The **Dixon Algebra** was originally put forward as a mathematical source
for the structure of the Standard Model of elementary particle physics.[1]
Beyond providing an explanation for the spinor fields and gauge symmetries of the Standard Model,
it also provides potential explanations for why our observable universe is dominantly matter
[2], and why there are three families of quarks and leptons
[3]. (A recent summary can be found here [4],
and here [5].)

The Dixon algebra is constructed from the Euclidean Hurwitz algebras. These are the real numbers, **R**,
the complex numbers, **C**, the quaternions, **H**, and the octonions, **O**.
They are, respectively, 1, 2, 4 and 8 dimensional over **R**, and they are
division algebras. There is a natural correspondence between these four algebras,
and the four parallelizable spheres, S^{0}, S^{1}, S^{3}, and S^{7}.

In short, define

**P** = **C**⊗**H**.

**T** = **C**⊗**H**⊗**O**.

**R**and**C**are both commutative [ab = ba] and associative [a(bc) = (ab)c].**C**and**P**are associative, but not commutative.**O**and**T**are neither associative nor commutative.

**P** and **T** are not division algebras. Their identities can be resolved into a collection
of orthogonal primitive idempotents (projection operators), 2 in the case of **P**, and 4 in the case of **T**.
Because **T** is not alternative, care must be taken to ensure that the 4 idempotents are projectors
(for example, each idempotent p must satisfy, p(pX) = pX for all X in **T**) [6].

Let **K** be any of the three division algebras, **C**, **H**, **O**,
or any of the combinations, **C**⊗**H**, **C**⊗**O**,
**H**⊗**O**, or **C**⊗**H**⊗**O**.
Let **K**_{L} be the algebra of all possible left actions of **K** on itself.
If **O** is part of **K**, then this requires nested actions (this idea is developed in elaborate detail
here). Each of these algebras, **K**_{L}, is isomorphic to an associative algebra
of square matrices over **R**, **C**, or **H**, and they each are isomorphic to ordinary Clifford algebras,
and, if **C** is involved, then they can be also be viewed as complex Clifford algebras.
In the case we view **K**_{L} as an ordinary Clifford algebra for some pseudo-orthogonal norm, the algebra
**K** plays the role of its spinor space. In this way these algebras can connect to physics.
(Complex Clifford algebras can also connect to physics [3].)

For example, the algebra of 2×2 matrices over **P**_{L} (denote **P**_{L}(2))
is isomorphic to the Dirac algebra (the complexification of the Clifford algebra of 1,3-spacetime), which is central to theoretical
descriptions of fermion particles, like the electron. Viewed in this way, **P**_{L}(2)) acts on
**P**^{2}, the space of 2×1 matrices over **P**. **P**^{2} is
the associated spinor space in this case, and it is where the fermion fields reside. A conventional Dirac spinor is 4×1
over **C**, which is 8-dimensional over **R**. **P**^{2} is 16-dimensional over
**R**, and it contains a doublet of Dirac spinors. This is a benefit of using **P** in this context,
for all the fermions in the Standard Model of elementary physics come in doublets. (These are associated with the Lie group, SU(2),
played in this context by the unit elements of **P**_{R} (**P** acting on **P**^{2}
from the right).

Generalizing to the case of the Dixon algebra, **T**, the algebra **T**_{L}(2) is isomorphic
to the complexification of the Clifford algebra of 1,9-spacetime (so, 6 extra space dimensions), and it acts on the spinor space,
**T**^{2}. Whereas **P**^{2} provides a description of doublets of Standard Model leptons,
**T**^{2} contains a full Standard Model family of leptons and quarks, and its antifamily.
Fundamentally, all of this is just a simple generalization of ordinary Dirac theory.

[1]
https://www.springer.com/gp/book/9780792328902

[2]
https://cmuc.karlin.mff.cuni.cz/pdf/cmuc1403/dixong.pdf

[3]
https://link.springer.com/article/10.1007%2FJHEP10(2014)046

[4]
https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/

[5]
https://www.wired.com/story/the-peculiar-math-that-could-underlie-the-laws-of-nature/

[6]
http://www.7stones.com/Homepage/octoIII.pdf

Wikipedia links are not included here. Although this is in fact not worthy of inclusion
into Wikipedia, I rather like it, and I don't want it to disappear from the universe entirely.
And that name - "Dixon Algebra" - is rather catchy. I find it strangely attractive.

Most of my work on **T** = **C**⊗**H**⊗**O**
has started from mathematics, then making obvious connections to physics. Here I'll start with physics
and use accepted ideas to motivate the use of **T**.
We start with the Dirac algebra and Dirac spinors, key elements of the Standard Model. The Dirac algebra is
**C**(4). It is the complexification of the Clifford algebra of ordinary 1,3-spacetime. Together
with spacetime differential operators, we get the Dirac operator, which acts on 4×1 column matrices
over **C**, each component of which is a complex function of spacetime. Schematically:

This column spinor could represent an electron field, or that of its associated neutrino. These two particle fields are part of a doublet with respect to the SU(2) part of the standard symmetry. So we get a new schematic:

In this case, the 4×4 matrix on the left represents the Dirac operator;
the 4×2 matrix to its right is the doublet of Dirac spinor fields, one column
for the electron, and one for its neutrino; and the 2×2 matrix on the right
is a element of U(1)×SU(2), represented in **C**(2). The electron
and neutrino parts of the 4×2 spinor doublet can be projected from the doublet
with the projection operators,

1 0

0 0

and
0 0

0 0

0 1

,
these two matrices forming a (non-unique) resolution of the identity of 0 1

Without going into details, the Dirac algebra has an element, γ_{5}, with which we
can construct other projection operators, most simply taking the forms

1 0 0 0

0 1 0 0

0 0 0 0

0 0 0 0

and
0 1 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 1 0

0 0 0 1

. 0 0 0 0

0 0 1 0

0 0 0 1

Acting from the left on the spinor doublet, these project the so-called left and right components of the spinors, which are observed to behave differently in physical interactions. At this point, to emphasize these distinct physical bits, we modify the schematic above as so:

The point of this exercise is to highlight the fact that at this level the physics is describable in terms of a bunch of 2×2 complex matrices. This is the Pauli algebra. And it is isomorphic to

**P** = **C**⊗**H**.

So, to reiterate, on the left we have the Dirac algebra, now represented by 2×2 matrices
over **P**. This acts from the left on the Dirac spinor doublet, now represented
as 2×1 matrices over **P**. And we top it off with U(1)×SU(2) acting
on the spinor doublet from the right, now represented in **P** alone. So,

Dirac algebra: **P**_{L}(2) .

Dirac spinor doublet: **P**^{2} .

U(1)×SU(2) in **P**_{R} .

I don't know how much more obvious the next step can be made. We have well-known and much loved physics represented using

**P** = **C**⊗**H**

But the automorphism group of

Dirac algebra of 1,9-spacetime: **T**_{L}(2) .

Dirac hyper-spinor doublet: **T**^{2} .

U(1)×SU(2) in **P**_{R} .

So, where's this SU(3)? Well, go read my books and papers. In short, the resolution of the identity
of **T** requires that we choose a direction in the imaginary part of **O**.
This picks out an SU(3) from G_{2}. In addition, our spacetime implied by the enhanced Clifford algebra,
**T**_{L}(2), has 6 extra space dimensions, and an associated SO(6) part of the overall SO(1,9).
The intersection of this SO(6) with G_{2} is the same SU(3).

The resolution of the identity of **T** consists of 4 orthogonal projection operators. With
these we can project from the hyper-spinor,
**T**^{2}, a complete family and anti-family of Dirac spinor fields for the first
generation of leptons and quarks, and their associated anti-particle spinors. In particular, we can
project down to the individual lepton Dirac spinor parts (and to their left and right parts, if need be),
but only to the SU(3) multiplet level (triplet or anti-triplet) for the quark sector. (This same process
projects from the 1,9-Dirac operator a 1,3-Dirac operator (one for matter, one for antimatter)). Therefore, we can
project down only to those bits that have the same mass: two lepton masses, and two quark masses (each
Dirac spinor in a quark triplet - i.e., each color of a given quark type, up or down - having the same
mass). These differences in the way SU(2) and SU(3) are manifested have been shown to be related to their
differences re chirality and symmetry breaking.

And, I would suggest, it is at this point that QFT notions should be incorporated - not before.
First you build the framework, then you add the walls. But of course, this will never be done,
because humanity is ruled by NPCs. And I'm too old and tired. Just bugger off. And get off of my lawn.

What's different? As a theoretical physicist interested in models designed to represent our
physical reality, you're likely familiar with the particle zoo. In particular, efforts to explain the zoo
of fermion fields (leptons and quarks) using (usually) some big algebraic structure, like a Lie group, rely
on its representations (often) to make sense of it all. That is, you've got your collection of fermion fields,
and you want to pin them to bits of your algebraic structure in some hopefully natural way. Do I know what I'm talking about?
Doubtful. I find most conventional notions boring, and I do not dig deep into things that bore me.

Anyway, the single algebraic bit I start with in the original 1-generation model is**T**^{2}.
This, it turns out, is not something to which to pin a generation of fermions and antifermions, it is that
generation. It is a single fermion field - hyper field. You get Lie groups, and multiplets within the hyperspinor,
falling out of the broader associated algebraic context to which **T** gives rise.

Ok, go away now. This is getting tedious. And don't follow me.

Anyway, the single algebraic bit I start with in the original 1-generation model is

Ok, go away now. This is getting tedious. And don't follow me.