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
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;
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.