# Single-pixel diffuser camera

Baolei Liu  Fan Wang<EMAIL_ADDRESS>Chaohao Chen  David McGloin
<EMAIL_ADDRESS>School of Electrical and Data Engineering, Faculty of
Engineering and IT, University of Technology Sydney, NSW 2007, Australia

(April 30, 2021)

###### Abstract

We present a compact, diffuser assisted, single-pixel computational camera. A
rotating ground glass diffuser is adopted, in preference to a commonly used
digital micro-mirror device (DMD), to encode a two-dimensional (2D) image into
single-pixel signals. We retrieve images with an 8.8% sampling ratio after the
calibration of the pseudo-random pattern of the diffuser under incoherent
illumination. Furthermore, we demonstrate hyperspectral imaging with line
array detection by adding a diffraction grating. The implementation results in
a cost-effective single-pixel camera for high-dimensional imaging, with
potential for imaging in non-visible wavebands.

††preprint: APS/123-QED

##  I Introduction

By encoding depth or spectral information, imaging assisted by a diffuser or
thin scatterer can retrieve a three-dimensional (3D) data cube rather than the
conventional 2D image obtained from an optical sensor. The concept of a
’diffuser camera’ (’DiffuserCam’) [1] has gained a lot of interest, due to its
compact and cost-effective approach. Retrieval of a three-dimensional (3D),
multi-view [2], multispectral [3] or hyperspectral image [4] via single-shot
computational imaging has recently been demonstrated. However, such approaches
are challenging for more esoteric wavelength bands, for instance, x-ray or
terahertz imaging. As the name suggests, single-pixel imaging [5, 6, 7, 8, 9],
produces images without the need for a 2D detector, making use of structured
detection or illumination of the object to computationally derive an image. A
such, single-pixel approaches are of interest in offering alternatives to
conventional imaging, both for applications in the visible, but also as a low-
cost alternative in regimes such as x-ray [10, 11], infrared [12] and
terahertz band [13], and even imaging atoms [14]. Additionally, single-pixel
approaches help to inform high-performance imaging techniques, for example, 3D
depth, time-resolved or multispectral imaging, in which CCD based systems
would be complicated or expensive to implement.

Typically, a single-pixel camera uses a digital micromirror device (DMD),
which is placed in the image plane, to modulate the image of objects with
different 2D structured sampling patterns. The single-pixel detector then
measures the corresponding total light intensity. By correlating the 1
dimensional (1D) single-pixel signals with the modulation patterns,
reconstruction algorithms such as compressed sensing can rebuild the 2D image.
Alternatively, the DMD can modulate the illumination of the object, an
approach commonly called computational ghost imaging (CGI) [15, 6, 16]. The
DMD can be replaced by a liquid crystal spatial light modulator [17], rotating
ground-glass diffusers [18] or LED arrays [19, 20]. Image retrieval can be
achieved using the same algorithms as in the structured detection scheme.

In cases where only passive imaging is needed, the structured detection scheme
offers the benefit of a more compact and cheaper imaging system, due to the
lack of a need for light sources. However, when using wavelengths such as
x-ray, conventional DMDs cannot be used for modulation. Thus, x-ray single-
pixel imaging was realized using the CGI scheme, using, for example, a
monochromatic x-ray beam passing through a slit array and a moving porous gold
film [10] or using polychromatic x-rays and a sheet of rotating sandpaper [11]
to generate pseudothermal illumination speckle patterns.

Here in this work, we present a single-pixel diffuser camera (SP-DiffuserCam)
that uses a low-cost rotating ground glass diffuser instead of a DMD for 2D
structured detection modulation with incoherent light. We show that the random
and fixed patterns of a simple diffuser placed in the image plane can serve as
2D light intensity modulation in single-pixel imaging. We refer to this
concept as a passive version of classical GI, but with no need for
simultaneous measurements of reference patterns. We show this concept is
readily extendable for hyperspectral imaging.

##  II Principle

Figure 1 presents the SP-DiffuserCam concept. The first procedure is a
calibration process to map the speckle-liked patterns of the diffuser, where
the intensity distributions $P(x,y,\theta)$ for each angle
$\theta(0^{\circ}\leq\theta\leq 360^{\circ})$ are acquired sequentially by
rotating the diffuser. Note that this could be achieved using an laterally
moving stage, but here we only consider rotation for system compactness. The
second step is the temporal single-pixel measurement. An object $O(x,y)$ is
illuminated by the same incoherent light source and imaged on the diffuser
plane by a lens. The transmitted light intensity passing through the object
$I(\theta)$, measured by a single-pixel detector under another repeated
rotation, is simply the integration of the light on the detector plane,

$I(\theta)=\int P(x,y,\theta)O(x,y)\,dx\,dy.$ (1)

We then can acquire the object image $R(x,y)$ by using the differential
correlation approach [21],

$R(x,y)=\left\langle I(\theta)P(x,y,\theta)\right\rangle-\frac{\left\langle
I(\theta)\right\rangle}{\left\langle I_{P}(\theta)\right\rangle}\left\langle
I_{P}(\theta)P(x,y,\theta)\right\rangle,$ (2)

where $\left\langle\right\rangle$ denotes the ensemble average over the
distribution of patterns and $I_{P}(\theta)=\int P(x,y,\theta)\,dx\,dy$
denotes the weights of the patterns.

Figure 1: Principle of SP DiffuseCam. (a) Initially, a calibration procedure
is conducted to acquire the structured patterns of the diffuser corresponding
to different angular positions. (b) Once calibration is finished, the 2D image
is retrieved by correlating the encoding patterns and the single-pixel
temporal signal in repeated rotation, during which the object is imaged on the
diffuser plane. (c) With a diffraction grating and line array detector, a
hyperspectral imaging setup can be established with single-pixel signals from
different wavelengths.

For hyperspectral imaging, the detection intensity $I(\theta,\lambda)$ is
simply the single-pixel signal corresponding to different wavelengths,
measured using a line detector placed after a grating. Thus, the spectral
image data cube can be reconstructed as

$R(x,y,\lambda)=\left\langle
I(\theta,\lambda)P(x,y,\theta)\right\rangle-\frac{\left\langle
I(\theta,\lambda)\right\rangle}{\left\langle
I_{P}(\theta)\right\rangle}\left\langle
I_{P}(\theta)P(x,y,\theta)\right\rangle,$ (3)

by only replacing the $I(\theta)$ with $I(\theta,\lambda)$ in Equation 2 and
without characterizing the surface roughness for the other wavelengths.

##  III Experiment

###  III.1 Single color SP-DiffuserCam

A simple proof-of-concept experimental realization of the approach is
presented in Figure 2. Figure 2 (a) depicts the optical configuration of a
single-color SP-DiffuserCam, where the light source is a monochromatic 530 nm
LED with a bandwidth of 33 nm (M530D2, Thorlabs). The object is imaged on the
diffuser plane, illuminated by the collimated light. The ground glass diffuser
($\varnothing$ = 24mm, DG10-120-MD, Thorlabs) is mounted on a motorized
rotation stage (PRM1/MZ8, Thorlabs), which has a 25 ∘/second rotation
velocity. A silicon amplified photodetector (PDA100A2, Thorlabs) is used to
measure the intensity fluctuations. For calibration, the speckle patterns
$P(x,y,\theta)$ of the diffuser are recorded by a camera (panda 4.2 bi, PCO
AG) through the same lens (f=50mm) used for the photodetector to keep the same
numerical aperture; the exposure time of 4ms is the same for both detectors. A
motor controller (KDC101, Thorlabs) and a low-cost DAQ device (USB-6002,
National Instruments) are used to synchronize the rotator and the detector or
the camera.

Figure 2: Demonstration of SP-DiffuserCam for monochrome imaging. (a)
Experimental setup. The object is imaged on the rotating diffuser plane, and
then the temporal signal is recorded by a free-space amplified photodetector
(PDA). (b) The retrieved images under increasing sampling ratios, compared to
the original object (left column). Scale bar: 3mm.

To ensure our patterns are sufficiently different, we use the edge of the
diffuser region(about $d=6mm$ from the center of the diffuser) as the image
plane area. Due to the small diameter of the diffuser, however, the
neighbouring patterns still have similar areas along the rotation direction.
The objects (Figure 2(b) left) are 3D printed transmission masks of ’U’, ’T’,
’S’ with a thickness of 2mm and the size about 3mm. The retrieved $64\times
64$ pixel images with sampling ratios of 2.2%, 4.4%, and 8.8% are shown in
Figure 2(b), corresponding to a minimum acquisition time of 14.4s, which is
limited by the maximum rotation velocity. Here we also use a simple block-
matching and the collaborative filtering algorithm [22] for noise suppression,
which takes about 0.02 seconds. Outlines of objects emerge at 2.2%, with image
quality increasing with improved sampling ratios to 8.8%. Further increasing
the number of scattering patterns would not improve the reconstruction
quality. The first reason is the ’imperfect’ reference patterns with overlap
regions between neighbouring ones. The second one is that a higher sampling
ratio leads to a finer angle variation of measurement within one cycle of
rotation, and increased overlap areas of the reference patterns. Thus, the
above challenges induce a phenomenon that the spatial information along the
vertical direction of rotation is reconstructed better than the parallel ones.
This is evident in Figure 2(b), in the top-right corner of ’U’, the bottom of
’T’ and the emerging noisy pixels in the left of ’S’. Note the rotation
directions are clockwise the three masks, with the sampling area in the lower
half of the circular diffuser.

###  III.2 Comparison of different diffusers

Figure 3: Comparison of diffusers with coarse and fine scattering: (a) Example
spatial images of 120 grit and 220 grit diffusers. Scale bar: 0.5mm. (b) The
2D correlation coefficients of corresponding two groups of diffuser patterns
range from $\theta$ = 0∘,1∘,$\cdots$, 360∘ are depicted in the bottom row.

The performance of our setup is largely dependent on the choice of diffuser.
Here we compare the 120-grit diffuser, which is used in Figure 2, and a
220-grit diffuser ($\varnothing$ = 24mm, DG10-220-MD, Thorlabs), which has
smaller grain size on the polished surface. Figure 3(a) shows typical
transmission images of the two diffusers, in which the former one shows a
higher contrast than the later one, due to the coarser polishing. We calculate
the correlation coefficient $C(I_{\theta 1},I_{\theta 2})$ of the two groups
of diffuser patterns, for a quantitative study, as defined by

$C(I_{\theta 1},I_{\theta 2})=\frac{\int(I_{\theta 1}-\bar{I}_{\theta
1})(I_{\theta 2}-\bar{I}_{\theta 2})\,dx\,dy}{\sqrt{(\int(I_{\theta
1}-\bar{I}_{\theta 1})^{2}\,dx\,dy)(\int(I_{\theta 2}-\bar{I}_{\theta
2})^{2}\,dx\,dy})},$ (4)

where $\theta$ = 0∘,1∘,$\cdots$, 360∘, $\bar{I}_{\theta 1}$ and
$\bar{I}_{\theta 2}$ denote the average values for two arbitrary patterns
$I_{\theta 1}$ and $I_{\theta 2}$, respectively. The mean correlation
coefficients between different angles for the 120 grit diffuser is
$C(I_{120})=0.55$, while for the 220 grit one we find $C(I_{220})=0.87$. Note
that these coefficients are based on $1024\times 1024$ pixels reference
patterns, and they would be further higher in the condition of Figure 3(b),
where the patterns are binned to $64\times 64$ matrices with lower contrasts.
A higher correlation means an improved intrinsic coherence between the
sampling basis and leads to a lower detection efficiency of the spatial
images. Key to contemporary single-pixel imaging is the use of high-efficiency
orthonormal patterns, such as Hadamard [12] or Fourier basis [7]. However,
since we adopt a commercial diffuser, the sampling patterns in our setup are
actually pseudo-random grayscale matrices, which are more related to classical
GI using laser and dynamic diffuser induced speckle patterns [18]. According
to a study of the influence of speckle size in GI [23], an optimal speckle
size exists in the range where the speckle size is comparable to the feature
size of the object. This is the reason that we choose to use the 120-grit
diffuser, as it is our coarsest diffuser and the one most approaches the
feature size of the masks (about $0.4mm$). Future work could focus on the
optimization of the diffuser or an integrated mini camera.

###  III.3 Hyperspectral SP-DiffuserCam

Figure 4: Demonstration of SP-DiffuserCam for hyperspectral imaging. (a)
Experimental setup. A white LED is used for illumination. A low-cost
transmission grating is used for light dispersion after the structured
modulation of a dynamic diffuser. (b) The retrieved images in the blue band
($461\pm 35$ nm) and red band ($627\pm 10$ nm) as well as their overlap,
compared with the original object. (c) Reconstructed spectrum of points A and
B in (b), compared with the conventional spectrometer measured ones. (d) The
reconstructed spectral images and data cube. The pseudo-colored spectral
images in (b) and (d) are converted from the spectra according to CIE color
matching functions.

One of the advantages of the SP-DiffuserCam is that it is easy to integrate
other functions within the platform, such as the hyperspectral imaging shown
in Figure 4. To do this, we can directly use the pre-recorded reference
patterns without additional calibration for specific wavelengths. Figure 4 (a)
illustrates the optical setup of the hyperspectral SP-DiffuserCam. Here we use
a cold white LED (MCWHLP1, Thorlabs) for a broad spectrum. Different from the
monochrome setup, a low-cost diffraction grating slide (1,000 Line/mm, Rainbow
Symphony, $0.40), is used to disperse the incident light. The object image on
the diffuser passes through the grating after the imaging lens (f=50mm) and is
focused onto a line array detector via the collection lens (f=35mm). Here a
compact COMS camera (acA1920-150um, Basler) serves as line-detector by binning
the pixels to a $16\times 1$ array in a sensor area of $9.5mm\times 3.3mm$. To
keep the same numerical aperture with the single-pixel measurement, we use the
same two lenses system for calibration.

The 1D spectral intensity at each degree (exposure time: 4ms) are recorded in
a 2D matrix $I(\theta,\lambda)$. Then this matrix is converted to a spectral
data cube $R(x,y,\lambda)$ by the reconstruction Equation 3. Here we apply
standardized [International Commission on Illumination (CIE)] color-matching
functions to the spectral cube to produce the pseudo-colored images to make
consistent with the original object (made by sticking the color filters to the
’T’ mask, a blue filter for the ’-’ and a red filter for the ’$|$’ of ’T’).
Figure 4 (b) shows the retrieved spectral image in the blue band ($461\pm
35nm$) and the red band ($627\pm 10nm$). Compared with the object photograph
in the lower right corner, the middle, overlap area in the reconstruction has
much lower intensities, due to refraction between the junction of the two
filters. The reconstructed spectra of two selected areas in Figure 4 (b) show
a similar trend when compared to measurements using a conventional
spectrometer (UV-2401 PC UV-VIS, Shimadzu) of the same two color filters.
Finally, we show the 3D spatio-spectral data cube in Figure 4 (d), as well as
the spatial mapping image at single wavebands ranging from 426nm to 637 nm.

##  IV conclusion

In conclusion, we have reported a passive single-pixel camera that employs a
rotating diffuser as the spatial modulator in the image plane and uses
incoherent light, for 2D imaging from 1D temporal signals. Note that classical
pseudothermal ghost imaging [18, 15, 21, 24] has also used a rotating ground-
glass diffuser illuminated by a laser beam to generate dynamic speckles, and
then split the beam into a reference and an object beam. After that, a CCD is
used to measure the diffracted speckles, and a single-pixel bucket detector is
used to measure the signals from the object, simultaneously, in which both CCD
and the object keep the same distances from the diffuser. As a comparison, our
setup uses pre-recorded reference patterns and does not need to measure these
in experiments. Another difference is the intensity distribution on the
surface of the diffuser is used here instead of the laser interference
speckles after the diffuser. In fact, our work can be considered as the
passive version of x-ray ghost imaging that uses pre-recorded patterns and an
incoherent source [10, 11]. However, we directly use the patterns of the
diffuser as the reference patterns, not the diffracted speckles after
propagation for a distance as in x-ray ghost imaging. Thus, our work would
help to compact these imaging systems. We also demonstrate our concept is
readily extended to achieve low-cost hyperspectral imaging, for 3D spatio-
spectral image retrieval with temporally 1D spectral signals. Furthermore, the
SP-DiffuserCam can be explored with coherent light, or other forms of imaging
techniques such as time-resolved imaging [25] using fast response detectors
and phase imaging [9, 24] by adopting phase engineered diffusers. We,
therefore, anticipate that this work will open opportunities for developing
cost-effective integrated single-pixel cameras, especially in exotic wavebands
and imaging under ultra-low illumination.

###### Acknowledgements.

Funding: Australian Research Council DECRA fellowship (DE200100074, F.W.),
Australian Research Council Discovery Project (DP190101058, F.W.). The authors
acknowledge financial support from the UTS Faculty of Engineering and IT, and
China Scholarship Council (B.L.: No.201706020170).

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# An Algebraic Roadmap of Particle Theories
Part I: General construction

N. Furey
Iris Adlershof, Humboldt-Universität zu Berlin,
Zum Grossen Windkanal 2, Berlin, 12489
<EMAIL_ADDRESS>
HU-EP-23/64

###### Abstract

It is a curious fact that our most well-studied BSM theories can inform us
about the Standard Model itself. Just over a decade ago, [1], Baez and Huerta
explained how the Standard Model’s gauge group can be seen to coincide with
the intersection of Georgi and Glashow’s SU(5), and Pati and Salam’s
$SU(4)\times SU(2)\times SU(2)$, inside Spin(10). More recently, [2], [3], it
was shown how (up to a factor of B-L) the Spin(10) model, the Pati-Salam
model, the Left-Right Symmetric model, the Standard Model pre-Higgs mechanism,
and the Standard Model post-Higgs mechanism are interlinked via a sequence of
division algebraic reflections. Between these three papers, [1], [2], [3], are
six theories that have been studied extensively by particle physicists. In
this article, we demonstrate how the full set of six familiar particle models
may be laid out in one connected algebraic particle roadmap.

The inclusion of a quaternionic reflection within the network further
differentiates $W^{\pm}$ bosons from the $Z^{0}$ boson in comparison to the
Standard Model. It may introduce subtle new considerations for the
phenomenology of electroweak symmetry breaking.

###### pacs:

112.10.Dm, 2.60.Rc, 12.38.-t, 02.10.Hh, 12.90.+b

##  I Introduction

You shall know a word by the company it keeps.

\- J.R. Firth, 1957.

If Quantum Mechanics, [4] and Special Relativity, [5], have taught us
anything, it is that objects are often best understood not in isolation, but
rather, in relation to their peers. Perhaps the Standard Model of particle
physics is no exception.

In this article, we identify a set of repeating connections, in series and in
parallel, between a number of neighbouring particle models. Among these nine
models, six are well-known. We pinpoint the post-Higgs Standard Model at the
cluster’s most highly constrained corner.

Figure 1:  Simplified preview of an algebraic particle roadmap. The detailed
version appears in Figure (6) at the end of this article.

For a glance ahead, see Figure (1). There, the network we are about to
assemble will be built up using the natural operator spaces of the division
algebras $\mathbb{R},\mathbb{C},\mathbb{H},$ and $\mathbb{O}.$ These operator
spaces (multiplication algebras) are readily described by Clifford algebras.

Since at least as early as the 1970s, physicists and mathematicians have been
identifying Clifford algebraic and division algebraic patterns within the
architecture of elementary particles. Now half a century ago, Günaydin and
Gürsey proposed a quark model based on octonions, [6]. Shortly thereafter came
a series of papers by Casalbuoni et al who proposed particle models based on a
large variety of Clifford algebras, [7], [8], [9]. Since these days, many
authors have invested a great deal of time and effort to the endeavour. In
what is apologetically far from an exhaustive list, we point out the Clifford
algebraic work of Trayling and Baylis (Cl(0,7)) [10], Barrett (via NCG) [11],
Zenczykowski ($\mathbb{C}l(6)$) [12], Connes et al (via NCG) [13], Gording and
Schmidt-May ($\mathbb{C}l(6)$) [14], Todorov ($\mathbb{C}l(10)$) [15],
Borštnik (Cl(p,q)) [16], the division algebraic work Adler ($\mathbb{H}$)
[17], De Leo ($\mathbb{H}$) [18], Toppan et al ($\mathbb{H},\mathbb{O}$) [19],
Baez ($\mathbb{R},\mathbb{C},\mathbb{H}$) [20], Duff et al
($\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$) [21], Hughes ($\mathbb{O}$)
[22], Catto et al ($\mathbb{O}$) [23], Gresnigt ($\mathbb{O}$) [24],
Asselmeyer-Maluga ($\mathbb{H},\mathbb{O}$) [25], Bolokhov ($\mathbb{H}$)
[26], Vaibhav and Singh (split-$\mathbb{H}$ and split-$\mathbb{O}$) [27],
Boyle ($J_{3}(\mathbb{O})$) [28], Jackson ($\mathbb{O}$) [29], Hunt
($\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$) [30], Lasenby ($\mathbb{O}$)
[31], Manogue et al ($E_{8}$) [32], [33], Hun Jang, (Hypercomplex) [35], Hiley
(split-$\mathbb{H}$) [36], and most recently Penrose (split-$\mathbb{O}$)
[34], the $\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$ work
of Dixon, [37], Perelman, [38], Chester et al, [39], Köplinger, [40].

This paper is the first of a series; see also [41], [42]. Although the model
described here was non-trivial to find, readers may appreciate that a large
number of its results can be confirmed easily.

##  II In context

The particle roadmap introduced in this article extends directly from previous
findings of Baez and Huerta [1], and Furey and Hughes [2], [3]. We summarize
these earlier findings here. This section provides the historical background
for this article, but is not strictly necessary to understand its results.

###  II.1 Algebra of grand unified theories

It is a surprisingly little-known fact amongst particle physicists that many
of our most well-studied theories are interrelated.

The birth of the Spin(10) model in the mid-1970s came only a few hours before
that of Georgi and Glashow’s SU(5) model [43], [44]. On its own, Georgi and
Glashow’s model posits a seemingly ad hoc fermionic particle content as the
$\mathbf{10}\oplus\mathbf{5^{*}}$ of SU(5). Why, one might wonder, this
curious combination of irreducible representations?

However, embedding these irreps inside the proposed $\mathbf{16}$ of Spin(10)
justifies the representation structure, while adding in a sterile neutrino in
the form of an $SU(5)$ singlet.

$\begin{array}[]{ccc}\mathfrak{so}(10)&\supset&\mathfrak{su}(5)\vspace{2mm}\\\
\mathbf{16}&\hskip
28.45274pt\mapsto&\mathbf{10}\oplus\mathbf{5^{*}}\oplus\mathbf{1}\end{array}$
(1)

Hence the Spin(10) model offers guidance for its younger SU(5) sibling.

In the same era, another (partially) unified theory was constructed by Pati
and Salam, [45]. Their intention was to capitalize on an observed pattern-
matching between quarks and leptons. Via the group $SU(4)\times SU(2)\times
SU(2)$, the Standard Model’s three “red, green, blue” quarks colours were
augmented to include a fourth “lilac” lepton colour. Furthermore, a symmetry
between left- and right-handed particles was conjectured. Curiously enough,
Pati and Salam’s model also fits neatly into the $\mathbf{16}$ of Spin(10):

$\begin{array}[]{ccc}\mathfrak{so}(10)&\supset&\mathfrak{su}(4)\oplus\mathfrak{su}(2)\oplus\mathfrak{su}(2)\vspace{2mm}\\\
\mathbf{16}&\hskip
28.45274pt\mapsto&\left(\mathbf{4},\mathbf{2},\mathbf{1}\right)\oplus\left(\mathbf{4^{*}},\mathbf{1},\mathbf{2}\right)\end{array}$
(2)

Again, we find that the Spin(10) model justifies for the Pati-Salam model an
otherwise arbitrary choice in fermion representations.

Now, more surprising than the kinship between the Spin(10) and SU(5) models,
or the kinship between Spin(10) and Pati-Salam models, is an unexpected
kinship between SU(5), Pati-Salam, and the Standard Model. That is, in [1],
Baez and Huerta report that the Standard Model’s gauge group coincides exactly
with the intersection between SU(5) and Pati-Salam symmetries. The following
Figure (2) relates these four well-studied particle models.

Figure 2:  Baez and Huerta, [1], explain how Standard Model symmetries (SM)
result from the intersection of Georgi and Glashow’s SU(5) and the Pati-Salam
(PS) symmetries SU(4)$\times$SU(2)$\times$SU(2). Spin(10) may be seen to break
to PS by preserving the splitting
$\mathbf{10}\mapsto\mathbf{6}\oplus\mathbf{4}$ of SO(10)’s defining
representation. Alternatively, Spin(10) may be seen to break to SU(5) by
preserving a grading and volume form of the spinor space
$\Lambda\mathbb{C}^{5}$.

But what could prompt the symmetries to break in this way? Reference [1] lists
three independent conditions:

1\. A requirement that the defining $\mathbf{10}$ of $\mathfrak{so}$(10)
splits into $\mathbf{6}\oplus\mathbf{4}$, (this breaks
$\mathfrak{so}(10)\mapsto\mathfrak{su}(4)\oplus\mathfrak{su}(2)\oplus\mathfrak{su}(2)$),

2\. The preservation of a $\mathbb{Z}$-grading on the spinorial
$\Lambda\mathbb{C}^{5}\simeq\mathbf{16}\oplus\mathbf{16^{*}}$ of
$\mathfrak{so}$(10), (this breaks $\mathfrak{so}(10)\mapsto\mathfrak{u}(5)$),

3\. The invariance of a volume form in the spinorial
$\Lambda\mathbb{C}^{5}\simeq\mathbf{16}\oplus\mathbf{16^{*}}$ of
$\mathfrak{so}$(10), (this breaks $\mathfrak{u}(5)\mapsto\mathfrak{su}(5)$).

Here, $\Lambda\mathbb{C}^{5}$ denotes the exterior algebra generated by
vectors of 5 complex dimensions. See also footnotes in [9].

From where do these conditions arise? Baez and Huerta encourage readers to
decipher the meaning behind these mysterious constraints.

###  II.2 Division algebraic symmetry breaking

That the first constraint,

$\begin{array}[]{ccc}\mathfrak{so}(10)&\mapsto&\mathfrak{so}(6)\oplus\mathfrak{so}(4)\vspace{2mm}\\\
\mathbf{10}&\hskip 28.45274pt\mapsto&\mathbf{6}\oplus\mathbf{4}\end{array}$
(3)

derives from the octonions was first discovered by Furey and Hughes.

These results were distributed widely amongst colleagues on the 16th of
February, 2021, [2], [3], and discussed extensively with some of them, [46].
In particular, Kirill Krasnov contributed to our efforts by following suit and
reproducing our results, thus confirming them.

This octonionic structure may be seen to be responsible for sending the
Spin(10) model to the Pati-Salam model, and the SU(5) model to the Standard
Model.

As will be explained in detail later on in this text, the Spin(10) $\mapsto$
Pati-Salam and SU(5) $\mapsto$ Standard Model transitions occur upon the
requirement that the symmetries be invariant under a certain type of
octonionic reflection. Please see Figure (3). Going beyond, it was found in
[2], [3] that invariance under a complementary octonionic reflection
furthermore sends this Pati-Salam model
($\mathfrak{so}(6)\oplus\mathfrak{so}(4)=\mathfrak{su}(4)\oplus\mathfrak{su}(2)\oplus\mathfrak{su}(2)$)
to the Left-Right Symmetric model
($\mathfrak{su}(3)\oplus\mathfrak{su}(2)\oplus\mathfrak{su}(2)\oplus\mathfrak{u}(1)$).
Then, invariance under a quaternionic reflection sends the Left-Right
Symmetric model to the Standard Model, augmented with a B-L symmetry
($\mathfrak{su}(3)\oplus\mathfrak{su}(2)\oplus\mathfrak{u}(1)\oplus\mathfrak{u}(1)$).
Finally, [2], invariance under a complex conjugate reflection sends the pre-
Higgs Standard Model
($\mathfrak{su}(3)\oplus\mathfrak{su}(2)\oplus\mathfrak{u}(1)$) to the post-
Higgs Standard Model ($\mathfrak{su}(3)\oplus\mathfrak{u}(1)$).

In short, invariance under $\mathbb{O},$ $\mathbb{H},$ and $\mathbb{C}$
reflections links five well-established particle models in a cascade of
breaking symmetries.

Figure 3:  References [2], [3] introduced a cascade of breaking symmetries.
The Spin(10) model breaks to the Pati-Salam model, (PS), via the octonions.
The Pati-Salam model breaks to the Left-Right Symmetric model, (LR), again via
the octonions. The Left-Right Symmetric model breaks to the Standard model +
B-L, (SM+), via the quaternions. The Standard model + B-L breaks to the
Standard Model’s unbroken symmetries, together with B-L, (UB+), via the
complex numbers. Two open questions remained in this model: (1) Why the
lingering B-L symmetry? (2) Could there be a more natural way to introduce
chirality in the quaternionic “$\mathbb{H}_{\textup{R}}$” step described in
[3]?

However, as with [1], some open questions remain. Namely, in the quaternionic
Left-Right Symmetric $\mapsto$ Standard Model step,

1\. An additional B-L symmetry persisted,

2\. Chirality was introduced by hand.

Could it be possible to tame these unruly features?

In this article, we demonstrate that both of these issues may be addressed
simultaneously. Furthermore, we offer a concise answer to those challenges
posed by Baez and Huerta as to where from their mysterious conditions may be
seen to originate.

As will be seen, the overarching theme may be described as the implementation
of simultaneous group actions on the fermionic representations. These
simultaneous group actions, or more precisely, Lie algebra actions, allow us
to merge the diagrams of Figures 2 and 3. The result (Figure 6) is a detailed
roadmap of particle models.

In order to keep the discussion as concise as possible, we will work on the
level of Lie algebras. Those interested in group theoretic details are
encouraged to consult [1].

###  II.3 A consistency condition

It is hard not to notice that, when augmented by a sterile neutrino, the
fermionic representations of the Standard Model materialize as broken pieces
of the $\mathbf{16}$ spinor representation of Spin(10). However, the
$\mathfrak{su}(3)\oplus\mathfrak{su}(2)\oplus\mathfrak{u}(1)$ Standard Model
symmetries acting on this representation space are unmistakably unitary. Could
there be more than one type of symmetry constraint working in concert?

On a not-obviously related front, let us point out the minimal left ideal
construction of spinors. As will be explained in detail shortly, such spinors
may be constructed from the $2n$ generating $\Gamma$-matrices of certain
$\mathbb{C}l(2n)$ Clifford algebras. Collectively, linear combinations of
these $\Gamma$-matrices form the defining $2n$ vector representation of the
Clifford algebra’s $\mathfrak{so}$(2n) symmetries. But this then begs the
question: If the spinor is constructed directly from these vectors, should it
not also transform under a group action induced by them?

In this article, we entertain the idea that Standard Model fermions should
simultaneously transform under both the known $\mathfrak{spin}(2n)$ spinor
action, and also under the known $\mathfrak{so}(2n)$ multivector action. This
consistency condition reduces $\mathfrak{spin}(2n)$ to $\mathfrak{su}(n)$. In
one stroke, this single requirement addresses both the chirality and B-L
issues plaguing [3]. It also fuses Baez and Huerta’s conditions 2. and 3. in
Section II.1 into a single constraint. It may be seen to be algebraically
equivalent to a statement made in a footnote of [9], where a volume form is
fixed. Here, we derive the result instead as a consistency condition in the
construction of fermions as multivectors.

The main results of these papers were first made public in an abstract
circulated for the Perimeter Institute Octonions and Standard Model conference
in 2021. We include the abstract here: [47].

###  II.4 Triality’s example

As an aside, we mention that the idea of simultaneous group actions is not new
to representation theory, [22], [33]. A phenomenon known as octonionic
triality has three copies of $\mathbb{O}$ each transforming under Spin(8).
Distinct $\mathfrak{so}$(8) actions on these 8D spaces allow one copy to be
identified with the $\psi$ spinor, another with the $\widetilde{\psi}$
conjugate spinor, and the final copy with the vector representation, $V$. Now,
introducing a requirement that each of the corresponding group actions
coincide then reduces the original $\mathfrak{so}$(8) symmetry to
$\mathfrak{g}_{2}$. While $\mathfrak{so}$(8) generated the triality symmetry
of the octonions, $\mathfrak{g}_{2}$ generates their automorphisms.

Parallels between automorphisms embedded inside triality symmetries, and the
standard model internal symmetries embedded inside $\mathfrak{spin}(10)$ will
be made in the third paper of this series, [42].

##  III Simultaneous group actions of the multivector type

In this section, we offer an explanation for the breaking of symmetry from
$\mathfrak{spin}(2n)\mapsto\mathfrak{su}(n)$, first proposed in [47]. In known
models, this corresponds to transitions

$\begin{array}[]{rcl}\textup{Spin(10)}&\mapsto&\textup{SU(5),}\vspace{2mm}\\\
\textup{Pati-Salam}&\mapsto&\textup{Standard Model,}\vspace{2mm}\\\
\textup{Left-Right Symmetric}&\mapsto&\textup{Standard Model.}\end{array}$

While algebraic single-step methods breaking
$\mathfrak{spin}(2n)\mapsto\mathfrak{u}(n)$ are ubiquitous in the literature,
algebraic single-step methods breaking
$\mathfrak{spin}(2n)\mapsto\mathfrak{su}(n)$ are not.

###  III.1 Three types of spinor

We argue in this subsection that not all spinor constructions are created
equal. We compare three inequivalent constructs with increasing complexity.

(I) The most common description of a spinor in the literature is simply as a
column vector (equivalently, single-indexed tensor) with entries in
$\mathbb{R}$ or $\mathbb{C}$.

(II) As an alternative to column vectors, certain spinors may be constructed
by upgrading these vector spaces to exterior algebras,
$\Lambda\mathbb{C}^{n}$. In this case, the spinor is endowed with a
$\mathbb{Z}$-grading and a concept of a wedge product on its elements.

(Aside) It has long been known that this construction leads to a complementary
description in terms of differential forms and vector fields, eg [9]. Namely,

$a_{j}=\textup{d}x^{j}\wedge\hskip
28.45274pta_{j}^{\dagger}=i\frac{\partial}{\partial x^{j}},$ (4)

where the $\\{a_{j}\\}$ represent the $n$ generating elements of
$\Lambda\mathbb{C}^{n}$.

Augmenting spinors-as-column-vectors to spinors-as-exterior-algebras
introduces a structural richness that would otherwise be invisible in the more
common column vector approach. However, this is not the end of the line.

(III) The construction of spinors as minimal left ideals (MLI) takes this
$\Lambda\mathbb{C}^{n}$, and adds another layer of structure beyond it. As
will be described shortly, this extra structure appears in the form of a non-
trivial vacuum state. The MLI spinor construction leads to a more unified
description by embedding the spinor directly into the Clifford algebra. We now
introduce this construction to readers not already familiar.

###  III.2 Minimal left ideal construction

Consider a Clifford algebra, $\mathbb{C}l(2n)$, over the complex numbers, and
generated by basis vectors $\Gamma_{k}$ for $k\in\\{1,\dots,2n\\}$, and
$n\in\mathbb{N}>0$. Let $\dagger$ denote an anti-linear involution such that
$(\Gamma_{k_{1}}\Gamma_{k_{2}})^{\dagger}=\Gamma_{k_{2}}^{\dagger}\Gamma_{k_{1}}^{\dagger}$.
In this setup, we are free to choose $\Gamma_{k}$ such that
$\Gamma_{k}^{\dagger}=-\Gamma_{k},$ and

$\\{\Gamma_{k_{1}},\Gamma_{k_{2}}\\}:=\Gamma_{k_{1}}\Gamma_{k_{2}}+\Gamma_{k_{2}}\Gamma_{k_{1}}=-2\delta_{k_{1}k_{2}},$
(5)

and we will indeed do so.

From here, we construct an alternative generating basis for $\mathbb{C}l(2n)$
as consisting of $n$ raising operators, $a_{j}$, and $n$ lowering operators,
$a^{\dagger}_{j}$, for $j\in\\{1,2,\dots n\\}$. (Naming conventions were
chosen so that this work matches previous articles, [2], [3].) Explicitly,

$a_{j}:=\frac{1}{2}\left(-\Gamma_{j}+i\Gamma_{n+j}\right)\hskip
28.45274pta^{\dagger}_{j}:=\frac{1}{2}\left(\Gamma_{j}+i\Gamma_{n+j}\right).$
(6)

From these definitions and equations (5), we have that

$\begin{array}[]{l}\\{a_{j_{1}},a_{j_{2}}\\}=\\{a_{j_{1}}^{\dagger},a_{j_{2}}^{\dagger}\\}=0\vspace{2mm}\\\
\\{a_{j_{1}},a^{\dagger}_{j_{2}}\\}=\delta_{j_{1}j_{2}}\end{array}$ (7)

for $j_{1},j_{2}\in\\{1,2,\dots n\\}$. We define $\Omega:=a_{1}a_{2}\cdots
a_{n}$. Then we form a hermitian vacuum state as

$v:=\Omega^{\dagger}\Omega=a_{n}^{\dagger}\cdots
a_{2}^{\dagger}a_{1}^{\dagger}a_{1}a_{2}\cdots a_{n}.$ (8)

We are now ready to define a minimal left ideal, $\Psi,$ as

$\Psi:=\mathbb{C}l(2n)\hskip 1.42262ptv.$ (9)

These raising and lowering operators allow us to make apparent a
$\mathbb{Z}$-grading within $\Psi$. It can be shown that

$\begin{array}[]{lc}\Psi=&c_{12\dots n}\hskip 1.42262pta_{1}a_{2}\cdots
a_{n}v\vspace{2mm}\\\ &\vdots\vspace{2mm}\\\ &+\hskip 2.84526ptc_{12}\hskip
1.42262pta_{1}a_{2}v+c_{13}\hskip 1.42262pta_{1}a_{3}v\dots+c_{n-1\hskip
1.42262ptn}\hskip 1.42262pta_{n-1}a_{n}v\vspace{2mm}\\\ &+\hskip
2.84526ptc_{1}\hskip 1.42262pta_{1}v+c_{2}\hskip
1.42262pta_{2}v\dots+c_{n}\hskip 1.42262pta_{n}v\vspace{2mm}\\\ &+\hskip
2.84526ptc_{0}\hskip 1.42262ptv\end{array}$ (10)

for $c_{0},c_{j},\dots,c_{12\dots n}\in\mathbb{C}$. From here, it is
straightforward to to see that $\Psi$ takes the form of a Fock space, familiar
to physicists.

It can be shown that the omission of $v$ in equation (10) leads to an exterior
algebra. Hence, the MLI construction may be seen to bestow upon an exterior
algebra the extra structure of a non-trivial vacuum state.

###  III.3 Adjoint, vector, and spinor representations

It is known, [48], that the Lie algebra $\mathfrak{so}(2n)$ may be represented
by bivectors $r_{k_{1}k_{2}}\Gamma_{k_{1}}\Gamma_{k_{2}}$ for
$r_{k_{1}k_{2}}\in\mathbb{R}$. Here, $k_{1},k_{2}\in\\{1,2,\dots 2n\\}$, and
$k_{1}\neq k_{2}$. Multiplication is given by the commutator,

$[r_{k_{1}k_{2}}\Gamma_{k_{1}}\Gamma_{k_{2}},\hskip
2.84526ptr_{k_{3}k_{4}}\Gamma_{k_{3}}\Gamma_{k_{4}}].$

Similarly, the defining $2n$-dimensional vector representation of
$\mathfrak{so}(2n)$ may be written as $V:=\sum_{k=1}^{2n}V_{k}\hskip
1.42262pt\Gamma_{k}$ for $V_{k}\in\mathbb{R}$. Its infinitesimal
transformations under $\mathfrak{so}(2n)$ are then given by

$\delta V=[r_{k_{1}k_{2}}\Gamma_{k_{1}}\Gamma_{k_{2}},\hskip 1.42262ptV].$
(11)

It is important to note that in this formalism, basis vectors are viewed as
carrying the transformations, not the coefficients.

Finally, infinitesimal transformations on the $2^{n}$ $\mathbb{C}$-dimensional
minimal left ideal $\Psi$ is given by

$\delta\Psi=r_{k_{1}k_{2}}\Gamma_{k_{1}}\Gamma_{k_{2}}\Psi.$ (12)

###  III.4 $\mathfrak{spin}(2n)$ $\mapsto$ $\mathfrak{su}(n)$ via the
multivector condition

In equation (10), we explained that spinors can be constructed purely from the
generating $\\{a_{1},a_{2},\dots a_{1}^{\dagger},a_{2}^{\dagger},\dots\\}$ of
the Clifford algebra. Using equation (6), we may then express this spinor
purely in terms of the $\\{\Gamma_{k}\\}$. Explicitly, these minimal left
ideals are composed of multivectors

$\Psi=c_{0}+c_{i_{1}}\Gamma_{i_{1}}+c_{i_{2}i_{3}}\Gamma_{i_{2}}\Gamma_{i_{3}}+\dots,$
(13)

for $i_{1},i_{2},i_{3},\dots\in\\{1,2,\dots 2n\\}$ and for some
$c_{0},c_{i_{1}},c_{i_{2}i_{3}},\dots$ $\in\mathbb{C}$.

As mentioned in equation (11), these $\\{\Gamma_{k}\\}$ form the vector
representation $V:=\sum_{k=1}^{2n}V_{k}\hskip 1.42262pt\Gamma_{k}$ of
$\mathfrak{so}(2n)$,

$\delta V=[r_{k_{1}k_{2}}\Gamma_{k_{1}}\Gamma_{k_{2}},\hskip 1.42262ptV].$
(14)

From this vantage point, it is then natural to wonder: Shouldn’t the
generating vectors induce a transformation rule on the minimal left ideals
that were built from them?

Equation (14) supplies a derivation. One would expect an induced infinitesimal
transformation of $\Psi$ to go as

$\begin{array}[]{l}\delta\Psi\vspace{2mm}\\\ =\delta c_{0}+c_{i_{1}}\hskip
1.42262pt\delta\Gamma_{i_{1}}+c_{i_{2}i_{3}}\hskip
1.42262pt\delta(\Gamma_{i_{2}})\Gamma_{i_{3}}+c_{i_{2}i_{3}}\Gamma_{i_{2}}\hskip
1.42262pt\delta(\Gamma_{i_{3}})+\dots\vspace{2mm}\\\ =c_{i_{1}}\hskip
1.42262pt[r_{k_{1}k_{2}}\Gamma_{k_{1}}\Gamma_{k_{2}},\hskip
2.84526pt\Gamma_{i_{1}}]+c_{i_{2}i_{3}}\hskip
1.42262pt[r_{k_{1}k_{2}}\Gamma_{k_{1}}\Gamma_{k_{2}},\hskip
2.84526pt\Gamma_{i_{2}}]\hskip 1.42262pt\Gamma_{i_{3}}\vspace{2mm}\\\ \hskip
7.11317pt+\hskip 2.84526ptc_{i_{2}i_{3}}\Gamma_{i_{2}}\hskip
1.42262pt[r_{k_{1}k_{2}}\Gamma_{k_{1}}\Gamma_{k_{2}},\hskip
2.84526pt\Gamma_{i_{3}}]+\dots,\end{array}$ (15)

which simplifies to

$\delta\Psi=[r_{k_{1}k_{2}}\Gamma_{k_{1}}\Gamma_{k_{2}},\hskip
1.42262pt\Psi]\hskip 2.84526pt.$ (16)

However, we already defined the infinitesimal transformation of spinors in
equation (12) as

$\delta\Psi=r_{k_{1}k_{2}}\Gamma_{k_{1}}\Gamma_{k_{2}}\Psi.$ (17)

So, which shall it be: equation (16) or equation (17)?

Let us entertain the idea that both symmetry actions (16) and (17) be
simultaneously obeyed. We will refer to this consistency condition as the
multivector condition.

Readers may confirm that only an $\mathfrak{su}(n)$ subalgebra of
$\mathfrak{so}(2n)$ survives the multivector condition.

$\begin{array}[]{c}\delta\Psi=[r_{k_{1}k_{2}}\Gamma_{k_{1}}\Gamma_{k_{2}},\hskip
1.42262pt\Psi]\hskip
2.84526pt=r_{k_{1}k_{2}}\Gamma_{k_{1}}\Gamma_{k_{2}}\Psi\vspace{2mm}\\\
\Rightarrow\vspace{2mm}\\\ \mathfrak{so}(2n)\hskip 5.69054pt\mapsto\hskip
5.69054pt\mathfrak{su}(n).\end{array}$ (18)

Explicitly, a generic element of the surviving $\mathfrak{su}(n)$ subalgebra
may be written in terms of raising and lowering operators as

$\begin{array}[]{lll}\ell_{n}&=&R_{j_{1}j_{2}}\left(a_{j_{1}}a^{\dagger}_{j_{2}}-a_{j_{2}}a^{\dagger}_{j_{1}}\right)\vspace{2mm}\\\
&+&R_{j_{1}j_{2}}^{\prime}i\left(a_{j_{1}}a^{\dagger}_{j_{2}}+a_{j_{2}}a^{\dagger}_{j_{1}}\right)\vspace{2mm}\\\
&+&R_{j}\hskip
1.42262pti\left(a_{j}a^{\dagger}_{j}-a_{j+1}a^{\dagger}_{j+1}\right),\end{array}$
(19)

where $j_{1}\neq j_{2}$ and
$R_{j_{1}j_{2}},R_{j_{1}j_{2}}^{\prime},R_{j}\in\mathbb{R}$. In terms of
$\\{\Gamma_{j}\\}$, these same $\mathfrak{su}(n)$ elements may be represented
as

$\begin{array}[]{lll}\ell_{n}&=&r_{j}\hskip
1.42262pt\Gamma_{j}\Gamma_{j+n}\vspace{2mm}\\\
&+&r_{j_{1}j_{2}}\left(\Gamma_{j_{1}}\Gamma_{j_{2}}+\Gamma_{j_{1}+n}\Gamma_{j_{2}+n}\right)\vspace{2mm}\\\
&+&r_{j_{1}j_{2}}^{\prime}\left(\Gamma_{j_{1}}\Gamma_{j_{2}+n}+\Gamma_{j_{2}}\Gamma_{j_{1}+n}\right),\end{array}$
(20)

where $r_{j},r_{j_{1}j_{2}},r_{j_{1}j_{2}}^{\prime}\in\mathbb{R},$ and
$\sum_{j=1}^{n}r_{j}=0.$

###  III.5 Significance

This result is significant for a number of reasons.

(+) It shows that a single consistency condition, (18), can offer an answer to
both of the open challenges 2. and 3. posed by Baez and Huerta in Section
II.1.

(+) As will appear later in this text, it eliminates the persistent unwanted
B-L symmetry from [3].

(+) We will also see that in the physically interesting cases of

$\begin{array}[]{rcl}\textup{Spin(10)}&\mapsto&\textup{SU(5),}\vspace{2mm}\\\
\textup{Pati-Salam}&\mapsto&\textup{Standard Model,}\vspace{2mm}\\\
\textup{Left-Right Symmetric}&\mapsto&\textup{Standard Model,}\end{array}$

this consistency condition supplies a natural explanation for the Standard
Model’s infamous chirality.

##  IV Simultaneous group actions of the reflective type

###  IV.1 Motivation for considering
$\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$

Recent work, [2], [3], introduced the result that one generation of
unconstrained fermions can be identified with one copy of
$\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$,

$\textup{one generation}\hskip 5.69054pt\leftrightarrow\hskip
5.69054pt\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}.$

Furthermore, the division algebraic substructure of this algebra
(surprisingly) led to a cascade of symmetry breakings in well-known particle
models:

$\begin{array}[]{l}\textup{Spin(10)}\mapsto\textup{Pati-
Salam}\mapsto\textup{Left-Right Symmetric}\mapsto\vspace{2mm}\\\ \hskip
21.33955pt\textup{Pre-Higgs Standard Model (+ B-L)}\mapsto\vspace{2mm}\\\
\hskip 42.67912pt\textup{Post-Higgs Standard Model (+ B-L).}\end{array}$

However, instead of identifying one generation with
$\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$, we will now
construct one generation as a minimal left ideal within
$\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$’s
multiplication algebra,

$\textup{one generation}\hskip 5.69054pt\leftrightarrow\hskip
5.69054pt\textup{MLI}.$

Doing so will allow us to implement the multivector condition from the
previous section.

$\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$’s
multiplication algebra, to be defined shortly, is isomorphic to its complex
endomorphisms,
$End\left(\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}\right)$.
It is also isomorphic as a matrix algebra to the complex Clifford algebra
$\mathbb{C}l(10)$. Hence it provides the facilities to build up a Spin(10)
model, and due to its division algebraic substructure, the facilities to break
that model down.

###  IV.2 The algebra
$\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$

The algebra $\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$ is
also known as the Dixon algebra, due to its independent implementation in
early particle models by Dixon. Readers are encouraged to see [37] for an
alternative perspective.

Throughout this article, all tensor products will be assumed to be over
$\mathbb{R}$ unless otherwise stated. We write the standard $\mathbb{R}$-basis
for $\mathbb{H}$ as
$\\{\epsilon_{0},\epsilon_{1},\epsilon_{2},\epsilon_{3}\\}$, where
$\epsilon_{0}=1$, $\epsilon_{j}^{2}=-1$ for $j\in\\{1,2,3\\}$, and
$\epsilon_{1}\epsilon_{2}=\epsilon_{3}$, with cyclic permutations. These
relations may be written more succinctly as
$\epsilon_{i}\epsilon_{j}=-\delta_{ij}+\varepsilon_{ijk}\hskip
1.42262pt\epsilon_{k}$ for $i,j,k\in\\{1,2,3\\}$, where $\varepsilon_{ijk}$ is
the usual totally anti-symmetric tensor with $\varepsilon_{123}=1$.

Similarly, we write the standard $\mathbb{R}$-basis for $\mathbb{O}$ as
$\\{e_{0},e_{1},\dots,e_{7}\\}$, where $e_{0}=1$ and
$e_{i}e_{j}=-\delta_{ij}+f_{ijk}\hskip 1.42262pte_{k}$ for
$i,j,k\in\\{1,2,\dots 7\\}$. Here, $f_{ijk}$ is a totally anti-symmetric
tensor with $f_{ijk}=1$ when $ijk\in\\{124,235,346,457,561,672,713\\}$. The
remaining values of $f_{ijk}$ are determined by anti-symmetry, and vanish
otherwise. We let $i$ denote a complex imaginary unit as usual.

Let
$\mathbb{A}:=\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}=\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$.
Note that $\mathbb{A}$ is naturally an algebra over $\mathbb{C}$: scalar
multiplication by $c\in\mathbb{C}$ is defined by setting $c(x\otimes y\otimes
w)=cx\otimes y\otimes w$ for $x\in\mathbb{C},y\in\mathbb{H},w\in\mathbb{O}$.
Multiplication of elements is defined by setting $(x_{1}\otimes y_{1}\otimes
w_{1})(x_{2}\otimes y_{2}\otimes w_{2})=x_{1}x_{2}\otimes y_{1}y_{2}\otimes
w_{1}w_{2}$ for all $x_{1},x_{2}\in\mathbb{C},y_{1},y_{2}\in\mathbb{H}$, and
$z_{1},z_{2}\in\mathbb{O}$. A $\mathbb{C}$-basis for $\mathbb{A}$ is given by
$\\{1\otimes\epsilon_{\mu}\otimes
e_{\nu}\mid\mu\in\\{0,1,2,3\\},\nu\in\\{0,1,\dots,7\\}\\}$. We see that
$\mathbb{A}$ is a 32 $\mathbb{C}$-dimensional non-commutative, non-associative
algebra.

From now on, we identify $\mathbb{C}$, $\mathbb{H}$, and $\mathbb{O}$ with
their images in $\mathbb{A}$ under the natural inclusion maps, thus writing
$c$ instead of $c\otimes\epsilon_{0}\otimes e_{0}$ for any $c\in\mathbb{C}$,
writing $\epsilon_{\mu}$ for $1\otimes\epsilon_{\mu}\otimes e_{0}$, and
writing $e_{\nu}$ for $1\otimes\epsilon_{0}\otimes e_{\nu}$ (here
$\mu\in\\{0,1,2,3\\},\hskip 2.84526pt\nu\in\\{0,1,\dots,7\\}$). Arbitrary
elements of $\mathbb{A}$ are then written as
$\sum_{\mu,\nu}c_{\mu\nu}\epsilon_{\mu}e_{\nu}$, where
$c_{\mu\nu}\in\mathbb{C},$ and we see that
$\epsilon_{\mu}e_{\nu}=e_{\nu}\epsilon_{\mu}$ for all $\mu,\nu$.

For those less comfortable with formal definitions, an example may help.
Suppose $a,b\in\mathbb{A}$ with $a=4\epsilon_{1}e_{2}$ and
$b=\left(5+i\right)\epsilon_{2}e_{4}$. Then
$ab=\left(20+4i\right)\epsilon_{3}e_{1}.$

###  IV.3 From $\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$
to its space of linear operators

In works dating back as early as [6], it was clear that the $\mathfrak{su}(3)$
adjoint part of gluons would be destined to reside not in $\mathbb{O},$ but
rather in the space of linear operators on $\mathbb{O}$. This space of linear
operators can be realized as $\mathbb{A}$’s multiplication algebra.

If bosons are to reside within $\mathbb{A}$’s multiplication algebra, then why
not include fermions there as well? One advantage of including fermions within
$\mathbb{A}$’s multiplication algebra is that, unlike with $\mathbb{A}$, this
space is now large enough to comfortably accommodate three generations. Hence,
we will position these fermions within $\mathbb{A}$’s multiplication algebra,
in the form of minimal left ideals.

First let us define what we mean by multiplication algebra. Suppose $x,y$ are
elements in an algebra $\mathbb{D}.$ We then define $L_{x}(y):=xy$. Similarly,
define $R_{x}(y):=yx$. Both $L_{x}$ and $R_{x}$ may then be seen as (possibly
distinct) linear maps sending $y$ to some new element in $\mathbb{D}.$ Hence,
$L_{x}$ and $R_{x}$ $\in End(\mathbb{D}).$

For $x,y,z\in\mathbb{D},$ these linear maps may be composed as in $L_{x}\circ
L_{y}(z)=x(y(z))$, $R_{x}\circ R_{y}(z)=((z)y)x,$ $L_{x}\circ
R_{y}(z)=x((z)y),$ $R_{x}\circ L_{y}(z)=(y(z))x$, etc. _Composition_ of these
maps, $\circ$, defines an _associative_ multiplication rule. Furthermore, maps
may be added in the obvious way. For example, $(L_{x}+L_{y})(z)=xz+yz.$

We define the left multiplication algebra of $\mathbb{D}$ as the subalgebra of
$End(\mathbb{D})$ generated by the $\\{L_{y}\hskip 1.42262pt|\hskip
1.42262pty\in\mathbb{D}\\}$. Similarly, the right multiplication algebra of
$\mathbb{D}$ is the subalgebra of $End(\mathbb{D})$ generated by
$\\{R_{y}\hskip 1.42262pt|\hskip 1.42262pty\in\mathbb{D}\\}$. We denote the
left multiplication algebra of an algebra $\mathbb{D}$ as
$\mathcal{L}_{\mathbb{D}},$ and its right multiplication algebra as
$\mathcal{R}_{\mathbb{D}}.$

As an example, consider the multiplication of two elements $A,B\in
End(\mathbb{A})$ with $A=4L_{\epsilon_{1}e_{2}}$ and
$B=\left(5+i\right)L_{\epsilon_{2}e_{4}}$. Then $AB:=A\circ
B=\left(20+4i\right)L_{\epsilon_{1}e_{2}}\circ
L_{\epsilon_{2}e_{4}}=\left(20+4i\right)L_{\epsilon_{3}e_{2}}\circ
L_{e_{4}}\neq\left(20+4i\right)L_{\epsilon_{3}e_{1}}.$

###  IV.4 Clifford factors

In this subsection, we examine the substructure of $\mathbb{A}$’s
multiplication algebra. We will find that it is this substructure that
ultimately leads the Spin(10) model to splinter into some of its broken
successors.

What are the multiplication algebras of $\mathbb{C},$ $\mathbb{H},$ and
$\mathbb{O}$? Of
$\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$? It can be
confirmed that the following isomorphisms hold (as matrix algebras):

$\begin{array}[]{lllll}&&\mathcal{L}_{\mathbb{C}}=\mathcal{R}_{\mathbb{C}}&\hskip
5.69054pt\leftrightarrow&Cl\left(0,1\right)\vspace{2mm}\\\
&&\mathcal{L}_{\mathbb{H}}&\hskip
5.69054pt\leftrightarrow&Cl\left(0,2\right)\vspace{2mm}\\\
&&\mathcal{R}_{\mathbb{H}}&\hskip
5.69054pt\leftrightarrow&Cl\left(0,2\right)\vspace{2mm}\\\
&&\mathcal{L}_{\mathbb{O}}\simeq\mathcal{R}_{\mathbb{O}}&\hskip
5.69054pt\leftrightarrow&Cl\left(0,6\right).\\\ \end{array}$ (21)

Since $\mathbb{C}$ is abelian, $\mathcal{L}_{\mathbb{C}}$ and
$\mathcal{R}_{\mathbb{C}}$ are equal elementwise. That is, $L_{c}=R_{c}\hskip
5.69054pt\forall c\in\mathbb{C}$. We identify these linear maps with
$Cl(0,1),$ where the vector generating $Cl(0,1)$ is given by multiplication by
the complex imaginary unit, $L_{i}=R_{i}$.

Unlike the complex numbers, $\mathbb{H}$ is non-abelian, and perhaps
unsurprisingly, $\mathcal{L}_{\mathbb{H}}$ and $\mathcal{R}_{\mathbb{H}}$
typically provide distinct linear maps on $\mathbb{H}$. We may identify each
of $\mathcal{L}_{\mathbb{H}}$ and $\mathcal{R}_{\mathbb{H}}$ with one copy of
$Cl(0,2)$, thereby identifying the combined multiplication algebras of
$\mathbb{H}$ with $Cl\left(0,2\right)\otimes Cl\left(0,2\right).$ For
concreteness, we will take $L_{\epsilon_{1}}$ and $L_{\epsilon_{2}}$ to
generate the first copy of $Cl(0,2),$ and $R_{\epsilon_{1}}$ and
$R_{\epsilon_{2}}$ to generate the second copy of $Cl(0,2),$ although there is
clearly a continuum of equivalent choices.

Finally, the left- and right-multiplication algebras of $\mathbb{O}$ may each
be identified with the Clifford algebra $Cl(0,6)$. However, unlike with the
quaternions, it is possible to show that each element of
$\mathcal{R}_{\mathbb{O}}$ gives the same linear map on $\mathbb{O}$ as some
element in $\mathcal{L}_{\mathbb{O}}.$ For example, $\forall f\in\mathbb{O}$,

$\begin{array}[]{lll}R_{e_{7}}f:=fe_{7}&=&\frac{1}{2}\left(-e_{7}f+e_{1}\left(e_{3}f\right)+e_{2}\left(e_{6}f\right)+e_{4}\left(e_{5}f\right)\right)\vspace{2mm}\\\
&=&\frac{1}{2}\left(-L_{e_{7}}+L_{e_{1}}L_{e_{3}}+L_{e_{2}}L_{e_{6}}+L_{e_{4}}L_{e_{5}}\right)f,\vspace{2mm}\end{array}$
(22)

where $e_{1},e_{2},\dots e_{7}$ represent octonionic imaginary units.
Therefore we see that although $L_{a}$ and $R_{a}$ are not equal for every
$a\in\mathbb{O}$, $\mathcal{L}_{\mathbb{O}}$ and $\mathcal{R}_{\mathbb{O}}$ do
provide the same set of linear maps on $\mathbb{O}.$ Hence, we write that
$\mathcal{L}_{\mathbb{O}}\simeq\mathcal{R}_{\mathbb{O}}.$ The set of
octonionic linear maps, identified with $Cl(0,6),$ may be generated by
$\\{L_{e_{j}}\\}$, or equivalently by $\\{R_{e_{j}}\\},$ where $j=1,\dots,6.$
Again, we emphasize that a continuum of equivalent choices exists (6-sphere
$S^{6}$) .

In short, we find that the linear maps coming from the multiplication algebras
of $\mathbb{C},$ $\mathbb{H},$ and $\mathbb{O}$ are given by $Cl(0,1),$
$Cl(0,2)\otimes Cl(0,2),$ and $Cl(0,6)$ respectively. The gradings for these
Clifford algebras can result from choosing the Clifford algebra’s generators
to be multiplication by an orthogonal set of imaginary units. $Cl(0,1)$ may be
generated by $L_{i}=R_{i};$ the two factors of $Cl(0,2)\otimes Cl(0,2)$ may be
generated by $\\{L_{\epsilon_{1}},L_{\epsilon_{2}}\\}$ and
$\\{R_{\epsilon_{1}},R_{\epsilon_{2}}\\};$ $Cl(0,6)$ may be generated by
$\\{L_{e_{j}}\\}$, or alternatively by $\\{R_{e_{j}}\\},$ where
$j\in\\{1,\dots,6\\}.$

Finally, combining $\mathcal{L}_{\mathbb{C}}=\mathcal{R}_{\mathbb{C}},$
$\mathcal{L}_{\mathbb{H}},$ and
$\mathcal{L}_{\mathbb{O}}\simeq\mathcal{R}_{\mathbb{O}},$ gives
$\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$’s left
multiplication algebra as

$\begin{array}[]{l}\mathcal{L}_{\mathbb{C}}\otimes\mathcal{L}_{\mathbb{H}}\otimes\mathcal{L}_{\mathbb{O}}\vspace{2mm}\\\
\simeq Cl(0,1)\otimes Cl(0,2)\otimes Cl(0,6)\simeq\mathbb{C}l(8).\end{array}$
(23)

Combining $\mathcal{L}_{\mathbb{C}}=\mathcal{R}_{\mathbb{C}},$
$\mathcal{L}_{\mathbb{H}},$ $\mathcal{R}_{\mathbb{H}},$ and
$\mathcal{L}_{\mathbb{O}}\simeq\mathcal{R}_{\mathbb{O}},$ gives
$\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$’s full
multiplication algebra as

$\begin{array}[]{l}\mathcal{L}_{\mathbb{C}}\otimes\mathcal{L}_{\mathbb{H}}\otimes\mathcal{R}_{\mathbb{H}}\otimes\mathcal{L}_{\mathbb{O}}\vspace{2mm}\\\
\simeq Cl(0,1)\otimes Cl(0,2)\otimes Cl(0,2)\otimes
Cl(0,6)\simeq\mathbb{C}l(10).\vspace{2mm}\end{array}$ (24)

###  IV.5 Hermitian conjugation

We define hermitian conjugation, $\dagger$, on elements of $End(\mathbb{A})$
as the involution such that $L_{e_{j}}^{\dagger}=-L_{e_{j}}$ $\forall
j\in\\{1,2,\dots 7\\}$, $L_{\epsilon_{m}}^{\dagger}=-L_{\epsilon_{m}}$ and
$R_{\epsilon_{m}}^{\dagger}=-R_{\epsilon_{m}}$ $\forall m\in\\{1,2,3\\}$, and
$L_{i}^{\dagger}=-L_{i}.$ As with matrix algebras, the hermitian conjugate
defined here obeys $\left(ab\right)^{\dagger}=b^{\dagger}a^{\dagger}$ $\forall
a,b\in End\left(\mathbb{A}\right).$

###  IV.6 Division algebraic reflections

We have now explored the substructure of
$\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$’s
multiplication algebra. Next, we would like to know how one goes about putting
this substructure to use.

In short, we will employ the $\mathbb{Z}_{2}$-gradings that are inherent to
each Clifford algebraic factor in equation (24). Or, said another way, we will
make use of generalized notions of reflection.

Referring to the first factor in equation (24), we know that
$\mathcal{L}_{\mathbb{C}}$ is isomorphic to the Clifford algebra $Cl(0,1)$. In
this case, complex conjugation, denoted
$*_{\overrightarrow{\mathbb{C}}}\vspace{.5mm}$, maps $L_{i}\mapsto-L_{i},$ and
defines the $\mathbb{Z}_{2}$-grading on this Clifford algebra. It also induces
a reflection of any complex number across the real axis.

It is straightforward to identify analog reflections in the subsequent
Clifford algebraic factors of equation (24). For example, a
$\mathbb{Z}_{2}$-grading on $\mathcal{L}_{\mathbb{H}}\simeq Cl(0,2)$ may be
induced by a quaternionic reflection
$*_{\overrightarrow{\mathbb{H}}}:L_{\epsilon_{m}}\mapsto-L_{\epsilon_{m}}$ for
$m\in\\{1,2\\}$. Similarly, a $\mathbb{Z}_{2}$-grading on
$\mathcal{L}_{\mathbb{O}}\simeq Cl(0,6)$ may be induced by an octonionic
reflection $*_{\overrightarrow{\mathbb{O}}}:L_{e_{j}}\mapsto-L_{e_{j}}$ for
$j\in\\{1,2,\dots 6\\}$.

Certain attentive readers might rightfully question why these octonionic and
quaternionic reflections involve only six and two imaginary units
respectively. Why not seven and three? This comes from the left multiplication
algebras being isomorphic to $Cl(0,6)$ and $Cl(0,2)$, not $Cl(0,7)$ and
$Cl(0,3)$. Nonetheless, rest assured that the final imaginary units become
reintegrated when these reflections are composed with Clifford algebraic
reversion: $(\Gamma_{i}\Gamma_{j})^{\rho}=\Gamma_{j}\Gamma_{i}.$ It is
tedious, but straightforward, to confirm that
$L_{e_{7}}=L_{e_{1}}L_{e_{2}}L_{e_{3}}L_{e_{4}}L_{e_{5}}L_{e_{6}}$. Hence,
$(L_{e_{7}}^{*_{\overrightarrow{\mathbb{O}}}})^{{\rho}}=L_{e_{7}}^{{\rho}}=-L_{e_{7}}.$
Similarly, $L_{\epsilon_{3}}=L_{\epsilon_{1}}L_{\epsilon_{2}}$. Hence,
$(L_{\epsilon_{3}}^{*_{\overrightarrow{\mathbb{H}}}})^{{\rho}}=L_{\epsilon_{3}}^{{\rho}}=-L_{\epsilon_{3}}.$

For ease of exposition, we have chosen specific imaginary units above.
However, readers should be aware that an entire 6-sphere of equivalent choices
exist for $\mathcal{L}_{\mathbb{O}}$, a 2-sphere of equivalent choices exist
for $\mathcal{L}_{\mathbb{H}}$, and a 0-sphere ($\mathbb{Z}_{2}$) of
equivalent choices exists for $\mathcal{L}_{\mathbb{C}}$.

###  IV.7 An explicit division algebraic representation of $\mathbb{C}l(10)$

There are many ways to generate $\mathbb{C}l(10)\simeq End(\mathbb{A})$ so
that the results in this paper materialize. One such set is given by

$\Gamma_{j}=iL_{e_{j}}L_{e_{7}}R_{\epsilon_{1}},\hskip
14.22636pt\Gamma_{m+6}=L_{e_{7}}L_{\epsilon_{m}}R_{\epsilon_{1}},\hskip
14.22636pt\Gamma_{10}=R_{\epsilon_{2}}$ (25)

for $j\in\\{1,\dots 6\\},$ and $m\in\\{1,2,3\\}.$ (Another, possibly simpler,
starting point has been found recently by John Barrett). We define

$\Gamma_{11}:=\prod_{p=1}^{10}\Gamma_{p},$ (26)

which in this case becomes

$\Gamma_{11}=-R_{\epsilon_{3}}.$ (27)

It allows us to define a chirality operator $-i\Gamma_{11}=iR_{\epsilon_{3}}$.
We specify raising operators $a_{c}$ and lowering operators $a_{c}^{\dagger}$
as

$\begin{array}[]{lll}a_{1}^{\dagger}:=\frac{1}{2}(\Gamma_{5}+i\Gamma_{4})&&a_{1}:=\frac{1}{2}(-\Gamma_{5}+i\Gamma_{4})\vspace{2mm}\\\
a_{2}^{\dagger}:=\frac{1}{2}(\Gamma_{3}+i\Gamma_{1})&&a_{2}:=\frac{1}{2}(-\Gamma_{3}+i\Gamma_{1})\vspace{2mm}\\\
a_{3}^{\dagger}:=\frac{1}{2}(\Gamma_{6}+i\Gamma_{2})&&a_{3}:=\frac{1}{2}(-\Gamma_{6}+i\Gamma_{2})\vspace{2mm}\\\
a_{4}^{\dagger}:=\frac{1}{2}(\Gamma_{8}+i\Gamma_{7})&&a_{4}:=\frac{1}{2}(-\Gamma_{8}+i\Gamma_{7})\vspace{2mm}\\\
a_{5}^{\dagger}:=\frac{1}{2}(\Gamma_{10}+i\Gamma_{9})&&a_{5}:=\frac{1}{2}(-\Gamma_{10}+i\Gamma_{9}),\vspace{2mm}\\\
\end{array}$ (28)

where readers may notice that we have made a permissible relabeling of indices
relative to equation (6) in order to connect with previous work, [2], [3].
Defining a nilpotent object

$\Omega:=\prod_{c=1}^{5}a_{c}$ (29)

then sets our hermitian vacuum state as $v:=\Omega^{\dagger}\Omega$. Finally,
we construct a minimal left ideal as

$\Psi:=\mathbb{C}l(10)v.$ (30)

This gives a $32\hskip 1.42262pt\mathbb{C}$ dimensional subspace of
$\mathbb{C}l(10)$. For this first article in the series, we will be
particularly interested in the $16\hskip 1.42262pt\mathbb{C}$ semi-spinor
given by

$\Psi_{\textup{L}}:=\frac{1}{2}(1+i\Gamma_{11})\Psi.$ (31)

In the second article of this series, [41], we will then extend the semi-
spinor in two inequivalent ways.

###  IV.8 Cascade of particle symmetries:
the $\mathfrak{spin}$(10) line

With our fermions now embedded within $End(\mathbb{A})$ we will demonstrate
how division algebraic reflections prompt them to fragment. Please see Figure
(4).

Figure 4:  Four types of division algebraic reflections are relevant in the
left-multiplication space. Invariance under the first octonionic reflection
restricts Spin(10) to Pati-Salam (PS). Invariance under the second octonionic
reflection restricts Pati-Salam to Left-Right Symmetric (LR). Invariance under
a quaternionic reflection restricts Left-Right Symmetric to the Standard
Model’s unbroken symmetries, together with a $\mathfrak{u}(1)$ related to B-L,
and another $\mathfrak{u}(1)$ related to the $Z^{0}$ boson, (UBZ+). Invariance
under complex conjugation eliminates the $\mathfrak{u}(1)$ related to the
$Z^{0}$, leaving the Standard Model’s unbroken symmetries, together with B-L,
(UB+). Readers can confirm that certain paths bifurcate.

A generic element of the 45-dimensional $\mathfrak{spin}$(10) may be written
as real linear combinations of bivectors:

$\begin{array}[]{ll}\ell_{10}^{D}:=&r_{ij}L_{e_{i}}L_{e_{j}}+r_{m}L_{\epsilon_{m}}+r^{\prime}_{m}L_{\epsilon_{m}}L_{e_{7}}R_{\epsilon_{3}}\vspace{2mm}\\\
&+\hskip
2.84526ptr_{mj}^{\prime}iL_{\epsilon_{m}}L_{e_{j}}+r_{j}^{\prime\prime}iL_{e_{j}}L_{e_{7}}R_{\epsilon_{3}}\end{array}$
(32)

for $i,j\in\\{1,\dots 6\\}$ with $i\neq j,$ $m\in\\{1,2,3\\},$ and
$r_{ij},r_{m},r^{\prime}_{m},r_{mj}^{\prime},r_{j}^{\prime\prime}\in\mathbb{R}$.
Here the ‘D’ in $\ell_{10}^{D}$ refers to ‘Dirac’.

When applied to our semi-spinor $\Psi_{\textup{L}}$, this
$\mathfrak{spin}$(10) action simplifies considerably:

$\begin{array}[]{lc}\ell_{10}^{D}\Psi_{\textup{L}}=&(r_{ab}L_{e_{a}}L_{e_{b}}+r^{\prime}_{mn}L_{\epsilon_{m}}L_{\epsilon_{n}}+r_{ma}^{\prime\prime}iL_{\epsilon_{m}}L_{e_{a}})\Psi_{\textup{L}},\vspace{2mm}\\\
&\mathfrak{spin}(10)\end{array}$ (33)

where this time $a,b\in\\{1,\dots 7\\}$ with $a\neq b,$ and
$m,n\in\\{1,2,3\\}$ with $m\neq n,$ while
$r_{ab},r^{\prime}_{mn},r_{ma}^{\prime\prime}\in\mathbb{R}.$ This Weyl action
matches that found earlier in [3]. Let us then define

$\begin{array}[]{lc}\ell_{10}:=&r_{ab}L_{e_{a}}L_{e_{b}}+r^{\prime}_{mn}L_{\epsilon_{m}}L_{\epsilon_{n}}+r_{ma}^{\prime\prime}iL_{\epsilon_{m}}L_{e_{a}}.\vspace{2mm}\\\
&\mathfrak{spin}(10)\end{array}$ (34)

With this representation of $\mathfrak{spin}(10)$ at our disposal, we may now
make use of the division algebraic reflections introduced in Section IV.6.
That these reflections would reduce the SO(10) grand unified theory to other
well-known particle models came originally as a surprise.

It is important to note that the $\mathfrak{spin}(10)$ element $\ell_{10}$
lives exclusively in the left-multiplication algebra of $\mathbb{A}$.
Therefore there are four types of division algebraic reflections that will be
relevant for us: those related to $\langle 1\rangle$
$\mathcal{L}_{\mathbb{O}},$ $\langle 2\rangle$ $\mathcal{R}_{\mathbb{O}},$
$\langle 3\rangle$ $\mathcal{L}_{\mathbb{H}},$ and $\langle 4\rangle$
$\mathcal{L}_{\mathbb{C}}.$

$\langle 1\rangle$ First of all, we would like to know: Which Lie subalgebra
of our $\mathfrak{spin}(10)$ symmetries is invariant under an octonionic
reflection? Readers can easily confirm that setting

${\ell}_{10}\Psi_{\textup{L}}={\ell}_{10}^{*_{\overrightarrow{\mathbb{O}}}}\Psi_{\textup{L}}\hskip
28.45274pt\forall\Psi_{L}$ (35)

restricts $\mathfrak{spin}(10)$’s $\ell_{10}$ to the Pati-Salam symmetries,
$\ell_{\textup{PS}}:$

$\begin{array}[]{rll}{\ell}_{\textup{PS}}\Psi_{\textup{L}}=(r_{ij}L_{e_{i}}L_{e_{j}}&+&r_{m}L_{\epsilon_{m}}+r^{\prime}_{m}iL_{\epsilon_{m}}L_{e_{7}})\Psi_{\textup{L}}\vspace{2mm}\\\
\mathfrak{spin}(6)&\oplus&\mathfrak{spin}(4)\end{array}$ (36)

for $i,j\in\\{1,\dots 6\\}$ with $i\neq j,$ and $m\in\\{1,2,3\\}.$ Note that
$\mathfrak{spin}(6)=\mathfrak{su}(4),$ and
$\mathfrak{spin}(4)=\mathfrak{su}(2)\oplus\mathfrak{su}(2)$.

Next, we will consider those symmetries furthermore invariant under an
analogous octonionic reflection, deriving from right multiplication. Namely
$*_{\overleftarrow{\mathbb{O}}}:R_{e_{j}}\mapsto-R_{e_{j}}$ $\forall
j\in\\{1,\dots,6\\}.$

$\langle 2\rangle$ Which Lie subalgebra of the Pati-Salam symmetries is
invariant under this complementary octonionic reflection? With a little work,
readers will find the Lie algebra of the Left-Right Symmetric model. Setting

${\ell}_{\textup{PS}}\Psi_{\textup{L}}={\ell}_{\textup{PS}}^{*_{\overleftarrow{\mathbb{O}}}}\Psi_{\textup{L}}\hskip
28.45274pt\forall\Psi_{L}$ (37)

restricts $\ell_{\textup{PS}}$ to the Left-Right Symmetric model’s gauge
symmetries, $\ell_{\textup{LR}}:$

$\begin{array}[]{rll}{\ell}_{\textup{LR}}\Psi_{\textup{L}}=(r_{ij}^{\prime\prime}L_{e_{i}}L_{e_{j}}&+&r_{m}L_{\epsilon_{m}}+r^{\prime}_{m}iL_{\epsilon_{m}}L_{e_{7}})\Psi_{\textup{L}},\vspace{2mm}\\\
\mathfrak{u}(3)&\oplus&\mathfrak{spin}(4)\end{array}$ (38)

where the coefficients $r_{ij}^{\prime\prime}\in\mathbb{R}$ are this time
restricted to give a $\mathfrak{u}(3)$ subalgebra of $\mathfrak{spin}(6)$.
Explicitly, we have eight $\mathfrak{su}(3)_{C}$ generators

$\begin{array}[]{lll}i\Lambda_{1}:=\frac{1}{2}\left(L_{34}-L_{15}\right)&&i\Lambda_{2}:=\frac{1}{2}\left(L_{14}+L_{35}\right)\vspace{2mm}\\\
i\Lambda_{3}:=\frac{1}{2}\left(L_{13}-L_{45}\right)&&i\Lambda_{4}:=-\frac{1}{2}\left(L_{25}+L_{46}\right)\vspace{2mm}\\\
i\Lambda_{5}:=\frac{1}{2}\left(L_{24}-L_{56}\right)&&i\Lambda_{6}:=-\frac{1}{2}\left(L_{16}+L_{23}\right)\vspace{2mm}\\\
i\Lambda_{7}:=-\frac{1}{2}\left(L_{12}+L_{36}\right)&&i\Lambda_{8}:=\frac{-1}{2\sqrt{3}}\left(L_{13}+L_{45}-2L_{26}\right),\end{array}$
(39)

where $L_{ij}$ is shorthand for the octonionic $L_{e_{i}}L_{e_{j}}$. The
generator for the $\mathfrak{u}(1)_{B-L}$ subalgebra of $\mathfrak{u}(3)$ may
be described as

$i(B-L):=\frac{1}{3}(L_{13}+L_{26}+L_{45}).$ (40)

An alternative route was found by Boyle in [28] that maps Spin(10) directly to
the Left-Right Symmetric model via $*_{\overleftarrow{\mathbb{O}}}$ in the
context of the exceptional Jordan algebra. This result likewise holds in our
current model.

$\langle 3\rangle$ Next, fixing

${\ell}_{\textup{LR}}\Psi_{\textup{L}}={\ell}_{\textup{LR}}^{*_{\overrightarrow{\mathbb{H}}}}\Psi_{\textup{L}}\hskip
28.45274pt\forall\Psi_{L}$ (41)

restricts ${\ell}_{\textup{LR}}$ to a
$\mathfrak{u}(3)\oplus\mathfrak{u}(1)\oplus\mathfrak{u}(1)$ subalgebra, which
we will describe by elements ${\ell}_{{\textup{UBZ+}}}$:

$\begin{array}[]{rllrll}{\ell}_{{\textup{UBZ+}}}\Psi_{\textup{L}}=(r_{ij}^{\prime\prime}L_{e_{i}}L_{e_{j}}&+&r_{3}L_{\epsilon_{3}}&+&r^{\prime}_{3}iL_{\epsilon_{3}}L_{e_{7}})\Psi_{\textup{L}}.\vspace{2mm}\\\
\mathfrak{u}(3)&\oplus&\mathfrak{u}(1)&\oplus&\mathfrak{u}(1)\end{array}$ (42)

This less recognizable Lie subalgebra describes the Standard Model’s unbroken
gauge symmetries, together with a B-L symmetry, and an additional
$\mathfrak{u}(1)$ symmetry corresponding to the $Z^{0}$ boson.

$\langle 4\rangle$ Finally, requiring this action’s invariance under complex
conjugation,

${\ell}_{{\textup{UBZ+}}}\Psi_{\textup{L}}={\ell}_{{\textup{UBZ+}}}^{*_{\overrightarrow{\mathbb{C}}}}\Psi_{\textup{L}}\hskip
28.45274pt\forall\Psi_{L}$ (43)

restricts ${\ell}_{{\textup{UBZ+}}}$ to a
$\mathfrak{u}(3)\oplus\mathfrak{u}(1)$ Lie subalgebra, described by elements
${\ell}_{{\textup{UB+}}}$:

$\begin{array}[]{rll}{\ell}_{{\textup{UB+}}}\Psi_{\textup{L}}=(r_{ij}^{\prime\prime}L_{e_{i}}L_{e_{j}}&+&r_{3}L_{\epsilon_{3}})\Psi_{\textup{L}},\vspace{2mm}\\\
\mathfrak{u}(3)&\oplus&\mathfrak{u}(1)\end{array}$ (44)

thereby eliminating the symmetry corresponding to the $Z^{0}$ boson. This
leaves us with the Standard Model’s unbroken gauge symmetries, after the Higgs
mechanism, in addition to a B-L symmetry.

###  IV.9 Cascade of particle symmetries: the Georgi-Glashow line

We will now mark a path of broken symmetries, parallel to that of the previous
section. This time, however, we will begin with Georgi and Glashow’s SU(5)
model. Please see Figure (5).

Figure 5:  The same division algebraic reflections as Figure (4) may be
applied to the Georgi-Glashow model. We find that invariance under either of
the octonionic reflections sends SU(5) to the Standard Model (SM). Unlike with
[3], though, chirality now arises organically, and the unwanted B-L symmetry
is eliminated. From here, invariance under a quaternionic reflection sends the
Standard Model’s symmetries to the Standard Model’s unbroken symmetries,
together with a $\mathfrak{u}(1)$ related to the $Z^{0}$ boson, (UBZ).
Invariance under the complex reflection finally eliminates this
$\mathfrak{u}(1)$ factor to give the Standard Model’s familiar unbroken
symmetries, (UB). Again, complex conjugation provides a bypass that maps SM
directly to UB.

The 24  dimensional $\mathfrak{su}$(5) Lie algebra may be represented as

$\begin{array}[]{lll}\ell_{5}&=&R_{cd}\left(a_{c}a^{\dagger}_{d}-a_{d}a^{\dagger}_{c}\right)\vspace{2mm}\\\
&+&R_{cd}^{\prime}\hskip
1.42262pti\left(a_{c}a^{\dagger}_{d}+a_{d}a^{\dagger}_{c}\right)\vspace{2mm}\\\
&+&R_{c}\hskip
1.42262pti\left(a_{c}a^{\dagger}_{c}-a_{c+1}a^{\dagger}_{c+1}\right),\end{array}$
(45)

for $c,d\in\\{1,\dots 5\\}$ and $c\neq d$. Here,
$R_{cd},R_{cd}^{\prime},R_{c}\in\mathbb{R}$. This representation of
$\mathfrak{su}(5)$ corresponds to the Lie subalgebra of $\mathfrak{spin}(10)$
surviving the multivector condition, as in equation (19).

Substituting in the division algebraic raising and lowering operators of
equation (28) into equation (45) gives an $\mathfrak{su}(5)$ element,
$\ell_{5}$, acting on $\Psi_{\textup{L}}$ as

$\begin{array}[]{lllll}&&\ell_{5}\Psi_{\textup{L}}=&&\vspace{2mm}\\\
&&\big{(}r_{j}i\Lambda_{j}+r^{\prime}_{m}L_{\epsilon_{m}}s+irY&&\vspace{2mm}\\\
&&\hskip 1.42262pt+\hskip
1.99168ptit_{1}\left(L_{\epsilon_{1}}L_{e_{4}}+L_{\epsilon_{2}}L_{e_{5}}\right)&+&it_{1}^{\prime}\left(L_{\epsilon_{1}}L_{e_{5}}-L_{\epsilon_{2}}L_{e_{4}}\right)\vspace{2mm}\\\
&&\hskip 1.42262pt+\hskip
1.99168ptit_{1}^{\prime\prime}\left(L_{\epsilon_{3}}L_{e_{4}}-iL_{e_{5}}L_{e_{7}}\right)&+&it_{1}^{\prime\prime\prime}(L_{\epsilon_{3}}L_{e_{5}}+iL_{e_{4}}L_{e_{7}})\vspace{2mm}\\\
&&\hskip 1.42262pt+\hskip
1.99168ptit_{2}\left(L_{\epsilon_{1}}L_{e_{1}}+L_{\epsilon_{2}}L_{e_{3}}\right)&+&it_{2}^{\prime}(L_{\epsilon_{1}}L_{e_{3}}-L_{\epsilon_{2}}L_{e_{1}})\vspace{2mm}\\\
&&\hskip 1.42262pt+\hskip
1.99168ptit_{2}^{\prime\prime}\left(L_{\epsilon_{3}}L_{e_{1}}-iL_{e_{3}}L_{e_{7}}\right)&+&it_{2}^{\prime\prime\prime}(L_{\epsilon_{3}}L_{e_{3}}+iL_{e_{1}}L_{e_{7}})\vspace{2mm}\\\
&&\hskip 1.42262pt+\hskip
1.99168ptit_{3}\left(L_{\epsilon_{1}}L_{e_{2}}+L_{\epsilon_{2}}L_{e_{6}}\right)&+&it_{3}^{\prime}(L_{\epsilon_{1}}L_{e_{6}}-L_{\epsilon_{2}}L_{e_{2}})\vspace{2mm}\\\
&&\hskip 1.42262pt+\hskip
1.99168ptit_{3}^{\prime\prime}\left(L_{\epsilon_{3}}L_{e_{2}}-iL_{e_{6}}L_{e_{7}}\right)&+&it_{3}^{\prime\prime\prime}(L_{\epsilon_{3}}L_{e_{6}}+iL_{e_{2}}L_{e_{7}})\big{)}\Psi_{\textup{L}}\vspace{2mm}\\\
&&\hskip 48.36958pt\mathfrak{su}(5)&&\end{array}$ (46)

where $j\in\\{1,\dots 8\\},$ $m\in\\{1,2,3\\}$, and
$r_{j},r^{\prime}_{m},r,t_{k},t_{k}^{\prime},\vspace{.5mm}$
$t_{k}^{\prime\prime},t_{k}^{\prime\prime\prime}\vspace{.5mm}\in\mathbb{R}$.
The idempotent $s$ is defined as $s:=\frac{1}{2}(1+iL_{e_{7}}),$ and preserves
$\mathfrak{su}(2)_{\textup{L}}$-active states. The weak hypercharge generator,
$iY$, is defined via

$iY\Psi_{\textup{L}}:=\left(\frac{1}{6}\left(L_{13}+L_{26}+L_{45}\right)-\frac{1}{2}L_{\epsilon_{3}}s^{*}\right)\Psi_{\textup{L}},$
(47)

following the conventions of [49].

With our $\mathfrak{su}(5)$ action now established, division algebraic
reflections may be deployed so as to break these symmetries sequentially.

$\langle 1\rangle$ Which Lie subalgebra of Georgi and Glashow’s
$\mathfrak{su}(5)$ exhibits immunity to an octonionic reflection? Readers may
easily confirm that fixing the constraint

${\ell}_{5}\Psi_{\textup{L}}={\ell}_{5}^{*_{\overrightarrow{\mathbb{O}}}}\Psi_{\textup{L}}\hskip
28.45274pt\forall\Psi_{L}$ (48)

restricts $\mathfrak{su}(5)$ to the Standard Model’s symmetries,
$\mathfrak{su}(3)\oplus\mathfrak{su}(2)\oplus\mathfrak{u}(1)$.

$\langle 2\rangle$ Interestingly, we find that the right-multiplication
condition

${\ell}_{5}\Psi_{\textup{L}}={\ell}_{5}^{*_{\overleftarrow{\mathbb{O}}}}\Psi_{\textup{L}}\hskip
28.45274pt\forall\Psi_{L}$ (49)

has an identical effect. In either case, we are left with the Standard Model’s
gauge symmetries acting on $\Psi_{\textup{L}}$ as

$\begin{array}[]{rccll}\ell_{\textup{SM}}\Psi_{\textup{L}}=\big{(}r_{j}i\Lambda_{j}&+&r^{\prime}_{m}L_{\epsilon_{m}}s&+&irY\big{)}\Psi_{\textup{L}}.\vspace{2mm}\\\
\mathfrak{su}(3)&\oplus&\mathfrak{su}(2)&\oplus&\mathfrak{u}(1)\end{array}$
(50)

It is worth taking a moment to point out that the result established here is
not only obtaining the correct Standard Model Lie algebras. Over the years, a
recurring challenge in algebraic models has been to also secure the correct
chiral fermion representations. That the correct representations are indeed
realized here will be established in [41].

$\langle 3\rangle$ It is perhaps this next quaternionic step that is of most
phenomenological interest. The condition

${\ell}_{\textup{SM}}\Psi_{\textup{L}}={\ell}_{\textup{SM}}^{*_{\overrightarrow{\mathbb{H}}}}\Psi_{\textup{L}}\hskip
28.45274pt\forall\Psi_{L}$ (51)

can be seen to restrict
$\mathfrak{su}(3)\oplus\mathfrak{su}(2)\oplus\mathfrak{u}(1)$ to
$\mathfrak{su}(3)\oplus\mathfrak{u}(1)\oplus\mathfrak{u}(1)$. This resulting
$\mathfrak{su}(3)\oplus\mathfrak{u}(1)\oplus\mathfrak{u}(1)$ action describes
the Standard Model’s unbroken gauge symmetries, together with an additional
$\mathfrak{u}(1)$ symmetry associated with the $Z^{0}$ boson. Explicitly, its
action on $\Psi_{\textup{L}}$ reduces to

$\begin{array}[]{rccll}\ell_{\textup{UBZ}}\Psi_{\textup{L}}=\big{(}r_{j}i\Lambda_{j}&+&r^{\prime}_{3}L_{\epsilon_{3}}s&+&irY\big{)}\Psi_{\textup{L}}.\vspace{2mm}\\\
\mathfrak{su}(3)&\oplus&\mathfrak{u}(1)&\oplus&\mathfrak{u}(1)\end{array}$
(52)

Its meaning for electroweak symmetry breaking will be discussed briefly later
in this text.

$\langle 4\rangle$ Finally we arrive at invariance under complex conjugation.
Applying the constraint that

${\ell}_{\textup{UBZ}}\Psi_{\textup{L}}={\ell}_{\textup{UBZ}}^{*_{\overrightarrow{\mathbb{C}}}}\Psi_{\textup{L}}\hskip
28.45274pt\forall\Psi_{L}$ (53)

leaves us with the Standard Model’s unbroken gauge symmetries,
$\mathfrak{su}(3)\oplus\mathfrak{u}(1)$. A generic element of this Lie algebra
acts on $\Psi_{\textup{L}}$ as

$\begin{array}[]{lrcl}\ell_{\textup{UB}}\Psi_{\textup{L}}=&\big{(}r_{j}i\Lambda_{j}&+&r^{\prime}iQ\big{)}\Psi_{\textup{L}},\vspace{2mm}\\\
&\mathfrak{su}(3)&\oplus&\mathfrak{u}(1)\end{array}$ (54)

where $r^{\prime}\in\mathbb{R},$ and $iQ$ turns out to be none other than the
electric charge generator,

$iQ:=\frac{1}{6}\left(L_{13}+L_{26}+L_{45}\right)-\frac{1}{2}L_{\epsilon_{3}}.$
(55)

Readers should take note that it is also possible to bypass the
$*_{\overrightarrow{\mathbb{H}}}$ step in $\langle 3\rangle$. In this case,
one may move from the Standard Model’s pre-Higgs
$\mathfrak{su}(3)\oplus\mathfrak{su}(2)\oplus\mathfrak{u}(1)$ directly to the
Standard Model’s post-Higgs $\mathfrak{su}(3)\oplus\mathfrak{u}(1)$, simply
via $*_{\overrightarrow{\mathbb{C}}}\vspace{1mm}$. Again, this outcome
originally came as a surprise.

Particularly satisfying about this last result is the fact that the familiar
complex conjugate is what dictates the Standard Model’s final unbroken gauge
symmetries. Conceivably, the Standard Model’s gauge group could have broken in
a different way, had it been paired with a different Higgs sector. Could the
complex conjugate ultimately be stewarding the symmetry breaking process?

##  V Roadmap

Now that we have charted out certain paths connecting familiar particle
models, we will consolidate them into one detailed roadmap.

In this paper, we have listed five different forms of symmetry breaking steps.
Namely, the multivector condition, $\Psi_{V}$, invariance under two types of
octonionic reflection, ${*_{\overrightarrow{\mathbb{O}}}}$ and
$*_{\overleftarrow{\mathbb{O}}}$, one quaternionic reflection,
$*_{\overrightarrow{\mathbb{H}}},\vspace{.5mm}$ and one complex reflection,
$*_{\overrightarrow{\mathbb{C}}}$. These four types of reflection are
precisely those that can be realized in terms of the left-multiplication
algebra of $\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}.$

The quaternionic reflection $*_{\overleftarrow{\mathbb{H}}},\vspace{.5mm}$
related to right multiplication, is isolated in the sense that, unlike the
others, it is not expressible in terms of the left multiplication algebra. In
the second article of this series, [41], we will see how it plays an important
role in the description of spacetime symmetries.

Our culminating diagram results when all five symmetry breaking steps are
amalgamated ($\Psi_{V}$, ${*_{\overrightarrow{\mathbb{O}}}}$,
$*_{\overleftarrow{\mathbb{O}}}$, $*_{\overrightarrow{\mathbb{H}}},$
$*_{\overrightarrow{\mathbb{C}}}$). For reflections, we begin with the
octonions, continue to the quaternions, and close with the complex numbers.
Please see Figure (6) at the end of this article.

Figure 6:  Six well-known models of elementary particles are interconnected
algebraically into a detailed particle roadmap. The “SO(10)” model (Spin(10)),
the Georgi-Glashow model (SU(5)), the Pati-Salam model (PS), the Left-Right
Symmetric model (LR), the Standard Model pre-Higgs-mechanism (SM), and the
Standard Model post-Higgs-mechanism (UB) are each interrelated via certain
algebraic constraints. The multivector condition is represented by edges
labeled as $\Psi_{V}$. Division algebraic reflection constraints are
represented by edges labeled as $*_{\mathbb{D}}$. Beyond these six well-known
models, there are three more. For the definition of UBZ+ see equation (42);
for the definition of UB+ see equation (44); for the definition of UBZ see
equation (52).

From the outset, readers may notice an interesting feature of this 5-step
network. Namely, not all pathways are unique; a number of them bifurcate.

As a first example, we identify two compatible symmetry breaking pathways from
the Spin(10) model to the Left-Right Symmetric model. One may first restrict
Spin(10) to those symmetries invariant under the octonionic reflection
$*_{\overrightarrow{\mathbb{O}}}$, resulting in the Pati-Salam model.
Subsequently imposing invariance under ${*_{\overleftarrow{\mathbb{O}}}}$ then
results in the Left-Right Symmetric model. As an alternate route, readers can
bypass directly to the Left-Right Symmetric model by simply restricting
Spin(10) once via $*_{\overleftarrow{\mathbb{O}}}$, as in [28].

Of more immediate experimental relevance is the bifurcated symmetry breaking
step from the Standard Model symmetries pre-Higgs mechanism,
$\mathfrak{su}(3)_{\textup{C}}\oplus\mathfrak{su}(2)_{\textup{L}}\oplus\mathfrak{u}(1)_{\textup{Y}}$,
to the Standard Model symmetries post-Higgs mechanism,
$\mathfrak{su}(3)_{\textup{C}}\oplus\mathfrak{u}(1)_{\textup{Q}}$. Readers
will encounter a direct path via
$*_{\overrightarrow{\mathbb{C}}}\vspace{.5mm}$ from
$\mathfrak{su}(3)_{\textup{C}}\oplus\mathfrak{su}(2)_{\textup{L}}\oplus\mathfrak{u}(1)_{\textup{Y}}\mapsto\mathfrak{su}(3)_{\textup{C}}\oplus\mathfrak{u}(1)_{\textup{Q}}$.
This algebraic transition coincides with the Standard Model’s famous
spontaneous symmetry breaking. On the other hand, these same endpoints may be
realized by first imposing an intermediate constraint under
$*_{\overrightarrow{\mathbb{H}}}$. This extra step acts to further break
$W^{\pm}$ bosons away from the $Z^{0}$ relative to the Standard Model. It will
be interesting to investigate whether or not such an intermediate symmetry
breaking step could alter the $W^{\pm}$ mass prediction, relative to that of
the Standard Model [54], [55].

##  VI Conclusion

In this article, we set out to understand the Standard Model of particle
physics in relation to a number of neighbouring particle models. We summarize
our findings:

In what may be viewed as a consistency condition, we introduced what we call
the multivector condition. This constraint dictates that minimal left ideal
fermions transform in accordance with the multivectors that comprise them. As
a result, for internal symmetries, we find that the adjoint, vector, and
spinor representations now each transform infinitesimally under the
commutator. The minimal left ideal spinors that we constructed constitute a
special case where the commutator and left multiplication coincide. The
condition is algebraically equivalent to, but conceptually distinct from,
constraints of earlier authors fixing the volume form of a
$\Lambda\mathbb{C}^{5}$ spinor.

The multivector constraint, [47], put an end to the lingering B-L and
chirality issues that once haunted [3]. It also offered one way to understand
and consolidate the Spin(10) $\mapsto$ SU(5) symmetry breaking steps explained
in [1].

The findings of Baez and Huerta, [1], and Furey and Hughes, [2], [3], are
fused into a single particle roadmap. A total of nine particle models are
shown to be interlinked via the multivector condition, and via constraints of
invariance under generalized division algebraic reflections. Six of these
models are familiar to particle physicists, namely the “SO(10)” model, the
Georgi-Glashow model, the Pati-Salam model, the Left-Right Symmetric model,
the Standard Model pre-Higgs mechanism, and the Standard Model post-Higgs
mechanism.

The algebraic relations between Standard Model pre- and post-Higgs may be of
special interest to phenomenologists. That is, we find it fortunate that the
complex conjugate singles out precisely those
$\mathfrak{su}(3)_{\textup{C}}\oplus\mathfrak{u}(1)_{\textup{Q}}$ gauge
symmetries found at low energies, including correct assignments of electric
charge, [41]. We find it interesting that an alternative symmetry breaking
path to this same endpoint may exist, bypassed, and perhaps overshadowed, by
the Higgs’ direct route. Could such a parallel path carry with it
phenomenological implications for electroweak physics?

###### Acknowledgements.

These manuscripts have benefitted from numerous discussions with Beth Romano.
The author is furthermore grateful for feedback and encouragement from John
Baez, Sukṛti Bansal, John Barrett, Latham Boyle, Hilary Carteret, Mia Hughes,
Kaushlendra Kumar, Agostino Patella, Shadi Tahvildar-Zadeh, Carlos Tamarit,
Andreas Trautner, and Jorge Zanelli. This work was graciously supported by the
VW Stiftung Freigeist Fellowship, and Humboldt-Universität zu Berlin.

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