Editing Grand Unified Theory (section)

=== Symplectic groups and quaternion representations ===
Symplectic gauge groups could also be considered. For example, {{math|Sp(8)}} (which is called {{math|Sp(4)}} in the article [[symplectic group]]) has a representation in terms of {{math|4 × 4}} quaternion unitary matrices which has a {{math|'''16'''}} dimensional real representation and so might be considered as a candidate for a gauge group. {{math|Sp(8)}} has 32 charged bosons and 4 neutral bosons. Its subgroups include {{math|SU(4)}} so can at least contain the gluons and photon of {{math|SU(3) × U(1)}}. Although it's probably not possible to have weak bosons acting on chiral fermions in this representation. A quaternion representation of the fermions might be:
: <math>
\begin{bmatrix}
e   + i\ \overline{e}            + j\ v             + k\ \overline{v} \\
u_r + i\ \overline{u}_\mathrm{\overline r} + j\ d_\mathrm{r} + k\ \overline{d}_\mathrm{\overline r} \\
u_g + i\ \overline{u}_\mathrm{\overline g} + j\ d_\mathrm{g} + k\ \overline{d}_\mathrm{\overline g} \\
u_b + i\ \overline{u}_\mathrm{\overline b} + j\ d_\mathrm{b} + k\ \overline{d}_\mathrm{\overline b} \\
\end{bmatrix}_\mathrm{L}
</math>

A further complication with [[quaternion]] representations of fermions is that there are two types of multiplication: left multiplication and right multiplication which must be taken into account. It turns out that including left and right-handed {{math|4 × 4}} quaternion matrices is equivalent to including a single right-multiplication by a unit quaternion which adds an extra SU(2) and so has an extra neutral boson and two more charged bosons. Thus the group of left- and right-handed {{math|4 × 4}} quaternion matrices is {{math|Sp(8) × SU(2)}} which does include the standard model bosons:
: <math> \mathrm{ SU(4,\mathbb{H})_L\times \mathbb{H}_R = Sp(8)\times SU(2) \supset SU(4)\times SU(2) \supset SU(3)\times SU(2)\times U(1) }</math>

If <math>\psi</math> is a quaternion valued spinor, <math>A^{ab}_\mu</math> is quaternion hermitian {{math|4 × 4}} matrix coming from {{math|Sp(8)}} and <math>B_\mu</math> is a pure vector quaternion (both of which are 4-vector bosons) then the interaction term is:
: <math>\ \overline{\psi^{a}} \gamma_\mu\left( A^{ab}_\mu\psi^b + \psi^a B_\mu \right)\ </math>