1.4.2 SU(2) as the Inscribed Sphere of the Light Cone

The $SU(2)$ matrix group plays a piv­otal role in physics and the Stan­dard Model and has a very nat­ur­al iso­mor­phic map­ping to the quater­nions. The group can be rep­re­sent­ed in its com­plex form, as the fol­low­ing $2 \times 2$ matrix

Its Lie al­ge­bra is com­posed of $2 \times 2$ skew-Her­mit­ian ma­tri­ces with traces equal to zero, rep­re­sent­ed as

and has a Lie al­ge­bra gen­er­at­ed by the fol­low­ing ma­tri­ces

These al­ge­bras can be mapped to the quater­nion form by chang­ing the sign of the $u_{2}$ matrix

The fol­low­ing quater­nion­ic re­la­tion­ships can be ver­i­fied for this map­ping by set­ting $u_{3} = \widehat{i},\ {- u}_{2} = \widehat{j},\ u_{1} = \widehat{k}$, for which the fol­low­ing prod­uct re­la­tion­ships are found to hold

In Figure 1.4.3, we see a sphere of unit radius, cen­tered at the point, $O' = \left( \sqrt{2},\ 0,0 \right)$, that touch­es the cone at a cir­cum­cir­cu­lar arc rep­re­sent­ed on the sphere by $BCB'C'$. The arc is com­plex-valued and the seg­ments $OB,\ OC,\ OB',\ OC'$, are all unit dis­tance, $\left| O(B,C,B',C') \right| = 1$, from the origin of the space, at $O = (0,0,0)$. Each point, $B,C,B',C'$, is a tan­gent point of the unit sphere where it touch­es the cone. The points $B = e^{\left( \frac{i\pi}{4} \right)},C = e^{\left( \frac{- i\pi}{4} \right)}$, lay in $X_{1}$-plane, and points $B' = e^{\left( \frac{i\pi}{4} \right)}$, $C' = e^{\left( 3\frac{i\pi}{4} \right)}$, lay in the $X_{3}$-plane, and all of these rep­re­sent the point of in­ter­sec­tion where the $D_{3\mathbb{z}}$ as­ymp­tot­ic rep­re­sen­ta­tions meet the $U(3)$ unit sphere in the $X_{1},X_{3}$-planes.

Since $B$ is a tan­gent point, the angle $\angle O'BO$ is a right-angle equal to $\frac{\pi}{2}$ ra­di­ans, as is the angle $\angle OrA$. We know the cone in­ter­sects the $U(3)$ unit sphere at the point $B$, which is $\frac{\pi}{4}$ ra­di­ans from the real-axis, there­fore angles $\angle BOO',\angle BO'O,\ \angle AOr,\ \angle rOA$ are also $\frac{\pi}{4}$ ra­di­ans in value. In terms of pro­por­tion­al­i­ty, $\overline{OO'} \propto \overline{AO}$ and $\overline{O'B} \propto \overline{Ar}$, which we ex­press as the re­la­tion­ship

For the chord dis­tance be­tween the points $B,B'$in the $X_{1}$-plane, we find imag­i­nary values for both di­rec­tion­al ori­en­ta­tions

which is due to them being sep­a­rat­ed by the imag­i­nary-valued por­tion of the $X_{1}$-plane. For points $C,C'$in the $X_{3}$-plane, we find real values for both

We define the seg­ments $\overline{BB'},$ $\overline{CC'}$ in matrix forms as $\Delta X_{1},\Delta X_{3}$, re­spec­tive­ly

which rep­re­sent the two points $B,B'$of the $X_{1}$-plane along the an­tidi­ag­o­nal, and points $C,C'$of the $X_{3}$-plane along the main di­ag­o­nal. The radial mod­u­lus is normed and rep­re­sent­ed as $\sqrt{r}$, since we will be taking prod­ucts of $\Delta X_{1},\Delta X_{3}$. These ma­tri­ces are arranged to have points that are clock­wise from the $X_{1}$-axis in the first row, and points coun­ter­clock­wise from the $X_{1}$-axis in the second row for each. We will con­tin­ue to main­tain this arrange­ment for our other in­scribed sphere groups for con­sis­ten­cy. In Figure 1.4.4, $BB',\ CC'$are rep­re­sent­ed as chords on the in­ter­sec­tion of the $SU(2),U(3)$ spheres.

Taking the ma­tri­ces $\Delta X_{1},\Delta X_{3},$and set­ting $r = 1 \rightarrow \sqrt{r} = 1$, there are four pos­si­ble and not nec­es­sar­i­ly com­mut­ing prod­uct com­bi­na­tions, which are

$\Delta X_{1}^{2}$=$\begin{pmatrix} 0 & e^{- \frac{i\pi}{4}} \\ e^{\frac{i\pi}{4}} & 0 \end{pmatrix}^{2} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \epsilon,$

We should im­me­di­ate­ly rec­og­nize these as the el­e­ments of the $SU(2)$ group which map to the quater­nions, and taking them as a sum of al­ge­bras, we find

yields a result that is iso­mor­phic to the quater­nions, which we denote as $SU_{H}(2)$.

$U(3)$ sphere cen­tered at the origin, $\Delta X_{1} = BB'$, $\Delta X_{3} = CC'$.

When taking prod­ucts of $\Delta X_{1},\Delta X_{3}$ with the $u_{j}$ group el­e­ments of $SU_{H}(2)$, we find there are eight dis­tinct ma­tri­ces of the form $\Delta X_{j,k}$, where the sub­script $(j,k)$ de­scribes the $X_{k}$-plane it rep­re­sents and the it­er­a­tion value. Below we can see that $\Delta X_{1,2},\Delta X_{3,2}$ are the re­flec­tions about the main di­ag­o­nal of $\Delta X_{1,1},\Delta X_{3,1}$, the ma­tri­ces $\Delta X_{1,3},\Delta X_{3,3}$, rep­re­sent the chords of the op­pos­ing side of the sphere for $\Delta X_{1,1},\Delta X_{3,1}$, and $\Delta X_{1,4},\Delta X_{3,4}$ as their re­spec­tive re­flec­tions. When taken to­geth­er, the com­po­si­tion of the dual re­flec­tions $\Delta X_{1,1},\Delta X_{3,1} \rightarrow$ $\Delta X_{1,2},\Delta X_{3,2}$, rep­re­sents a ro­ta­tion of by $\pi$ ra­di­ans of the group. This ro­ta­tion, is es­sen­tial­ly, a flip­ping of the sphere from the up/down to down/up po­si­tion, seen in Figure 1.4.5, where the point po­si­tions for $E$ and $G$ have been swapped. This will be the same case for the swap­ping of the point po­si­tions, $H$ and $I$

Ex­plor­ing fur­ther, when taking per­mu­ta­tion el­e­ments from co­in­cid­ing it­er­a­tion pairs $\Delta X_{1,k},\Delta X_{3,k}$, and eval­u­at­ing their prod­ucts $\Delta X_{1,k}^{2},\ \Delta X_{3,k}^{2},\Delta X_{1,k}\Delta X_{3,k},\Delta X_{3,k}\Delta X_{1,k},$we find the fol­low­ing $SU(2)$-type group forms

In Eq. 1.4.26, there are two dis­tinct group rep­re­sen­ta­tions, the first is our $SU_{H}(2)$ form

with in­di­vid­ual ma­tri­ces of the form

which are con­sis­tent with the Lie al­ge­bra $\mathfrak{su}(2)$, as $SU_{H}(2)\rightarrow \mathfrak{s}\mathfrak{u}_{H}(2) = \begin{pmatrix} ia & z \\ - \overline{z} & - ia \end{pmatrix}.$The other group we denote, as $SU_{H}(2)',$is ro­tat­ed from $\Delta X_{1,1},\Delta X_{3,1} \rightarrow \Delta X_{1,2},\Delta X_{3,2}$, and the con­ju­gate mirror $\ \Delta X_{1,1},\Delta X_{3,1} \rightarrow \Delta X_{1,3},\Delta X_{3,3}$, where

has for its matrix el­e­ments

and we denote the Lie al­ge­bra $\mathfrak{s}\mathfrak{u}_{H}^{'}(2)$, as

For both forms, the fol­low­ing prod­uct re­la­tion­ships hold

which are the quater­nion prod­uct re­la­tion­ships for $u_{3} = \widehat{i},\ u_{2} = \widehat{j},\ u_{1} = \widehat{k}.$We can con­tin­ue this process around the $U(3)$ sphere to find the other re­la­tion­ships as to how the planes $X_{1},X_{2},X_{3}$ are arranged with re­spect to one an­oth­er, where the $\Delta X_{1,k},\Delta X_{3,k}$ groups are like the $\Delta X_{3,k},\Delta X_{2,k}$ chord matrix groups, which we rep­re­sent, as

The $\Delta X_{2,k}$ chords are re­flect­ed in the $\Delta X_{2,k},\Delta X_{1,k}$ prod­ucts, when com­pared to the prod­uct ex­am­ples for$\ \Delta X_{1,k}\Delta X_{3,k}$

The $SU(2)$ forms of the $\Delta X_{3,k},\Delta X_{2,k}$ prod­ucts are

while the $\Delta X_{2,k},\Delta X_{1,k}$ groups have an anti-sym­met­ric re­la­tion­ship to the pre­vi­ous groups in terms of the prod­ucts of their $u_{j}$ el­e­ments, and are de­not­ed in terms of their $SU(2),SU(2)'$group forms, as

The arrange­ment of in­scribed spheres is shown in Figure 1.4.7, where the red spheres are in­ter­sect­ed by the $X_{1}$-axis and rep­re­sent $\Delta X_{1,k},\Delta X_{3,k}$, green spheres are in­ter­sect­ed by the $X_{2}$-axis and rep­re­sent $\Delta X_{2,k},\Delta X_{1,k}$, and the blue spheres are in­ter­sect­ed by the $X_{3}$-axis and rep­re­sent $\Delta X_{3,k},\Delta X_{2,k}$. Other forms of the $SU(2)$ group can be found to exist with com­bi­na­tions of $\Delta X_{j,k},\Delta X_{j \pm 1,k \pm n}$, for$\ n= \mathbb{Z}$, where taking the ex­am­ple $\Delta X_{1,1},\Delta X_{3,4}$, we find the prod­uct pairs are

which yields an $SU(2)$ group of the form

Eq. 1.4.37 rep­re­sents an $SU(2)$ group of the form

for which all the fol­low­ing still hold

Cal­cu­lat­ing $\Delta X_{1,1},\Delta X_{3,3}$, would equate to find­ing the prod­ucts of the matrix chord $\Delta X_{1,1}$ with the mirror of $\Delta X_{3,1}$, yield­ing the al­ter­nate al­ge­bras

which com­pris­es an $SU(2)$ group of the form

for which the fol­low­ing still hold

This group dif­fers, in that the prod­ucts of the $u_{j}$ el­e­ments have become neg­a­tive/re­versed

Sim­i­lar prod­uct cal­cu­la­tions for $\Delta X_{1,2},\Delta X_{3,4}$, result in the group

com­prised of the el­e­ments

whose inter-prod­ucts will have the same re­ver­sals of the afore­men­tioned $\Delta X_{1,1},\Delta X_{3,2}$ group

From the stand­point of the $u_{j}$ ma­tri­ces, we see that they are both pos­i­tive and neg­a­tive in their $SU(2)$ group re­sults, and we will have a total of $2^{3} = 8$ dif­fer­ent pos­si­ble forms of $SU(2)$, which we denote as the $\mathcal{SU}(2)$ set

all of which have de­ter­mi­nants equal to $a^{2} + b^{2} + c^{2} + d^{2}$. The groups with as­ter­isks in the $\mathcal{SU}(2)$ set hold to the re­verse prod­ucts

and the groups in bold font denote the in­scribed spheres in the cones that we have been con­sid­er­ing. The other groups $\in \mathcal{SU}(2)$ can be gen­er­at­ed by chords that span from one cone to an­oth­er, and two ex­am­ples are

which yield groups $\in \mathcal{SU}(2)$, not from the in­scribed sphere groups. We can also note that the $\mathcal{SU}(2)$ sub­group

is formed as the result of chords on op­po­site sides of the sphere