$(e_{R},d_{R})$, but the number of fermion multiplets remains the same,
$n_{\mkern-1.8muf}=4\times 3=12$.

  * •

Electroweak scalars in the standard-model regime. In the standard model, the
electroweak multiplet transforms as a pair of 2-component ${\rm SU}(2)_{L}$
spinors with opposite hypercharges $Y=\pm 1$, so $n_{s}=2$ for each of ${\rm
U}(1)_{Y}$ and ${\rm SU}(2)_{L}$, while $n_{s}=0$ for ${\rm SU}(3)_{c}$. This
is the same as in the “minimal” standard model of the electroweak Higgs field.

  * •

Electroweak scalars in the Pati-Salam regime. In the Pati-Salam regime, the
electroweak Higgs multiplet (28) transforms as a 4-component vector under
${\rm Spin}(4)_{w}$. But ${\rm Spin}(4)_{w}$ is isomorphic to ${\rm
SU}(2)_{R}\times{\rm SU}(2)_{L}$, and the electroweak multiplet effectively
transforms as a pair of 2-component spinors under each of ${\rm SU}(2)_{R}$
and ${\rm SU}(2)_{L}$, so $n_{s}=2$ for each of ${\rm SU}(2)_{R}$ and ${\rm
SU}(2)_{L}$. The scalar Dynkin index is the spinor index $S_{2}=\tfrac{1}{2}$.

  * •

Adjoint Higgs scalars in the standard-model regime. In the standard model
regime, the Higgs multiplet $\bm{E}$ that breaks the ${\rm
Spin}(4)_{w}\times{\rm Spin}(6)_{c}$ symmetry, equation (27), transforms as a
bivector under each of ${\rm SU}(2)_{L}$ and ${\rm Spin}(6)_{c}$. The
bivectors carry zero $Y$ charge. Thus the $\bm{E}$ multiplet contributes
$n_{s}=1$ for each of ${\rm SU}(2)_{L}$ and ${\rm Spin}(6)_{c}$, with adjoint
Dynkin index $S_{2}=2$ for ${\rm SU}(2)_{L}$, and adjoint index $S_{2}=3$ for
${\rm Spin}(6)_{c}$.

  * •

Adjoint Higgs scalars in the Pati-Salam regime. In the Pati-Salam regime, the
Higgs multiplet $\bm{E}$ that breaks the ${\rm Spin}(4)_{w}\times{\rm
Spin}(6)_{c}$ symmetry transforms as a bivector under each of ${\rm
SU}(2)_{R}$, ${\rm SU}(2)_{L}$, and ${\rm Spin}(6)_{c}$. Thus the $\bm{E}$
multiplet contributes $n_{s}=1$ for each of ${\rm SU}(2)_{R}$, ${\rm
SU}(2)_{L}$, and ${\rm Spin}(6)_{c}$, with adjoint Dynkin index $S_{2}=2$ for
${\rm SU}(2)_{R}$ and ${\rm SU}(2)_{L}$, and adjoint index $S_{2}=4$ for ${\rm
Spin}(6)_{c}$.

The above counting of scalars includes those from the adjoint Higgs scalar
multiplet $\bm{E}$, equation (27). Correctly, only scalars whose masses are
less than the running energy scale $\mu$ should be included. In the
literature, in calculating the running of coupling parameters it is common to
ignore Higgs scalars other than the known electroweak Higgs scalar, on the
grounds that any additional Higgs scalars may well be massive. The calculation
shown in Figure 3 assumes contrariwise that the Higgs scalars coming from
$\bm{E}$ are less than the running energy scale $\mu$, because this yields the
largest energy of grand unification, $10^{15}\,{\rm GeV}$, equations (86),
more consistent with the constraint estimated by [9].

To summarize, in the standard model, the factor in parentheses on the right
hand side of equation (78) for the running of coupling parameters is

$\displaystyle{\rm U}(1)_{Y}:\ $ $\displaystyle\ -\ $
$\displaystyle\tfrac{2}{3}\times\tfrac{5}{6}\times 12$ $\displaystyle\ -\ $
$\displaystyle\tfrac{1}{6}\times\tfrac{1}{2}\times 2$ $\displaystyle\ =\ $
$\displaystyle{-}\tfrac{41}{6}\ ,$ (83a) $\displaystyle{\rm SU}(2)_{L}:\ $
$\displaystyle\tfrac{11}{3}\times 2$ $\displaystyle\ -\ $
$\displaystyle\tfrac{2}{3}\times\tfrac{1}{2}\times 12$ $\displaystyle\ -\ $
$\displaystyle\tfrac{1}{6}\times(\tfrac{1}{2}\times 2+2\times 1)$
$\displaystyle\ =\ $ $\displaystyle\tfrac{17}{6}\ ,$ (83b) $\displaystyle{\rm
SU}(3)_{c}:\ $ $\displaystyle\tfrac{11}{3}\times 3$ $\displaystyle\ -\ $
$\displaystyle\tfrac{2}{3}\times\tfrac{1}{2}\times 12$ $\displaystyle\ -\ $
$\displaystyle\tfrac{1}{6}\times 3\times 1$ $\displaystyle\ =\ $
$\displaystyle\tfrac{13}{2}\ .$ (83c)

The sum over fermions of squared hypercharge in equation (79) is
$\sum_{f}\left(\tfrac{1}{2}Y\right)^{2}=\tfrac{10}{3}$ per generation, or an
average of $\tfrac{5}{6}$ per ${\rm SU}(2)_{L}$ or ${\rm SU}(3)_{c}$ fermion
multiplet, which accounts for the factor $\tfrac{5}{6}$ in the ${\rm
U}(1)_{Y}$ expression (83a) (a common practice, not followed here, is to
multiply the hypercharge coupling parameter $g_{Y}^{2}$ by $\tfrac{5}{3}$ so
that the fermionic factor $\tfrac{5}{6}$ in (83a) becomes $\tfrac{1}{2}$, the
same as the fermionic factors in ${\rm SU}(2)_{L}$ and ${\rm SU}(3)_{c}$; the
scalar factor in (83a) would then change from $\tfrac{1}{6}$ to
$\tfrac{1}{10}$, and the overall value would change from $-\tfrac{41}{6}$ to
$-\tfrac{41}{10}$). In the Pati-Salam regime before ${\rm
Spin}(4)_{w}\times{\rm Spin}(6)_{c}$ symmetry breaking, the factor in the
running of coupling parameters is

$\displaystyle{\rm SU}(2)_{R},\ {\rm SU}(2)_{L}:\ $
$\displaystyle\tfrac{11}{3}\times 2$ $\displaystyle\ -\ $
$\displaystyle\tfrac{2}{3}\times\tfrac{1}{2}\times 12$ $\displaystyle\ -\ $
$\displaystyle\tfrac{1}{6}\times(\tfrac{1}{2}\times 2+2\times 1)$
$\displaystyle\ =\ $ $\displaystyle\tfrac{17}{6}\ ,$ (84a) $\displaystyle{\rm
Spin}(6)_{c}:\ $ $\displaystyle\tfrac{11}{3}\times 4$ $\displaystyle\ -\ $
$\displaystyle\tfrac{2}{3}\times\tfrac{1}{2}\times 12$ $\displaystyle\ -\ $
$\displaystyle\tfrac{1}{6}\times 4\times 1$ $\displaystyle\ =\ $
$\displaystyle 10\ .$ (84b)

The factors for ${\rm SU}(2)_{R}$ and ${\rm SU}(2)_{L}$ are the factors for
each of the groups individually (and are, naturally, the same); the factors do
not add.

If the right-handed weak coupling parameter $g_{r}$ is treated as unknown,
then the condition (77) leads to no prediction for the energy scale of ${\rm
Spin}(4)_{w}\times{\rm Spin}(6)_{c}$ unification. However, in a scenario where
${\rm Spin}(4)_{w}$ is the broken remnant of a higher-dimensional spin group,
as considered in this paper, the right- and left-handed coupling parameters
$g_{r}$ and $g_{w}$ of ${\rm Spin}(4)_{w}$ are same,

$g_{r}=g_{w}\ ,$ (85)

in which case the condition (77) leads to a definite prediction.

The left panel of Figure 3 shows the combination $gg_{Y}/(g_{w}g_{c})$,
equation (77), which is predicted to be 1 at ${\rm Spin}(4)_{w}\times{\rm
Spin}(6)_{c}$ symmetry breaking. The right panel of Figure 3 shows the running
of the hypercharge, weak, and colour coupling parameters $g_{Y}$, $g_{w}$, and
$g_{c}$ as a function of the renormalization cutoff energy $\mu$. The energy
scale of ${\rm Spin}(4)_{w}\times{\rm Spin}(6)_{c}$ symmetry breaking, where
$gg_{Y}/(g_{w}g_{c})=1$, and the energy scale of grand unification, where
$g_{w}=g_{c}$, are predicted to be

$\mu_{\tfrac{gg_{Y}}{g_{w}g_{c}}=1}=1.4\times 10^{12}\,{\rm GeV}\
,\quad\mu_{g_{w}=g_{c}}=1.0\times 10^{15}\,{\rm GeV}\ .$ (86)

The energy scales (86) assume that the adjoint Higgs scalars are all light
compared to the scale $\mu$. If instead the adjoint Higgs scalars are all
heavy compared to $\mu$, then

$\mu_{\tfrac{gg_{Y}}{g_{w}g_{c}}=1}=4.3\times 10^{11}\,{\rm GeV}\
,\quad\mu_{g_{w}=g_{c}}=3.4\times 10^{14}\,{\rm GeV}\ .$ (87)

The predicted energy $\approx 10^{12}\,{\rm GeV}$ of Pati-Salam unification is
comparable to (slightly greater than) that of the most energetic cosmic rays
observed [104, 105].

The predicted energy $\approx 10^{15}\,{\rm GeV}$ of grand unification falls
below the lower limit

$\mu_{{\rm Spin}(10)}\gtrsim 4\times 10^{15}\,{\rm GeV}$ (88)

inferred for ${\rm Spin}(10)$ models by [9] from the Super-Kamiokande lower
limit of $\gtrsim 1.6\times 10^{34}\,{\rm yr}$ on the proton lifetime [8].
Thus the ${\rm Spin}(11,1)$ model may already be ruled out. However, it should
be borne in mind, first, that the prediction of $10^{15}\,{\rm GeV}$ is based
on the simplifying assumptions of 1-loop renormalization and an abrupt
transition at the Pati-Salam energy, and, second, that the ${\rm Spin}(11,1)$
symmetry breaking chain differs from the ${\rm Spin}(10)$ chains considered by
[9]. An improved computation would be desirable.

The upper limit on the energy scale of cosmological inflation inferred from
the upper limit to $B$-mode polarization power in the cosmic microwave
background measured by the Planck satellite is [106, eq. (26)]

$\mu_{\textrm{inflation}}\lesssim 2\times 10^{16}\,{\rm GeV}\ .$ (89)

The predicted energy scale (86) of grand symmetry breaking is within the upper
limit (89), although it is not clear that the grand symmetry breaking and
inflationary scales can be compared directly.

The ${\rm Spin}(4)_{w}\times{\rm Spin}(6)_{c}$ mixing angle $\theta_{r}$
defined by equations (61) is predicted to be

$\theta_{r}=0.73=42^{\circ}\ ,\quad\tan\theta_{r}=0.90\ ,$ (90)

insensitive to whether adjoint scalars are taken to be light or massive in the
running of coupling parameters.

##  10 Summary of predictions, and potential flaws

Subsection 10.1 summarizes the predictions of the proposed ${\rm Spin}(11,1)$
model. Subsection 10.2 highlights possible flaws that merit scrutiny.

###  10.1 Predictions

  1. 1.

${\rm Spin}(11,1)$, like ${\rm Spin}(10)$, predicts a right-handed neutrino,
with consequences that have been well explored in the literature.

  2. 2.

The tight fit of the Dirac and standard-model algebras in the ${\rm
Spin}(11,1)$ geometric algebra admits a unique minimal symmetry breaking chain
(1), proceeding via the Pati-Salam group ${\rm Spin}(4)_{w}\times{\rm
Spin}(6)_{c}$.

  3. 3.

${\rm SU}(5)$ cannot be on the path to grand unification, because its
generators fail to commute with the spacetime generators (25) in ${\rm
Spin}(11,1)$.

  4. 4.

All the Higgs fields involved in symmetry breaking, including grand, Pati-
Salam, and electroweak symmetry breaking, lie in a common 66-component ${\rm
Spin}(11,1)$ scalar multiplet $\bm{E}$ of bivectors, equation (27). The Higgs
fields are the scalar (spin 0) counterparts of the vector (spin 1) gauge
fields of ${\rm Spin}(11,1)$.

  5. 5.

The grand Higgs field $\langle\bm{T}\rangle$, equation (46), is available to
drive cosmological inflation.

  6. 6.

The generators of the electroweak Higgs field $\bm{H}$, equation (28), fail to
commute with the grand Higgs field $\langle\bm{T}\rangle$. To allow the
electroweak Higgs field to persist without being absorbed at grand symmetry
breaking, we argue, §7.4, that the 11th spatial dimension
$\bm{\gamma}^{+}_{t}$ behaves as a scalar dimension, as discussed in §4.4 of
[66]. Consequently the grand unified group is ${\rm Spin}(10,1)$ rather than
${\rm Spin}(11,1)$.

  7. 7.

Grand symmetry breaking is predicted to occur at $10^{15}\,{\rm GeV}$, and
Pati-Salam symmetry breaking at $10^{12}\,{\rm GeV}$, equations (86).

  8. 8.

The grand Higgs field $\langle\bm{T}\rangle$ breaks $t$-symmetry, which
generates a Majorana mass term (53) for the right-handed neutrino, although
the right-handed neutrino acquires that Majorana mass only at Pati-Salam
symmetry breaking. Later, after electroweak symmetry breaking, the left-handed
neutrino can acquire a small mass by the standard see-saw mechanism [7].

  9. 9.

The Pati-Salam group ${\rm Spin}(4)_{w}\times{\rm Spin}(6)_{c}$ predicts 8
vector gauge boson fields beyond those of the standard model, comprising 2
right-handed weak gauge bosons $R$, and 6 leptoquark gauge bosons $C$. At
Pati-Salam symmetry breaking, the weak and leptoquark bosons acquire masses in
the ratio $m_{R}/m_{C}=1.1$, equation (71).

  10. 10.

The $\langle\bm{P}\rangle$ Higgs field, equation (65), that breaks Pati-Salam
symmetry carries zero standard model charge, and in particular does not flip
the Dirac boost bit, so, unlike the electroweak Higgs field, cannot give
masses to fermions, §8.5. Fermions other than the right-handed neutrino remain
massless until electroweak symmetry breaking.

  11. 11.

As remarked at the end of §8.4, in the ${\rm Spin}(11,1)$ model the only
evident candidate for cosmological dark matter is an ultralight scalar [93]
with vanishing standard model charge, the scalar counterpart of either the
photon or the $Z$-boson.

  12. 12.

In the model proposed in this paper, supersymmetry [76] is not needed to
achieve grand unification.

###  10.2 Places to pick holes

Here we bring attention to areas that the critical reader might examine for
flaws.

> “We are very much aware that we are exploring unconventional ideas and that
> there may be some basic flaw in our whole approach which we have been too
> stupid to see.” — Sidney Coleman & Erick Weinberg [89].

  1. 13.

The adjustment (17) of ${\rm Spin}(10)$ bivectors is essential to permit
unification of ${\rm Spin}(10)$ and the Lorentz group ${\rm Spin}(3,1)$ in
${\rm Spin}(11,1)$. Without this adjustment, the ${\rm Spin}(11,1)$ idea
fails.

  2. 14.

To allow the electroweak Higgs bivectors to survive grand symmetry breaking,
it appears necessary to postulate, §7.4, that the 11th dimension
$\bm{\gamma}^{+}_{t}$, the spatial partner of the time vector
$\bm{\gamma}^{-}_{t}$, is a scalar that does not generate any gauge symmetry,
in which case the symmetry at grand symmetry breaking is ${\rm Spin}(10,1)$
rather than the parent symmetry ${\rm Spin}(11,1)$. We have not speculated how
${\rm Spin}(11,1)$ might break to ${\rm Spin}(10,1)$.

  3. 15.

The 12 (modified) bivectors $\bm{\gamma}^{\pm}_{t}\bm{\gamma}^{\pm}_{k}$,
$k=r,g,b$, are unbroken by the grand Higgs field $\langle\bm{T}\rangle$, but
fail to commute with several Lorentz bivectors (24), so cannot persist after
grand symmetry breaking, §7.1. We suggest that these bivectors are broken and
become massive as the dimensionality of spacetime drops from 10+1 to 3+1, but
we have not offered a detailed model of this process.

  4. 16.

The grand Higgs field $\langle\bm{T}\rangle$ breaks $t$-symmetry, thereby
generating a Majorana mass term $\bar{\nu}\cdot\langle\bm{T}\rangle\nu$,
equation (53), for the right-handed neutrino. The result is that $B{-}L$ is
violated even though $\langle\bm{T}\rangle$ commutes with $B{-}L$.

  5. 17.

The right-handed neutrino can acquire its Majorana mass only after Pati-Salam
symmetry breaking, when the neutrino is isolated into a standard-model
singlet. But the Pati-Salam Higgs field $\langle\bm{P}\rangle$ can acquire its
expectation value only after $B{-}L$ is violated by the right-handed neutrino
acquiring its Majorana mass. The two processes must be tightly coupled, §8.3,
but we have not offered a detailed model.

  6. 18.

The predicted energy scale of grand unification is $10^{15}\,{\rm GeV}$,
equations (86), which is less than the lower limit of $4\times 10^{15}\,{\rm
GeV}$ inferred for ${\rm Spin}(10)$ models by [9] from the Super-Kamiokande
lower limit on the proton lifetime [8]. A more careful calculation needs to be
done that follows the symmetry breaking chain of the ${\rm Spin}(11,1)$ model
rather than the ${\rm Spin}(10)$ models of [9].

##  11 Conclusions

${\rm Spin}(10)$ is well-known as a promising candidate for a grand unified
group that contains the standard-model group ${\rm U}(1)_{Y}\times{\rm
SU}(2)_{L}\times{\rm SU}(3)_{c}$. But the full elegance of ${\rm Spin}(10)$
emerges only when its elements are recast in terms of spinors [65, 69, 66].
Spinors in ${\rm Spin}(N)$ are indexed by a bitcode with $[N/2]$ bits, each
bit representing a conserved charge, and each bit taking the values
$+\tfrac{1}{2}$ ($\uparrow$) or $-\tfrac{1}{2}$ ($\downarrow$). The 5 charges
of the standard model — hypercharge, weak isospin, and three colours — are
harmoniously re-expressed in terms of a 5-bit bitcode $yzrgb$, with 2 weak
bits $y$ and $z$, and 3 colour bits $r$, $g$, $b$, equations (2).

This paper is motivated by some striking (to the authors) features, §2.3, of
the ${\rm Spin}(10)$ chart (6) of standard model fermions, which seem to
suggest that the ${\rm Spin}(10)$ algebra and the Dirac algebra, which is the
geometric algebra of the Lorentz group ${\rm Spin}(3,1)$, might unify
nontrivially in a common spin algebra. The standard treatment of ${\rm
Spin}(10)$ as a grand unified group takes each of the $2^{5}=32$ fermions of a
generation to be a Weyl fermion, a 2-component chiral (massless) fermion
transforming under ${\rm Spin}(3,1)$, for a total of $2^{6}$ degrees of
freedom. If the ${\rm Spin}(10)$ and ${\rm Spin}(3,1)$ algebras unify
nontrivially, then the unified algebra must have $2^{6}$ spinor degrees of
freedom. Given that there must be 1 time dimension, the spinor algebra must be
that of ${\rm Spin}(11,1)$. This requires adding a 6th bit, the $t$-bit, or
time bit, to the 5 bits $yzrgb$ of ${\rm Spin}(10)$.

The usual assumption, motivated by the Coleman-Mandula no-go theorem [60, 61,
62], is that the generators of a grand unified group such as ${\rm Spin}(10)$
commute with those of spacetime. But that assumption is unnecessarily strong:
the Coleman-Mandula theorem requires only that the generators of unbroken
internal symmetries commute with those of spacetime. In the ${\rm Spin}(11,1)$
model, before grand symmetry breaking the internal group and the spacetime
group are one and the same, so the Coleman-Mandula theorem is satisfied
trivially. After grand symmetry breaking, the ${\rm Spin}(11,1)$ algebra
contains the internal (initally Pati-Salam, then standard-model) and Dirac
algebras as commuting subalgebras, in accordance with the Coleman-Mandula
theorem.

The main result of this paper is the expressions (25) for the vectors of the
Dirac algebra in terms of multivectors of ${\rm Spin}(11,1)$. The Dirac
vectors (25) satisfy the Dirac algebra, and they all commute with all standard
model bivectors modified per (17), in accordance with the Coleman-Mandula
theorem. The modification (17), which is the key trick of this paper, is to
multiply all imaginary standard model bivectors by the colour chiral operator
$\varkappa_{rgb}$, which leaves the standard-model algebra unchanged, but
allows the Dirac algebra to be embedded in the ${\rm Spin}(11,1)$ algebra in a
nontrivial way.

The remainder of the paper, §5–§9, identifies the symmetry breaking chain and
associated Higgs sector that breaks ${\rm Spin}(11,1)$ to the standard model.
The tight fit of the Dirac and standard-model algebras in the ${\rm
Spin}(11,1)$ algebra leaves little freedom of choice. The minimal symmetry
breaking chain (1) is unique, proceeding via the Pati-Salam group. The minimal
Higgs sector is similarly unique, consisting of a 66-component ${\rm
Spin}(11,1)$ scalar multiplet transforming in the adjoint represention. The
same Higgs multiplet mediates all symmetry breakings, including grand, Pati-
Salam, and electroweak. The running of coupling parameters predicts that grand
symmetry breaking occurs at $10^{15}\,{\rm GeV}$, and Pati-Salam symmetry
breaking at $10^{12}\,{\rm GeV}$, equations (86).

The grand Higgs field $\langle\bm{T}\rangle$, equation (46), breaks $t$
symmetry, is available to drive cosmological inflation at the grand unified
scale, and generates a large Majorana mass for the right-handed neutrino by
flipping its $t$-bit. The electroweak Higgs field $\langle\bm{H}\rangle$,
equation (29), breaks $y$ symmetry, and generates fermion masses by flipping
their $y$-bit.

Although the unified algebra is that of ${\rm Spin}(11,1)$, the gauge field
before grand symmetry breaking cannot be the full group ${\rm Spin}(11,1)$,
because if it were, then the subset of ${\rm Spin}(11,1)$ bivectors comprising
the 4 electroweak Higgs bivectors $\bm{\gamma}^{+}_{t}\bm{\gamma}^{\pm}_{k}$,
$k=y,z$, equation (28), would be absorbed into generating masses for the
corresponding bivector gauge fields, whereas in practice those 4 electroweak
Higgs bivectors persist, remaining available to mediate electroweak symmetry
breaking. The solution proposed in §7.4 is that the spatial dimension
$\bm{\gamma}^{+}_{t}$, the partner of the time dimension
$\bm{\gamma}^{-}_{t}$, behaves as a scalar dimension, not participating in any
gauge symmetry, as discussed in §4.4 of [66]. The grand unified group is then
${\rm Spin}(10,1)$, not ${\rm Spin}(11,1)$.

Section 10.1 summarizes the various predictions of the ${\rm Spin}(11,1)$
model, while section 10.2 highlights its possible flaws.

It is beyond the scope of this paper to address whether the proposed ${\rm
Spin}(11,1)$ model might be accommodated in string theory. It can scarcely
escape notice that ${\rm Spin}(10,1)$ has the same number 11 of spacetime
dimensions as maximal supergravity [107, 108, 109, 110], the low-energy limit
of M theory [111, 112], which is the conjectured extension of string theory to
include higher-dimensional objects, branes. Extensions of string theory to 12
spacetime dimensions, F-theory, have also been proposed [113, 114, 115, 116].
String-theory-inspired models usually assume that the parent spacetime is, at
least locally, a product space, consisting of 4 large dimensions multiplied by
a space of compactified or hidden dimensions. By contrast, in the ${\rm
Spin}(11,1)$ geometric algebra, although the time dimension is a vector, each
of the 3 spatial Dirac dimensions is a pentavector, a 5-dimensional
multivector, equations (25). The spatial dimensions share a common
2-dimensional factor, and beyond that are each 3-dimensional. Is this
arrangement viable in string theory?

## Data statement

No new data were created or analysed in this study.

We thank Pierre Ramond, John Baez, John Huerta, and Kirill Krasnov for helpful
conversations and correspondence. This research was supported in part by FQXI
mini-grant FQXI-MGB-1626.

##  Appendix A Dirac algebra

The Dirac algebra is the geometric (Clifford) algebra associated with the
group ${\rm Spin}(3,1)$ of Lorentz transformations in 3+1 spacetime
dimensions. In the chiral representation, the 4 chiral basis spinors
$\bm{\epsilon}_{a}\equiv\\{\bm{\epsilon}_{{{\Uparrow}{\uparrow}}},\,\bm{\epsilon}_{{{\Downarrow}{\downarrow}}},\,\bm{\epsilon}_{{{\Downarrow}{\uparrow}}},\,\bm{\epsilon}_{{{\Uparrow}{\downarrow}}}\\}$
of the Dirac algebra are the column spinors111 The ordering of the Dirac bits
is opposite from that in the construction in §3.1 of [66], and the sign of one
of the spinors is flipped,
$\\{\bm{\epsilon}_{{{\Uparrow}{\uparrow}}},\,\bm{\epsilon}_{{{\Downarrow}{\downarrow}}},\,\bm{\epsilon}_{{{\Downarrow}{\uparrow}}},\,\bm{\epsilon}_{{{\Uparrow}{\downarrow}}}\\}=\\{\bm{\epsilon}_{{{\uparrow}{\uparrow}}},\,\bm{\epsilon}_{{{\downarrow}{\downarrow}}},\,\bm{\epsilon}_{{{\uparrow}{\downarrow}}},\,-\bm{\epsilon}_{{{\downarrow}{\uparrow}}}\\}\
.$ (91) The reordering of bits and the sign flip ensures the chiral and Dirac
representations of the orthonormal Dirac $\gamma$-matrices take a standard
form, equations (95) and (97)

$\bm{\epsilon}_{{{\Uparrow}{\uparrow}}}=\left(\begin{array}[]{c}1\\\ 0\\\ 0\\\
0\end{array}\right)\
,\quad\bm{\epsilon}_{{{\Downarrow}{\downarrow}}}=\left(\begin{array}[]{c}0\\\
1\\\ 0\\\ 0\end{array}\right)\
,\quad\bm{\epsilon}_{{{\Downarrow}{\uparrow}}}=\left(\begin{array}[]{c}0\\\
0\\\ 1\\\ 0\end{array}\right)\
,\quad\bm{\epsilon}_{{{\Uparrow}{\downarrow}}}=\left(\begin{array}[]{c}0\\\
0\\\ 0\\\ 1\end{array}\right)\ .$ (92)

The indices
$\\{{{\Uparrow}{\uparrow}},{{\Downarrow}{\downarrow}},{{\Downarrow}{\uparrow}},{{\Uparrow}{\downarrow}}\\}$
signify the transformation properties of the basis spinors: $\Uparrow$ and
$\Downarrow$ signify boost weight $+\frac{1}{2}$ and $-\frac{1}{2}$, while
$\uparrow$ and $\downarrow$ signify spin weight $+\frac{1}{2}$ and
$-\frac{1}{2}$. The first two spinors,
$\bm{\epsilon}_{{{\Uparrow}{\uparrow}}}$ and
$\bm{\epsilon}_{{{\Downarrow}{\downarrow}}}$, have right-handed Dirac
chirality (even number of up-bits), while the last two,
$\bm{\epsilon}_{{{\Downarrow}{\uparrow}}}$ and
$\bm{\epsilon}_{{{\Uparrow}{\downarrow}}}$, have left-handed Dirac chirality
(odd number of up-bits). Chiral spinors are natively massless.

Dirac basis spinors
$\bm{\epsilon}_{a}\equiv\\{\bm{\epsilon}_{{\sf{T}}{\uparrow}},\,\bm{\epsilon}_{{\sf{T}}{\downarrow}},\,\bm{\epsilon}_{{\mathchoice{\raisebox{0.0pt}{\rotatebox{180.0}{\mbox{$\displaystyle\sf{T}$}}}}{\raisebox{0.0pt}{\rotatebox{180.0}{\mbox{$\textstyle\sf{T}$}}}}{\raisebox{0.0pt}{\rotatebox{180.0}{\mbox{$\scriptstyle\sf{T}$}}}}{\raisebox{0.0pt}{\rotatebox{180.0}{\mbox{$\scriptscriptstyle\sf{T}$}}}}}{\uparrow}},\,\bm{\epsilon}_{{\mathchoice{\raisebox{0.0pt}{\rotatebox{180.0}{\mbox{$\displaystyle\sf{T}$}}}}{\raisebox{0.0pt}{\rotatebox{180.0}{\mbox{$\textstyle\sf{T}$}}}}{\raisebox{0.0pt}{\rotatebox{180.0}{\mbox{$\scriptstyle\sf{T}$}}}}{\raisebox{0.0pt}{\rotatebox{180.0}{\mbox{$\scriptscriptstyle\sf{T}$}}}}}{\downarrow}}\\}$
are massive linear combinations of chiral spinors,

$\underset{\textrm{Dirac}}{\bm{\epsilon}_{a}}=X_{ab}\underset{\textrm{chiral}}{\bm{\epsilon}_{b}}\
,$ (93)

where $X$ is the symmetric ($X=X^{\top}$), unitary ($X^{-1}=X^{\dagger}$)
matrix

$X\equiv{1\over\sqrt{2}}\left(\begin{array}[]{cccc}1&0&-i&0\\\ 0&1&0&-i\\\
-i&0&1&0\\\ 0&-i&0&1\end{array}\right)\ .$ (94)

The Dirac basis spinors $\bm{\epsilon}_{{\sf{T}}{\uparrow}}$ and
$\bm{\epsilon}_{{\sf{T}}{\downarrow}}$ represent massive spinors in their rest
frames, while
$\bm{\epsilon}_{{\mathchoice{\raisebox{0.0pt}{\rotatebox{180.0}{\mbox{$\displaystyle\sf{T}$}}}}{\raisebox{0.0pt}{\rotatebox{180.0}{\mbox{$\textstyle\sf{T}$}}}}{\raisebox{0.0pt}{\rotatebox{180.0}{\mbox{$\scriptstyle\sf{T}$}}}}{\raisebox{0.0pt}{\rotatebox{180.0}{\mbox{$\scriptscriptstyle\sf{T}$}}}}}{\uparrow}}$
and
$\bm{\epsilon}_{{\mathchoice{\raisebox{0.0pt}{\rotatebox{180.0}{\mbox{$\displaystyle\sf{T}$}}}}{\raisebox{0.0pt}{\rotatebox{180.0}{\mbox{$\textstyle\sf{T}$}}}}{\raisebox{0.0pt}{\rotatebox{180.0}{\mbox{$\scriptstyle\sf{T}$}}}}{\raisebox{0.0pt}{\rotatebox{180.0}{\mbox{$\scriptscriptstyle\sf{T}$}}}}}{\downarrow}}$
represent massive antispinors in their rest frames.

Orthonormal vectors
$\\{\bm{\gamma}_{0},\,\bm{\gamma}_{1},\,\bm{\gamma}_{2},\,\bm{\gamma}_{3}\\}$
in the chiral representation are the four $4\times 4$ unitary matrices

$\bm{\gamma}_{0}=\left(\begin{array}[]{cc}0&1\\\ -1&0\end{array}\right)\
,\quad\bm{\gamma}_{a}=\left(\begin{array}[]{cc}0&\sigma_{a}\\\
\sigma_{a}&0\end{array}\right)\ ,$ (95)

where $\sigma_{a}$ are Pauli matrices, and 1 is the $2\times 2$ unit matrix.
The advantage of the representation (95) compared to some others commonly
found in the literature is that the chiral basis vectors in the chiral
representation are purely real, equations (100). The Dirac representation of
orthonormal or chiral basis vectors may be obtained from the chiral
representation by the transformation

$\underset{\textrm{Dirac}}{\bm{\gamma}_{m}}=X\underset{\textrm{chiral}}{\bm{\gamma}_{m}}X^{-1}\
,$ (96)

with $X$ from equation (94). The orthonormal vectors in the Dirac
representation are

$\bm{\gamma}_{0}=i\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right)\
,\quad\bm{\gamma}_{a}=\left(\begin{array}[]{cc}0&\sigma_{a}\\\
\sigma_{a}&0\end{array}\right)\ .$ (97)

In the notation $\bm{\gamma}^{\pm}_{k}$ of paired orthonormal vectors used
elsewhere in this paper, equations (10), the orthonormal vectors
$\bm{\gamma}_{a}$ are

$\\{\bm{\gamma}_{1},\,\bm{\gamma}_{2},\,\bm{\gamma}_{3},\,\bm{\gamma}_{0}\\}=\\{\bm{\gamma}_{1}^{+},\,\bm{\gamma}_{1}^{-},\,\bm{\gamma}_{2}^{+},\,i\bm{\gamma}_{2}^{-}\\}\
.$ (98)

The bivectors $\bm{\sigma}_{a}$ and $I\bm{\sigma}_{a}$ ($\bm{\sigma}_{a}$ are
$4\times 4$ matrices satisfying the same algebra as Pauli matrices) and the
pseudoscalar $I$ in the chiral representation are

$\bm{\gamma}_{0}\bm{\gamma}_{a}=\bm{\sigma}_{a}\equiv\left(\begin{array}[]{cc}\sigma_{a}&0\\\
0&-\sigma_{a}\end{array}\right)\
,\quad\tfrac{1}{2}\varepsilon_{abc}\bm{\gamma}_{b}\bm{\gamma}_{c}=I\bm{\sigma}_{a}=i\left(\begin{array}[]{cc}\sigma_{a}&0\\\
0&\sigma_{a}\end{array}\right)\ ,\quad I=i\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)\ .$ (99)

Chiral basis vectors in the chiral representation, commonly called Newman-
Penrose basis vectors, are the real matrices

$\bm{\gamma}_{v}=\left(\begin{array}[]{cc}0&\sigma_{v}\\\
-\sigma_{u}&0\end{array}\right)\
,\quad\bm{\gamma}_{u}=\left(\begin{array}[]{cc}0&\sigma_{u}\\\
-\sigma_{v}&0\end{array}\right)\
,\quad\bm{\gamma}_{+}=\left(\begin{array}[]{cc}0&\sigma_{+}\\\
\sigma_{+}&0\end{array}\right)\
,\quad\bm{\gamma}_{-}=\left(\begin{array}[]{cc}0&\sigma_{-}\\\
\sigma_{-}&0\end{array}\right)\ ,$ (100)

where $\sigma_{m}$ are the Newman-Penrose Pauli matrices

$\displaystyle\sigma_{v}\equiv{1\over\sqrt{2}}\left(1+\sigma_{3}\right)=\sqrt{2}\left(\begin{array}[]{cc}1&0\\\
0&0\end{array}\right)\ $
$\displaystyle,\quad\sigma_{u}\equiv{1\over\sqrt{2}}\left(1-\sigma_{3}\right)=\sqrt{2}\left(\begin{array}[]{cc}0&0\\\
0&1\end{array}\right)\ ,$ (101e)
$\displaystyle\sigma_{+}\equiv{1\over\sqrt{2}}\left(\sigma_{1}+i\sigma_{2}\right)=\sqrt{2}\left(\begin{array}[]{cc}0&1\\\
0&0\end{array}\right)\ $
$\displaystyle,\quad\sigma_{-}\equiv{1\over\sqrt{2}}\left(\sigma_{1}-i\sigma_{2}\right)=\sqrt{2}\left(\begin{array}[]{cc}0&0\\\
1&0\end{array}\right)\ .$ (101j)

The labelling $v$, $u$, $+$, $-$ for the chiral basis vectors is conventional
in the Newman-Penrose community. In the notation $\bm{\gamma}_{k}$ and
$\bm{\gamma}_{\bar{k}}$ of paired chiral vectors used elsewhere in this paper,
equations (10), the chiral (Newman-Penrose) basis vectors $\bm{\gamma}_{a}$
are

$\\{\bm{\gamma}_{+},\,\bm{\gamma}_{-},\,\bm{\gamma}_{v},\,\bm{\gamma}_{u}\\}=\\{\bm{\gamma}_{1},\,\bm{\gamma}_{\bar{1}},\,\bm{\gamma}_{2},\,-\bm{\gamma}_{\bar{2}}\\}\
.$ (102)

The spinor metric $\varepsilon$ in the chiral representation is [66]

$\varepsilon=I\bm{\sigma}_{2}=\left(\begin{array}[]{cccc}0&1&0&0\\\
-1&0&0&0\\\ 0&0&0&1\\\ 0&0&-1&0\end{array}\right)\ .$ (103)

The expression for $\varepsilon$ shows that the spinor metric flips boost and
spin, leaving chirality unchanged. The conjugation operator is related to the
spinor metric by $C=-i\varepsilon\bm{\gamma}_{0}^{\top}$. The conjugation
operator in the chiral representation is

$C=iI\bm{\gamma}_{2}=\left(\begin{array}[]{cccc}0&0&0&i\\\ 0&0&-i&0\\\
0&-i&0&0\\\ i&0&0&0\end{array}\right)\ .$ (104)

The expression for $C$ shows that the conjugation operator flips chirality and
spin, leaving boost unchanged. The expressions for the spinor metric and
conjugation operator in terms of multivectors are to be understood as holding
in the particular representation. The spinor metric and conjugation operators
do not transform as multivectors under Lorentz transformations; rather, they
are Lorentz invariant, unchanged under Lorentz transformations.

##  Appendix B Naming of bits

This paper adopts the notation $t,y,z,r,g,b$ for the 6 bits of ${\rm
Spin}(11,1)$ spinors, consisting of the time bit $t$, two weak bits $y,z$, and
three colour bits $r,g,b$. [65] used $r,w,b$ (red, white, blue) for the colour
bits, and (arbitrarily) $p,g$ (purple, green) for the weak bits. [69] adopted
$r,g,b$ (red, green, blue) for the colour bits, and $d,u$ (down, up) for the
weak bits, on the grounds that down and up quarks $d^{c}_{L}$ and $u^{c}_{L}$
have up bits respectively $dc$ and $uc$ (with $c$ one of $r,g,b$), per the
${\rm Spin}(10)$ chart (6).

The present paper follows [69] for the colour bits, since it is consistent
with the elegant mental picture that equal amounts of the three colours
$r,g,b$ are colourless, as are all free fermionic bound states at low energy
(leptons, baryons, mesons). However, [69]’s adoption of $d,u$ for the weak
bits is potentially confusing, first because each bit $d$ or $u$ can itself be
either down or up, and second because $d$ and $u$ might be misinterpreted as
measuring the number of $d$ and $u$ quarks, potentially confusing weak isospin
with isospin.

The present paper chooses $y$, $z$ for the weak bits in part because it
reduces the potential for misinterpretation, and in part because $y$ and $z$
are infrared bands to be used by the Vera Rubin Observatory (formerly the
LSST) [117], for which astronomical first light is expected in 2024. The
sequence $yzrgb$ is in (inverse) order of wavelength,

$\\{y,z,r,g,b\\}\sim\\{1000,900,600,500,400\\}\,{\rm nm}\ .$ (105)

Note that hypercharge $Y$ carries one unit of $y$-charge, in addition to other
charges, equation (2a).

##  Appendix C Gauge fields of subgroups of ${\rm Spin}(10)$

The standard-model gauge group ${\rm U}(1)_{Y}\times{\rm SU}(2)_{L}\times{\rm
SU}(3)_{c}$ is a subgroup of the ${\rm SU}(5)$ subgroup of ${\rm Spin}(10)$.
The gauge fields (generators) of ${\rm SU}(5)$ comprise the subset of gauge
fields of ${\rm Spin}(10)$ that leave the number of up-bits of a spinor
unchanged. The gauge bivectors of ${\rm SU}(n)$ with $n=5$ constitute
$(n{+}1)(n{-}1)=24$ bivectors comprising the $n(n-1)=20$ off-diagonal
bivectors

$\displaystyle\tfrac{1}{2}(1-\varkappa_{kl})\bm{\gamma}^{+}_{k}\bm{\gamma}^{+}_{l}$
$\displaystyle=\tfrac{1}{2}(\bm{\gamma}^{+}_{k}\bm{\gamma}^{+}_{l}+\bm{\gamma}^{-}_{k}\bm{\gamma}^{-}_{l})$
$\displaystyle=\tfrac{1}{2}(\bm{\gamma}_{k}\operatorname{\wedge}\bm{\gamma}_{\bar{l}}+\bm{\gamma}_{\bar{k}}\operatorname{\wedge}\bm{\gamma}_{l})\
,$ (106a)
$\displaystyle\tfrac{1}{2}(1-\varkappa_{kl})\bm{\gamma}^{+}_{k}\bm{\gamma}^{-}_{l}$
$\displaystyle=\tfrac{1}{2}(\bm{\gamma}^{+}_{k}\bm{\gamma}^{-}_{l}-\bm{\gamma}^{-}_{k}\bm{\gamma}^{+}_{l})$
$\displaystyle=\tfrac{i}{2}(\bm{\gamma}_{k}\operatorname{\wedge}\bm{\gamma}_{\bar{l}}-\bm{\gamma}_{\bar{k}}\operatorname{\wedge}\bm{\gamma}_{l})\
,$ (106b)

and the $n{-}1=4$ diagonal bivectors

$\tfrac{1}{2}\,\bm{\gamma}^{+}_{k}\bm{\gamma}^{-}_{k}=\tfrac{i}{2}\,\bm{\gamma}_{k}\operatorname{\wedge}\bm{\gamma}_{\bar{k}}\quad\mbox{modulo}\quad\tfrac{1}{2}\sum_{k}\bm{\gamma}^{+}_{k}\bm{\gamma}^{-}_{k}=\tfrac{i}{2}\sum_{k}\bm{\gamma}_{k}\operatorname{\wedge}\bm{\gamma}_{\bar{k}}\
,$ (107)

with indices $k$ and $l$ running over $y$, $z$, $r$, $g$, $b$. The quantity
$\varkappa_{kl}\equiv\bm{\gamma}_{k}\operatorname{\wedge}\bm{\gamma}_{\bar{k}}\operatorname{\wedge}\bm{\gamma}_{l}\operatorname{\wedge}\bm{\gamma}_{\bar{l}}=-\bm{\gamma}^{+}_{k}\bm{\gamma}^{-}_{k}\bm{\gamma}^{+}_{l}\bm{\gamma}^{-}_{l}$
in equations (106) is the $kl$ chiral operator. The factor
$\tfrac{1}{2}(1-\varkappa_{kl})$ is a projection operator, whose square is
itself, which serves to project its argument into the space where the sum of
$k$ and $l$ charges is zero. The S in ${\rm SU}(5)$ restricts to ${\rm U}(5)$
matrices of unit determinant, effectively removing the bivector
$\tfrac{1}{2}\sum_{k}\bm{\gamma}^{+}_{k}\bm{\gamma}^{-}_{k}$ that rotates all
spinors in an ${\rm SU}(5)$ multiplet by a common phase.

The bivectors of ${\rm Spin}(10)$ that are not in ${\rm SU}(5)$ are the 20
off-diagonal bivectors

$\displaystyle\tfrac{1}{2}(1+\varkappa_{kl})\bm{\gamma}^{+}_{k}\bm{\gamma}^{+}_{l}$
$\displaystyle=\tfrac{1}{2}(\bm{\gamma}^{+}_{k}\bm{\gamma}^{+}_{l}-\bm{\gamma}^{-}_{k}\bm{\gamma}^{-}_{l})$
$\displaystyle=\tfrac{1}{2}(\bm{\gamma}_{k}\operatorname{\wedge}\bm{\gamma}_{l}+\bm{\gamma}_{\bar{k}}\operatorname{\wedge}\bm{\gamma}_{\bar{l}})\
,$ (108a)
$\displaystyle\tfrac{1}{2}(1+\varkappa_{kl})\bm{\gamma}^{+}_{k}\bm{\gamma}^{-}_{l}$
$\displaystyle=\tfrac{1}{2}(\bm{\gamma}^{+}_{k}\bm{\gamma}^{-}_{l}+\bm{\gamma}^{-}_{k}\bm{\gamma}^{+}_{l})$
$\displaystyle=-\tfrac{i}{2}(\bm{\gamma}_{k}\operatorname{\wedge}\bm{\gamma}_{l}-\bm{\gamma}_{\bar{k}}\operatorname{\wedge}\bm{\gamma}_{\bar{l}})\
,$ (108b)

and the 1 diagonal bivector

$iX\equiv\tfrac{1}{2}\sum_{k}\bm{\gamma}^{+}_{k}\bm{\gamma}^{-}_{k}=\tfrac{i}{2}\sum_{k}\bm{\gamma}_{k}\operatorname{\wedge}\bm{\gamma}_{\bar{k}}\
,$ (109)

with indices $k$ running over $y,z,r,g,b$. The bivector $X$ measures total
$yzrgb$ charge $y+z+r+g+b$, and $iX$ is the generator of the ${\rm U}(1)$
factor that would complete ${\rm SU}(5)$ to ${\rm U}(5)$.

The 6 bivectors involving the weak charges $y$ and $z$ generate the weak group
${\rm Spin}(4)_{w}={\rm SU}(2)_{R}\times{\rm SU}(2)_{L}$, a product of right-
and left-handed components. The subgroup of ${\rm Spin}(4)_{w}$ that preserves
the number of $yz$ up-bits is the standard-model left-handed component ${\rm
SU}(2)_{L}$. The bivectors of the left-handed weak group ${\rm SU}(2)_{L}$
comprise the $2{+}(2{-}1)=3$ bivectors (106) and (107) with $k$ and $l$
running over $y$ and $z$. The bivectors are equivalent to $i\tau_{k}$ with
$\tau_{k}$ satisfying the Pauli algebra,

$\displaystyle i\tau_{1}$
$\displaystyle\equiv\phantom{-}\tfrac{1}{2}(1-\varkappa_{yz})\bm{\gamma}^{+}_{y}\bm{\gamma}^{-}_{z}$
$\displaystyle=\phantom{-}\tfrac{1}{2}(\bm{\gamma}^{+}_{y}\bm{\gamma}^{-}_{z}-\bm{\gamma}^{-}_{y}\bm{\gamma}^{+}_{z})$
$\displaystyle=\phantom{-}\tfrac{i}{2}(\bm{\gamma}_{y}\operatorname{\wedge}\bm{\gamma}_{\bar{z}}+\bm{\gamma}_{z}\operatorname{\wedge}\bm{\gamma}_{\bar{y}})\
,$ (110a) $\displaystyle i\tau_{2}$
$\displaystyle\equiv-\tfrac{1}{2}(1-\varkappa_{yz})\bm{\gamma}^{+}_{y}\bm{\gamma}^{+}_{z}$
$\displaystyle=-\tfrac{1}{2}(\bm{\gamma}^{+}_{y}\bm{\gamma}^{+}_{z}+\bm{\gamma}^{-}_{y}\bm{\gamma}^{-}_{z})$
$\displaystyle=-\tfrac{1}{2}(\bm{\gamma}_{y}\operatorname{\wedge}\bm{\gamma}_{\bar{z}}-\bm{\gamma}_{z}\operatorname{\wedge}\bm{\gamma}_{\bar{y}})\
,$ (110b) $\displaystyle i\tau_{3}$
$\displaystyle\equiv-\tfrac{1}{2}(1-\varkappa_{yz})\bm{\gamma}^{+}_{y}\bm{\gamma}^{-}_{y}$
$\displaystyle=-\tfrac{1}{2}(\bm{\gamma}^{+}_{y}\bm{\gamma}^{-}_{y}-\bm{\gamma}^{+}_{z}\bm{\gamma}^{-}_{z})$
$\displaystyle=\phantom{-}\tfrac{i}{2}(\bm{\gamma}_{z}\operatorname{\wedge}\bm{\gamma}_{\bar{z}}-\bm{\gamma}_{y}\operatorname{\wedge}\bm{\gamma}_{\bar{y}})\
,$ (110c)

where
$\varkappa_{yz}\equiv\bm{\gamma}_{y}\operatorname{\wedge}\bm{\gamma}_{\bar{y}}\operatorname{\wedge}\bm{\gamma}_{z}\operatorname{\wedge}\bm{\gamma}_{\bar{z}}$
is the weak chiral operator. The left-handed projection operator
$\tfrac{1}{2}(1-\varkappa_{yz})$ equals 1 acting on left-handed weak chiral
states, and vanishes acting on right-handed weak chiral states. The bivectors
of the right-handed weak group ${\rm SU}(2)_{R}$ comprise 3 bivectors
equivalent to $i\rho_{k}$ with $\rho_{k}$ satisfying the Pauli algebra,

$\displaystyle i\rho_{1}$
$\displaystyle\equiv-\tfrac{1}{2}(1+\varkappa_{yz})\bm{\gamma}^{+}_{y}\bm{\gamma}^{-}_{z}$
$\displaystyle=-\tfrac{1}{2}(\bm{\gamma}^{+}_{y}\bm{\gamma}^{-}_{z}+\bm{\gamma}^{-}_{y}\bm{\gamma}^{+}_{z})$
$\displaystyle=\tfrac{i}{2}(\bm{\gamma}_{y}\operatorname{\wedge}\bm{\gamma}_{z}-\bm{\gamma}_{\bar{y}}\operatorname{\wedge}\bm{\gamma}_{\bar{z}})\
,$ (111a) $\displaystyle i\rho_{2}$
$\displaystyle\equiv\phantom{-}\tfrac{1}{2}(1+\varkappa_{yz})\bm{\gamma}^{+}_{y}\bm{\gamma}^{+}_{z}$
$\displaystyle=\phantom{-}\tfrac{1}{2}(\bm{\gamma}^{+}_{y}\bm{\gamma}^{+}_{z}-\bm{\gamma}^{-}_{y}\bm{\gamma}^{-}_{z})$
$\displaystyle=\tfrac{1}{2}(\bm{\gamma}_{y}\operatorname{\wedge}\bm{\gamma}_{z}+\bm{\gamma}_{\bar{y}}\operatorname{\wedge}\bm{\gamma}_{\bar{z}})\
,$ (111b) $\displaystyle i\rho_{3}$
$\displaystyle\equiv\phantom{-}\tfrac{1}{2}(1+\varkappa_{yz})\bm{\gamma}^{+}_{y}\bm{\gamma}^{-}_{y}$
$\displaystyle=\phantom{-}\tfrac{1}{2}(\bm{\gamma}^{+}_{y}\bm{\gamma}^{-}_{y}+\bm{\gamma}^{+}_{z}\bm{\gamma}^{-}_{z})$
$\displaystyle=\tfrac{i}{2}(\bm{\gamma}_{y}\operatorname{\wedge}\bm{\gamma}_{\bar{y}}+\bm{\gamma}_{z}\operatorname{\wedge}\bm{\gamma}_{\bar{z}})\
.$ (111c)

The right-handed projection operator $\tfrac{1}{2}(1+\varkappa_{yz})$ equals 1
acting on right-handed weak chiral states, and vanishes acting on left-handed
weak chiral states. The expressions for the bivectors $i\rho_{1}$ and
$i\rho_{2}$ are equivalent to equations (108b) and (108a) with indices $k$ and
$l$ equal to $y$ and $z$. The diagonal bivector $i\rho_{3}$ measures the total
$yz$ charge $y+z$,

$i\rho_{3}=\tfrac{1}{2}\sum_{k=y,z}\bm{\gamma}^{+}_{k}\bm{\gamma}^{-}_{k}=\tfrac{i}{2}\sum_{k=y,z}\bm{\gamma}_{k}\operatorname{\wedge}\bm{\gamma}_{\bar{k}}\
.$ (112)

The 15 bivectors involving the colour charges $r$, $g$, and $b$ generate the
colour group ${\rm Spin}(6)_{c}$. The subgroup of ${\rm Spin}(6)_{c}$ that
preserves the number of $rgb$ up-bits is the standard-model colour group ${\rm
SU}(3)_{c}$. The bivectors of the colour group ${\rm SU}(3)_{c}$ comprise the
$6{+}(3{-}1)=8$ bivectors (106) and (107) with $k$ and $l$ running over $r$,
$g$, and $b$. The 7 other bivectors of ${\rm Spin}(6)_{c}$ consist of the 6
leptoquark bivectors given by equations (108) with $k,l$ drawn from pairs of
distinct colour indices $r,g,b$, and the 1 diagonal bivector that measures the
total $rgb$ charge $r+g+b$,

$-i\tfrac{3}{2}(B-L)=\tfrac{1}{2}\sum_{k=r,g,b}\bm{\gamma}^{+}_{k}\bm{\gamma}^{-}_{k}=\tfrac{i}{2}\sum_{k=r,g,b}\bm{\gamma}_{k}\operatorname{\wedge}\bm{\gamma}_{\bar{k}}\
.$ (113)

The 1 hypercharge bivector, the generator of ${\rm U}(1)_{Y}$, is defined to
be the bivector whose eigenvalue is $iY$ where $Y$ is the hypercharge,
equation (2a),

$iY\equiv\tfrac{1}{2}\\!\sum_{k=y,z}\bm{\gamma}^{+}_{k}\bm{\gamma}^{-}_{k}-\tfrac{1}{3}\\!\sum_{k=r,g,b}\bm{\gamma}^{+}_{k}\bm{\gamma}^{-}_{k}=i\Biggl{(}\tfrac{1}{2}\\!\sum_{k=y,z}\bm{\gamma}_{k}\operatorname{\wedge}\bm{\gamma}_{\bar{k}}-\tfrac{1}{3}\\!\sum_{k=r,g,b}\bm{\gamma}_{k}\operatorname{\wedge}\bm{\gamma}_{\bar{k}}\Biggr{)}\
.$ (114)

The 1 electromagnetic charge bivector, the generator of ${\rm U}(1)_{Q}$, is
defined to be the bivector whose eigenvalue is $iQ$ where $Q$ is the electric
charge, equation (5),

$iQ\equiv\tfrac{1}{2}\bm{\gamma}^{+}_{z}\bm{\gamma}^{-}_{z}-\tfrac{1}{6}\\!\sum_{k=r,g,b}\bm{\gamma}^{+}_{k}\bm{\gamma}^{-}_{k}=i\Biggl{(}\tfrac{1}{2}\bm{\gamma}_{z}\operatorname{\wedge}\bm{\gamma}_{\bar{z}}-\tfrac{1}{6}\\!\sum_{k=r,g,b}\bm{\gamma}_{k}\operatorname{\wedge}\bm{\gamma}_{\bar{k}}\Biggr{)}\
.$ (115)

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