Journal of Modern Physics, 2013, 4, 94-120
http://dx.doi.org/10.4236/jmp.2013.44A011 Published Online April 2013 (http://www.scirp.org/journal/jmp)
Copyright © 2013 SciRes. JMP
Grand Unified SU(8) Gauge Theory Based on Baryons
which Are Yang-Mills Magnetic Monopoles
Jay R. Yablon
Schenectady, New York, USA
Email: jyablon@nycap.rr.com
Received January 15, 2013; revised April 22, 2013; accepted April 27, 2013
Copyright © 2013 Jay R. Yablon. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
Based on the thesis that baryons including protons and neutrons are Yang-Mills magnetic monopoles which the author
has previously developed and which has been confirmed by over half a dozen empirically-accurate predictions, we de-
velop a GUT that is rooted in the SU(4) subgroups for the proton/electron and neutron/neutrino which were used as the
basis for these predictions. The SU(8) GUT group so-developed leads following three stages of symmetry breaking to
all known phenomenology including a neutrino that behaves differently from other fermions, lepto-quark separation,
replication of fermions into exactly three generations, the Cabibbo mixing of those generations, weak interactions which
are left-chiral, and all four of the gravitational, strong, weak, and electromagnetic interactions. The next steps based on
this development will be to calculate the masses and energies associated with the vacuum terms of the Lagrangian, to
see if additional empirical confirmations can be achieved, especially for the proton and neutron and the fermion masses.
Keywords: GUT; SU(8); Yang-Mills; Baryons; Magnetic Monopoles; Nuclear Physics; Binding Energy; Protons;
Neutrons; Fermions; Quarks; Electrons; Neutrinos; Leptons; Generations; Cabibbo Mixing; Chirality;
Gravitation; QCD; Electroweak
1. Introduction
In a recent paper [1], the author introduced the thesis that
baryons, including protons and neutrons, are Yang-Mills
magnetic monopoles. Based on this thesis, it was possi-
ble to predict that the electron rest mass is related to the
masses of the up and down quarks according to


3
2
2π
edu
mmm ((11.22) of [1]), with the factor
of

3
2
2π emerging following a Gaussian integration
over three space dimensions. Subsequent calculations
showed that the best known values of the up and down
masses in turn lead a binding energy of 7.667 MeV per
the proton and 9.691 MeV per neutron yielding an aver-
age binding energy of 8.679 MeV per nucleon ((12.6)
through (12.8) of [1]), very much in accord with what is
empirically observed, and to binding energies for 56Fe
which were predicted to be extremely close to what is
observed for that nuclide. Noting also that the deuteron
binding energy is extremely close to what is known from
best available data to be the mass of the up quark, we
further hypothesized that these might be one and the
same, which could be explained if the energies released
during nuclear fusion are based on some form of “reso-
nant cavity” analysis in which the discreet energies
which are observed to be released are based on the
masses of the quarks contained within the nucleons and
nuclides. This led to a prediction that 56Fe has a latent
available binding energy of 493.028394 MeV ((12.14) of
[1]), which we then contrasted to the empirical binding
energy of 492.253892 MeV. This small difference was
understood as indicating that 99.8429093% of the avail-
able binding energy predicted by this model of nucleons
as Yang-Mills magnetic monopoles goes into binding
together the 56Fe nucleus, and that the remaining
0.1570907% goes into confining the quarks within the
nucleons. This in turn, lead us by the conclusion of [1] to
a deepened understanding of how quark confinement is
intimately related to nuclear binding and fission and fu-
sion and the peak in per nucleon binding energies at 56Fe,
and perhaps to an understanding of the so-called First
ECM effect (see [1], pp. 62 and 66).
A second paper [2] extended this analysis, and showed
that based on this same “resonant cavity” analysis, the
binding energies of the remaining 1s nuclides, namely 3H,
3He and 4He, can be predicted to at least parts per hun-
J. R. YABLON
Copyright © 2013 SciRes. JMP
95
dred thousand and in most cases parts per million. This
latter paper also showed that the observed neutron-pro-
ton mass difference is predicted by the relationship
 


3
2
32 32π
ud du
MnM pmmmmm
 
(in (7.2) of [2]) to better than 1 part per million. In Sec-
tion 10 of [2], we explained why this should be regarded
as an exact relationship, and therefore modified our ear-
lier hypothesis that the deuteron binding energy is ex-
actly equal to the up quark mass, into one in which these
energies are very close—to just over 8 parts in ten mil-
lion—but not exactly the same. In Section 9 of [2] we
used these results to predict solar fusion energies solely
from up and down quark masses, and found the results to
also be in very tight accord with the observed data.
The lesson taken from [1,2] together, is that empirical
evidence strongly supports the thesis that Yang-Mills
magnetic monopoles are in fact baryons on the basis of
seven independent predictions which closely match the
experimental data, specifically: 1) the electron mass in
relation to the up and down masses, 2) the 56Fe binding
energy specifically, and the per-nucleon binding energies
on the order of 8.68 MeV in general, 3) the proton minus
neutron mass difference, and 4-7) the four distinct nu-
clide binding energies predicted for 4) 2H, 5) 3H, 6) 3He
and 7) 4He. The study of solar fusion in Section 9 of [2]
does not contain anything independent of the predictions
1) through 7), but rather applies several of these predict-
tions in combination, and underscores that a “resonant
cavity” analysis of nucleons and nuclides does consis-
tently lead to empirically-accurate binding energies, evi-
denced by all of predictions 3) through 7) above.
While the theoretical foundation for all of these suc-
cessful predictions was laid throughout [1], it was the
field strength tensors for the proton and neutron, (11.3)
and (11.4) of [1], reproduced below:
P
Tr
2,
"" ""
dduu
dd uu
F
imm

 
 



 
 

 




(1.1)
N
Tr
2,
""" "
uudd
uu dd
F
imm

 
 



 
 

 




(1.2)
when used to calculate the energy according to
3
1Tr d
2
EFFx



((11.7) of [1]), which formed the specific basis for the
calculations that led to all of these predictions. These
field strength tensors, in turn, emerged as stable magnetic
monopoles following specification of the SU(4)P “pro-
tium” and SU(4)N “neutrium” gauge groups in Section 7
of [1], followed by breaking the symmetry of these
groups using the baryon minus lepton number generator
B L via
GUT
vBL
 ((8.1) of [1]). So we take
the thesis presented in Sections 7 and 8 of [1] that the
protons and neutrons emerge following the B L break-
ing of the SU(4)P and SU(4)N groups to be supported by
the compelling evidence of predictions 1) through 7), and
so regard SU(4)P and SU(4)N as subgroups that do de-
scribe the real physical universe, not just some arbitrary
groups that may or may not appear in the natural world.
In short, we take accurate empirical predictions 1)
through 7) above as direct evidence of the physical real-
ity of SU(4)P and SU(4)N.
Based on all of the foregoing, we shall in this paper
take SU(4)P and SU(4)N as physically-validated, reliable
building blocks for developing a “Grand Unified Theory”
(GUT) based on the empirically-confirmed thesis that
baryons, including protons and neutrons, are Yang-Mills
magnetic monopoles.
2. Unification and Grand Unification in
Physical Science
At least since the time when Isaac Newton hypothesized
that the terrestrial “force” which caused an apple to fall
from a tree was the same as the celestial “force” which
guided the movements of the planets, unification has
been a central objective of physical science. The pre-
eminent scientist, entrepreneur and statesman Benjamin
Franklin catapulted to fame when he realized that the
terrestrial sparks he was creating in his laboratory were
of a unified piece with the lightning from the heavens,
and applied that understanding in a very practical way to
develop lightning rods which cured an epidemic of mid-
18th century electrocutions throughout Europe brought
about by the superstition of sending church bellringers to
steeples at the highest place in town to clang large metal-
lic bells to ward off the anger of the Gods every time a
lightning storm approached. James Clerk Maxwell in
1873 elaborated what to that date was, and perhaps even
to today’s date is, the preeminent physical unification
and at least the very paradigm of unification, as he pulled
together the disparate threads of Gauss, Faraday and
Ampere into a unifying set of equations for electricity
and magnetism. This was deepened a generation later
with Einstein and Minkowski’s Lorentz-invariant unifi-
cation of space and time. In these and similar endeavors
the underlying theme has always been the same: to take
what appear on their surface to be disparate natural phe-
nomena, and acquire a deeper understanding which
shows them to be governed by a single, common prince-
ple. The success of past unifications leaves today’s gen-
J. R. YABLON
Copyright © 2013 SciRes. JMP
96
eration of physicists with the firm conviction that further
unifications can still be achieved, and that one day in the
future, all of the laws of nature can and will be deduced
from one common vantage point. After all, what is natu-
ral science other than an endeavor to explain what is ob-
served through our direct senses and the clever instru-
mentation that extends our senses, by relating those ob-
servations to mathematically precise laws of nature
which apply consistently, uniformly and replicably,
without exception, in the broadest possible range of cir-
cumstances?
So-called “Grand Unified Theories,” or GUTs, are part
and parcel of this esteemed tradition, and are based spe-
cifically on the advent of Yang-Mills gauge theories and
the realization that these Yang-Mills theories have a re-
markable capacity to explain what is observed in nature
as evidenced though their already-successful application
to weak and strong and electroweak interactions. The
Georgi-Glashow SU(5) model [3] which was reviewed at
some length in Section 8 of [1] was one of the first
“GUTs” and is perhaps the best known. The basic idea of
Georgi-Glashow and any other GUT is to be able to rep-
resent all of the fermions which are observed in nature,
and all of their interactions, using a single, simple gauge
group with a symmetry which is then broken in one or
more stages to arrive at the particle and interaction phe-
nomenology observed in a laboratory setting. The fer-
mions are the up and down quarks, the electron and neu-
trino leptons, and ideally their higher-generational carbon
copies distinguished from the first generation solely by
larger mass. The generators of the gauge group represent
“interactions” of which there are understood to be four:
gravitational, strong, weak and electromagnetic. The ei-
genvalues of the diagonalized generators of the gauge
group, which are linearly related to discrete natural
numbers such as 2
3 and 1
3
and 1
2
and 1 and 0,
represent the “charges” of these fermions with respect to
these interactions. A particular fermion may be associ-
ated with a particular eigenstate (eigenvector) of a repre-
sentation of the GUT gauge group if all of its eigenvalues
for all of the generators match up with what are known to
be the charges of that fermion with respect to all of these
interactions. So, for example, an electron is by definition
the fermion eigenstate for which the lepton number ei-
genvalue L = 1, the baryon number eigenvalue B = 0, and
the electric charge eigenvalue Q = 1. And the transi-
tions/decays of a fermion from one eigenstate into an-
other, or its interactions in a given eigenstate, lead to the
mediating vector bosons of the theory. The trick in any
GUT, is to characterize all of the interrelated charges of
all of the fermions in the “simplest” way possible, to
understand the stages and ways in which the symmetry of
the group is broken starting at ultra-high energies and
working down to energies which can be reached in a
laboratory setting, and of course, to end up with some-
thing that accurately comports with all observed empiri-
cal data.
With this in mind, and as used in the discussion here,
we distinguish “GUTs” from “unified field theories”
more generally, as that subset of unified field theories
which is specifically centered on understanding fermions
and their interactions via their discrete charges using
Yang-Mills gauge groups, and on making whatever ob-
servable predictions can be made based on such an un-
derstanding. So, for example, Kaluza-Klein theory,
which to this day represents an exceedingly elegant clas-
sical unification of general relativity with Maxwell’s
electrodynamics using a fifth spacetime dimension that
from today’s vantage point is best understood as the
“matter dimension” [4], is most certainly a form of “uni-
fied field theory” (and one which in the view of this au-
thor warrants more universal acceptance than it has at
present, especially given that what we know of Yang-
Mills gauge theory should permit both gravitation and
electromagnetism in Kaluza-Klein form to be extended
into non-Abelian domains). But Kaluza-Klein is not a
GUT in the sense that GUTs are focused on the use of
Yang-Mills gauge groups to represent fermions and their
interactions, and Kaluza Klein, at least absent a Yang-
Mills extension, has nothing to say about fermions.
While one may define the term “GUT” more expansively
to also include so-called “supersymmetric” theories, the
foregoing defines by example, what we have in mind in
this paper when referring to a “GUT”, as opposed to a
“unified field theory” without the GUT qualifier.
The Galilean foundation for all of modern science is
that theory must be the confirmed by observation, and
that the goal or at least an important by-product of theory
is to systematically explain observation. For physical
theorists, the pursuit is about systematically compre-
hending nature and confirming that comprehension based
on experimental data, or as Hawking and Einstein have
more loftily put it, “reading the mind of God.” Because
GUTs necessarily theorize about the behavior of nature
at ultra-high energies such as 1015 GeV and even higher
that are unlikely to ever be reached by human experi-
mentation under any foreseeable circumstances (with the
possible exception of what we can learn by peering back
billions of years through astronomical telescopes), such
GUT theories necessarily opine on physics that may for-
ever be beyond the reach of direct experimental confir-
mation. So the only way to discern the primacy of one
GUT over another is indirectly, by virtue of what it pre-
dicts about low energy phenomenology that we can or
may soon be able to observe. So as we consider how to
construct the “puzzle” which is a GUT and decide what
“pieces” to use in that puzzle, we want to start with puz-
J. R. YABLON
Copyright © 2013 SciRes. JMP
97
zle pieces that already are solidly-grounded in empirical
observation.
Based on the seven independent predictions enumer-
ated in the last section which closely match the empirical
nuclear binding and related data based on the thesis that
baryons are Yang-Mills magnetic monopoles, the GUT
that we develop here will start with the SU(4)P and
SU(4)N gauge groups developed in Sections 7 and 8 of
[1], knowing that these groups now have been validated
by over half a dozen independent pieces of empirical data
from nuclear and particle physics. Additionally, because
we have shown in [1,2] how to connect these gauge
groups to energy numbers which can be and indeed have
been empirically confirmed, an important objective in
developing a GUT on the basis of SU(4)P and SU(4)N is
to lay the foundation for perhaps obtaining additional,
similar, successful predictions of other known energies
which have been crying out for theoretical understanding
for decades, most particularly, and most importantly, the
free proton and neutron masses, and the observed fer-
mion masses.
If it should be possible on the basis of a particular
GUT to make accurate predictions of the proton and neu-
tron and/or fermion masses, then even absent the ability
to ever directly observe the 1015 GeV and higher energy
phenomena which lead to these predictions, such predict-
tions would certainly be solid evidence, albeit through
indirect inference rather than direct observation, that
such a GUT has also explained to us how nature behaves
behind the veil of energies that we shall most certainly
never get to directly observe (again, with possible astro-
nomical caveat).
In other words, because a GUT, by its very nature,
seeks to reach into energy domains that will likely be
forever beyond human reach, it must fulfill the Galilean
project by accurately explaining all of the masses and
energies that we do observe through the instrumentation
that does rest within our grasp, while at the same time
teaching us about physics at energies that we shall likely
never have the capacity to see directly. It is the prediction
of the energies and masses we do observe, that gives us
some measure of confidence that we are not being led
astray by what the GUT tells us about the physics of un-
reachable energies. To use a different metaphor, GUTs
seek to teach us about an entire iceberg, most of which
we shall never be able to observe. So what the GUT
teaches us about the tip of that iceberg which we can see,
must be solidly-confirmed by empirical data every step of
the way for us to have some confidence in what it teaches
us about the rest of the iceberg which will forever remain
out of sight.
Based on the foregoing, the purpose of this paper is to
develop a GUT rooted in the thesis that baryons are
Yang-Mills magnetic monopoles and the seven success-
ful predictions which have already emanated from that
thesis in [1,2], and to lay the foundation for additional
mass and energy predictions, including those of the free
proton and neutron and the fermion masses.
3. Some Clues for Pursuing the Proton,
Neutron and Fermion Masses
Before we can make predictions of the proton and neu-
tron and fermion masses, we must construct a reliable,
empirically-grounded GUT, and we must know how to
break its symmetry. Why do we say this?
We have already shown in [2] how the nuclear binding
energies in the 1s shell arise from using the field strength
tensors (1.1) and (1.2) to calculate an energy
3
gaugedEx
L via the pure gauge terms in the La-
grangian (3.8) of [2]:

3
2
binding
12π
2ABBAAA BB
FF FF




L, (3.1)
together with components ,
ud
mm and ud
mm of the
outer product
P
ABCD
E ((4.9) through (4.11) of [2]). But
these binding energies are calculated using only the pure
gauge field terms (3.1) of the Lagrangian developed in
(3.12) of [2], written with the terms slightly regrouped:







 
3
2
3
2
322
2
322
2
12π
2
2π
2π
11
2π.
22
ABBAAA BB
AB AA
B
ABB
AB BAAA BB
AB BAAA BB
FF FF
DD DD
 
 











L
(3.2)
We have not yet even begun to develop these other
terms at all, yet it is made very clear by the development
in [1,2] that additional energy numbers can and will arise
from complete development of these terms. So, we must
develop these additional terms and we will look to them
to perhaps lead us to the proton and neutron and fermion
masses. But because all of these terms contain the vac-
uum
, the actual numeric energy values we obtain
from these
-containing terms will depend upon the
GUT gauge group we choose, and upon its vacua
and how these vacua are used to break symmetry. (We
use the plural vacua because we have in mind breaking
symmetry in sequence using the Planck vacuum on the
order of 1019 GeV, so called GUT vacuum on the order
of 1015 GeV, and the Fermi vacuum vF = 246.219651
GeV used to break electroweak interactions to electro-
magnetic interactions via
 
21 1
WY EM
SU UU .)
For example, given from (3.11) of [2] that:
J. R. YABLON
Copyright © 2013 SciRes. JMP
98


,
EF
EF
E
F
DiG
 

 

, (3.3)
we see that terms in (3.2) with


DD
 will mix
gauge fields G
with vacuum fields . So whereas
the pure gauge terms (3.1) led to expressions such as (4.9)
and (4.10) of [2], namely:
 
33
3
22
1
2π2πd
2
00 00
0000,
00 00
PABCDPAB PCD
dd
uu
uu
EFFx
mm
mm
mm









(3.4)
 
33
3
22
1
2π2πd
2
00 00
0000,
0000
NABCDNAB NCD
uu
dd
dd
EFFx
mm
mm
mm









(3.5)
we should be alert to opportunities to develop mixed
gauge field/vacuum terms where one of these matrices is
replaced by a vev, especially the Fermi vev vF =
246.219651 GeV, so we can develop an energy “tool-
box” with such expressions as
F
u
vm and
F
d
vm
.
Why the Fermi vev? And why these square root expres-
sions? Because numerical inspection of the square roots
of the three main masses in (4.11) of [2] used to calculate
binding energies throughout [2], times the square root of
the Fermi vev, shows that:
739.960397 MeV
u
vm , (3.6)
1099.12211 MeV
d
vm , (3.7)
901.835259 MeV
ud
vmm . (3.8)
These clearly are at exactly the right order of magni-
tude to explain the free proton and neutron masses M(p)
= 938.272046(21) MeV and M(n) = 939.565379(21)
MeV, if and when we can put (3.6) through (3.8) and like
expressions into the right context and obtain the right
coefficients. And where do such coefficients come from?
The generators of a GUT! (The author’s subsequent pa-
per in this same special issue of JMP starts with (3.8) to
indeed successfully explain the free neutron and proton
rest masses.)
So the proton and neutron masses, via the order of
magnitude analysis above, straddle right down the mid-
dle of the Fermi vev and the masses of the quarks. One
should therefore be on the lookout for ways to exploit
this via the “mixed” gauge field vacuumG
terms in
Lagrangian (3.2). And as noted at the end of Section 10
of [2], one should keep in mind that relation
 


3
2
32 32π
ud du
MnM pmmmmm
 
for the free neutron-proton mass difference now allows
us to find the neutron and proton masses individually, so
long as we can find sum expression which involves the
sum of these masses. So it may well be that our target
should be
M
nMp or some multiple thereof
(perhaps the 4He alpha nucleus studied extensively in
[2]?) rather than either of these masses individually.
For another example, we go all the way back to (2.1)
of [1], Maxwell’s charge equation:

2
,
,
JF DG
GiGG J
mGG iGG
 

 



 

 



(3.9)
with DiG

 , and where in the final term we
have hand-added a “Proca mass.” Based on (3.3), we can
readily specify an analogous field equation:

2
,
,
JF DiG
JmiG
 
 






 

(3.10)
for a Yang-Mills (non-commuting) scalar field
with
a scalar source J. In fact, this is just the Klein-Gordon
equation for a non-Abelian (non-commuting) Yang-Mills
scalar field with a non-zero scalar source, into which we
have hand-added a Proca mass in the usual way. The rea-
son this is of interest is that a central step in Section 2 of
[1] was to develop the inverse GIJ

and then intro-
duce fermion field wavefunctions
via J


,
so that we went from GIJ I

 

 . But we
can follow an analogous path with (3.10) by building
scalar source J out of fermions via J
. Then we
can develop an inverse via
I
J and follow the
analogous progression IIJ
 . Consequently,
the terms of the Lagrangian (3.2) quadratic in the scalar
field can be developed as 22 2
IIJ

 , and
then we can follow the path of section 3 of [1] by em-
ploying spin sums

2
uu Nmm

, with the
full progression



22 2
22
II
I
J
Nm m


 

Then, if we pursue the same course of development as
in [1] from start to finish, when we finally reach the
counterpart of (11.19) of [1] and collapse the propagators
so that interactions occur essentially at a point, we will
end up with a Lagrangian term of the schematic form:

22
22
II
I
f
J
Nm
mm

 
 


L
(3.11)
But this is the form of a Fermion mass term in a La-
J. R. YABLON
Copyright © 2013 SciRes. JMP
99
grangian, with the mass of the fermion specifically iden-
tified with

22
I
f
mNmm. Concurrently, the vev
v should also enter into this when we break symmetry
with a generator G by setting vG
. So this is a pos-
sible prescription, using the  terms in (3.2), for
revealing a fermion rest mass out a Lagrangian while
preserving gauge symmetry and thus maintaining renor-
malizability!
But because the specifics of all of this center around
the vacua , it becomes essential to have the right GUT
gauge group, and to know how to break its symmetry in
appropriate sequence. As noted above, to do this, we
begin to develop a GUT gauge group using the SU(4)P
and SU(4)N gauge groups developed in Sections 7 and 8
of [1], knowing that these groups now have been vali-
dated by over half a dozen independent pieces of empiri-
cal evidence from nuclear and particle physics. We build
upon these empirically-validated puzzle pieces in the
hope that this run of positive empirical predictions will
continue with the masses and energies predicted by the
terms in (3.2) which include the vacua .
4. An Unbroken SU(8) GUT Group which
Accommodates All Fermions and Left and
Right-Chiral States, All Interactions,
Three Generations, and an Idiosyncratic
Neutrino, with Nothing Missing and
Nothing Superfluous
The proton and the neutron, of course, form an SU(2)
weak isospin doublet

,pn with 311
,
22
I
 


, re-
spectively. But both the proton and the neutron are com-
posite entities comprising three quarks, and as we have
argued and indeed supported with empirical nuclear
binding data, they are Yang-Mills magnetic monopoles.
So really, the proton/neutron doublet is a doublet of trip-
lets,


,,, ,,duuudd . And the left-chiral weak iso-
spin quantum numbers 3
L
I
associated with this triplet
doublet are 111111
,,,,
2222 2 2

 


 
 

.
In Section 7 of [1], we demonstrated that at ultra-high
GUT energies the proton was part of a larger gauge
group which we dubbed the SU(4)P “protium” group
which includes the proton and the electron, and that the
neutron was similarly part of a larger gauge group we
dubbed the SU(4)N “neutrium” group which includes the
neutron and the neutrino. As we then showed in Section 8
and specifically (8.1) of [1], these two groups are broken
by a vacuum

GUT
vBL of a “baryon minus lep-
ton number” generator 15
8
3
BL
 such that a pro-
ton triplet ,,
R
GB
duu is separated from the electron for
SU(4)P and a neutron triplet ,,
R
GB
udd is separated
from the neutron for SU(4)N, and each triplet becomes
part of a topologically-stable magnetic monopole. It was
on the basis of these proton and neutron triplets broken
out from SU(4), that we successfully rendered the seven
predictions summarized in Section 1, and also correctly
derived the fusion energy released during the solar fis-
sion cycle strictly as a function of the up, down and elec-
tron fermion masses. So these triplets and the SU(4)P and
SU(4)N groups in which they are embedded would appear
to be very solid puzzle pieces for constructing a larger
GUT which is well-grounded empirically. That is exactly
what we shall do here.
Normally, one works from the phenomenological
gauge group

321
CWY
SU SUU and tries to
find larger simple groups G which embed all of these and
their associated fermions. The SU(5) model of Georgi-
Glashow [3] reviewed at some length in Section 8 of [1]
is a paradigmatic example. Here, we shall start with
SU(4)P and SU(4)N which we know lead to accurate
binding energy predictions, and seek to construct a larger
simple gauge group which includes these two groups,
and which also encompasses the usual phenomenological
gauge group

321
CWY
SU SUU
. The group we
shall choose?

84 4
N
P
SU SUSU. This is a
larger group than SU(5), but as we shall see, it brings
with it numerous benefits including 1) the ability to ac-
commodate a non-zero neutrino mass and thus right-
handed chiral neutrinos which are omitted from SU(5); 2)
the ability to accommodate all flavors and colors of fer-
mion, as well as protons and neutrons, all in the funda-
mental group representation (SU(5) splits the fermions
into a fundamental 5 and a non-fundamental 10 repre-
sentation while omitting the right-chiral neutrino); 3) the
ability to accommodate different left and right-handed
chiral projections with respect to weak hypercharge Y
and weak isospin I3, for all fermions; 4) a solution, at
long last, to the mystery of fermion replication into ex-
actly three generations, and 5) interaction generators that
may well be associated with gravitation based on the
manner in which the elusive neutrino stands alone with
respect to all other fermions by having an exceedingly
tiny neutrino mass that is orders of magnitude smaller
than the masses of the other fermions, and based on the
ability to finally understand the origins of fermion gen-
eration replication.
We construct this SU(8) group by establishing a fun-
damental representation containing the fermion octuplet
,,, ,,,,
RGB RGB
udd eduu
. The neutron triplet
,,
R
GB
udd and proton triplet

,,
R
GB
duu are design-
nated in separate parenthesis for visual emphasis, and as
we can see, the neutrium group

,,,
R
GB
ud d
occupies
the first four members of this octuplet and the protium
group
,,,
R
GB
eduu occupies the last four members.
Of course, what really counts are the quantum numbers,
J. R. YABLON
Copyright © 2013 SciRes. JMP
100
so let’s now turn to those.
In (7.1) of [1], we established that for the protium
quadruplet, the electric charge generator could be speci-
fied by

8158
22
2
33
QBL

 . But in
(7.3) of [1], we were required to use a different electric
charge generator, namely 8
2
3
Q
for the neutrium
quadruplet. This if OK when the proton and electron are
treated separately from the neutron and neutrino, and this
was good enough to get us over half a dozen good bind-
ing energy predictions and other empirically-supported
relationships. But once we put all these fermions to-
gether into one octuplet representation of a unifying
group this is no longer OK, and we need to define a new
electric charge generator that works uniformly for all of
the fermions in the group.
So let us now see exactly how we can put these two
groups together and what this implies for the nature of
the GUT that does so. SU(8) of course contains seven
diagonalized 8 × 8 generator matrices, so rather than take
up visual space with seven 88
matrices in which all
but the diagonal elements are zero, let us construct this
group using the tables below which convey the same
information more compactly in an easier-to-follow form.
First, as just noted, the electric charge generator is

8158
22
2
33
QBL


for SU(4)P, while it is 8
2
3
Q
for SU(4)N. So if we
lay out the eight fermions of this octuplet in a vertical
column and show the three generators 15 8 3
,,

of
SU(4) in the first three columns (we actually show
15
8
3
BL
 which is merely a linear multiple of
15
), and then show generators for electric charge Q,
right-chiral weak hypercharge 2
R
YQ, left chiral weak
hypercharge L
YBL
, and right-chiral weak isospin
30
R
I
, which are all linear combinations of one or more
of the three generators 158 3
,,

, we end up with Ta-
ble 1 below.
In Table 1 below, we have segregated the SU(4)P and
SU(4)N generators

158 3
3,,
8BL

  from the
remaining generators, so that we can clearly see that
there are three linearly-independent degrees of freedom.
The remaining generators for Q, YR, YL, are all linear
combinations of the first three generators, and so provide
no additional degrees of freedom, while 30
R
I
can be
trivially obtained from any other generator using the co-
efficient 0. We shall wish, in the course of our analysis,
to maintain a focus on the independent degrees of free-
dom. What makes the upper neutrium quadruplet not
unified with the lower protium quadruplet is the fact, as
mentioned above, that although all the other generators
have the same form (i.e., are invariant) as between the
Table 1. Fermions and generators of SU(4)N and SU(4)P.
Linearly Independent Degrees of Freedom Linear Combinations
15
8
3
BL
 8
3
8
2
3
Q
2
R
YQ L
YBL 3
R
I
ν 1 0 0 0 0 1 0
uR
1
3 1
3 0 2
3 4
3 1
3 0
dG 1
3 1
23
1
2 1
3
2
3
1
3 0
dB
1
3 1
23
1
2
1
3
2
3
1
3 0
" " " 8
2
3
QBL
 " " "
e 1 0 0 1
2
1 0
dR 1
3 1
3 0 1
3
2
3
1
3 0
uG 1
3 1
23
1
2 2
3 4
3 1
3 0
uB 1
3 1
23
1
2
2
3 4
3 1
3 0
J. R. YABLON
Copyright © 2013 SciRes. JMP
101
upper and lower quadruplets as denoted by the “dittos”,
the electric charge generators are defined by different
linear combinations. So electric charge Q is not an in-
variant as between these two quadruplets. It is worth not-
ing that for all of these fermions, L
YBL, so
L
Y is
not itself a linearly-independent generator from 15
.
The one generator that we do not see explicitly repre-
sented in the above, of course, is the generator 31
2
L
I
of left-chiral weak interaction, and this is related very
intimately to the different Q generators as highlighted
above. So, let us now a) introduce 3
L
I
and b) use this
3
L
I
in combination with YL which happens to be equal in
all cases to BL, to specify 3
2
L
L
QY I
as is ordi-
narily done in electroweak theory. Then, having Q in
hand, and given 30
R
I, we may further specify
2
R
YQ if we insist on the chiral-invariant relationship
33
22
L
LR R
QYI YI
.
So we now take Table 1 above, introduce all seven of
the SU(8) diagonalized generators with the normalization

21
Tr 2
i
, and specify suitable linear combinations of
these. Then, we review not only how this accommodates
the fermions and generators in Table 1 above, but also
the new interaction generators that are introduced and
their possible physical significance.
Aesthetically, is very simple and natural for the eight
fundamental flavors and colors of fermion

,,,,,,,
R
GB RGB
uddeduu
to each be made a member
of the fundamental representation of SU(8). And, be-
cause one does have eight fermions in nature (per gen-
eration), a natural question is, why not use SU(8)? Some-
times, what appears to be the simplest approach really is
the simplest approach, and leads to the best results, and
we don’t have to try to unnaturally “squish” eight fer-
mions into a smaller group like SU(5) and then lose the
right-chiral neutrino and split the representations.
In this regard, the question we shall explore largely
throughout the rest of this paper—which is one of the
reasons why one might not use SU(8)—is whether SU(8)
is simply too large and can or ought to be made smaller.
(We shall answer this question, “no”!) By “too large,” we
refer not to aesthetics, but to superfluity: does this group
introduce any extra, superfluous particles or interactions
which simply do not appear anywhere in the natural
world. Put concisely, the underlying question is this: is
SU(8) sufficient, and is everything in SU(8) necessary?
Does it yield everything, and not one iota more? (We
shall answer these questions, “yes”!)
Specifically, in going from two disjoint SU(4) groups
in Table 1 to one unified SU(8) group in Table 2, we
have gone from three independent generators 15 8 3
,,

to seven. Out of these four new generators, we have left
three of these, 6348 35
,,

, in their “native” form with-
out alteration, pending further exploration of these gen-
erators below. The fourth new generator, 24
, we do not
show explicitly. Rather, we use the degree of freedom
provided by 24
to introduce the left-chiral weak isospin
generator 3
L
I
, which we define as a linear combination
of six of the seven “native” generators according to:
Table 2. Fermions and generators of SU(8).
Linearly Independent Degrees of Freedom Linear Combinations
63
48
35
3
L
I
B
L
8
3
L
Y Q
R
Y 3
R
I
ν 17
228
0 0 1
2 1
0 0 1 0 0 0
uR
1
228
16
221
0 1
2 1
3 1
3 0 1
3 2
3 4
3 0
dG 1
228
1
221
15
215
1
2
1
3 1
23
1
2 1
3 1
3
2
3
0
dB
1
228
1
221
1
215
1
2
1
3 1
23
1
2
1
3 1
3
2
3
0
e 1
228
1
221
1
215
1
2
1
0 0 1 1 2 0
dR 1
228
1
221
1
215
1
2
1
3 1
3 0 1
3 1
3
2
3
0
uG 1
228
1
221
1
215
1
2 1
3 1
23
1
2 1
3 2
3 4
3 0
uB 1
228
1
221
1
215
1
2 1
3 1
23
1
2
1
3 2
3 4
3 0
J. R. YABLON
Copyright © 2013 SciRes. JMP
102
36348 35
24 158
24 2
72115
222
.
53
3
L
I


 

(4.1)
One can readily check that

311111111
diag,,,,,,,
22222222
L
I



as in Table 2.
Now, for the bottom quadruplet with

,,,
R
GB
ed u u,
we have 15
2
23
BL
 as before. But this relation-
ship needs to be replicated out of the native generators
for the top quadruplet

,,,
R
GB
udd
as well. This is
realized by the following linear combination of native
generators:
6348 35
24 15
4443
95
7321
22 2
2
35 3
BL




(4.2)
So we use (4.1) and (4.2) above to account for the two
linearly-independent degrees of freedom in 24
and
15
. It is easy to check as in Table 2, that

111 111
diag 1,,,,1,,,
333 333
BL 
 


.
Similarly, we cannot use 8
alone, but must dupli-
cate this as well for the top quadruplet

,,,
R
GB
ud d
.
This is achieved by defining a 8
generator:
848 35248
72 2
315
35

 
. (4.3)
As required from Table 2, a check finds that
8
diag
11 111 1
0,, , ,0,, ,
3232332323




Finally, and similarly, we need to define a 3
ac-
cording to:
335243
32
55


. (4.4)
This yields

31111
diag0, 0,,, 0, 0,,
2222

 


as in Table 2. The foregoing, (4.1) through (4.4), account
for four of the seven linearly-independent degrees of
freedom in SU(8). We have yet to explore the three na-
tive-form generators 63 48 35
,,

.
From here, we define several other generators which
are linear combinations of (4.1) through (4.4). First, via
(4.2), we define:
63 48
3524 15
44
7321
43 228
95 353
L
YBL

 

(4.5)
which happens to be exactly equal to BL in (4.2) and
so is not linearly independent. But
L
Y is non-chiral, i.e.,
it only applies to left-chiral projections. Next, we use
(4.5) and (4.1) to define the electric charge generator in
the usual manner, via:

333
483524
15 8
11
22
27 422
33 35
315
22
233
LL LR
QYIBLII



 

(4.6)
One can check to see that

211 122
diag0,, , ,1, ,,
333 333
Q



,
as required by Table 2. In the third expression we make
use of 30
R
I
, to show by way of contrast that Volo-
vok’s Equation (12.8) in [5] also leads via a different
route to the exact same

33
1
2
L
R
QBLII
.
Next, we formally specify that the right-chiral genera-
tor
30
R
I
(4.7)
is to be zero for all the fermions so that only left-chiral
particles will interact weakly. At the same time we insist
that the electric charge generator
33
11
22
L
LRR
QYIYI
  (4.8)
is to be defined as chiral symmetric for all fermions. This
chiral insistence together with (4.6) and (4.7) finally
leads to:
4835 24
15 8
47 842
233 35
315
24
433
R
YQ


 

(4.9)
So at this point, all of the known quantum numbers of
the fermions are fully specified, including the left and
right chiral projections for Y and I3. The fermions all re-
side in the fundamental representation of SU(8), and the
proton and neutron are represented as well in the way
J. R. YABLON
Copyright © 2013 SciRes. JMP
103
that we have ordered the fundamental representation.
And, while all of the foregoing certainly accounts for the
observed fermions and their quantum numbers, we still
have three extra linearly-independent degrees of freedom,
which we can and do choose to associate with the gen-
erators 6348 35
,,

we have left in their native state.
Now we return to the critical question: With these
three apparently superfluous degrees of freedom, does
SU(8) provide too much freedom? Does SU(8) provide
more than what is necessary? Might we find some way,
in the spirit of Georgi Glashow SU(5), to “squish” these
fermions into a smaller group and take away some of this
apparently-superfluous freedom? The answer is, no! And
the reason is that this extra freedom is not superfluous,
but is actually fully accounted for in the known particle
phenomenology, and particularly, in the odd quirks of the
neutrino and in the replication of fermion generations.
Let us see how.
First, the neutrino. One of the very perplexing features
of the neutrino is that it has almost no mass, and is mad-
deningly-elusive. While the electron and the quarks do
have different masses from one another, the neutrinos are
in a league of their own, by orders of magnitude. The
neutrino mass is almost zero, which means that it travels
at very close to the speed of light. Because of the
equivalence of gravitational and inertial mass, the fact
that the mass of the neutrino is so very different from that
of all the other fermions means that in some rough man-
ner of speaking, it is gravitating differently as well. For
example, the mass of the electron’s neutrino is less than 2
eV [6], while the electron itself has a mass of about 511
KeV, which is over 250,000 times as large. This is of a
totally different nature, involving completely different
orders of magnitude, than 4 351853369
ue
mm . and
9 601723351
de
mm . which are the relationships be-
tween the quark masses and the electron masses based on
the quark masses arrived at in (10.3) and (10.4) of [2].
This appears to make the neutrino qualitatively different
from all the other fermions, and we need to pinpoint the
origins of this difference.
Now consider the 63
in Table 2 and the fact that
63 1
47
 for all of the up and down quarks and the
electron, but that 63 7
47
 has a completely differ-
ent value for the neutrino. Moreover, not only is the
magnitude different by 7 to 1, but even more importantly,
the sign is different. Indeed, that is why we chose to
place the neutrino as the very top member of the SU(8)
fermion octuplet. That means that the neutrino will in-
teract completely differently under the interaction asso-
ciated with 63
—whatever that interaction may be—
from any other fermion. But if there is any interaction
under which the neutrino behaves differently than all the
other fermions, it is the gravitational interaction, because
the most pronounced way in which the neutrino differs
from the other fermions is via its ghostly mass and thus
its ghostly way of gravitating. Further, we know on gen-
eral principles that for any Yang-Mills gauge group
which unifies gravitation with the other three interactions,
there will have to be at least one degree of freedom given
to the gravitational interaction. The only question is
where and how this appears.
So, we now make a preliminary association of the
63
generator with a degree of freedom for a gravita-
tional interaction, and we do so in a way that bakes in for
the neutrino, an entirely different way of gravitating and
thus displaying its mass, than any other fermion.
So, now we have accounted at least in a general way
(which we shall seek to deepen in the upcoming discus-
sion) for all four of the known interactions, but we still
have two more degrees of freedom unaccounted for,
namely, those provided by 48 35
,
. What are we to
make of these? This brings us again to the question: does
this not give us too much freedom? And again, the an-
swer is, no!
We still have to account for the replication of fermions
into three generations, which is another oddity of the
material world almost as mysterious as the oddities of the
neutrino just discussed. Let’s ask the question directly:
even if 63
is related to gravitation and can explain why
the neutrino behaves so differently from all the other
fermions, 4835
,
still give us two apparently super-
fluous degrees of freedom. What does this mean? What
can we do with those extra two degrees of freedom? And
specifically, might they be origin of generation replica-
tion?
Any time we have two degrees of freedom such as are
provided by 48 35
,
, it is possible to construct three
eigenstates out of those degrees of freedom. So, let us do
just that, and label these states ,,e
as in Figure 1
below.
We use “primes” in these generators, because if they
do represent degrees of freedom associated with genera-
tion replication, they do not act in same way as the re-
maining generators 6324 15 83
,,,,

in Table 2.
These other five generators represent a “vertical” sym-
metry wherein each of the eight fermions is distin-
guished from one another by different quantum numbers.
But the appearance of three generations in which the
quantum numbers are identical from one generation to
the next, and in which for a given fermion, generation is
distinguished exclusively by rest mass, is a horizontal
symmetry. And it is for and to this horizontal symmetry,
that we shall develop and devote the generators
48 35
,
.
Now, in the forthcoming discussion, we shall seek to
uncover in detail, the particular mechanism by which
J. R. YABLON
Copyright © 2013 SciRes. JMP
104
these two generators 4835
,
separate themselves from
the remaining vertical generators 63 383
,,, ,IBL

to situate themselves horizontally. The only point being
made at the moment, is that two extra generators in
Yang-Mills theory, such as 48 35
,
, provide enough
freedom to support three distinct states as in Figure 1.
And these three states will come equipped with their own
3 × 3 unitary matrices U to mix these states. And, if we
are asking ourselves whether the extra two generators
48 35
,
provide too much freedom at the same time that
we are seeking an explanation of the three fermion gen-
erations, and given that those two extra generators pro-
vide precisely the freedom needed to allow each particle
to exist in one of three additional horizontal generational
states, then perhaps these are not superfluous after all,
but are instead the source of the generations. In that case,
SU(8) becomes a perfect fit, large enough to accommo-
date all that is observed including the idiosyncratic be-
havior of the neutrino and the replication of fermion
generations, and not one bit larger so as to contain any-
thing superfluous that is not observed.
So in Table 3 below, we shall use the schematic sym-
bol
to denote a visual shorthand for Figure 1 below:
a condensed symbol that represents two degrees of free-
dom which are used to provide three distinct states
,,e
which appear in Figure 1. And, let us replace the
generators 4835
,
with this schematic to represent the
horizontal symmetry of generation replication. Thus, we
now rewrite Table 2 in the form of Table 3 as shown
below.
Figure 1. Three generation eigenstates constructed from
48
and 35
.
Table 3. Fermions and generators of SU(8), with generation replication.
Linearly Independent Degrees of Freedom Linear Combinations
63
48 35
,
3
L
I
B
L
8
3
L
Y Q
R
Y 3
R
I
ν 17
228
1
2 1
0 0 1
0 0 0
uR
1
228
1
2 1
3 1
3 0 1
3 2
3 4
3 0
dG 1
228
1
2
1
3 1
23
1
2 1
3 1
3
2
3
0
dB
1
228
1
2
1
3 1
23
1
2
1
3 1
3
2
3
0
e 1
228
1
2
1
0 0 1
1 2 0
dR 1
228
1
2
1
3 1
3 0 1
3 1
3
2
3
0
uG 1
228
1
2 1
3 1
23
1
2 1
3 2
3 4
3 0
uB 1
228
1
2 1
3 1
23
1
2
1
3 2
3 4
3 0
J. R. YABLON
Copyright © 2013 SciRes. JMP
105
Now, in Table 3, SU(8) has nothing superfluous, all
eight fermions are represented with both left and
right-chiral states, and each can exist in one of the three
,,e
 horizontal generation eigenstates. We see that
there are now four vertical interactions: 1) the strong
QCD interaction with three color states and two genera-
tor degrees of freedom 83
,

; 2) the weak isospin in-
teraction represented by 3
L
I
; 3) a BL interaction to
which the electromagnetic interaction of (4.6) is linearly
related by

33
11
22
L
LL
QYIBLI; and 4) a final
63
providing a degree of freedom for a gravitational
interaction, under which all fermions except the neutrino
interact in one way, and under which the neutrino acts in
a very different way, in a league by itself. This is the
unbroken GUT group that seems best situated to fully
accommodate not only all the known fermions and inter-
actions and their key phenomenological properties, but
the Yang-Mills magnetic monopoles which we now
know are baryons, and which are very naturally grouped
in this way of representing SU(8).
5. Spontaneous Symmetry Breaking of SU(8)
at the Planck and GUT Energy Scales,
and the Emergence of Fermion
Generations and Fermion Mass Degrees of
Freedom
In Section 8 of [1], we reviewed spontaneous symmetry
breaking in the Georgi-Glashow SU(5) model, to provide
a backdrop for breaking the protium group via

431
P
PCB L
SUSUU
 and the neutrium group
via
 
431
N
NCB L
SUSU U
. This of course led
to stable protons and neutrons and later to the several
accurate empirical binding energy predictions already
noted. Here, we review a similar symmetry breaking
based on the SU(8) group developed in the previous sec-
tion. Specifically, we review three symmetry breaking
operations: a first symmetry breaking operation using the
contemplated “gravitational” generator 63
at or near
the Planck scale; a second symmetry breaking operation
using the L
YBL generator at an ultra-high GUT
energy perhaps in the 1015 GeV vicinity, and a third
break of the electroweak symmetry at the Fermi scale
using the electric charge generator Q. It is this third
symmetry breaking that we hope to use to accurately
predict the proton and neutron masses as discussed in
Section 3 and highlighted in (3.6) to (3.8). But to set the
context, let us start with the first two high-energy sym-
metry breaking operations using 63
and L
YBL.
If 63
is indeed a gravitational generator, then its
mass scale will be at or near (within an order of magni-
tude of) the Planck mass which is defined by 2
P
GM c,
where G is the gravitational constant and c contains
the Planck constant and the speed of light. In terms of
energies that we have been discussing here,
19
1 22110
P
M.
GeV is nineteen orders of magnitude
larger than the proton mass. It is theorized that at this
energy, there is a violent sea of vacuum perturbations,
and two of the best references to review this understand-
ing are [7,8]. We shall examine all of this more closely
here as well, in the next section.
Without yet going through all the details in this pass, if
we employ the Lagrangian (3.2) and specify a Planck
vacuum ;1,,63
i
PPi
i

 , we may break symme-
try at or near
P
P
vM
using the 63
generator such
that:



63 , . .,diagdiag
17,1,1,,1,1,1,1,1
228
i
PPP Pi
P
vie
v

 
 (5.1)
with 0;1,, 62
Pi i
otherwise. (Again, we are not
concerned here with the exact relationship which why we
use
rather than =, but rather an order of magnitude
examination of the qualitative features of this symmetry
breaking.) This would immediately set the neutrino
which is the top member of the elementary fermion octu-
plet
,,,,,,,
R
GB RGB
uddeduu
on a course to behave
differently from all the other particles. If 63
is indeed
a gravitational degree of freedom which notion we began
to entertain in the last section, then it makes sense to re-
gard the degree of freedom that 63
provides to be a
freedom associated with the rest mass of the fermion, i.e.,
to be a vertical mass degree of freedom. So with symme-
try breaking of the neutrino from all the other fermions at
the Planck scale, right below the Planck scale all of the
fermions except the neutrino would have one mass, and
the neutrino would have a different mass. Most notably,
the neutrino would have an oppositely-signed generator
from all of the other seven fermions, which we shall re-
visit in the next section. Thus, the neutrino can be ex-
pected right from the start, to behave very uniquely as
regards its mass, and as regards to how it gravitates. This
could be a root cause of why the quark mass to electron
mass ratios are
4 351853369,9 601723351
ue de
mm .mm .
,
while 250,000
e
mm
. One can envision that masses
which are equal at the Planck scale might separate so that
they differ from one another by factors of 4.35 to 1 or
9.60 to 1 at observable energies. But for a ratio
250,000
e
mm
we expect this to be more than just
“screening adjustments” as we go from high to low ener-
gies. We expect this to be “baked in” to the underlying
structure of the GUT gauge group right from the start.
Moving on, we now venture down to the vicinity of a
second 15
10 GeV
GUT
v, where we break the symmetry
J. R. YABLON
Copyright © 2013 SciRes. JMP
106
with L
YBL. Again, we are simply for the moment
talking about orders of magnitude for this energy scale.
In fact, we have already discussed BL symmetry
breaking at some length in Section 8 of [1]. But in that
earlier discussion, we regarded

4
P
SU and
4
N
SU
as disjoint groups each breaking down via


431,
431
P
PCB L
N
NCB L
SUSU U
SUSU U


to produce a





111
π3π1π1
CBB
SUU U

homotopy group with stable magnetic monopoles, essen-
tially based on the disjointed groups of Table 1. Now, in
contrast, we have conjoined these groups into SU(8) as
represented by Table 2 above. So the symmetry breaking
we are about to explore is a “wholesale” breaking of

4
P
SU and

4
N
SU together at once in SU(8), ver-
sus the parallel, but “retail” symmetry breaking of

4
P
SU and

4
N
SU conducted in Section 8 of [1].
It is also worth noting as reviewed in Section 8 of [1],
that Georgi and Glashow also break symmetry using the
Y generator, albeit such that


11111
diagdiag, , ,,
33322
i
iGUT
Tv

  


for a right-chiral quintuplet

,,,,
RGBC C
R
ddde v of
fermions. So here, we are doing exact same thing as
Georgi and Glashow insofar as using a Y generator to
break the GUT symmetry circa 1015 GeV, but we are
merely using a different group SU(8) versus SU(5), with
all the fermions in the fundamental representation as
shown in Table 2. Now let’s proceed.
The group is now SU(8). Exactly as in (8.1) of [1], the
vacuum we use is:

GUTGUTGUT L
vBLvY (5.2)
Here, however, because of the SU(8) group, we have:



L
diag
111 111
diag 1,,,,1,,,
333 333
diag diag
GUT
i
iGUT
GUT GUT
Tv
vBLvY





(5.3)
Unlike Section 8 of [1], we no longer have
15
8
3
BL
 from which we set 15
2
23GUT
v

and so obtain the Clebsch-Gordon coefficient via
22 22
15
8
3GUT GUT
vCv
 .
Rather, here we have a BL specified in (4.2)
which is a linear combination of five generators. Thus, to
break symmetry here, picking off the coefficients in (4.2),
we now must set:
63 4835
24 15
4443
;;;
95
7321
22 2
;2
35 3
GUTGUT GUT
GUT GUT
vvv
vv
 

 

(5.4)
with all the remaining 0
i
. The invariant scalar:
22222
63483524 15
2
222
1616 1634242
792181595 3
80
21
GUT
GUT GUT
v
vCv




 




(5.5)
yields a Clebsch-Gordon coefficient 280
21
C (Note the
28
3
C
from the earlier 15
8
3
BL
 included in
the calculation of the above). One may then employ the
procedure such as is outlined in (11.5) and (11.6) of [1]
to obtain gauge bosons masses in the usual way, and
these will have masses on the order of GUT
v.
But our interest here is in what happens at lower ener-
gies, after this symmetry has been broken, because that
brings us into energy ranges with are experimentally ob-
servable.
First, by breaking symmetry via
GUTGUT L
vBLvY ,
which for which the generator eigenvalues are
111 111
1,,,,1,,,
333 333




,
we “fracture” the eight fermions in Tables 2 and 3 into a
1
L
YBL
 hypercharge doublet of leptons
,e
and a 1
3
L
YBL
 hypercharge sextuplet of quarks
,, ,,
RGBRG B
duu udd. Of course, we know that
,,
R
GB
duu is a proton and

,,
R
GB
udd is a neutron,
so this sextuplet may also be viewed as a 1
L
YBL

proton/neutron doublet
,pn . Referring to Tables 2
and 3, the weak isospin for each doublet

,, ,epn
is
given by 311
,
22
I



. Of course for the proton this is
arrived at by adding 31 111
2 222
I
   for its three
quarks, and for the neutron similarly via
31111
2 222
I
.
J. R. YABLON
Copyright © 2013 SciRes. JMP
107
Note also that by virtue of how the triplets in

,, ,,
RGBRG B
duu udd are ordered, each entry in

,,
R
GB
duu forms a weak isospin doublet with respect
to its corresponding same-colored entry in

,,
R
GB
udd .
Each of the three quarks also enjoys two color degrees of
freedom R, G, B associated with the SU(3)C’ generators
83
,

, see (4.3) and (4.4). So the group arrived at fol-
lowing BL symmetry breaking is schematically rep-
resented by:
 
 
862
321
L
BL
CWYBL
SUSU SU
SUSU U


 (5.6)
The

GUTGUT L
vBLvY  symmetry breaking
has fractured the quarks from the leptons into a sextuplet
of quarks each with 1
3
B and a doublet of leptons
each with 1L. Just as in Georgi/Glashow, this breaks
a lepto-quark symmetry. This is the origin of the
 
62
B
L
SU SU factor. But the quarks are grouped
into a proton and neutron doublet with 311
,
22
I



,
and of course the two members of the lepton doublet also
both have 311
,
22
I



. This is the well-known “iso-
spin redundancy” that exists and between quarks/baryons
and leptons and leads some to consider “preon” models
such as that discussed in Section 12 of [5]. For
quarks/baryons, we use

321
L
CWYBL
SUSU

to represent their status following L
YBL symmetry
breaking. That is, the proton and neutron each containing
an

3C
SU color triplet of quarks, form an
2W
SU
weak doublet


,321
L
CWYBL
pn SUSU

with every single fermion containing an identical
1
3
L
YBL, hence the
1L
YBL
U factor. For lep-
tons, the neutrino and electron form an

2W
SU weak
doublet
 
,21
L
WYBL
eSU

 with each contain-
ing an identical 1
L
YBL, hence the
1L
YBL
U
factor, albeit for a different value of L
YBL than
that of the quarks/baryons. Overall, with the detailed
interrelationships just noted, we reproduce the pheno-
menological product group
321
CWY
SU SUU
.
Given that we have used

GUTGUT L
vBLvY
for symmetry breaking at GUT
v, all that we have just
described should be readily apparent from Tables 2 and
3. But a bonus that we obtain here, which is not obtained
in Georgi-Glashow SU(5), is the fermion generation rep-
lication. This is how:
In SU(5) which is broken using GUT
vY , there are
four degrees of freedom based on the linearly-indepen-
dent generators 24 15 83
,,,TTTT
. After symmetry break-
ing there are still four degrees of freedom; they are
merelyreshuffled into 83
,
for

3
3,
C
SUI for
2W
SU , and Y for
1Y
U. None of these degrees of
freedom disappear after symmetry breaking; they simply
sit across one another in several “irregular” linear com-
binations.
Here, however, in going from

862
B
L
SUSU SU,
two “vertical” degrees of freedom “disappear”, because
SU(8) has seven diagonalized generators while SU(6) has
only five, and the separate B and L subscripts in
862
B
L
SUSU SU are all part of a single de-
gree of freedom represented by L
YBL. But this
reduction-by-two in the degrees of freedom cannot van-
ish into thin air; it must show up in some other way. That
is, following symmetry breaking using
GUTGUT L
vBLvY , there are two-free floating
degrees of freedom from 48 35
,
that have become
decoupled from the remaining five vertical degrees of
freedom. But, as shown in Figure 1, these free-floating
degrees of freedom have precisely the properties needed
to create a new horizontal freedom with exactly three
states. So we label these three states ,,e
as in Figure
1, we associate this with the fermion generation replica-
tion, and we therefore make a carbon copy of each fer-
mion in triplicate, using the conventional symbols
,,,,,udcstb for the quarks, ,,e
for the electrons,
and ,,
e

for the neutrinos. The vertical quantum
numbers associated with each type of fermion ,,;uct
,,;, ,dsbe
and ,,
e

are identical for each triplet.
The fermions across generations are distinguished only
by the mass values, and so apparently, it is the free-
floating generators 4835
,
which provide the horizon-
tal fermion mass degrees of freedom to enable each fer-
mion of a given type to take on one of three mass values.
Thus we may formulate Table 4 below.
Studying Table 4 and the above comments about the
generational mass freedom, we now can better develop
our understanding of the so-called gravitational degree of
freedom 63
which we discussed a short while ago in
relation to (5.1). Whereas 48 35
,
provide freedom for
the fermions of any given type to take on one of three
mass values, we also need a degree of freedom for each
of the four basic fermion “prototypes” ,,,eu d
to have
different masses within a single generation, as is also
clearly observed. This, in fact, is the role of 63
. While
the neutrino is set on a different mass trajectory at the
outset at the Planck scale because its 63
generator ei-
genvalue is 17
228
while that for all of the other
fermions is the oppositely signed 1
228
with 1/7 the
J. R. YABLON
Copyright © 2013 SciRes. JMP
108
Table 4. Quarks and leptons with generation replication following B L = YL GUT symmetry breaking.
Linearly Independent Degrees of Freedom Linear Combinations
63
3
L
I
B
L
8
3
L
Y Q
R
Y 3
R
I
Quarks
1
228
1
2 1
3 1
3 0 1
3 2
3 4
3 0
1
228
1
2
1
3 1
23
1
2 1
3 1
3
2
3
0



,,
,,
,,
R
G
B
uct
ndsb
dsb






1
228
1
2
1
3 1
23
1
2
1
3 1
3
2
3
0
1
228
1
2
1
3 1
3 0 1
3 1
3
2
3
0
1
228
1
2 1
3 1
23
1
2 1
3 2
3 4
3 0



,,
,,
,,
R
G
B
dsb
puct
uct






1
228
1
2 1
3 1
23
1
2
1
3 2
3 4
3 0
Leptons

,,
e
vvv

17
228
1
2 1
0 0
1
0 0 0

,,e
1
228
1
2
1
0 0
1
1 2 0
magnitude, the fact that all fermions but the neutrino
have the same 63
tells us that at the Planck scale all of
the ,,eud have the same mass, and that the differences
among these masses that we detect at observable energies
stems from the differences introduced by the other verti-
cal generators 3,,
L
I
BLQ. So we now see that collec-
tively speaking, the three generators 6348 35
,,

are all
responsible for providing the mass degrees of freedom to
the fermions, with 63
providing a vertical freedom to
differentiate among ,,,,eu d
and with 4835
,
pro-
viding two more horizontal degrees of freedom to dif-
ferentiate the mass spectrum for a given fermion type
into three permitted generational values. To the extent
that one regards the quantum degrees of freedom that
lead to discrete fermion masses as related to gravitational
interactions given that mass and gravitation are inextri-
cably linked, we now conclude that all three of
63 4835
,,

are the quantum generators of gravita-
tional interactions, similarly to how 83
,
generate
strong interactions. But these 63 48 35
,,

act differ-
ently from 83
,
insofar as 1) 63
acts vertically
while 48 35
,
act horizontally, and 2) 48 35
,
only
start to act horizontally after they decouple from the
other vertical generators at GUT
v as a consequence of
the lepto-quark symmetry breaking using the vacuum

GUT
vBL . This is examined further in Table 5
below.
Finally, what this tells us is that in order to ascertain
an answer to the question “why do the fermions have the
masses they have?”, the theoretical answer is this: follow
the 6348 35
,,

generators, understand how 48 35
,
separate out and start to act horizontally at vP and GUT
v,
and understand how the masses evolve as one moves
downward in energy from there toward the masses we do
observe in the laboratory. In this regard, if 63
is used
to break symmetry at or near the Planck scale as in (5.1),
then we immediately see a break via

87SU SU
1U with the neutrino fractured from all the other
fermions. So, we already lose one vertical generator,
which we take to be 48
, which decouples and becomes
horizontal. Thus, below the Planck scale but above the
GUT scale, we would expect to see two fermion genera-
tions. Then, as we pass downward through the GUT scale
and break the lepto-quark symmetry as in (5.2), we drop
down to
62
B
L
SU SU and now two of the gen-
erators have decoupled from vertical to horizontal giving
rise to a third generation. It would therefore make sense
to believe that the observed substantial variation from
first to second generation masses, and then again from
second to third generation, has it origin in this sequential
breaking of symmetry that starts with one generation at
the Planck scale, turns into two generations between the
Planck scale and the GUT lepto-quark scale, and turns
into three generations below the GUT scale. At each
scale as one “cools down,” the masses become “frozen”
in a manner of thinking. And it would seem to make
sense due to their relatively larger masses that the high
mass fermions, namely the ,,,tb
, are the ones that
already exist in precursor form at the Planck scale, that
the ,,,cs

arise between the Planck scale and the
J. R. YABLON
Copyright © 2013 SciRes. JMP
109
Table 5. Mass degrees of freedom afforded by the gravitational interaction, below GUT energy.
48 35
1,0
3



48 35
11
,2
23


 48 35
11
,2
23



17
228
G νe νμ ντ
1
228
G uR cR tR
1
228
G dG sG bG
1
228
G dB sB bB
1
228
G e μ τ
1
228
G dR sR bR
1
228
G uG cG tG
1
228
G uB cB tB
GUT scale, and that the ,,,
eeu d
which predominate
and are the ground states at observable energies are the
last generation to emerge, below the 15
~10 GeV scale
at which the lepto-quark symmetry is broken and the
48 35
,
decouple from other generators. Perhaps what
happens at each symmetry breaking stage is that the one
(or two) generations which exist before symmetry
breaking “spin off” a portion of their mass to make two
(or three) fermions when the generators decouple. That is,
for example, what is “one electron” above the Planck
scale has to become “two electrons” below the Planck
scale, and these then have to further turn into three elec-
trons below the GUT scale, at the same time that the
generators are decoupled.
One final point before concluding this section pertains
to chiral symmetry. Because the left-chiral generator
L
YBL for all fermions, at the same time that we
break symmetry at the GUT energy using (5.2) and (5.3),
we have also forced a breaking of chiral symmetry. That
is, the weak interactions start to become chiral non-
symmetric at the GUT scale, as part and parcel of the
L
YBL symmetry breaking. As discussed briefly at
the end of Section 5 of [1], baryon and meson physics is
endemically, organically non-chiral, which is consistent
with what is experimentally observed, all with
50123
i

being the mainspring. Via what may be
thought of as Dirac’s “quinternian” progression
50123
i

from Hamilton’s quaternion 22
ij
21kijk
, any time one has what looks like a “vec-
tor” object from one viewpoint, one can use
50123
i

to create an “axial” object from another
“dual” viewpoint, and “vector” and “axial” turn out to
have a duality relationship that is integral to the Dirac
algebra, all using “duality” based on the work of Reinich
[9] later elaborated by Wheeler [10] which uses the
Levi-Civita formalism (see [11] at pages 87-89). So
given the degree to which baryon physics is fundamen-
tally non-chiral courtesy of a Dirac algebra for which
50123
i

is as integral to fermion physics as
1ijk
is to spatial rotations, it makes perfect sense that
as soon as protons and neutrons are crystalized into being
as stable magnetic monopoles by L
YBL symmetry
breaking, we also bring about the non-chiral nature of the
weak and weak hypercharge interactions.
6. The Geometrodynamic Planck Vacuum,
and What Makes the Neutrino Different
(or, Let’s Finally Catch that Mischievous
Neutrino)
With all that we have learned in Section 5, let us make a
second pass through the Planck scale, and to see what
else we may be able to learn.
It has long been believed, and experimentally given
credence by the Lamb-Retherford shift in electromag-
netic phenomenon, that near the Planck length,
35
1.61624 10
meters, and over Planck time scales of
44
5.39121 10
sec, there is a violent sea of vacuum
perturbations near the Planck energy 1.221 × 1019 GeV,
see the earlier referenced [7,8] where this is developed in
detail. It is also well-understood that energy fluctuations
J. R. YABLON
Copyright © 2013 SciRes. JMP
110
of this magnitude on such a small scale do have the effect
of topologically creating microscopic black holes, also
called wormholes, with a Schwarzschild radius at or near
the Planck length. Let us now take a closer look at ex-
actly what is believed to occur at this scale. Again, along
the lines discussed in Section 2, it is unlikely that humans
will ever be able to directly observe physics at the Planck
length, but the development of such physics in the con-
text of a GUT may lead us to low energy mass and en-
ergy predictions which—if they accord with empirical
data—could then give us some confidence that the GUT
which leads to such accord is also describing the
Planck-length physics “behind the veil” with some sem-
blance of accuracy.
When Wheeler talks in his seminal work [8] about the
geometrodynamic Planck vacuum, the vacuum he envi-
sions is constructed from a series of simple algebraic
calculations with which it is important to be familiar. So
let us review those here. First, Newton’s law of gravita-
tion 2
12
F
Gm mr contains a numerator 12
Gm m
which has the same dimensions as the natural constant
c. So the Planck mass 2
P
M
is defined as the unique,
natural mass unit formed out of the Newtonian numerator
from G, and c, namely:
2
P
GM c. (6.1)
The above means that P
M
cG so that the
Planck energy 25
PP
EMc cG. The Fermi vev
energy vF is similarly defined using the Fermi constant via
24
2FF
Gv cc, with the 2 having historical origins
based on how
F
G was first defined before electroweak
interactions were well-understood. Comparing “apples to
apples” the correspondence is 2
F
GG.
The reduced Compton wavelength of a Planck mass
(6.1) is easily calculated to be:
3
PP
G
Mc c

 . (6.2)
Now we consider a large collection of Planck masses
P
M
separated from one another by
P
, in what would
be a natural state of resonance. The negative gravitational
potential energy EG between any two
P
M
separated by
P
is easily calculated to be:
25
P
GP
PP
GM cc
EE
G
 

 . (6.3)
But this is simply the negative of the Planck energy!
So as Wheeler first surmised, a collection of Planck mass
fluctuations (on average) separated by the Planck length
(on average) averages out to be a vacuum because the
negative gravitational energy precisely cancels the posi-
tive Planck energies which are posited in the first place,
on average. Nonetheless, in very localized regions on the
order of
P
, there are very violent fluctuations of very
high energy occurring. This is the so-called “geometro-
dynamic vacuum.”
It is also important to note that the Schwarzschild
“black hole” radius for a (non-rotating) Planck mass may
be calculated to be:
22 3
2222
P
SP
GM Gc G
rG
cc c
 

. (6.4)
Because the black hole radius is twice as large as the
Planck length, this means that all of these fluctuations are
occurring out of sight, behind a black hole horizon.
On top of this, Hawking [12] teaches seventeen years
after Wheeler’s initial elaboration of the geometrody-
namic vacuum, based on general relativistic gravitational
theory, that black holes emit a blackbody radiation spec-
trum. So if we recognize that the Planck vacuum is a
vacuum in which the masses on average are Planck
masses separated on average by the Planck length, and
then like any good student of statistics we ask the natural
follow up question “what is the actual statistical distribu-
tion of these energies about the average?” Hawking pro-
vides a clear answer: because these fluctuations are oc-
curring behind an event horizon, the distribution is ob-
served externally to the event horizon as a thermody-
namic, blackbody spectrum. It would also make sense,
therefore, to consider the prospect that when we observe
blackbody radiation in the natural world, we are in fact
observing a gravitational phenomenon from the Planck
vacuum screened through over twenty orders of magni-
tude, which would render the blackbody spectrum that
kicked off the quantum revolution in 1901 [13], a cones-
quence of gravitational theory. So much for disunion
between gravitational theory and quantum theory!
But returning to GUTs, the Wheeler vacuum also
teaches us something about the generator 63
with


63 1
diag7, 1, 1, 1, 1, 1,1,1
228

which we are associating on a preliminary basis with
gravitation, which is this: One may look at the Planck
vacuum in one of two entirely equivalent ways: First, one
can say that there are a tremendous number of fluctua-
tions with positive energy
P
E on average, separated
by
P
on average, thus giving rise to an equal amount
of negative gravitational energies
P
E on average, thus
resulting in a vacuum on average, which has a gravita-
tional blackbody distribution of energy when viewed
from outside the event horizon, and which is redshifted
as our observational perch recedes to that from which
Planck first characterized this distribution. Second, one
can start with negative energy fluctuations, separate them
by
P
, and they will gravitate to produce positive en-
ergy fluctuations. Each way of looking at this is equally
J. R. YABLON
Copyright © 2013 SciRes. JMP
111
valid. It is a “chicken and the egg” question. One can
develop an equally sensible description of the exact same
physics no matter where one starts: positive Planck
masses producing negative gravitational energies, or
negative Planck masses producing positive gravitational
energies. It does not matter. These are two alternative
descriptions of exactly the same thing.
Now, let’s talk about specific fermions, such as the

,,,,,,,
R
GB RGB
uddeduu
of our SU(8) GUT group.
How do these actually take root in the vacuum? How are
they “conceived” and “born”? Through the lens of 1957,
referring to electromagnetic charge Q, Wheeler says in [8]
that “classical charge appears as the flux of lines of force
trapped in a multiply connected metric ... trapped by the
topology of the space.” In other words, charge gets
“trapped” in the black hole wormholes. Updating this
with all that we have learned in the intervening half cen-
tury especially about Yang-Mills gauge theories and how
charges such as the electric charge arise from the gen-
erators of Yang-Mills theory, we might say that these
Planck-mass fluctuations “trap” the Yang-Mills internal
symmetries (which include the electric charge), and that
this is how particles are “born.” Or, in parlance we in-
troduce here, the physical fermions
,,,,,,
R
GB R
udded
,
GB
uu arise when a Planck-scale fluctuation is “fertile-
ized” by the Yang-Mills generators of internal symmetry.
So a neutrino
is conceived when a fluctuation with
Planck mass magnitude is fertilized by the generator ei-
genvalues in Table 2 corresponding to the neutrino. The
same holds true for the up quark (in three colors), the
down quark (in three colors) and the electron. Then, as
Wheeler points out, the particles we observe from 20
orders of magnitude lower, have had all but the most
miniscule portion of their original ~MP masses can-
celled/averaged out by the positive and negative energy
fluctuations of the vacuum, leaving behind only a small
mass residue which results from the trapping of the field
lines, i.e., from the fertilization. Those are the particles
and masses we observe.
But if the Planck vacuum raises a chicken and the egg
question, the next question is this: how does nature de-
cide whether the egg comes first or the chicken comes
first? Does nature fertilize the positive energy fluctua-
tions into observed particles, or the negative energy ones?
Or, might she fertilize both? And what would a fertilized
positive energy fluctuation look like, versus a fertilized
negative energy fluctuation? And, fundamentally, how is
this precisely-balanced positive versus negative energy
symmetry in the Planck vacuum broken, in favor of the
very miniscule (relative to the Planck vacuum) prepon-
derance of positive energy over negative energy that we
observe in the material universe?
Now our


63 1
diag7, 1, 1, 1, 1, 1, 1, 1
228

generator provides the critical clue: If this is a gravita-
tional generator as we have begun to surmise, and if this
generator is actually used to break symmetry at or near
the Planck energy as in (5.1), and given that this is the
energy at which gravitation is dominant as is clear from
(6.1) through (6.4), then this generator will have a great
deal to do with how the Planck vacuum first gets fertile-
ized to produce what we observe. So the gravitational
charge of the neutrino being of opposite sign from the
gravitational charges of all the other fermions suggests
that perhaps neutrinos are fertilized negative energy
Planck vacuum fluctuations and the up and down quarks
and the electron are all fertilized positive energy Planck
vacuum fluctuations. Not only would this neatly resolve
the chicken and egg problem, but it would explain many
other things as well, especially about the ever-elusive
neutrino.
First, this would truly place neutrinos in a class by
themselves. They would be born of negative energy
Planck scale fluctuations, brought about via the gravita-
tional interactions of positive energy Planck scale fluc-
tuations. Other fermions are rooted in “Planck matter”;
neutrinos are rooted in “Planck gravitation.” Second,
above the Planck energy, behind the event horizon, we
would expect there to be a complete symmetry among all
of the octuplet members

,,,,,,,
R
GB RGB
uddeduu
.
Any one fermion can readily decay into any other, and all
would exist in equal numbers as part of an octuplet set.
Thus, any time there is a neutrino, there are also seven
other fermions to go along with that neutrino. Then, after
we break the symmetry and the neutrino hooks up with
negative energy fluctuations and the other seven fer-
mions hook up with positive energy fluctuations, we
would have a seven-to-one ratio of fermions which are
rooted in positive energy fluctuations over fermions
rooted in negative energy fluctuations. So as we reached
lower and lower energies, there would be a net domi-
nance of positive energy-rooted fermions over negative
energy-rooted fermions. As such, this could help to ex-
plain how the positive versus negative energy symmetry
of the Planck vacuum becomes broken. This is especially
so given the fact that at low energies the neutrino masses
become so very much smaller than all the other fermion
masses.
Third, while we conventionally hold to the view that
all matter gravitates the same way as all other matter, this
would tell us that this conventional wisdom holds true for
all matter except the neutrino. Below the Planck scale,
the neutrino would fundamentally be a fermion rooted in
negative energy fluctuations, while all of the other fer-
mions would be rooted in positive energy fluctuations.
This could certainly provide some degree of confidence
that as we start to trace the development of the fermions
from the Planck scale down to the laboratory scale, we
J. R. YABLON
Copyright © 2013 SciRes. JMP
112
may come to understand why 4 351853369
ue
mm .
and 9 601723351
de
mm ., while 250,000
e
mm
.
The neutrino would start off in the Planck vacuum with a
negative energy ~P
M
 where
represents the
alteration in energy due to the fertilization of the negative
energy gravitational fluctuation, while all the other fer-
mions f would start off with a positive energy
~
P
f
M

rooted in the matter fluctuations. Then,
after screening of twenty orders of magnitude, the neu-
trino mass would end up very close to, and slightly larger
than zero, and the rest of the fermion masses would end
up more substantially above zero, with the observed
masses between 5
2.5 10 and 6
2.5 10 times as large
as what is observed for the neutrino.
Further, if the neutrino gravitates differently from
every other fermion (which we shall explore even further
in the next section), then its elusive, idiosyncratic be-
haveiors may be much better understood. From a tech-
nology viewpoint, this also suggests that if one ever
hopes to develop technologies to “shield” gravitation or
overcome gravitational attraction other than by the brute
force of rocket propulsion, the neutrino would be central
to that undertaking. Harvesting and controlling the elu-
sive neutrino, however, would be the core technology
challenge. And, since neutrinos do exist throughout the
universe as elusive as they may be, this would also mean
that cosmological theories based on the supposition that
all matter gravitates in relation to all other matter in
exactly the same way would have to be modified to rec-
ognize that the neutrino defies this supposition.
As a consequence of the forgoing, let us now choose a
negative gravitational charge for the neutrino to go with
the negative energy fluctuations, as a matter of conven-
tion. Then, let us introduce the hypothesis—which needs
to be borne out through detailed calculation of its cones-
quences—that the neutrinos are in fact conceived at or
near the Planck scale when negative energy gravitational
fluctuations in the Planck vacuum become fertilized with
the negative gravitational charge of the neutrino
63 17
228

 , and that quarks and electrons
are born at or near the Planck scale when positive energy
gravitational fluctuations in the Planck vacuum become
fertilized with the positive gravitational charge of a quark
or an electron 63 1
,, 228
ude
 .
And in this regard, choosing the convention of a nega-
tive gravitational charge for the neutrino to go with the
negative Planck energy fluctuations, we now explicitly
define a gravitational interaction generator:
 
63 1
;diag 7,1,1,1,1,1,1,1
228
GG
 . (6.5)
We may find occasion to adjust this coefficient
1
228
as we calculate from this point forward, but this
sign reversal, and the identification of 63
with a
gravitational generator G, makes clear 1) that the neu-
trino is understood to gravitate differently than all the
other fermions as we shall further examine in a moment,
and 2) that the neutrino is rooted in negative energy
Planck fluctuations while all the other fermions are rooted
in positive fluctuations. Or, as Wheeler might say, the neu-
trino lines of force are trapped in negative energy topo-
logical wormholes, and the quark and electron lines of
force are trapped in positive energy topological wormholes.
7. Spontaneous Symmetry Breaking,
Fermion and Generator Fractures, and
Intergenerational Cabibbo Mixing of
Left-Chiral Hypercharge Doublets
As we now return to spontaneous symmetry breaking, it
will be important to develop an understanding of what
we shall call “fermion fractures” and “generator frac-
tures.” While the fermion fracturing we are about to de-
scribe may already be implicitly understood as a feature a
spontaneous symmetry breaking, it is important to make
this understanding explicit, as this will play a crucial role
in understanding generation replication, and especially,
the Cabibbo mixing which for leptons leads to so-called
neutrino oscillations (which have been largely response-
ble for demonstrating that the neutrino does have some
tiny mass, contrary to what may have been believed two
or three decades ago).
When a gauge group has not been broken at all, and
assuming that fermions have been assigned to the fun-
damental representation of that gauge group, then any
one fermion is completely free to decay into any other
fermion. SU(3)QCD provides a good example of this. As
we can see from Table 1, or as will be understood in any
event, there are three color eigenstates
838 3
111
,0, ,2
323
RG

 ,
83
11
,2
23
B

 .
The symmetry is not broken, so any of these eigen-
states may freely decay into any other one of these ei-
genstates, even though their quantum numbers are dif-
ferent. For example, all three color states R, G, B have
completely different 3
, namely, 311
0, ,
22
, yet
they freely transition among themselves, which is central
to QCD interactions. Similarly, as just discussed, above
the Planck scale any fermion may transition into any
J. R. YABLON
Copyright © 2013 SciRes. JMP
113
other fermion.
Once a symmetry is broken, however, some fermions
become “fractured” from some other fermions, and they
are forbidden from decaying into one another except un-
der very limited conditions. It is these limited conditions
which are of central interest in the discussion following.
Let us first break the symmetry of SU(8) at the Planck
scale using (5.1), which we recast in light of (6.5) as:



,..,
diag diag
17,1,1,,1,1,1,1,1
228
PP
i
PPi
P
vGie
v




(7.1)
What then happens? Of course, similarly to what was
discussed in Section 8 of [1], the vacuum commutes such
that ,0,1,,48
i
Pi


 .
It also self-commutes with G, that is,
,,0
PP
GvGG 
.
But our real interest here is to look at the fermions
themselves.
The neutrino, with 17
228
G
 ,becomes
fractured from all the other fermions with 1
228
G,
and can no longer decay into any of these other states via
the generator G that was used to break the symmetry. It
would be as if the red quarks in QCD were suddenly for-
bidden from decaying into green or blue quarks—but of
course they can do so because the QCD symmetry is
never broken. If G is a gravitational generator, then the
neutrino can no longer undergo a gravitational decay
through G into any other fermion. What does that mean?
The neutrino will no longer gravitate with any other fer-
mion except for another neutrino! But—and this is criti-
cal—it may still undergo other types of decay through
the generators of other interactions. Let’s elaborate:
If the neutrino is to decay into any other fermion after
the symmetry is broken via (7.1), it must decay into a
fermion via an interaction governed by an interaction
generator other than 63
gravitation such that the fer-
mion has the same charge value under that other inter-
action generator as that of the neutrino. Referring to
Table 2 to make this clear, this means that the neutrino
still can undergo a 35
decay into a
R
u quark because
each has 35 0
. And it can still undergo a 3
L
I
decay
into any up quark, because these and the neutrino all
have 31
2
L
I. Most importantly, as will become central
in the discussion be- low, the neutrino can still undergo
L
BLY decay into an electron because both the neu-
trino and the electron have the same 1
L
BLY 
and so form a doublet under
L
BLY . This latter abil-
ity for the neutrino and the electron to decay into one
another as like-charge members of a 1
L
BLY 
doublet, lasts until the electroweak symmetry is finally
broken at much lower (Fermi vev) energies into the elec-
tromagnetic interaction.
Now let’s look at the remaining seven fermions. Even
after the symmetry breaking (7.1), these fermions are
completely free to decay into one another via the gravi-
tational generator G, because they are all like-valued
1
228
G eigenstates of G. They all continue to
gravitate with one another, while the neutrino steps aside
and stops gravitating with them. Indeed, starting at the
Planck scale, and until one drops down to GUT energies
on the order of 1015 GeV, these seven other fermions
remain part of an SU(7) septuplet. Since all of these fer-
mions are united by the common characteristic that they
are born through the fertilization of positive (+) energy
vacuum fluctuations, we shall refer to this group as
SU(7)+. Thus, between the Planck scale and the GUT
scale, the gauge group is
 
71
G
SU U
, and the
topologically-stable SU(7) magnetic monopoles with all
the fermions of a 2H atom are
 
111
π(7) π(1)π(1)
GG
SU UU
. The
1G
U
emanates from the commutation of
,,0
PP
GvGG
, and is based on a neutrino
singlet 17
228
G
 plus a septuplet of the
remaining fermions all of which are in 1
228
G
states. Again, importantly, the neutrino can no longer
interact gravitationally via 63
with any of the remain-
ing seven fermions. If it is to interact with them, it must
do so via other non-gravitational interactions.
Now let’s progress down the energy scale and break
symmetry with
L
BLY
in the vicinity of
15
~10 GeV
GUT
v using (5.2). The residual gauge groups
are now those shown and discussed in (5.6), but let’s
again look closely at how the fermions are fractured, and
let’s also look at the loss of two generators going from
86SUSU.
Referring now to Table 4, the 48 35
,
generators are
no longer in play as vertical generators, because two gen-
erators are lost going from SU(8) to SU(6). These do not
disappear entirely, but become horizontal as already dis-
cussed, in a manner we shall momentarily develop fur-
ther. As to the remaining five linearly-independent verti-
cal generators in Table 4, the electrons and the quarks
still remain a gravitational septuplet and so can still in-
teract gravitationally with one another (while the neu-
trino does not)! Following the rule that after symmetry
J. R. YABLON
Copyright © 2013 SciRes. JMP
114
breaking the only decays which are permitted are decays
under a given generator for which the decaying fermions
have a like-charge, the remaining decays options are as
among members of the quark sextuplet of fermions with
1
3
BL, and between the lepton doublet of fermions
with 1BL
. The former decays among fermions in
the 1
3
BL sextuplet, consist of QCD strong interact-
tions decays among the R, G, B color eigenstates based
on the 83
,

generators, and weak decays between
states with 3
11
,
32
L
L
YI due to the common
1
3
L
Y. The latter decays between the two fermions in
the 1BL lepton doublet, consist of weak decays
between the neutrino and the electron with
31
1, 2
L
L
YI due to the common 1
L
Y .
Now, however, most importantly, the quarks have be-
come fully fractured from the leptons, and there is no
more decay permitted between quarks and leptons. This
is because, referring to Table 4, there is not a single ver-
tical generator other than 63
for which any quark
shares the same charge as any lepton, so hereafter, the
only way for a quark to interact with a lepton is gravita-
tionally. And the neutrino—the odd man out—does not
interact gravitationally with any other fermions besides
another neutrino, because its gravitational charge is dif-
ferent from that of all the other fermions and that gravi-
tational generator was used to break the Planck symmetry.
Further, as was developed in detail in Section 8 of [1],
the breaking of BL also creates stable magnetic
monopoles



111
π3π1π1
CBB
SUU U

which manifest as protons and neutrons forming
,pn
doublets with 1B. So this is also the symmetry break
at which protons and neutrons are born. And, with
L
BLY , as noted at the end of Section 5, the weak
interaction becomes non-chiral to go along the with
chiral non-symmetry of baryon interactions as discussed
in Section 5 of [1].
So the BL symmetry breaking is responsible for
several interrelated phenomena: it brings about the three
generations observed at low energy, it brings about pro-
tons and neutrons, it forecloses lepto-quark decays, and
because
L
BLY , it brings about the broken chiral
symmetry of the weak interactions.
Now, at some level, everything discussed so far in this
section about fermion fracturing due to symmetry break-
ing restates what is likely obvious, because it is known
that one of the very basic consequences of symmetry
breaking is that it forecloses certain decays which are
permitted to occur in the higher state of symmetry before
the symmetry is broken. From a thermodynamic view, it
“freezes out” certain transitions below a certain critical
temperature (recognizing too that some symmetries are
not broken but are actually restored on the opposite end
of the scale, near absolute zero, where electrons are su-
perconducted freely without any apparent friction from
the protons and neutrons from which they separate at
GUT energies, which suggests that superconductivity
may well be a phenomenon at which the SU(7) symmetry
between electrons and quarks is restored so electrons can
flow through rather than a round protons and neutrons).
But the reason for focusing on fermion fracturing in this
way, is because we will now venture into the not-obvious
realm of generation replication and apply these observa-
tions to understand what happens there as well.
If the rule is that after symmetry breaking fermions
can only decay into other fermions with like-charges
under some interaction that was not used to break the
symmetry, then what happens to the horizontal genera-
tors 48 35
,
after
L
BLY
symmetry breaking? Not
only have quarks become fractured from leptons, but
48 35
,
have themselves become fractured from the
other generators! So we not only have fermion fracturing,
we have generator fracturing. If we follow suit, then it
would seem that a similar set of rules may well apply.
Let’s explore.
First, referring to Table 2,


48 1
diag0,6, 1, 1, 1,1,1, 1
221

and


35 1
diag0,0,5, 1,1,1,1, 1
215

are the two fractured generators. Because these no longer
differentiate an observable vertical symmetry, but still do
provide two degrees of freedom as illustrated in Figure 1
in section 5, let us transform these two generators into
48 35
,
with the eigenvalues shown in Figure 1. No
new calculation is required: we simply use (4.3) and (4.4)
but without 8
and 3
, and so redefine 48 35
,

48 35
,
according to:
48 483524
72 2
315
35

, (7.2)
353524
32
55

. (7.3)
It is readily seen that


48 1
diag0, 2,1,1, 0, 0,0,0
23

and


35 1
diag0, 0,1,1, 0,0,0, 0
2
 .
J. R. YABLON
Copyright © 2013 SciRes. JMP
115
So these generators now do yield the SU(3) configure-
tion shown in Figure 1, albeit with eight eigenstates, five
of which are all zero-valued and trivial, and three of
which are not. We can now label these three non-trivial
eigenstates as:
48 35
1,0
3
e


 
, (7.4)
48 35
11
,2
23
 

 
, (7.5)
48 35
11
,2
23
 

 , (7.6)
just as illustrated in Figure 1. However, these are now
free-floating generators once the
L
BLY symmetry
is broken, so they no longer provide vertical symmetry
quantum numbers for any of the fermions, as illustrated
in Tables 3 and 4. Rather, they appear to provide a repli-
cation of each fermion into three generations. But if this
is the case, then they should lead to other facets of gen-
eration replication as well, including Cabibbo-type mix-
ing, and to the observation that the only way a particle
from one generation can transform into a particle of an-
other generation is via left-chiral weak interaction decays
from one weak isospin to a different weak isospin, and
not directly. As we shall now see, this is a consequence
of the fermion and generator fracturing highlighted above
and the “freezing” restrictions that come into play after
symmetry breaking.
Because the generators 4835
,
have become frac-
tured from the other generators, and given what we know
about the fermion generations from experimental obser-
vations, it appears that each of the ,,e
eigenstates is
fractured from one another so that it is now forbidden for
a direct transition to take place between any of the three
states (7.4), (7.5), (7.6), i.e., no decays may take place
any longer via the 48 35
,

(or 48 35
,
) interaction
generators. Symbolically, e


. Any decays that
do take place, must occur via another generator for
which the charges are the same as among the fermions
involved in the decay. The fermion has to find a “loop-
hole.” This is exactly like the discussion we had at the
beginning of this section about the neutrino in relation to
the remaining fermions from which it becomes fractured
at
P
v, or the fracturing of the quarks from the leptons at
GUT
v. In order to undergo decay into a different fermion,
a fermion must find a different generator and a different
fermion which has the same charge as the original fer-
mion with respect to that different generator.
So referring to Table 4, if a first-generation e fermion
is to decay into a second generation μ fermion or a
third-generation τ fermion, it must to do so via a genera-
tor other than 48 35
,

, into a fermion for which it
shares an identical charge for that other generator. For
the leptons, this is straightforward: the electron and the
neutrino share a common charge 1
L
BLY , and
so for a first generation electron to become a second
generation electron, it must go from e


or
e
e

, all of which have the same
L
BL Y
1
. This is the only remaining “decay loophole.”
Again this is exactly what was discussed earlier with
regard to fermion fracturing. And so, for the first time,
we see Cabibbo mixing and neutrino oscillations, be-
cause that is exactly how these work as well. This also
explains flavor non-conservation as regards the genera-
tions: at the end of e


or e
e

, what
started as a first generation electron is now a second
generation electron and neither 48
nor 35
is con-
served, and this is because the generators are fractured.
For the quarks it is a little more complicated, because
this transition rule needs to be strengthened due to strong
interactions. In particular, if a fermion can undergo a
e
transition by decay through at least one
generator that is the same for both, then, for example,
referring to Table 4, one could observe a
R
G
uc
transition, because both the
R
u and the G
c have the
same 1
3
L
BL Y

. This would imply that Cabibbo
mixing can occur not only via weak but also via strong
interactions, and the latter, of course, is not observed.
So for horizontal symmetry transitions, it appears that
we have to tighten the rules even further. Specifically, it
appears that for a horizontal transition to be permitted,
not just one, but all of the vertical degrees of freedom in
Tables 3 and 4 must be the same as between the two fer-
mions involved in the decay. Table 3 actually illustrates
this rule the best, because this rule says that a horizontal
e
transition must occur either as a transition
between the first and fifth, second and sixth, third and
seventh, or fourth and eighth fermions in Table 3. These
are the fermion doublets which share a common:
83
,1,0,0,1
L
eBL Y


  , (7.7)

83
111
,,,0,
33
3
RR L
ud BLY


, (7.8)
83
,
1111
,,,,
323
23
GG
L
ud
BL Y


 (7.9)
83
,
11 11
,,,.
323
23
BB
L
ud
BL Y


 (7.10)
So in sum, one can have neither e
, nor
e

, nor uct
, nor dsb tran-
J. R. YABLON
Copyright © 2013 SciRes. JMP
116
sitions, because each of these has different 4835
,
eigenvalues. These states are all fractured from one an-
other. One cannot have intergenerational transitions be-
tween

,e
and any of the quark doublets because
these have been fractured from one another by BL
breaking. One cannot have intergenerational RG
B transitions among (7.8), (7.9) and (7.10) because
although QCD is never broken, the QCD generators are
different as among red, green and blue states. If any ver-
tical generators, or any horizontal generators are dif-
ferent as between two fermions, then based on what we
observe, the apparent rule is that the horizontal transi-
tion is not permitted. So all that is permitted—the only
“loophole” left for decay—are the e
, ,
R
RG
udu
G
d and
B
B
ud transitions, because these are the
only transitions for which all of the generators listed are
the same for both fermions. And here, because of the
tightened rules when it comes to horizontal transitions
based on fractured generators, even the right-chiral gen-
erator
R
Y is excluded, because this too is not the same
as between the members of each of the above doublets.
This is why we show
L
Y in the above but not
R
Y. This
means only the left-chiral states may participate in tran-
sitions among the e
 states in (7.4) to (7.6).
Observationally, we know that this is also a characteristic
of left-chiral weak generational interactions.
These stronger rules for the horizontal generators may
at first seem arbitrary, but they are not. They may be un-
derstood because for the horizontal generators, not only
are some fermions fractured from other fermions, but the
horizontal generators themselves are fractured from the
vertical generators. It is the fracturing of both generators
and fermions which leads to such stringency. So for a
vertical generator that breaks symmetry but is not itself
fractured from the other vertical generators, transitions
are permitted so long as at least one other vertical gen-
erator provides the same charge as between the two tran-
sition states. But for a generator which has itself been
fractured from the other generators, the rule is even more
restrictive. Now, transitions are permitted only if all of
the involved vertical generators provide the same charge
as between the two transition states.
Now, the astute reader may notice that the electric
charge Q and left-chiral weak isospin 3
L
I
are also not
the same as between the two fermions in any of the dou-
blets in (7.7) through (7.10) above.
,0,1Qe
and

21
,,
33
Qud



as between the members of
these doublets, as well as

311
,,
22
L
Ie




and

311
,,
22
L
Iud



. And so, the question might be
asked, why are even these interactions permitted? After
all, this changes the generators also, so by these rules,
shouldn’t this be forbidden also? But further reflection
makes this answer clear: the electric charge does not
emerge as a physically-preclusive generator until it is
used to break the electroweak symmetry at much lower
energies determined by the Fermi vacuum F
v
246.219651 GeV . This is the same way in which BL
is not a preclusive generator until its breaks symmetry at
GUT energies. So indeed, once we break electroweak
symmetry, no transitions are permitted between genera-
tions. But at the same time, neiter will e
or
ud
be permitted, but this is because weak interact-
tions are no longer permitted either (in the historical
sense that the weak interaction becomes “weak”). So
what we learn from this, is that the ability of fermions to
change generations will wax and wane in lock step with
the weak interaction itself and the breaking of elec-
troweak symmetry, just as is observed!
By imposing the more stringent rule that once the
48 35
,
interaction generators have become fractured
from the other generators by BL symmetry breaking
at 15
~10 GeV
GUT
v, no horizontal transitions are permit-
ted among the (7.4) to (7.6) states unless all of the re-
maining vertical generators—chiral symmetric or not—
are the same as between the fermions involved in the
transition, we arrive at precisely the type of mixing that
is observed in nature as among the three generations.
This makes generation mixing part and parcel of weak
interactions, while excluding the strong interactions and
even the right-chiral states from participation in genera-
tional mixing.
So, now we take the final, formal steps to mathemati-
cally represent all of these decay restrictions. Referring
to Section 12.12 of [14], the two generators 48 35
,
introduce two degrees of freedom and so define three-
non-trivial horizontal eigenstates ,,e
in (7.4) through
(7.6) and Figure 1, representing eigenstates of SU(3),
which states are precluded from direct transformation
into one another according to the rules just outlined be-
cause they are fractured generators. SU(3) can be used to
form unitary matrices U with 933 components.
Because the only permitted transitions are (7.7) through
(7.10), we can alter the phase of any of the 23 6
quark states which we designate

,ud
,,, ,,uctdsb following Table 3, without altering the
physics. Similarly for leptons. But one may omit an
overall phase change which still leaves the physics in-
variant. This means that U must be a function of
933
minus 623
plus 1 parameters, i.e., 4 pa-
rameters. But an orthogonal 33 matrix only has
3, 23C
real parameters, which leaves one residual
phase. So for the leptons l, we may choose to form this
matrix in the representation:
J. R. YABLON
Copyright © 2013 SciRes. JMP
117
111 1313
22 1133121232312323
223312123 23123 23
1000100
000 ee
000e0e e
iδiδ
l
iiδiδ
ll
csc scss
Ucssc cssccccssccssc
scscss csccscsscc
  
  
 
 
 
 

 
(7.11)
and for the quarks q we form the analogous:
111 1313
22 1133121232312323
223312123 23123 23
1000100
000 ee
000e0e e
iδiδ
q
iiδiδ
qq
cscscss
Ucssc cssccccssccssc
scscss csccscsscc
 

 

 
 




(7.12)
To implement the lepton mixing, we keep in mind
from (7.7) that for a e
 transition to take
place which alters the quantum numbers in (7.4) through
(7.6), we cannot go directly from e
, but must
engage in a vertical transition between the states
,e
in which all of the generators 8
1, 0,BL

30,1
L
Y

do not change. The only permitted tran-
sition is e
. Now, one can always apply (7.11) to
both of
,e
, but then one of them can always be
transformed into a pure state while the other is similarly
transformed, without changing the physics. In other
words, all that is observable is the relative transition as
between
,e
. So following the usual conventions, we
use (7.11) to transform the lower members of the
,e
doublet, that is, we define:
113 13
12123 23123 23
12123 2312323
ee
ee
iδiδ
ili
iδiδ
l
ec scss e
esccccssccsscUe
sscs cc scsscc



 

 

 
 
 

 
 
(7.13)
Similarly for the quarks of each color C = R, G, B, we define:
11313
12123 2312323
12123 23123 23
ee
ee
C C
iδiδ
CiCCqCi
iδiδ
C C
q
dc scssd
dssccccss ccsscsUd
bsscsccscssccb
 
 


 
 
 
 
(7.14)
Because
R
Y is not the same as between the members
of each of the (7.7) through (7.10) doublets, right-chiral
transitions are also precluded, and the only permitted
transitions are for left-chiral states. So these will be pro-
jected with

5
11
2
. Further, because 83
,

are not
the same except as between members of the four distinct
doublets in (7.7) through (7.10), the only permitted tran-
sitions will be between one lepton and another lepton,
and between a first quark of a given color and a second
quark of the same color ,,CRGB. This keeps the
strong QCD interaction out of generation-changing tran-
sitions (and also out of any CP violation), and makes this
an exclusively weak, left-handed chiral phenomenon. So
for leptons, the transition currents will be:


 
11313
5
12123 23123 23
12123 23123 23
55
1
ee1
2
ee
11
11
22
iδiδ
le
iδiδ
l
il iiiiLiL
csc sse
Jsccccssccssc
sscscc scs scc
Ue ee



 
  







 



(7.14)
And for quarks of each color ,,CRGB, they will be:


 
11313
5
12123 23123 23
12123 23123 23
55
1
ee1
2
ee
11
11
22
C
iδiδ
qCCC C
iδiδ
C
q
iCqiC iCiC iLC iLC
csc ssd
J
uctsccccssccsscs
s
scsccscssccb
uUdudud



 





 



(7.15)
J. R. YABLON
Copyright © 2013 SciRes. JMP
118
This is exactly what the phenomenology demonstrates!
So, returning to the question posed at the very outset
of the discussion following Table 2, not only does SU(8)
not provide too much freedom, but upon careful consid-
eration and development, it provides exactly the right
amount of freedom to explain the precisely observed
fermion phenomenology of three generations. Further, by
applying the rule that fermions which are fractured from
one another after symmetry breaking cannot decay into
one another except by a vertical interaction other than the
vertical interaction that was used to break symmetry, and
that decay with regards to a fractured generator which
thereafter becomes a free-floating horizontal degree of
freedom is only permitted between fermion eigenstates
for which all of the surviving vertical generators are the
same, we can use SU(8) to explain everything that we
know about the qualitative features of the interactions we
observe, from generation replication to weak chiral
non-symmetry to Cabibbo mixing to the fact that this
mixing occurs only via weak isospin decays between
left-handed states. And in the process we have perhaps
found that neutrinos do not gravitate with any fermions
aside from other neutrinos, which is likely to be of tre-
mendous consequence as this is better developed and
understood and especially if it can ever be exploited.
Before concluding this section, let us now return to the
first three generators 63 4835
,,

of SU(8). Based on
the earlier review of how 63
breaks symmetry near the
gravitational Planck scale and sets the neutrino on a tra-
jectory to have a mass orders of magnitude smaller than
that of any other fermion; given how the 4835
,
frac-
ture from the other vertical generators and form the basis
for two horizontal degrees of freedom that underlie three
fermion generations in which one fermion is distin-
guished from one another solely by mass and not by any
other quantum numbers from a vertical degree of free-
dom, and given that mass and gravitation are inextricably
linked such that gravitation is the “mass interaction,” we
now formally associate these three generators 63 48
,,
35
with the gravitational interaction, at the elementary
particle level, below the GUT energy. Using (7.4) to (7.6)
and (6.5), we highlight this connection in Table 5 of
Section 5.
The horizontal degrees of freedom from 48 35
,
which to enable the fermions in each generation to have
distinct masses in relation to their counterparts in the
other two generations are shown horizontally, while the
vertical degree of freedom G enabling each fermion
within a generation to have a distinct mass is shown ver-
tically. Of course, with SU(3)C remaining unbroken, dif-
ferent colors of the same flavor of quark within one gen-
eration have the same mass. As noted earlier, using the
notation, the vertical gravitational generator G does
not distinguish the ,,ud e
masses from one an-
other within a generation. So at high energies, as noted,
the fermions (other than neutrinos) within a generation
all have the same mass. It is only through the stages of
symmetry breaking and the remaining generators
L
BL Y
, 3
L
I
and Q, that the mass spectrum within a
generation separates. This may be thought of as mass/
energy differences emanating from strong, weak, and
electromagnetic interactions, i.e., one may regard quark
masses to differ from electron masses because they are
quarks not leptons, and up and down quark masses to
differ because their weak isospins and electric charges
are different. Gravitational generators provide the free-
dom for these differences to occur.
As to interactions, after all symmetry breaking in-
cludeing electroweak symmetry breaking is completed,
the seven generators of SU(3) now are allocated as fol-
lows: three degrees of freedom go to gravitation in the
form of 63 48 35
,,

, two degrees of freedom go to
strong QCD interactions via 83
,

, one degree of
freedom goes to left-chiral weak interactions via 3
L
I
, and
the final degree of freedom goes to electromagnetic in-
teractions via Q. Seven linearly-independent degrees of
freedom, and eight vertical fermion eigenstates, thus ac-
count perfectly, with nothing missing and nothing super-
fluous, for the observed phenomenology of the fermions
and their interactions, including generation replication
and Cabibbo mixing, left-chiral weak interactions, and
the elusive and perhaps gravitationally-defiant behavior
of the neutrino.
8. Summary and Conclusion
We have in the foregoing focused on the breaking of
symmetry at the Planck scale and the GUT scale, which,
astronomical observation aside, is many orders of mag-
nitude beyond what we may ever hope to observe di-
rectly. The final stage of symmetry breaking is elec-
troweak symmetry breaking at the Fermi vev F
v
246.219651 GeV . This is in the realm of observation,
and the generator used to break this symmetry is the
electric charge generator Q. This final symmetry break
gives rise to the electromagnetic interaction which
dominates atomic and chemical structure and much of
what is most directly observed in the natural world be-
yond gravitational interactions. That is, beyond objects
falling to earth and planets wandering the heavens along
prescribed trajectories, electromagnetic phenomena in
electromagnetic and chemical and atomic form are our
first line of direct experience of the natural world. Our
experience of nuclear phenomena—based on the protons
and neutrons which come to life as stable magnetic
monopoles at the GUT scale as has been reviewed here
and in [1]—comes to us through the laboratory instru-
mentation that we used to extend the range of our physi-
J. R. YABLON
Copyright © 2013 SciRes. JMP
119
cal senses, and gives rise to the vast preponderance of the
matter that populates and animates the universe.
When we break the electroweak symmetry we make
use of the electric charge generator (4.6), and analo-
gously to (5.1) through (5.3), employ the Fermi vacuum:
i
FiFF
vQ

, (8.1)
which specifically means that:


diagdiag
211122
0,,,,1,,, diag
333 333
i
FiF
FF
T
vvQ





(8.2)
Picking off the coefficients from the generators in
(4.6), for each non-zero component of the vacuum we
then have:
48 3524
15 8
27 422
;;;
33 35
315
22
2;
33
F
FFFF F
FFFF
vvv
vv
 

 
 
(8.3)
which leads to:

22 222
483524 15 8
2
222
471642 8 4
939159533
16
3
FFFFF
F
FF
v
vCv

 







(8.4)
and consequently an electroweak Clebsch-Gordon coef-
ficient:
4
3
C. (8.5)
This is how the electroweak symmetry is broken for
the SU(8) group that we have developed throughout this
paper. This final symmetry break fractures all fermions
of different electric charges from one another, and so
precludes their decay into one another. Referring to Ta-
ble 4, weak isospin transitions between up and down
quarks with differing charges

21
,,
33
Qud



are
now precluded, as are similar transitions between elec-
trons and neutrinos with
 
,0,1Qe
. This shuts
down the weak interaction (in the historical view, renders
it “weak”; in hindsight it is probably better called the
“faint” interaction), and because weak isospin decays as
reviewed in the last section are the only avenues permit-
ted for generation-changing transitions, generational
transitions also are turned off in lock step. The only tran-
sitions still permitted after electroweak symmetry break-
ing, given that Q is a vertical symmetry generator and so
not subject to the very stringent rules laid out in the last
section for horizontal transitions, are the vertical, color-
changing R, G, B transitions of QCD, which are still al-
lowed to occur because the quarks involved in these
interactions are part of a triplet in which 1
3
BL
is
the same for each, and the QCD symmetry remains un-
broken. That is, the only permitted decays once elec-
troweak symmetry is broken, are decays along the
BL
generator for particles of like BL with un-
broken 83
,
generators, which, of course, are strong
QCD interactions. With the exception of the RG
B
transitions of QCD, no fermion may transform
into any other different type of fermion.
Now, following three stages of symmetry breaking—at
the Planck scale, the GUT scale and the Fermi scale—all
of the fermions have become fractured from one another,
generation transitions cease, and the particles are frozen
into the configurations of our everyday experience. The
SU(8) symmetry with seven generator degrees of free-
dom that we started with in Table 2 still does exist, but it
has become hidden and distorted behind twenty orders of
magnitude of vacuum screening and three stages of
symmetry breaking that have fractured neutrinos from
the other fermions and broken off their gravitational
communication, broken the Planck symmetry between
positive and negative energy fluctuations, fractured
quarks from leptons, fractured two generators from the
remaining five to provide horizontal generational replica-
tion, brought about Cabibbo-type mixing among these
generations for left-handed chiral projections only, and
finally, fractured the upper and lower members of the
like-hypercharge YL (weak isospin) doublets from one
another, turned off the weak interactions, and frozen the
particles in place so that all we observe at the lowest en-
ergies are electromagnetic and strong interactions, as
well as the bulk interaction of gravitating masses which
is eluded by the neutrino.
This GUT, which is based on the hypothesis that bary-
ons are Yang-Mills magnetic monopoles and is rooted in
the SU(4)P and SU(4)N subgroups developed in Section 7
of [1] which yielded over half a dozen accurate predict-
tions in [1,2] as reviewed in Section 1 here, leads system-
atically to all of the qualitative particle and interaction
phenomenology which we are able to observe with our
senses and the extension of our senses through experi-
mental apparatus. But the confirmation of the particular
GUT proposed here, versus other possible GUTs which
reproduce similar phenomenally, needs to come through
mass and energy predictions which continue the suc-
cessful empirical matches developed in [1,2]. As dis-
cussed in Section 3, one would expect that these energy
predictions should come about by developing the re-
maining
-containing terms in the Lagrangian density
(3.2) which we have not yet developed, and then making
J. R. YABLON
Copyright © 2013 SciRes. JMP
120
use of these to calculate various energies 3
dEx
L
to be matched up with empirical data. Along the way, the
development should proceed on a parallel course to that
of Sections 2 through 11 of [1], making use of the
non-Abelian Klein-Gordon Equation (3.10), representing
scalar sources as J
, employing the same sort spin
sums and the same Gaussian ansatz modeling of fer-
mions that was developed respectively in Sections 3 and
9 of [1], and keeping in mind the clues we have elabo-
rated in (3.6) through (3.8) and (3.11) here, all while em-
ploying the GUT and symmetry breaking that has been
elaborated here.
It is clear from [1,2] that it will be possible via this ap-
proach to calculate and predict definitive mass and en-
ergy values, just as has been done previously in [1] and
[2]. It will then be left to interpret those values as we did
in Sections 11 and 12 of [1] and throughout [2], and to
compare them with experimental data to try to ascertain
the meaning of those calculations and predictions to ob-
tain sensible numerical matches to observed energy data.
That is, we clearly will be able to calculate energies. The
question will be whether the energies we are able to cal-
culate will match and make sense in relation to the em-
pirical data as well as they did in [1,2].
Success in this endeavor, if it should arrive, would
validate that this particular GUT may indeed be the one
that nature has selected to govern the phenomenology of
the material universe, and would provide some confi-
dence that the development elaborated here does reach
“behind the veil” to explain how nature really does oper-
ate in energy domains likely to forever remain beyond
the reach of our direct senses and the extension of our
senses gained through experimental devices and meth-
ods.
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