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[Phys-l] The Pati Salam Model And The Fitzpatrick Two Space Charge Structure. .




The Pati Salam Model And The Fitzpatrick Two Space Charge Structure. .
For over one hundred years as Physicists have penetrated the secrets of
nature they have discovered that matter exists as layers of structure. We know
matter is made of atoms which in turn are composed of electrons, protons
and neutrons. The protons and neutrons in turn are composed of quarks. Have
we reached the end of the road in discovering the underlying structure of
matter? The purpose of this post is to suggest that the answer to this
question may be no.

The impetus for still considering another layer of matter is the
conviction that nature has used only a few fundamental building blocks to construct
the Universe. What I will be proposing here is just one particular model
which attempts to do this. My choice is based on two criteria. One, a sense
of esthetics, which of course can be subjective and two, I feel any model
that describes a deeper structure of matter must provide a possible answer to
the puzzle of Fermion triplication. (The Family Problem)
This is why I will not be discussing the Harari prequark theory and its
latest manifestation as a dyon / monopole model. This is because, in my view,
this model fails badly in dealing with the family problem.
I will begin by describing the Fitzpatrick two space model. This model is
an attempt to find the underlying structure and origin of the so called
accidental symmetries of the Lagrangian. These are the global charges
associated with fermions: For example, Baryon number, Lepton number, strangeness,
charm, truth, beauty, electron, muon and tau numbers.
This model starts with the Fermion number and Fermion number operator
which define Fermion matter and antimatter states in a Hilbert space. We can
write the eigenfunction equations as
F(op)*[p>=f_m*[p> and F(op)*[pbar>= f_a*[pbar>
Here p and pbar are the matter and antimatter states respectively, f_m and
f_a are matter and antimatter Fermion charges and F(op) is a Hermitian
matrix which is the linear operator of the Eigenfunction. In any interaction
we can expect that
f_m-f_a= constant.
The matrix values of F(op) equal
F(op)_11= cos[theta] F(op)_12= sin[theta]*exp[-i*phi]
F(op)_21= sin[theta]*exp[i*phi] F(op)_22= -cos[theta]
The condition of the F(op) eigenfunction is consistent with normalization
and orthogonally since
<p I p >= <pbar[ pbar > = 1 & <p Ipbar>= <pbar[p>= 0
And
F(op)_11= < p[ F(op)] p> F(op_12= <p [ F(op)]pbar>
F(op)_21= <pbar[F(op)]p> F(op)_22= <pbar [F(op)]pbar>
When cos[theta]=1 and phi=0 then
F(op)=sigma_z and
F(op)_11=f_a F(op)_22=f_a All other terms zero.
Our interest here is expanding the Fermion charge number to generate a
richer structure which will allow us to accommodate the other global charges.
We use the group structure
Z_2 (f) =Z_2 (sigma_z) X SO (1, 1)
This provides a new linear operator in a complex two space. We do this by
performing an analytic continuation on F(op). Our goal is to generate an
operator F(v) such that
F(v)= sigma_z*T(v)
Where
T (v)_11=coshv T(v)_22= sinhv
T (v)_21= sinhv T(v)_22=coshv
We define
Theta= i*v and phi=pi/2 which gives us
F (v) _11= coshv F(v)_22= sinhv
F (v) _21=-sinhv F(v)_22=-coshv
This gives us the Eigenfunction
F(v)*Q=f*Q
We can see that Q is a vector is a complex (Lorentz) two space. To see how
this vector is defined, we look a little closer at the electric charge
structure for Fermions in terms of flavor doublets. According to the standard
model, all left spinning fermions are members of a SU (2) weak isospin
doublet. For example the states [u> and [p> are differentiated by their isospin
charge. The states [u> and d> are said to span a two dimensional Hilbert
space of SU (2). We form an isospin operator as
Tau_i = (1/2)*sigma_i so
Tau_3= (1/2)*sigma_z
Giving us the Eigenfunctions
tau_3* [u>= I_3*[u> and tau_3* [d > =I_3*[d>
Where I_3 the isospin eigenvalue.
We can form an electric charge operator as
Q(op)= tau_3+ (1/6)*I_2
Giving us the Eigenfunctions
Q (op)*[u>=q_1*[u> and Q(op)*[q_2>= q_2*[d>
This suggests that is might be fruitful to describe the Fermion electric
charges as a two space vector.
Q= [q_1 q_2]
.
Since we know the electric charge structure for Fermions is a function of
the degrees of freedom of the color charge we define v as:
v =lnN_c
Where N_c is the color degrees of freedom.
Considering the eigenvectors Q for fundamental Fermions in our two space
we can resolve our vector into two (no more no less) linearly independent
vectors such that
Q=U+V
This allows us to define any Fermion in terms of its charge state [u v>
.This relates to SU(5) symmetry where we can define all fermions in terms of
their five color charges.
F = [c_1 c_2 c_3 c_4 c_5 >
Where Q_em = c_4-(1/3)*(c_1+c_2+c_3) which conforms to the Gell Mann
Nishijima formula:
Q=I_3+Y/2
Where I_3 is the third component of isospin and Y is hypercharge.
We can view
- (1/3)*(c_1+c_2+c_3)
as the electrostrong component of electric charge and
c_4
as the electroweak component of electric charge.
Therefore we can define the two space charges as
u=c_4 v= -(1/3)*(c_1+c_2+c_3)
By thus defining the metric of the complex two space we can expand this “
vector triad” to account for all the Fermion families. We can generate a
family raise operator as
R(op)_11= -q’_2 R(op)_12=-3*q_1
R(op)_21=3*q_2 R(op)=22=q_1
Where primed values are charges in the Lepton sector.
The raise operator for the antimatter Fermions is
R(op)^c=(-sigma_x*R(op)*(-sigma_x))
R(op) and R(op)^c operate on the U vector to generate the vector charge
structure for the upper Fermion families. By defining the metric of the two
space we can define the various doublet U^2 charges which will make
clearer the family structure. In the two space the metric is:
G=sigma_z= diag[ f_m f_a ]
Therefore we see that
U^2=g_ij*u_i*u_j
This gives us
U^2 (first family) =0
U^2(second family) =1
U^2 (third family) =-1
Therefore we can define the vector triad charges for all three families as

Matter Charge Structure
{u d}= {1/2 -1/2} + {1/6 1/6} {e v_e} = {1/2 -1/2} + {-1/2
-1/2}
{C S}= {1 0} + {-1/3 -1/3} { mu v_mu}= { -1 0}+ { 0 0}
{T b}= {0 -1} + {2/3 2/3} {tau v_tau} = {0 1} + {-1
-1}
Anti Matter Charge Structure
{d u}= {1/2 -1/2} + {-1/6 -1/6} {v_e e} = {1/2 -1/2} +
{1/2 1/2}
{S C}= {0- 1} + {1/3 1/3} {v_mu mu} = {0 1} + {0
0}
{b T }= {1 0} + {-2/3 -2/3} {v_tau tau} = {-1 0) + {1
1}
We will fine that this charge structure is well accommodated by the Pati
Salam prequark model.
It has been suspected for some time that the symmetries of SU(5)/ SO(10)
and hence by the same logic the Fitzpatrick vector triad charge model are
the result of topological charge structure. The Pati Salam prequark theory
is one such theory. In this theory all fermions are composed of three
fundamental particles.
Flavons which come in two types F_1 & F_2
Chromons which come in four types C_0 C_1 C_2 C_3
Somons which come in three types S_1 S_2 S_3
The Flavons and Chromons are spinor dyons carrying both electric and magne
tic charge. The Somons are spinor monopoles. Their charge structure is
defined as
F_1 Q=1/2 g=1 F_2 Q=-1/2 g= 1
C_0= -1/2 g=2 C_1& C_2 & C_3 Q=1/6 g=2
S_1 & S_2 Q=0 g=-3
Where g is the quantized magnetic charge. Of course the antiparticles of
these prequarks would have all charges reversed.
Since Fermions do not have monopole magnetic charge, every Fermion must be
composed of one of each type of prequark to cancel out this monopole
magnetic charges. By utilizing magnetic charge rather than the hyper color
charge (which was used by earlier models) it's possible account better for the
very small masses of the Leptons as composites due the much greater
binding energy expected with dyons and monopoles as building blocks.
We can easily define these prequarks in terms of the Fitzpatrick two space
vector triad charges.
F_1= {½ 0} F_2= {-1/2 0}
C_0= {0 -1/2} C_1 & C_2 & C_3 = {0 1/6}
S_1= {0 0} S_2= {1/2 -1/2} S_3= {-1/2 1/2}
Given the five color charge structure of SU(5) we can generate all the
charges of the standard model.
Q_U (1) = c_4-(1/3)*(c_1+c_2 c_3)
Q_SU (2) = ½*(c_4-c_5 c_5-c_4)
Q_SU (3) =1/2*(c_1-c_3 c_2-c_1 c_3-c_2)
Q_U (1) = c_4+c_5-(2/3)*(c_1+c_2+c_3)
Just as the quark model accounts for all the Baryons, composite vector
and Pseudoscalars of the standard model, the Pati Salam prequark theory
provides a structure which accounts for all the vector and scalar particles
associated with the electroweak and the Higgs interactions. In this model ALL
invariant mass charge would be due the dynamics of particle confinement.
In essence the Higgs mechanism would just be another way of describing a
more fundamental particle structure.
The Bosons of the standard model consist of three prequarks and three
antiquarks. The Standard model interactions would generate rotation in one or
more of the internal charge spaces of the fermion. For example, the charged
weak force would generate rotations in the internal flavon space, while
gluons would generate rotations in the Chromon internal space. Interestingly,
this model suggest additional gauge or scalar fields such as an
interaction to generate rotation in the Soman internal space, which would be a
special version of the Higgs scalar field known as a familon field. These would
be the Nambu-Goldstone bosons associated with the broken symmetry of
fermion family charges.
In addition, we might expect an interaction to rotate the particle from
the colored Chromon space to the uncolored space, in other words, an
interaction that would change a quark to a lepton or visa versa, a leptoquark boson.

While this has been a very simple overview of this topic and many
questions remain to before this theory can be a serious contribution in the quest
for Unification, I hope I have provided enough detail to see the very app
ealing aspects of the Pati Salam prequark theory and have illustrated how
compatible this theory is with the Fitzpatrick two space model.
Bob Zannelli



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