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*From*: spinozalens@aol.com*Date*: Wed, 6 Jan 2010 10:00:01 EST

Electric Charge as a Vector in Lorentz Space.

In this post based on the work of O. Bar and U J Wiese of MIT, Afsar Abbas

at the Center for Theoretical Physics, JMI New Deli and Gerald Fitzpatrick

formerly of PRI Research and Development Corp I will take a look at

electric charge structure as a function of color multiplicity. The basic premise

here is that electric charge can be represented as a real internal vector in

a lorentz two space. In addition, I will adopt the assumption that the

fundamental Quantum entities of all matter are composed of are Spinors and

that Bosons are the result of composite Fermion structure. If this is true it

would seem to have interesting consequences for gravity, though in this

post gravity won't be discussed. Nevertheless this assertion and its

consequences have been proposed in several papers by C. Weterrich and others. Based

on this idea, space time might be thought of as an emergent property of

nature at some given Energy Scale. This idea also has some support from ideas

expressed by F Wilczek is his book "The Lightness of Being, Mass, Ether and

the Unification of Forces"

According to the Standard model, all left handed Fermions (And right

handed Anti Fermions) are members of SU(2) weak isospin doublets. These states

may be properly thought of as being two different states of a single Fermion

Field. Using conventional Isospin language there exist an Isospin

operator;

tau= (1/2)*Sigma

Where Sigma are the familiar Pauli Spin Matrices. We can represent

Fermions in a two dimensional Hilbert Space, the two dimensional

representation of SU(2). This gives us the Spinor Eigenstates

[ U> = { 1,0} and {d> = { 0,1}

Where

tau_3 = (1/2)*sigma_3

As the Isospin operator so that

tau_3*[U>= +(1/2)*[U> and tau_3*[d> =-(1/2)*[d>

Where +- 1/2 eigenvalues are T_3 is the third component of Global

Isospin.

This allows us to construct an Electric Charge operator

Q(op)= tau_3 + 1/(2*N_c)*I_2

Where N_c are the color degrees of freedom and I_2 is the 2 X 2

identity matrix.

Giving us the Eigen function

Q(op)*[U>= q_1*{U> and Q(op)*[d> = q_2*[d>

Here the Eigenvalues are electric charge, the scalar values of the two

space vector

Q= { q_1, q_2}

The metric for this two space can be defined as

G__i,j = sigma_3

This gives us a global Charge

Q_T= g_i,j*q_1*q_2

Q_T is in fact , what is normally defined as Lepton number for

color degrees of freedom 1 and Baryon number for color degrees of freedom of

3.

By defining the color degrees of freedom with an entropy like parameter

v where

v= lnN_c

We can derive a general two space Electric Charge Eigen function

F(v)*Q= f*Q

Where f, the Eigenvalue, is the Fermion number.

With

F(v)_11 = - F(v)_22 = coshv & F(v)_12= - F(v)_21= sinhv

Which gives us ( N_c > 1) Quarks

q_1= sinhv/(exp[v]-f)

q_2= q_1-1

N_c=1 Leptons

q_1= - sinhv/(exp[v]-f}

q_2= q_1 +1

All the particles of the Standard model are defined by N_c = 1 and N_c

=3. But perfectly consistent models for other values of N_c are possible.

Assuming the requirement to cancel Witten Global anomaly is a single

generation all positive odd value of N_c produces consistent physics. Based on

this we can have;

N_c= 5 q_1= 3/5 q_2= -2/5

N_c =7 q_1 = 4/7 q_2= -3/7

N_c=9 q_1 =5/9 q_2 = -4/9

and so on.

Lim N_c approaches infinity q_1=1/2 and q_2 = -1/2

For any finite value of N_c the result is both composite integer charged

Fermions and Bosons. In fact a subset of the resulting particles would look

like ordinary Hadrons of the standard model. For example the rate of pions

would be unaffected were N_c > 3. This is because the decay rate is not

solely a function of N_c but also the electric charges values q_1 and q_2. We

have;

Gamma( pion=2*photon)= N_c^2*(

g_ij*q_1*q_2)^2*alpha^2M_pi^3/(64*pi^3*F_pi^2)

Here we see that

N_c^2*( g_ij*q_1*q_2)^2 = 1

For any values of q_1 and q_2 which cancel the Witten Anomaly.

However, if we relax the requirement to cancel anomalies within a

single generation and we have an even number of Fermion generation, N_c need

not be odd.

However, this would give us a very strange Universe. We would have color

neutral fractional charged particles and no free Fermion particle except for

N_c =1. Such a Universe would most likely preclude any complex

structures.

In addition, based on this model of fundamental charge structure we might

think of Fermions as the fundamental quanta and Bosons the result of

composite Fermions structure. All Bosonic fields, can in fact, be expressed

Mathematically as composite Spinor fields.

We have

Scalar= psibar*psi

Pseudoscalar= psibar*gamma^5psi

Vector= psibar*gamma_mu*psi

Pseudovector= psibar*gamma^mu*gamma^5*psi

Tensor= psibar*sigma^mu^nu*psi

Where sigma^mu,nu= (i/2)*[gamma^mu*gamma^nu- gamma^nu*gamma^mu]

Where gamma_i are the Dirac matrices and psibar is the adjoint Spinor

psibar= psi^dag*gamma^0

All this raises some interesting questions. Why did nature choose N_c=1 and

N_c=3 for our Universe? Or did it? Might we find N_c> 3 at higher

energies? Could these higher multiplicity Hadrons relate to upper KK modes in M

Theory? This is impossible in the old String theory models but the addition

of Branes to the mix relaxes the possible charge structure which would

seem to allow these higher color multiplicity charges. Most likely this isn't

true but it does seem strange that there exist a consistent anomaly free

electric charge structure that nature ignored. Coincidentally the spin

Eigenstates of the upper multiplicity match up one to one with the spin states of

the higher spectrum String modes. However, this hardly constitutes very

convincing evidence, it may just a coincidence. But who knows.

Bob Zannelli

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