; 0 ▓  0 ▓ ]

where the orders of the columns, from left to right, corresponds

to the set of base vectors, (q^out_r,  q_c,    _~q^out_r,    _~q_c).

The physical matrix

  q⁺ⁱⁿ_r η  q^out_r’ = _~S_rr’ .

The zeros in У are due to the orthogonality conditions and the

Shaded areas represent, say a matrix of the type σ = [0 1;1 0].

              Thus the matrix η would have the form

η = [1 0  0  0;  0  ▓  0  0; 0  0  1  0; 0  0  0   ▓   ]

upon substitution, the above relations in the equation

У⁺ η У   = η  we get S⁺S = _~S⁺_~S =1 and   _~S⁺ S = S⁺ _~S =0.

MODIFIED QUANTAL EM FIELD:

Presence of the gravitational field introduces a modification of the photon field. Since we expect the fields described

by    q_r  , _~q_r   to interact via massive Boson fields of mass

m_B  ,   ̰m_B   and m_B⁰ .

The EM interaction is, then

H_EM = j_μ (eA _μ + i g⁺ B⁺_μ   + i g⁰ B⁰ _μ    +   i g^- B^-_μ   )

where i is the usual electromagnetic current and

B⁺_μ   = B_μ  + i   ̰B_μ   

B^-_μ   = B_μ  -  i   ̰B_μ

and a neutral field B⁰  which is introduced for the sake of iso-spin symmetry completion of the new fields.

Hence the Riemannian curvature of space-time itself may be the

source of a corresponding iso-spin current. The nature of this obviously is expected to be different from that of similar iso-spin currents of hadron spectroscopy essentially because we consider the coupling constants 

                                               ( g⁺ , g^-   , g⁰) .

 In the eventuality of   g⁺ , g^-   , g⁰ approximately the same, (iso-

Gravitational charge independence hypothesis), we have then to

Lowest order, the sum of the two diagrams,

Pic1

These yield the term, instead of the usual e ²/ k² term,

[ e²/k²  -   q²/( k²+ m ⁺_B²) –   q²/ (k² + m^- _B²)    - q²/ (k² + m⁰ _B²)]

which is O(k^-8) as k² à ∞.    Thus   in the presence of the

Gravitational ( EM type) interaction makes it possible to remove

all infinities from the EM mass difference between hadrons, as

well as those associated with the radiative corrections to weak

decays.

Essential feature of this theory is that H _EM ≠ H⁺ _EM.

The interaction Hamiltonian is not Hermitian.

MODIFIED ELECTRO-WEAK CURRENTS:

Abdus alam (1968) and Wienberg (1967) have formulated

a renormalizable theory based essentially on the Higg’s mechanism. They treated the muon and its neutrino in exactly the same way as

an electron and its neutrino. They treated these doublets to transform as spinor representations of the weak group SU_L(2) where L stands for the left-handedness. 

  The Hadronic part of electric current is,

J _λ ≃  V_λ³ + 1/3  V_λ ⁸

And L _λ ^α =   ½ (V_λ^α +A _λ^α)  and R _λ ^α= ½ ⁽V_λ^α - A _λ^α)

α= 1, 2, 3, 4, 5, 6, 7, 8. In a quark model these may be written as,

L _λ ^α =   ½   ̅ψ_q    γ_λ   λ_1/2 ^α ( 1+ γ₅) ψ_q

and R _λ ^α =   ½   ̅ψ_q    γ_λ   λ_1/2 ^α ( 1- γ₅) ψ_q

with λ ^α as the SU(3) group generators. Hadronic weak current

accordingly becomes

L’ _λ ^α =   ½   ̅ψ_q’    γ_λ   λ_1/2 ^α ( 1+ γ₅) ψ_q’

With p’  =  p, and Θ_c  the Cabibbo’s angle.

(n’; λ) = [cos Θ_c   sin  Θ_c   ;  - sin  Θ_c    cos Θ_c]

Universal strength of the weak interactions of lepton and of

Hadrons becomes therefore apparent when one compares this with

the leptonic part of the fundamental tensor H_ijk components satisfying a spinor form of equations, allows me now to define a neutral current J_λ³ which also includes these new spinors, with a new Quantum Number called, “GRAVY” G=1, (and Y=0, C=0, T=1/2

respectively for the hypercharge, Charm and iso-spin). Thus the

neutral current is given by,

J_λ³ = ½ [ ̅P_L  γ_λ  P_L    -    ̅n¹_L γ_λ  n¹_{L  +}  ̅P_L  γ_λ  P_L

                    -  ̅λ¹_L γ_λ    λ¹_{L  +}  ̅q_L  γ_λ  q_L    -  ̅ ̰q_L  γ_λ   ̰q_L]

which has zero strangeness. The charged currents J_λ^1,2 have the

Gravy-Changing factors.  The bad high-energy behavior of

Pic 2

is cancelled not by a graph like

pic 3

but by graphs like,

pic 4 

and pic5

in which , Gravy-Quarks are exchanged.

                Incidentally these pair of spinor fields  q,   ̰q  on a

Space-Time curvature normalized to q_A   ̰q^A =1, forming a dyad

or Gravy-Spin frame, allows to define a unique null tetrad

( l^a, m^a , ̅m^a , n^a )  at each point. Conversely any null tetrad

defines a dyad uniquely to the ambiguous sign. We may adopt,

l^a   =  q^A q*^A’ ,   m^a = q^A  ̰q*^A’   ,

̅m^a =   ̰q^A q*^A’   , n^a   =   ̰q^A  ̰q*^A’

which are clearly unaffected by the symmetry operation,

(q^A, ̰q*^A’ ) à ( - q^A , -  ̰q*^A’  )

It is apparent that a canonically, an orthonormal tetrad is

chosen,

T^a = 1/√2 (l^a + n^a );   X^a = 1/√2 (m^a  + ̅m^a);

Y^a =  1/√2 (m^a  -  ̅m^a);  Z^a  =  1/√2 (l^a - n^a );

The twelve spin coefficients and their association with the

Ricci coefficients for the complete null tetrad can easily be

given. Those are analogous to the Newman-Penrose spin

coefficients and accordingly may be denoted by

κ, σ, ρ,τ, -ν,-λ,-μ,-π,α,β,γ,ε

Note: These are also known as Ricci Tensor coefficients defined by

γ_ijk  =  Z _ib;a Z_j ^b Z_k^a which are antisymmetric in first two indices.

       The choice of the null tetrad l^an_a= 1 with unit space- 

like vectors  X^a  and Y^a orthogonal to the null vectors( future-

pointing null directions at each point of space-time) implies a

two-dimensional “gauge”, freedom which is the 2-parameter sub-

group of the Lorentz group. The group has the boosts

l^a à r l^a   and   n^a   à r^-1 n^a  and the spatial rotations

m^a à e^iΘ  m^a . The gauge group is multiplicative group of complex numbers Z= r e^iΘ.

                        Obviously the Spin and Boost weighted scalar of type

{P, Q} arise with spin-weight of ½ (P-Q) and boost weight of

½ (P+Q), where the scalars undergo transformations, such as

Φ à Z^P Z^*Q Φ.

In terms of these we have the effect derivative operators

represented by the diagram below.

Fig.1

                                 

With the diagrams of derivative operations and the

directions of Boost weight and Spin-weight changes, given in

Fig.2(a) 

and Fig. 2(b).

     

Derivative operations are defined analogous to the Geroch space-

time null directional formulation.(R. Geroch, A. Held, R. Penrose)

Thus, we contend here, only by giving the basic formulations of

our spinors calculus.

The essential difference of the present renormalized theory of

electro-weak interactions is the inclusion of Gravy-changing parts

and as well the neutral Gravy currents by a spinor doublet instead

of, such as singlet Charm spinor, made in the simplest schemes by

S. L. Glashow, Illiopoulos, J and L. Maini (1970).

ACKNOWLEDGEMENT

  The author is deeply indebted to Late Prof. K. R. Rao, D.Sc. (Madras), D.Sc. (London) for his sustained interest in my expertise in Theoretical Physics and encouraging comments to further the cause of my research endeavor.

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