gill1109 wrote:Nice! The standard model is derived by playing with symmetries of solutions of fancy systems of differential equations. The Dirac equations. So it is plausible that some of its features come from playing with symmetries of much more simple objects. I hypothesize that that is what you are doing. I hope the experts will take notice.
As far as the Standard Model symmetries are involved, I believe I found some parts and pieces that might help. I go through the math here rather quickly. So if I made a mistake, please correct me.
The Standard Model has the symmetry of U(1)XSU(2)XSU(3). The U(1) symmetry describes the electromagnetic interaction. The SU(2) symmetry describes the Isospin symmetry of the Weak Interaction. And the SU(3) symmetry describes the Color symmetry of the Strong Interaction. See page 2 of
https://uw.physics.wisc.edu/~himpsel/449group.pdf.
However, U(1) is described by the the complex numbers. And SU(2) is described by the quaternions. And SU(3) is described by octonions. See these references. If you know of any better references, let me know.
https://en.wikipedia.org/wiki/Symmetry_in_quantum_mechanics#U(1),
https://en.wikipedia.org/wiki/Representation_theory_of_SU(2),
https://golem.ph.utexas.edu/category/2020/07/octonions_and_the_standard_mod.html.
hypercomplex numbersThe quaterions and octonions are hypercomplex numbers that can be derived from the complex numbers using the Cayley-Dickson construction. See
https://math.ucr.edu/home/baez/octonions/node5.html. This construction is achieved by an iteration process that's similar to the iteration process used in my efforts.
Very briefly, the hypercomplex numbers can be achieved by re-inserting complex numbers into the definition of a complex number that exist in a higher dimension. So, start with a complex number in general such as
, where
.
Then to create a quaternion number let,
and
. And we insert these into
, where
is another complex number different than
. This gives us,
 + (e + fi)j)
, or,
or,
,
where
, and
, and where
,
, and
.
Notice that since
, for example, the quaternions no longer commute under multiplication.
The quaternion also have a matrix representation, with
,
,
,
. See
https://mathworld.wolfram.com/Quaternion.html.
These are called the Pauli matrices used in QFT for the Weak interaction. And they have the same multiplication properties as
. Notice that these matrices also have the complex numbers inside them.
And then to create the octonions, we can insert quaternion numbers into the definition of a complex number. So let
and
be the quaternions:
, and let
. And insert these into the complex number
, where
is a different complex number other than
. This gives us,
 + \left( {q + ri + sj + tk} \right)\ell)
, or
, or
,
where
. The multiplication table for the octonions gets a little more complicated and can be seen here with a change of notation:
https://en.wikipedia.org/wiki/Octonion. The translation to the notation used here is:
,
,
,
,
,
,
,
.
But notice now, for example, in the notation used here that
\delta \ne i(k\delta ))
. So the octonions no longer have the associative property of multiplication.
And the octonions also have a matrix representation in the form of the Gell-mann matrices used in QFT for the Strong interaction, see page 24 of
http://people.oregonstate.edu/~drayt/MTH679/handouts/notes06.pdf, and see page 3 of
https://arxiv.org/pdf/1907.12520.pdf. See also,
http://benasque.org/2011qfext/talks_contr/2034_Bisht.pdf. Notice that the Gell-mann matrices for octonions seem to contain the Pauli matrices for quaternions.
I'm sure you are aware that,
. This is because
can be expanded into a power series, and any power of a complex number is also a complex number. So likewise, I take it that
, and
. And since
is unitary, I take it that
and
are likewise unitary.
So we can express this iterative process in terms of exponents as,
}} = \,\,\,complex\,\,number)
,
}} = \,\,\,quaternion\,\,number)
,
}} = \,\,\,octonion\,\,number)
.
SU(2) Weak bosonsAnd this all fits nicely into the iteration process I use in my thesis. See my previous post in this thread for more details. But in summary of my framework, I start with the principle that a conjunction implies both a forward and reverse implication,
 \to (p \to q) \wedge (q \to p))
.
I describe each point in space with individual coordinates. And I then mathematize logical implication by going through set theory, then using the Dirac measure to get the Kronecker delta, and then getting the Dirac delta function from that. I use the Gaussian exponential to represent the Dirac delta. But the assignment of coordinates to each point is arbitrary. And one implication should not be considered more important or weighted any differently than another. So in order to ensure no preferential treatment of one implication over another, and to maintain coordinate independence, I assigned the exponent of the Gaussian representation of each implication to be a complex number. That way the absolute value of each implication is the same; the only difference is a phase. All of this enables an implication to be mapped to a transition amplitude of a particle.
\,\,\, \mapsto \,\,\, < {x_q}|U(t)|{x_p} > \,\, = \,\,{(\frac{m}{{2\pi \hbar i({t_q} - {t_p})}})^{1/2}}{e^{im{{({x_q} - {x_p})}^2}/2\hbar ({t_q} - {t_p})}})
.
And the reverse implication is the complex conjugate of this and represents the antiparticle. There are only two possible conjugates in the complex numbers; so they would account for the
electron and
positron. And we see that this exponent to a complex number gives us the U(1) symmetry of the
electromagnetic interaction.
Then to iterating we apply the conjunction to implications principle again to get,
 \wedge (t \to u)] \to [(r \to s) \to (t \to u)] \wedge [(t \to u) \to (r \to s)])
All I did was insert and implication into each proposition. In the math, since the transition amplitude is a function of
and
, this means I would iterate by replacing these by transition amplitudes themselves. Then I will have amplitudes inside amplitudes, exponents inside exponents. So I make the following substitutions:
}})^{1/2}}{e^{ + im{{({x_s} - {x_r})}^2}/2\hbar ({t_s} - {t_r})}})
, and
}})^{1/2}}{e^{ + im{{({x_u} - {x_t})}^2}/2\hbar ({t_u} - {t_t})}})
.
This gives us,
(If you cannot see all of the math in this picture, use ctrl-mouse-wheel to decrease the size of text.)
.
This is of the form
}})
, which makes this a quaternion. And this gives us the SU(2) symmetry of the Weak interaction.
There are four possibilities with the equation above. What is shown is a
positron connected to a
positron as indicated by the plus sign in the (s,r) exponent and the plus sign in the (u,t) exponent. So this would represent a
boson. But there are three more possibilities. We could have an
electron with a minus sign in the (s,r) exponent and a
positron with a plus sign in the (u,t) exponent. Or we could have a
positron with a plus sign in the (s,r) exponent and an
electron with a minus in the (u,t) exponent. But since the difference between these exponents is squared (the purple 2), there is no mathematical difference. They are the same particle. They have zero
electrical charge, since the
electron charge cancels with the
positron charge. So this would be the
boson. And lastly, we could have an
electron with a minus in the (s,r) exponent and an
electron with a minus in the (u,t) exponent. This is a different particle than the other two and would be the
boson.
The next question is whether it makes any difference if the
j is conjugated all by itself. In other words, is the forward implication the same or different in this second iteration. I assume that if we were to derive a real value from a quaternion (for example, an expectation value), then the quaternion conjugate would reverse the sign of the
j and then be multiplied by the original. This would cancel the quaternion out of the result. So the expectation value would be the same for both forward and reverse implication in this iteration. So in essence the forward particle is identical to the reverse particle in the real world.
Before moving on, let's employ some abbreviations for simplicity. With the
A's and
B's and the
W defined appropriately, the previous equation can be expressed as,
.
This allows us to see the complex numbers without the clutter of unneeded math.
SU(3) quarksIn the next iteration, implications of implications are substituted for each proposition in an implication. The conjunction rule then give us,
 \to (t \to u)] \wedge [(v \to w) \to (x \to y)]\} \to)
 \to (t \to u)] \to [(v \to w) \to (x \to y)]\} \wedge)
 \to (x \to y)] \to [(r \to s) \to (t \to u)]\})
.
In the math, this means I would substitute two instances of the previous math equation into the definition of a transition amplitude. I abbreviate with the added
C and
Q to get,
(If you cannot see all of the math in this picture, use ctrl-mouse-wheel to decrease the size of text.)
.
We have an exponent inside an exponent inside an exponent. This is of the form,
}})
, and so it is an octonion, which give us the SU(3) symmetry of the Strong interaction.
And now there are five unique, possible configuration since because of the square (blue 2) a (+W1,-W2) is the same as a (-W1,+W2). These are (+W1,+W2), (+W1,0W2), (-W1,-W2), (-W1,0W2), (+W1,-W2)=(-W1,+W2)=(0W1,0W2), and these correspond to the first mass generation
quarks. The zero quark (the last 3 in the list), is not part of the Standard Model and has yet to be observed; perhaps it is dark matter.
So let's take a moment to consider what happens when you have exponentials inside exponentials. In the first iteration we have wavefunctions in the form,
)
. This is just a sinusoidal waveform; it propagates forever as does the EM field. However, in the second iteration, we have wavefunctions in the form,
])
. When
is small this would propagate like a normal wavefunction. But as
gets large, the wavefunction will no longer propagate as usual; it will break apart. Since this is meant to represent the propagation of the Weak bosons, this might be why these bosons do not propagate very far before breaking up. And in the third iteration, we have wavefunctions in the form,
]\})
. These don't propagate very well at all before breaking up. I have to wonder if they break up by becoming other quarks.
Before moving on, it should be noted that the real, complex, quaternion, and octonion numbers each form a normed division algebra. This means that multiplication and division have unique values. But as we iterate the Cayley-Dickson construction, we lose multiplication properties. I've already shown that the quaternions no longer commute under multiplication, thus neither do the octonions. The octonions lose the associative property of multiplication. And if we were to iterate again we would get the sedenions, but these do not form a division algebra since multiplication can form zero-divisors. This means we lose the property of multiplication altogether, which means we can no longer represent logical conjunction with multiplication. So my formalism stops with the octonions.
gluonsBut there might be a way of connecting
quarks without iterating to sedenions. If we were to iterated again, we would get,
^{1/2}}{e^{\kappa G{{(\ell {Q'_1} - \ell {Q'_2})}^2}}})
,
where
and is one of the many bases of the hypercomplex, sedenion numbers. But note that,
^{1/2}} = {e^{ - \kappa \pi /4}})
. So the above becomes,
}^2}}})
, which is equal to,
}^2} - \kappa \pi /4]}})
.
Here, the
are for the amplitudes of the
quarks that have exponentials inside exponentials inside an exponential, with leading factors included.
Now, if
^2} = - \pi /4)
, then there would be no
and no sedenions to consider. The connection between
quarks, the
gluons, requires that,
.
Or, the two
quarks in a
gluon must differ by a plus or minus factor of
, then
quarks can interact. Or if the
^2})
term were to be zero, then there would be no
quark terms connected to the sedenion term,
.This would be yet another type of
gluon to connect
quarks. So there are three possible values that a
gluon can have. Could this possibly account for the red, green, blue
color charge of a
gluon? This sedenion exercise only supplies a restriction in the octonions in order to make a connection. So the gluons belong to the octonion SU(3) symmetry. Here, there is no counting how many
gluons have various
electric charge. So these
gluons do not have an
electric charge.
What I think I've shown is that the normed division algebras of real, complex, quaternion, and octonion numbers can be used to explain the particles of the Standard Model and why they have the symmetries they have. Since any further iterations give zero divisors in the multiplication, the formalism stops with octonions. As I will try to show in another post, this may explain the reason that there are only 3 mass generations as well.
Feel free to comment as you please on substance or style. Thanks.
which is the same as 
, 
, and 
.
, 
, 
