1.1.27 Projective Forms

In this sec­tion, we will use the ex­am­ple of the $S_{3\mathbb{C}}/D_{3\mathbb{C}}$ group and its el­e­ments to in­ves­ti­gate the ex­po­nen­tial map onto $U(3).$By ex­po­nen­ti­a­tion, we take the simple ro­ta­tions and re­flec­tions of the Lie al­ge­bra el­e­ments and trans­form these into a con­tin­u­ous man­i­fold in $\mathbb{C}^{3}$ space. In layper­son's terms, we are taking the group el­e­ments from ro­tat­ing/re­flect­ing the tri­an­gle's ver­tices to an in­fi­nite number of po­ten­tial vertex points aligned with the com­plex axes. This makes it pos­si­ble to rotate and re­flect con­tin­u­ous­ly to any points in the $\mathbb{C}^{3}$ space. Again, we show in Figure 1.1.20 below, the iden­ti­ty $\epsilon$ ori­ent­ed along the pos­i­tive real axes for our equi­lat­er­al tri­an­gle $ABC$.

When con­sid­er­ing per­mu­ta­tions in matrix form, we can gain more geo­met­ric in­sight when com­pared to per­mu­ta­tions de­pict­ed in two-rowed forms. To clar­i­fy the rep­re­sen­ta­tion of per­mu­ta­tions in these stan­dard forms, we denote the $X_{1}$ per­mu­ta­tion from $S_{3\mathbb{C}}/D_{3\mathbb{C}}$ in a two-rowed per­mu­ta­tion form as

where we have re­flect­ed the second and third el­e­ments, while keep­ing the first fixed. Al­though it is clear to see what is hap­pen­ing in the two-rowed per­mu­ta­tion no­ta­tion, we are miss­ing that any ro­ta­tion or re­flec­tion occurs in some geo­met­ric space. In the $X_{j},Y_{j}$ matrix no­ta­tion, it is well-de­fined that we are using ma­tri­ces in a $\mathbb{C}^{3}$ geo­met­ric space. The entry $X_{1}(1,1)$ is the axis and re­mains fixed, while there is a re­flec­tion in the $X_{2}/X_{3}$-plane that is rep­re­sent­ed by the $2 \times 2$ matrix in the bottom-right corner. The crit­i­cal point to re­al­ize is these are Lie al­ge­bra gen­er­a­tors ori­ent­ed in a geo­met­ric space.

In Figure 1.1.21, we rep­re­sent the $S_{3\mathbb{C}}/D_{3\mathbb{C}}$ group's pos­i­tive imag­i­nary axes in three di­men­sions to orient the ob­serv­er. We define a pos­i­tive ro­ta­tion as a coun­ter­clock­wise linear trans­for­ma­tion, which im­plies the $iX_{2}$-axis is lo­cat­ed coun­ter­clock­wise from $iX_{1}$, for $iX_{3}$ de­fined as the axis of ro­ta­tion, and the $iX_{3}$ axis is lo­cat­ed clock­wise from the $iX_{1}$ axis for $iX_{2}$, de­fined as the axis of ro­ta­tion. To remind the reader, the $X_{j},Y_{j}$-type per­mu­ta­tions are shown again below

We can rep­re­sent the $X_{j}$ per­mu­ta­tion el­e­ments as ori­entable forms, where the matrix entry along the main di­ag­o­nal is the fixed axis for each el­e­ment and can be thought of as a vector on its tail, coming out of the page. We rep­re­sent this in Figure 1.1.22 using points going from left to right to rep­re­sent $iX_{3},iX_{2},iX_{1}$, as seen on the bottom-right of each form.

The other two vec­tors are pro­ject­ed in the re­spec­tive $X_{j}$-planes, point­ing up for the pos­i­tive/coun­ter­clock­wise ori­en­ta­tion and to the left for the neg­a­tive/clock­wise ori­en­ta­tion. Fur­ther, we can rep­re­sent the $X_{j}$ per­mu­ta­tion el­e­ments in no­ta­tion to en­com­pass this, where $\mathbf{\bigodot}$ sig­ni­fies the vector or­thog­o­nal to the pro­ject­ed form, $\circlearrowleft$ as the axis which is ori­ent­ed coun­ter­clock­wise from $\mathbf{\bigodot}$, and $\circlearrowright$ as the axis ori­ent­ed clock­wise from $\mathbf{\bigodot}$. Using this no­ta­tion, the Lie al­ge­bras can be rep­re­sent­ed in the fol­low­ing matrix forms

The reader may recall, the columns in ma­tri­ces rep­re­sent the vec­tors in the $x_{j}$ di­rec­tions, while the rows of the ma­tri­ces are the $x_{k}$ com­po­nents of each column vector. So, taking the per­mu­ta­tion $X_{3}$ as an ex­am­ple, we have

We can use the $Y_{j}$ Lie al­ge­bras to form prod­ucts with the $X_{1}$ matrix, while re­main­ing aware of the pro­ject­ed ori­en­ta­tions. The first prod­uct $X_{1},$rep­re­sents the iden­ti­ty acting on $X_{1}$, which main­tains $X_{1}$ as un­changed

The second prod­uct $Y_{2}X_{1}$, trans­forms the $X_{1}$ al­ge­bra to the $X_{2}$ al­ge­bra

where a cyclic modulo shift of all the el­e­ments in $X_{1}$ has oc­curred by one entry to the left (Figure 1.1.23). The red $Y_{2}$ vector has a right cir­cu­lar arrow next to it, in­di­cat­ing the trans­for­ma­tion action it has on $X_{1}$, ro­tat­ing the $X_{1}$ ver­tices in a clock­wise fash­ion around the axis of the $Y_{2}$ vector.

The third prod­uct, $Y_{3}X_{1}$, trans­forms the $X_{1}$ al­ge­bra in the op­pos­i­tive pro­jec­tive ori­en­ta­tion com­pared to the $X_{3}$ al­ge­bra

where a cyclic modulo shift of all the el­e­ments in $X_{1}$ has oc­curred by one entry to the right (Figure 1.1.24). The red $Y_{3}$ vector with a left cir­cu­lar arrow next to it. $Y_{3}$ ro­tates the $X_{1}$ ver­tices in a coun­ter­clock­wise manner about the $Y_{3}$ vector axis.

These ex­am­ples serve to un­der­stand better the geo­met­ric ro­ta­tions that result from the $Y_{j}X_{k}$ com­po­si­tion of func­tions. As ro­ta­tions, they occur in all three of the com­plex di­men­sions in the $\mathbb{C}^{3}$ space, dif­fer­ing from the more cus­tom­ary two-di­men­sion­al ro­ta­tions. The $Y_{j}X_{k}$ com­po­si­tions are not nec­es­sar­i­ly com­mu­ta­tive, with the ex­cep­tion of the iden­ti­ty, $Y_{1}$,

Using the al­ter­na­tive prod­uct orders for the pre­vi­ous ex­am­ples, we find

where these re­sults in­di­cate this change in prod­uct order has changed the di­rec­tion of ro­ta­tion for each. In the fol­low­ing figure, we con­sid­er the three pos­i­tive ori­en­ta­tions of the $X_{j}$ al­ge­bras, which pro­ceed­ing from left to right are $X_{1},X_{2},X_{3}$, and the $ix_{j}$ com­po­nents from the $X_{j}$ al­ge­bras rep­re­sent ori­ent­ed vec­tors on the $\mathbb{C}^{3}$-sphere.

The $Y_{j}$ al­ge­bras act to trans­form one $X_{j}$ al­ge­bra to an­oth­er on the $\mathbb{C}^{3}$-sphere, and we can rep­re­sent each $Y_{j}$ in sym­bol­ic form as

where $\epsilon$ sym­bol­izes the iden­ti­ty trans­for­ma­tion, and the left and right cir­cu­lar arrows for $Y_{2},Y_{3}$ in­di­cate the di­rec­tion of ro­ta­tion for the com­po­nent ver­tices about the $Y_{j}$ axis acting on $X_{j}$.

The geo­met­ric and ro­ta­tion­al spaces we per­ceive in space­time are con­tin­u­ous at the macro­scop­ic level, and many argue that spaces become pix­e­lat­ed at quan­tum dis­tances. Ro­ta­tions and boosts seem con­tin­u­ous at the macro­scop­ic level, un­der­scor­ing that these forms must have a math­e­mat­i­cal frame­work for pro­jec­tion to become con­tin­u­ous, or near-con­tin­u­ous, as geo­met­ric man­i­folds. The re­spon­si­ble frame­work is ex­po­nen­ti­a­tion, and for higher-di­men­sion­al Lie al­ge­bras, matrix ex­po­nen­ti­a­tion. The ex­po­nen­tial func­tion, it turns out, par­al­lels the in­te­gra­tion process, and we use their like­ness to con­sid­er how a form might project from a "zero-di­men­sion­al" point to man­i­fest in the form of a con­tin­u­ous man­i­fold. We

start by taking zero to rep­re­sent the "zero-di­men­sion­al" point, per­form­ing in­te­gra­tion in the $X_{1}$-space

as­sum­ing a con­stant of in­te­gra­tion $\mathcal{c}$. We take this con­stant $\mathcal{c}$ to be equal to unity, rep­re­sent­ing the value of a one-di­men­sion­al Lie al­ge­bra form, stem­ming from a volume in­te­gral of a "zero-di­men­sion­al" point. As the Lie al­ge­bra, let $\mathcal{c}$ be un­chang­ing and always equal to one

Con­tin­u­ing this process, while keep­ing to $\forall\ \mathcal{c}_{k} = 1$, we in­te­grate suc­ces­sive­ly six more times, find­ing the re­sult­ing par­tial series

In­te­grat­ing an in­fi­nite number of times will yield the ex­po­nen­tial series $e^{x_{1}}$

To define this math­e­mat­i­cal process, we in­tro­duce an in­te­gral like the $k^{th}$ de­riv­a­tive no­ta­tion of a func­tion $\left( \frac{\partial^{n}}{\partial x^{n}} \right),$as

which rep­re­sents the $k^{th}$ in­te­gral of $f(x)$ with re­spect to $x$, with the con­tin­u­ous con­stant of in­te­gra­tion $\mathcal{c}_{k}$. As an ex­am­ple, for the in­te­gra­tion of zero from above, we can state math­e­mat­i­cal­ly that the fifth in­te­gral of 0 $\forall\ \mathcal{c}_{k} = 1$, is

De­f­i­n­i­tions (1.1.125-126) – For the vector $\mathcal{c}x_{j}$, where $x_{j}$ is a unit basis vector of some $n$-di­men­sion­al vector space and $\mathcal{c}$ is a scalar con­stant$,$ the In­fi­nite In­te­gra­tion Func­tions${\ f}_{\mathbb{R}}^{- k}\left( \mathcal{c}x_{j} \right)\ ,\ f_{\mathbb{C}}^{- k}\left( \mathcal{c}ix_{j} \right)$ start­ing at the zero point, for $\mathcal{c}x_{j}\in \mathbb{R},\ and\ \mathcal{c}ix_{j}\in \mathbb{C}$, are de­fined as

The­o­rem (1.1.127) – Taking a unit basis vector $x_{j},$ as a fixed form el­e­ment, the al­ge­bra func­tions $f_{\mathbb{R}}\left( x_{j} \right),f_{\mathbb{C}}\left( ix_{j} \right)$ are equal to the ex­po­nen­tial map­ping $exp:\mathfrak{g} \rightarrow G\in \mathbb{R},\mathbb{C}$ onto a con­tin­u­ous man­i­fold $\forall\ \mathfrak{g} =x_{j}$.

Proof –

$\square$

We apply this to the two el­e­ments $X_{3},Y_{3}$ from the $S_{3\mathbb{C}}/D_{3\mathbb{C}}$ group, defin­ing scalar con­stants equal to $\mathcal{c}$, where for $X_{3}$, we find

and for $Y_{2}$

The value for $\mathcal{c}$ re­lates to the mag­ni­tude of the ex­po­nen­tial series re­sult­ing from the in­te­gra­tion func­tions above. When $\mathcal{c} = 1$, this fits the frame­work of a unit vector's ex­po­nen­tial map­ping. It is ap­pro­pri­ate to denote the com­bi­na­tion of mag­ni­tude and unit vector ori­en­ta­tion as $\mathcal{c}e_{1}$, where $e_{1}$ is the unit vector and $\mathcal{c}$ is a scalar mul­ti­ple, or mag­ni­tude, of this $\mathcal{c}e_{1}$ vector. The ex­po­nen­tial map­ping trans­forms the Lie al­ge­bra el­e­ments of a group from simple ro­ta­tions and re­flec­tions to in­fin­i­tes­i­mal trans­for­ma­tions on a con­tin­u­ous man­i­fold in $\mathbb{C}^{3}$ space. We have de­fined the in­fi­nite in­te­gra­tion func­tions $f_{\mathbb{R}}^{- \infty}\left( \mathcal{c}x_{j} \right)$ and $f_{\mathbb{C}}^{- \infty}\left( \mathcal{c}ix_{j} \right)$, point­ing out the par­al­lel these have to the ex­po­nen­tial func­tion. The ori­ent­ed pro­jec­tion of a polyg­o­nal form through in­fi­nite in­te­gra­tion/ex­po­nen­ti­a­tion ex­tends that form into a con­tin­u­ous geo­met­ric space. These con­sid­er­a­tions are es­sen­tial in es­tab­lish­ing how some­thing out­side of a geo­met­ric space can give rise to the said space, in­ter­ject­ing the notion of causal­i­ty into our pro­ceed­ings.