1.4.10 The Pauli Matrices Subgroup of the Symmetric Conic Group

In physics, the Pauli ma­tri­ces are rou­tine­ly used in quan­tum me­chan­ics to rep­re­sent the spin of fun­da­men­tal par­ti­cles like the elec­tron, and they are used in the Pauli equa­tion which con­sid­ers the in­ter­ac­tion of a par­ti­cle in the elec­tro­mag­net­ic field. The Pauli ma­tri­ces are typ­i­cal­ly de­scribed as being re­lat­ed to the $u_{j}$ ma­tri­ces of the $SU(2)$ group by the fol­low­ing

where we see the $u_{j}$ ma­tri­ces are mul­ti­plied by $\pm i$ to arrive at the Pauli ma­tri­ces $\sigma_{j}$. The ma­tri­ces are in­vo­lu­to­ry, mean­ing that each matrix, when squared, equals the iden­ti­ty matrix $\sigma_{j}^{2} = I$. The ma­tri­ces are trace­less and have de­ter­mi­nants equal to $- 1.$Math­e­mat­i­cal­ly, it would seem quite plau­si­ble that, given the re­la­tion­ship the Pauli ma­tri­ces have with the $u_{j}$ ma­tri­ces, there exists analog rep­re­sen­ta­tions of them in the sym­met­ric conic group. This is indeed the case. Like the way el­e­ment prod­ucts gen­er­at­ed the $SU(2)$ sub­groups, taking the same ap­proach ac­com­plish­es this for the sym­met­ric conic group's Pauli matrix rep­re­sen­ta­tions. Using de­fined ma­tri­ces from the sym­met­ric conic group

sim­i­lar prod­ucts from the $SU(2)$ group cases result in al­ge­bras con­sis­tent with the Pauli ma­tri­ces. This can be seen using prod­ucts of the $\sigma X_{1,1}$,$\sigma X_{1}X_{2}$ ma­tri­ces

which, once again, differ from the $2 \times 2$ Pauli ma­tri­ces by the ro­ta­tion­al axis of the al­ge­bra in the $a_{1,1}$ entry. The above rep­re­sen­ta­tions are the Pauli ma­tri­ces for the bottom-right $2 \times 2$ sub­ma­tri­ces. All ma­tri­ces above $\mathcal{o}_{j}$ have de­ter­mi­nants equal to $- 1$, and are in­vo­lu­to­ry. They are not trace­less ma­tri­ces, but they have real eigen­val­ues. By set­ting

which is common in physics, we find the nor­mal­ized eigen­vec­tors using the con­ven­tion that $1$ is placed in the second entry for the pos­i­tive and neg­a­tive wave­func­tions $\psi_{x},\psi_{y}$

which can be com­pared to the common nor­mal­ized eigen­vec­tors of the $2 \times 2$ Pauli ma­tri­ces

These are the same, except for the uni­tary third eigen­vec­tor in the sym­met­ric conic groups. In terms of the eigen­val­ues, each $2 \times 2$ Pauli matrix has two values equal to $\pm 1$, while for the $3 \times 3$ cases, there are three values also equal to $\pm 1$ for the co­in­cid­ing Pauli eigen­vec­tors, with the ad­di­tion of the entry $1$ in the third axis. Ro­tat­ing the gen­er­a­tors by $i$, re­sults in the fol­low­ing ma­tri­ces

whose prod­ucts, like the Pauli ma­tri­ces, have the neg­a­tive forms

These ma­tri­ces all have de­ter­mi­nants of $1$, are Her­mit­ian, and remain in­vo­lu­to­ry. For the prod­uct

this rep­re­sents the neg­a­tive result, as com­pared to the orig­i­nal prod­uct form

It is ap­par­ent that chiral forms of the Pauli ma­tri­ces exist in the sym­met­ric cone group $S_{3\mathbb{C}}^{\Delta}$, much the same as the $SU(2)$ group. There are sev­er­al other Pauli matrix groups that can be made by vary­ing the values for the $a_{j,j}$ matrix entry along the main di­ag­o­nal, rep­re­sent­ing the axis of ro­ta­tion for both forms of the gen­er­a­tors and their prod­ucts, as well as by ro­tat­ing these forms to find other chiral part­ners. Taking the ex­am­ple

we con­sid­er the fol­low­ing matrix prod­ucts

whose re­sults are con­sis­tent in form to the neg­a­tive Pauli ma­tri­ces, dif­fer­ing only in the $a_{1,1}$ matrix en­tries. Other matrix forms may act as gen­er­a­tors for the Pauli ma­tri­ces, which can have cone num­bered de­ter­mi­nants

These yield the orig­i­nal Pauli ma­tri­ces, with the ad­di­tion­al en­tries $a_{1,1} = i$, by way of the prod­ucts

With an imag­i­nary-valued $a_{1,1}$ entry, we find two real-valued eigen­val­ues of $\pm 1$ for the 2^nd and 3^rd columns of $\sigma_{i}^{'}$, with the third eigen­val­ue $i$, for the 1^st columns of $\sigma_{i}^{'}$.