4-Component Spinors for SL(4,C) and Four Types of Transformations

1. Introduction

Let $L_A$ be an element of $SL(2,C)$. In an exponential form with parameters $\theta$ and $\eta$:

$$L_A=exp(-\frac{i}{2}(\overrightarrow{\theta }·\overrightarrow{\sigma }+i\overrightarrow{\eta }·\overrightarrow{\sigma }))$$

(1)

$\overrightarrow{\sigma }$ is the Pauli vector with ${\sigma }_1={\sigma }_x,{\sigma }_2={\sigma }_y,{\sigma }_3={\sigma }_z$. The subscript ${}_A$ is introduced in order to distinguish the other forms of L that will be introduced subsequently.

We rewrite $L_A$ and its complex conjugate in the following compact forms:

$$L_A=exp(-\frac{i}{2}{\overrightarrow{\pi }}_A·\overrightarrow{\sigma }),L_A^{*}=exp(\frac{i}{2}{\overrightarrow{\pi }}_A^{*}·{\overrightarrow{\sigma }}^{*})$$

(2)

${\left({\pi }_A\right)}_i={\theta }_i+i{\eta }_i$ and ${}^{*}$ denotes complex conjugation. $L_A$ corresponds to the Lorentz transformation with ${\theta }_i$ and ${\eta }_i$ being the rotation and boost parameters, respectively.

It is well known that the complex version of the $4\times 4$ Lorentz transformation matrix can be written as a matrix direct product of $L_A$ and $L_A^{*}$:

$$\lambda =L_A\otimes L_A^{*}$$

(3)

In order to obtain the familiar real matrix form of the Lorentz transformation it is enough to change the basis:

$$\Lambda =A(L_A\otimes L_A^{*})A^{-1}$$

(4)

where

$$A=\frac{1}{\sqrt{2}}\left(\begin{array}{cccc}1& 0& 0& 1\\ 0& 1& 1& 0\\ 0& i& -i& 0\\ 1& 0& 0& -1\end{array}\right),A^{-1}=A^{†}=\frac{1}{\sqrt{2}}\left(\begin{array}{cccc}1& 0& 0& 1\\ 0& 1& -i& 0\\ 0& 1& i& 0\\ 1& 0& 0& -1\end{array}\right)$$

(5)

Now, it is straightforward to show that $SO(3,1)$ can be written as a commutative product of $SL(4,C)$ and $SL{(4,C)}^{*}$ by simply rewriting Eq.(4) in a factorized form:

$$\Lambda =\left[A(L_A\otimes I)A^{-1}\right]\left[A(I\otimes L_A^{*})A^{-1}\right]=Z_A{Z}_A^{*}=Z_A^{*}{Z}_A,$$

(6)

$$Z_A=A(L_A\otimes I)A^{-1},Z_A^{*}=A(I\otimes L_A^{*})A^{-1}.$$

(7)

$Z_A$ and $Z_A^{*}$ are the $4\times 4$ versions of $L_A$ and $L_A^{*}$ matrices. They can be expressed in terms of ${\Sigma }_i$ matrices:

$$Z_A=exp(-\frac{i}{2}{\overrightarrow{\pi }}_A·\overrightarrow{\Sigma }),Z_A^{*}=exp(\frac{i}{2}{\overrightarrow{\pi }}_A^{*}·{\overrightarrow{\Sigma }}^{*}).$$

(8)

$\overrightarrow{\Sigma }=({\Sigma }_1,{\Sigma }_2,{\Sigma }_3)$ and ${\Sigma }_i$ are $4\times 4$ versions of Pauli matrices:

$${\Sigma }_1=\left(\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& -i\\ 0& 0& i& 0\end{array}\right),{\Sigma }_2=\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& i\\ 1& 0& 0& 0\\ 0& -i& 0& 0\end{array}\right),{\Sigma }_3=\left(\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& -i& 0\\ 0& i& 0& 0\\ 1& 0& 0& 0\end{array}\right)$$

(9)

These are traceless Hermitian matrices and they satisfy the same commutation relations as ${\sigma }_i$ matrices

$$\left[\frac{1}{2}{\Sigma }_i,\frac{1}{2}{\Sigma }_j\right]=\frac{i}{2}{ϵ}^{ijk}{\Sigma }_k.$$

(10)

By definition, ${\Sigma }_{\mu }=A({\sigma }_{\mu }\otimes I)A^{-1}$, $(\mu =0,1,2,3)$, ${\Sigma }_0$ is the $4\times 4$ identity. ${\Sigma }_{\mu }$ basis do not form a complete set for $4\times 4$ matrices, but the set of ${\Sigma }_{\mu }{\Sigma }_{\nu }^{*}$ does.

From the Eq.(7), $Z_A$ can be found in terms of the elements of $L_A$:

$$Z_A=\left(\begin{array}{cccc}{\alpha }_0& {\alpha }_1& {\alpha }_2& {\alpha }_3\\ {\alpha }_1& {\alpha }_0& -i{\alpha }_3& i{\alpha }_2\\ {\alpha }_2& i{\alpha }_3& {\alpha }_0& -i{\alpha }_1\\ {\alpha }_3& -i{\alpha }_2& i{\alpha }_1& {\alpha }_0\end{array}\right)$$

(11)

where ${\alpha }_0=\frac{1}{2}(L_{11}+L_{22})$,${\alpha }_1=\frac{1}{2}(L_{12}+L_{21})$,${\alpha }_2=\frac{i}{2}(L_{12}-L_{21})$, and ${\alpha }_3=\frac{1}{2}(L_{11}-L_{22})$. Hence, $L_A$ can be written in terms of ${\alpha }_{\mu }$ as

$$L_A=\left(\begin{array}{cc}{\alpha }_0+{\alpha }_3& {\alpha }_1-i{\alpha }_2\\ {\alpha }_1+i{\alpha }_2& {\alpha }_0-{\alpha }_3\end{array}\right)$$

(12)

We can write $L_A$ and $Z_A$ in terms of ${\sigma }_{\mu }$ and ${\Sigma }_{\mu }$ matrices:

$$L_A={\alpha }_0{\sigma }_0+{\alpha }_1{\sigma }_1+{\alpha }_2{\sigma }_2+{\alpha }_3{\sigma }_3$$

(13)

$$Z_A={\alpha }_0{\Sigma }_0+{\alpha }_1{\Sigma }_1+{\alpha }_2{\Sigma }_2+{\alpha }_3{\Sigma }_3$$

(14)

Or, simply

$$L_A={(++++)}_{\sigma }.$$

(15)

$$Z_A={(++++)}_{\Sigma }.$$

(16)

We also define the spinor metric g for $SL(4,C)$ that corresponds to the spinor metric $ϵ$ of $SL(2,C)$:

$$g=g^{\mu \nu }=i\eta {\Sigma }_2^{*}=\left(\begin{array}{cccc}0& 0& i& 0\\ 0& 0& 0& -1\\ -i& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right);g^{-1}=g_{\mu \nu }=g^{†}.$$

(17)

$\eta$ is the mostly minus Minkowski metric1.

$Z_A$ preserves the Minkowski metric:

$$Z_A^T\eta Z_A=\eta$$

(18)

Since $\eta$ is real, $Z_A^T\eta Z_A=\eta$ directly entails ${\Lambda }^T\eta \Lambda =\eta$. In an analogy with $ϵ{\sigma }_i{ϵ}^{-1}=-{\sigma }_i^{*}$, we have the following very useful relation:

$$g{\Sigma }_i{g}^{-1}=-{\Sigma }_i^{*}$$

(19)

In this note we will show that there are eight generalized eigenvectors of ${\Sigma }_3^{*}$ matrix that can be interpreted as 4-component covariant spinors. The generalized eigenvectors can be pairwise grouped into four categories. The first pair transforms in the usual way, but the other three transform in different ways.

In the following we will study the first and the second pairs in detail, and we will introduce the remaining two in the subsequent sections.

2. The First and the Second Pairs and Their Transformation Properties

Let $L_A=exp(-\frac{i}{2}{\overrightarrow{\pi }}_A·\overrightarrow{\sigma })$ be the $(\frac{1}{2},0)$ representation of the Lorentz group that acts on the 2-component left-chiral spinor ${\xi }_{{}_L}$:

$${\xi }_{{}_L}\to {\xi }_{{}_L}^{\prime }=L_A{\xi }_{{}_L}.$$

(20)

where

$$L_A=\left(\begin{array}{cc}{L}_{11}& L_{12}\\ L_{21}& L_{22}\end{array}\right)$$

(21)

In terms of the components $u,v$ of ${\xi }_{{}_L}$:

$$u\to u^{\prime }=L_{11}u+L_{12}v,v\to v^{\prime }=L_{21}u+L_{22}v.$$

(22)

Let us call this transformation scheme $T_A$.

Let ${\dot{L}}_A=exp(-\frac{i}{2}{\overrightarrow{\pi }}_A^{*}·\overrightarrow{\sigma })$ be the dotted version corresponding to the $(0,\frac{1}{2})$ representation of the Lorentz group. ${\dot{L}}_A={\left(L_A^{-1}\right)}^{†}$. Let ${\xi }_{{}_R}$ be the 2-component right-chiral spinor. ${\xi }_{{}_R}=ϵ{\xi }_{{}_L}^{*}$, where

$$ϵ={ϵ}^{ab}=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right),{ϵ}^{-1}={ϵ}_{ab}={ϵ}^{†}.$$

(23)

${\xi }_{{}_R}$ transforms as

$${\xi }_{{}_R}\to {\xi }_{{}_R}^{\prime }={\dot{L}}_A{\xi }_{{}_R}$$

(24)

In terms of the components $u,v$, Eq.(24) is equivalent to the scheme $T_A$ given in Eq.(22).

What happens when $L_A$ acts on $ϵ{\xi }_{{}_L}$? In this case, in terms of the components

$$u\to u^{\prime }=L_{22}u-L_{21}v,v\to v^{\prime }=-L_{12}u+L_{11}v$$

(25)

Let us call this transformation scheme $T_B$. We can write $T_B$ in a matrix form:

$$\left(\begin{array}{c}u\\ v\end{array}\right)\to \left(\begin{array}{c}{u}^{\prime }\\ v^{\prime }\end{array}\right)=\left(\begin{array}{cc}{L}_{22}& -L_{21}\\ -L_{12}& L_{11}\end{array}\right)\left(\begin{array}{c}u\\ v\end{array}\right)$$

(26)

Let us name this transformation matrix as $L_B$. Note that, $L_B={\left({\dot{L}}_A\right)}^{*}$, and Eq.(26) is nothing but the transformation of ${\xi }_{{}_L}$ under the action of $L_B$, which is a type $T_B$ transformation.

Now, let $Z_A=exp(-\frac{i}{2}{\overrightarrow{\pi }}_A·\overrightarrow{\Sigma })$ be the $(\frac{1}{2},0)$ representation of $SL(4,C)$ that acts on the first pair of the 4-component undotted covariant spinors:

$${\chi }_{{}_{\left(1\right)}}\to Z_A{\chi }_{{}_{\left(1\right)}},{\chi }_{{}_{\left(2\right)}}\to Z_A{\chi }_{{}_{\left(2\right)}}$$

(27)

where ${\chi }_{{}_{\left(1\right)}}$ and ${\chi }_{{}_{\left(2\right)}}$ are the generalized eigenvectors of ${\Sigma }_3^{*}$ 2:

$${\chi }_{{}_{\left(1\right)}}=\frac{1}{\sqrt{2}}\left(\begin{array}{c}u\\ v\\ -iv\\ u\end{array}\right),{\chi }_{{}_{\left(2\right)}}=\frac{1}{\sqrt{2}}\left(\begin{array}{c}-v\\ -u\\ -iu\\ v\end{array}\right)$$

(28)

Indices in the parentheses are simply labels for 4-component spinors.

Now consider the second pair of the generalized eigenvectors of ${\Sigma }_3^{*}$:

$${\chi }_{{}_{\left(3\right)}}=\frac{1}{\sqrt{2}}\left(\begin{array}{c}-v\\ u\\ -iu\\ -v\end{array}\right),{\chi }_{{}_{\left(4\right)}}=\frac{1}{\sqrt{2}}\left(\begin{array}{c}u\\ -v\\ -iv\\ -u\end{array}\right)$$

(29)

Transformation scheme of ${\chi }_{{}_{\left(3\right)}}$ and ${\chi }_{{}_{\left(4\right)}}$ is different from that of ${\chi }_{{}_{\left(1\right)}}$ and ${\chi }_{{}_{\left(2\right)}}$. Under the action of $Z_A$, ${\chi }_{{}_{\left(1\right)}}$ and ${\chi }_{{}_{\left(2\right)}}$ transform according to the scheme $T_A$, but ${\chi }_{{}_{\left(3\right)}}$ and ${\chi }_{{}_{\left(4\right)}}$ transform according to the scheme $T_B$. However, we may think in an alternative way: Suppose that ${\chi }_{{}_{\left(3\right)}}$ and ${\chi }_{{}_{\left(3\right)}}$ are different kind of objects with different transformation properties, such that another transformation matrix, $Z_B$, acts on them and under the action of $Z_B$ they transform according to the scheme $T_A$:

$${\chi }_{{}_{\left(3\right)}}\to Z_B{\chi }_{{}_{\left(3\right)}},{\chi }_{{}_{\left(4\right)}}\to Z_B{\chi }_{{}_{\left(4\right)}},$$

(30)

By definition $Z_B=A(L_B\otimes I)A^{-1}$:

$$Z_B=\left(\begin{array}{cccc}{\alpha }_0& -{\alpha }_1& {\alpha }_2& -{\alpha }_3\\ -{\alpha }_1& {\alpha }_0& i{\alpha }_3& i{\alpha }_2\\ {\alpha }_2& -i{\alpha }_3& {\alpha }_0& i{\alpha }_1\\ -{\alpha }_3& -i{\alpha }_2& -i{\alpha }_1& {\alpha }_0\end{array}\right)={\alpha }_0{\Sigma }_0-{\alpha }_1{\Sigma }_1+{\alpha }_2{\Sigma }_2-{\alpha }_3{\Sigma }_3.$$

(31)

Or, simply

$$Z_B={(+-+-)}_{\Sigma }$$

(32)

Now let ${\dot{Z}}_A=exp(-\frac{i}{2}{\overrightarrow{\pi }}_A^{*}·\overrightarrow{\Sigma })$ be the $(0,\frac{1}{2})$ representation. ${\dot{Z}}_A={\left(Z_A^{-1}\right)}^{†}$. We regard the generalized eigenvectors of ${\Sigma }_3$ as 4-component undotted contravariant spinors and we define the first pair as follows:

$${\chi }^{\left(1\right)}=\frac{1}{\sqrt{2}}\left(\begin{array}{c}v\\ -u\\ -iu\\ v\end{array}\right),{\chi }^{\left(2\right)}=\frac{1}{\sqrt{2}}\left(\begin{array}{c}u\\ -v\\ iv\\ -u\end{array}\right)$$

(33)

Under the action of ${\dot{Z}}_A$, dotted versions of ${\chi }^{{}^{\left(1\right)}}$ and ${\chi }^{{}^{\left(1\right)}}$ transform according to the scheme $T_A$.

$${\dot{\chi }}^{{}^{\left(1\right)}}\to {\dot{Z}}_A{\dot{\chi }}^{{}^{\left(1\right)}},{\dot{\chi }}^{{}^{\left(2\right)}}\to {\dot{Z}}_A{\dot{\chi }}^{{}^{\left(2\right)}}$$

(34)

The second pair of the generalized eigenvectors of ${\Sigma }_3$ is defined as

$${\chi }^{\left(3\right)}=\frac{1}{\sqrt{2}}\left(\begin{array}{c}u\\ v\\ iv\\ u\end{array}\right),{\chi }^{\left(4\right)}=\frac{1}{\sqrt{2}}\left(\begin{array}{c}v\\ u\\ -iu\\ -v\end{array}\right)$$

(35)

Under the action of ${\dot{Z}}_A$, the dotted versions of ${\chi }^{{}^{\left(3\right)}}$ and ${\chi }^{{}^{\left(4\right)}}$ transform according to the scheme $T_B$. But, they transform according to the scheme $T_A$ under the action of ${\dot{Z}}_B$:

$${\dot{\chi }}^{{}^{\left(3\right)}}\to {\dot{Z}}_B{\dot{\chi }}^{{}^{\left(3\right)}},{\dot{\chi }}^{{}^{\left(4\right)}}\to {\dot{Z}}_B{\dot{\chi }}^{{}^{\left(4\right)}}$$

(36)

where ${\dot{Z}}_B={\left(Z_B^{-1}\right)}^{†}$ by definition. ${\chi }^{{}^{\left(a\right)}}$ is related to ${\chi }_{{}_{\left(a\right)}}$ by the $SL(4,C)$ metric, ${\chi }^{\left(a\right)}=g{\chi }_{{}_{\left(a\right)}}$, and its dotted version is defined as 3.

$${\dot{\chi }}^{\left(a\right)}={\left(g{\chi }_{{}_{\left(a\right)}}\right)}^{*}.$$

(37)

We write various forms of Z and L matrices in compact notation to manifest the parallelism between them:

$$L_A={(++++)}_{\sigma },L_B={(+-+-)}_{\sigma },{\dot{L}}_A={(+---)}_{\sigma }^{*},{\dot{L}}_B={(++-+)}_{\sigma }^{*}.$$

(38)

$$Z_A={(++++)}_{\Sigma },Z_B={(+-+-)}_{\Sigma },{\dot{Z}}_A={(+---)}_{\Sigma }^{*},{\dot{Z}}_B={(++-+)}_{\Sigma }^{*}.$$

(39)

3. Outer Products of 4-Component Spinors and Null 4-Vectors

Let us define the outer product $W_{{}_L}={\xi }_{{}_L}{\xi }_{{}_L}^{†}$ which transforms as

$$W_{{}_L}\to W_{{}_L}^{\prime }=\left(L_A{\xi }_{{}_L}\right){\left(L_A{\xi }_{{}_L}\right)}^{†}=L_A{W}_{{}_L}{L}_A^{†}$$

(40)

This is a type $T_A$ transformation. Determinant of $W_{{}_L}$ is zero, hence $W_{{}_L}$ can be associated with a null 4-vector through the substitutions, $t=\frac{1}{2}(u\dot{u}+v\dot{v}),x=\frac{1}{2}(u\dot{v}+v\dot{u}),y=\frac{i}{2}(u\dot{v}-v\dot{u}),z=\frac{1}{2}(u\dot{u}-v\dot{v})$:

$$W_{{}_L}=\left(\begin{array}{cc}t+z& x-iy\\ x+iy& t-z\end{array}\right)$$

(41)

We also define the outer product $W_{{}_R}={\xi }_{{}_R}{\xi }_{{}_R}^{†}$ which transforms as

$$W_{{}_R}\to W_{{}_R}^{\prime }=\left({\dot{L}}_A{\xi }_{{}_R}\right){\left({\dot{L}}_A{\xi }_{{}_R}\right)}^{†}={\dot{L}}_A{W}_{{}_R}{\dot{L}}_A^{†}$$

(42)

This is also a type $T_A$ transformation. Determinant of $W_{{}_R}$ is zero and $W_{{}_R}$ can be associated with a null 4-vector:

$$W_{{}_R}=\left(\begin{array}{cc}t-z& -x+iy\\ -x-iy& t+z\end{array}\right)$$

(43)

Note that $W_{{}_R}$ can be obtained from $W_{{}_L}$ by parity inversion.

There are outer product forms of 4-component spinors that can be associated with null 4-vectors. ${\mathcal{W}}_{{}_{\left(11\right)}}={\chi }_{{}_{\left(1\right)}}{\chi }_{{}_{\left(1\right)}}^{†}$ and ${\mathcal{W}}_{{}_{\left(22\right)}}={\chi }_{{}_{\left(2\right)}}{\chi }_{{}_{\left(2\right)}}^{†}$ transform in a similar way with $W_{{}_L}$:

$${\mathcal{W}}_{{}_{\left(11\right)}}=\left(\begin{array}{cccc}u\dot{u}& u\dot{v}& iu\dot{v}& u\dot{u}\\ v\dot{u}& v\dot{v}& iv\dot{v}& v\dot{u}\\ -iv\dot{u}& -iv\dot{v}& v\dot{v}& -iv\dot{u}\\ u\dot{u}& u\dot{v}& iu\dot{v}& u\dot{u}\end{array}\right),{\mathcal{W}}_{{}_{\left(22\right)}}=\left(\begin{array}{cccc}v\dot{v}& v\dot{u}& -iv\dot{u}& -v\dot{v}\\ u\dot{v}& u\dot{u}& -iu\dot{u}& -u\dot{v}\\ iu\dot{v}& iu\dot{u}& u\dot{u}& -iu\dot{v}\\ -v\dot{v}& -v\dot{u}& iv\dot{u}& v\dot{v}\end{array}\right)$$

(44)

For $a=1$ and $a=2$, ${\mathcal{W}}_{{}_{\left(aa\right)}}$ transform according to the scheme $T_A$ as

$${\mathcal{W}}_{{}_{\left(aa\right)}}\to {\mathcal{W}}_{{}_{\left(aa\right)}}^{\prime }=\left(Z_A{\chi }_{{}_{\left(a\right)}}\right){\left(Z_A{\chi }_{{}_{\left(a\right)}}\right)}^{†}=Z_A{\mathcal{W}}_{{}_{\left(aa\right)}}{Z}_A^{†}.$$

(45)

This equation is equivalent to the Eq.(40), and it is the main motivation behind the interpretation of ${\chi }_{{}_{\left(a\right)}}$ as 4-component spinors for $SL(4,C)$.

${\dot{\mathcal{W}}}^{{}^{\left(11\right)}}={\dot{\chi }}^{{}^{\left(1\right)}}{\dot{\chi }}^{{}^{\left(1\right)†}}$ and ${\dot{\mathcal{W}}}^{{}^{\left(22\right)}}={\dot{\chi }}^{{}^{\left(2\right)}}{\dot{\chi }}^{{}^{\left(2\right)†}}$ transform in a similar way with $W_{{}_R}$:

$${\dot{\mathcal{W}}}^{{}^{\left(11\right)}}=\left(\begin{array}{cccc}v\dot{v}& -u\dot{v}& -iu\dot{v}& v\dot{v}\\ -v\dot{u}& u\dot{u}& iu\dot{u}& -v\dot{u}\\ iv\dot{u}& -iu\dot{u}& u\dot{u}& iv\dot{u}\\ v\dot{v}& -u\dot{v}& -iu\dot{v}& v\dot{v}\end{array}\right),{\dot{\mathcal{W}}}^{{}^{\left(22\right)}}=\left(\begin{array}{cccc}u\dot{u}& -v\dot{u}& iv\dot{u}& -u\dot{u}\\ -u\dot{v}& v\dot{v}& -iv\dot{v}& u\dot{v}\\ -iu\dot{v}& iv\dot{v}& v\dot{v}& iu\dot{v}\\ -u\dot{u}& v\dot{u}& -iv\dot{u}& u\dot{u}\end{array}\right)$$

(46)

For $a=1$ and $a=2$, ${\dot{\mathcal{W}}}^{{}^{\left(aa\right)}}$ transform according to the scheme $T_A$ as

$${\dot{\mathcal{W}}}^{{}^{\left(aa\right)}}\to {\dot{Z}}_A{\dot{\mathcal{W}}}^{{}^{\left(aa\right)}}{\dot{Z}}_A^{†}$$

(47)

${\mathcal{W}}_{{}_{\left(aa\right)}}$ and ${\dot{\mathcal{W}}}^{{}^{\left(aa\right)}}$ are Hermitian and zero determinant matrices, hence they correspond to null 4-vectors.

We also have outer products of 4-component spinors of the other kind. For $a=3$ and $a=4$, ${\mathcal{W}}_{{}_{\left(aa\right)}}$ and ${\dot{\mathcal{W}}}^{{}^{\left(aa\right)}}$ transform according to the scheme $T_A$ under the action of $Z_B$ and ${\dot{Z}}_B$:

$${\mathcal{W}}_{{}_{\left(aa\right)}}\to Z_B{\mathcal{W}}_{{}_{\left(aa\right)}}{Z}_B^{†},{\dot{\mathcal{W}}}^{{}^{\left(aa\right)}}\to {\dot{Z}}_B{\dot{\mathcal{W}}}^{{}^{\left(aa\right)}}{\dot{Z}}_B^{†}$$

(48)

These are also Hermitian and zero determinant matrices and they correspond to null 4-vectors.

4. Quaternion Forms and 4-Vectors

In general, we can treat $t,x,y$ and z as variables that do not depend on u and v. Then, we can associate the following matrices $X_{{}_L}$ and $X_{{}_R}$ with 4-vectors, which are not necessarily null:

$$W_{{}_L}\to X_{{}_L}=\left(\begin{array}{cc}t+z& x-iy\\ x+iy& t-z\end{array}\right)=t{\sigma }_0+x{\sigma }_1+y{\sigma }_2+z{\sigma }_3={(++++)}_{\sigma }$$

(49)

$$W_{{}_R}\to X_{{}_R}=\left(\begin{array}{cc}t-z& -x+iy\\ -x-iy& t+z\end{array}\right)=t{\sigma }_0-x{\sigma }_1-y{\sigma }_2-z{\sigma }_3={(+---)}_{\sigma }$$

(50)

det$X_{{}_L}=$det$X_{{}_R}=t^2-x^2-y^2-z^2$ and in general not zero. $X_{{}_L}$ and $X_{{}_R}$ transform as

$$X_{{}_L}\to X_{{}_L}^{\prime }=L_A{X}_{{}_L}{L}_A^{†},X_{{}_R}\to X_{{}_R}^{\prime }={\dot{L}}_A{X}_{{}_R}{\dot{L}}_A^{†}$$

(51)

These are matrix representations of quaternions, because $-i{\sigma }_1,-i{\sigma }_2,-i{\sigma }_3$ matrices have the same properties as the Hamilton’s quaternion basis, $\mathbf{i},\mathbf{j},\mathbf{k}$:

$$X_{{}_L}=t{\sigma }_0+ix(-i{\sigma }_1)+iy(-i{\sigma }_2)+iz(-i{\sigma }_3)=t\mathbb{1}+ix\mathbf{i}+iy\mathbf{j}+iz\mathbf{k}.$$

(52)

Similarly,

$$X_{{}_R}=t\mathbb{1}-ix\mathbf{i}-iy\mathbf{j}-iz\mathbf{k}.$$

(53)

In order to make the analogy with $SL(4,C)$ we consider the following two column objects that are pairwise combinations of 4-component spinors:

$${\chi }_{{}_A}=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}u& -v\\ v& -u\\ -iv& -iu\\ u& v\end{array}\right),{\chi }_{{}_B}=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}-v& u\\ u& -v\\ -iu& -iv\\ -v& -u\end{array}\right),{\chi }^{{}^A}=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}v& u\\ -u& -v\\ -iu& iv\\ v& -u\end{array}\right),{\chi }^{{}^B}=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}u& v\\ v& u\\ iv& -iu\\ u& -v\end{array}\right)$$

(54)

where ${\chi }_{{}_A}=({\chi }_{{}_{\left(1\right)}},{\chi }_{{}_{\left(2\right)}})$, ${\chi }_{{}_B}=({\chi }_{{}_{\left(3\right)}},{\chi }_{{}_{\left(4\right)}})$, ${\chi }^{{}^A}=({\chi }^{{}^{\left(1\right)}},{\chi }^{{}^{\left(2\right)}})$, ${\chi }^{{}^B}=({\chi }^{{}^{\left(3\right)}},{\chi }^{{}^{\left(4\right)}})$.

We define an outer product of 4-component spinor pair in the form ${\mathcal{W}}_{{}_A}={\chi }_{{}_A}{\chi }_{{}_A}^{†}$, which is formally a quaternion:

$${\mathcal{W}}_{{}_A}=\frac{1}{2}\left(\begin{array}{cccc}u\dot{u}+v\dot{v}& u\dot{v}+v\dot{u}& iu\dot{v}-iv\dot{u}& u\dot{u}-v\dot{v}\\ v\dot{u}+u\dot{v}& v\dot{v}+u\dot{u}& iv\dot{v}-iu\dot{u}& v\dot{u}-u\dot{v}\\ -iv\dot{u}+iu\dot{v}& -iv\dot{v}+iu\dot{u}& v\dot{v}+u\dot{u}& -iv\dot{u}-iu\dot{v}\\ u\dot{u}-v\dot{v}& u\dot{v}-v\dot{u}& iu\dot{v}+iv\dot{u}& u\dot{u}+v\dot{v}\end{array}\right)$$

(55)

${\mathcal{W}}_{{}_A}$ can be written as a sum of two basic forms: ${\mathcal{W}}_{{}_A}={\mathcal{W}}_{{}_{\left(11\right)}}+{\mathcal{W}}_{{}_{\left(22\right)}}$. In its present form det${\mathcal{W}}_{{}_A}=0$ and ${\mathcal{W}}_{{}_A}$ corresponds to a null 4-vector, but we can associate ${\mathcal{W}}_{{}_A}$ with an arbitrary 4-vector in terms of the variables $t,x,y$ and z:

$${\mathcal{W}}_{{}_A}\to {\mathcal{Q}}_{{}_A}=\left(\begin{array}{cccc}t& x& y& z\\ x& t& -iz& iy\\ y& iz& t& -ix\\ z& -iy& ix& t\end{array}\right)=t{\Sigma }_0+x{\Sigma }_1+y{\Sigma }_2+z{\Sigma }_3.$$

(56)

${\mathcal{Q}}_{{}_A}={(++++)}_{\Sigma }$ and it is the $4\times 4$ version of $X_{{}_L}$:

$${\mathcal{Q}}_{{}_A}=A(X_{{}_L}\otimes I)A^{-1}$$

(57)

Similarly, we define ${\dot{\mathcal{W}}}^{{}^A}$:

$${\dot{\mathcal{W}}}^{{}^A}={\dot{\chi }}^{{}^A}{\dot{\chi }}^{{}^{A†}}={\dot{\mathcal{W}}}^{{}^{\left(11\right)}}+{\dot{\mathcal{W}}}^{{}^{\left(22\right)}}=\frac{1}{2}\left(\begin{array}{cccc}\dot{v}v+\dot{u}u& -\dot{v}u-\dot{u}v& -i\dot{v}u+i\dot{u}v& \dot{v}v-\dot{u}u\\ -\dot{u}v-\dot{v}u& \dot{u}u+\dot{v}v& i\dot{u}u-i\dot{v}v& -\dot{u}v+\dot{v}u\\ i\dot{u}v-i\dot{v}u& -i\dot{u}u+i\dot{v}v& \dot{u}u+\dot{v}v& i\dot{u}v+i\dot{v}u\\ \dot{v}v-\dot{u}u& -\dot{v}u+\dot{u}v& -i\dot{v}u-i\dot{u}v& \dot{v}v+\dot{u}u\end{array}\right)$$

(58)

In terms of the variables $t,x,y$ and z:

$${\dot{\mathcal{W}}}^{{}^A}\to {\dot{\mathcal{Q}}}^{{}^A}=\left(\begin{array}{cccc}t& -x& -y& -z\\ -x& t& iz& -iy\\ -y& -iz& t& ix\\ -z& iy& -ix& t\end{array}\right)=t{\Sigma }_0-x{\Sigma }_1-y{\Sigma }_2-z{\Sigma }_3.$$

(59)

${\dot{\mathcal{Q}}}^{{}^A}={(+---)}_{\Sigma }$ and it is the $4\times 4$ version of $X_{{}_R}$:

$${\dot{\mathcal{Q}}}^{{}^A}=A(X_{{}_R}\otimes I)A^{-1}.$$

(60)

${\dot{\mathcal{Q}}}^{{}^A}$ can be obtained from ${\mathcal{Q}}_{{}_A}$ by parity inversion and they transform as

$${\mathcal{Q}}_{{}_A}\to Z_A{\mathcal{Q}}_{{}_A}{Z}_A^{†},{\dot{\mathcal{Q}}}^{{}^A}\to {\dot{Z}}_A{\dot{\mathcal{Q}}}^{{}^A}{\dot{Z}}_A^{†}$$

(61)

These are type $T_A$ transformations, hence these forms correspond to 4-vectors.

The outer product ${\mathcal{W}}_{{}_B}={\chi }_{{}_B}{\chi }_{{}_B}^{†}$ is also a quaternion:

$${\mathcal{W}}_{{}_B}=\frac{1}{2}\left(\begin{array}{cccc}v\dot{v}+u\dot{u}& -v\dot{u}-u\dot{v}& -iv\dot{u}+iu\dot{v}& v\dot{v}-u\dot{u}\\ -u\dot{v}-v\dot{u}& u\dot{u}+v\dot{v}& iu\dot{u}-iv\dot{v}& -u\dot{v}+v\dot{u}\\ iu\dot{v}-iv\dot{u}& -iu\dot{u}+iv\dot{v}& u\dot{u}+v\dot{v}& iu\dot{v}+iv\dot{u}\\ v\dot{v}-u\dot{u}& -v\dot{u}+u\dot{v}& -iv\dot{u}-iu\dot{v}& v\dot{v}+u\dot{u}\end{array}\right)$$

(62)

In terms of variables $t,x,y$ and z:

$${\mathcal{W}}_{{}_B}\to {\mathcal{Q}}_{{}_B}=\left(\begin{array}{cccc}t& -x& y& -z\\ -x& t& iz& iy\\ y& -iz& t& ix\\ -z& -iy& -ix& t\end{array}\right)=t{\Sigma }_0-x{\Sigma }_1+y{\Sigma }_2-z{\Sigma }_3.$$

(63)

${\mathcal{Q}}_{{}_B}={(+-+-)}_{\Sigma }$.

We also write ${\dot{\mathcal{W}}}^{{}^B}$:

$${\dot{\mathcal{W}}}^{{}^B}={\dot{\chi }}^{{}^B}{\dot{\chi }}^{{}^{B†}}=\frac{1}{2}\left(\begin{array}{cccc}\dot{u}u+\dot{v}v& \dot{u}v+\dot{v}u& i\dot{u}v-i\dot{v}u& \dot{u}u-\dot{v}v\\ \dot{v}u+\dot{u}v& \dot{v}v+\dot{u}u& i\dot{v}v-i\dot{u}u& \dot{v}u-\dot{u}v\\ -i\dot{v}u+i\dot{u}v& -i\dot{v}v+i\dot{u}u& \dot{v}v+\dot{u}u& -i\dot{v}u-i\dot{u}v\\ \dot{u}u-\dot{v}v& \dot{u}v-\dot{v}u& i\dot{u}v+i\dot{v}u& \dot{u}u+\dot{v}v\end{array}\right)$$

(64)

$${\dot{\mathcal{W}}}^{{}^B}\to {\dot{\mathcal{Q}}}^{{}^B}=\left(\begin{array}{cccc}t& x& -y& z\\ x& t& -iz& -iy\\ -y& iz& t& -ix\\ z& iy& ix& t\end{array}\right)=t{\Sigma }_0+x{\Sigma }_1-y{\Sigma }_2+z{\Sigma }_3.$$

(65)

${\dot{\mathcal{Q}}}^{{}^B}={(++-+)}_{\Sigma }$ and it can be obtained from ${\mathcal{Q}}_{{}_B}$ by parity inversion. ${\mathcal{Q}}_{{}_B}$ and ${\dot{\mathcal{Q}}}^{{}^B}$ transform with $Z_B$ and ${\dot{Z}}_B$:

$${\mathcal{Q}}_{{}_B}\to Z_B{\mathcal{Q}}_{{}_B}{Z}_B^{†},{\dot{\mathcal{Q}}}^{{}^B}\to {\dot{Z}}_B{\dot{\mathcal{Q}}}^{{}^B}{\dot{Z}}_B^{†}$$

(66)

These transformations obey the scheme $T_A$ also, hence they correspond to 4-vectors.

With the compact notation we can show a very nice symmetry: The form of the transformation matrix matches the form of the transformed object. For example, $Z_A={(++++)}_{\Sigma }$ acts on the form ${\mathcal{Q}}_A={(++++)}_{\Sigma }$, $Z_B={(+-+-)}_{\Sigma }$ acts on the form ${\mathcal{Q}}_B={(+-+-)}_{\Sigma }$, ${\dot{Z}}_A={(+---)}_{\Sigma }^{*}$ acts on the form ${\dot{\mathcal{Q}}}^A={(+---)}_{\Sigma }$, and ${\dot{Z}}_B={(++-+)}_{\Sigma }^{*}$ acts on the form ${\dot{\mathcal{Q}}}^B={(++-+)}_{\Sigma }$.

5. Two More Pairs of Spinors

There are four eigenvectors of ${\Sigma }_3^{*}$ that constitute a complete orthonormal set of basis:

$$e_1=\left(\begin{array}{c}1\\ 0\\ 0\\ 1\end{array}\right),e_2=\left(\begin{array}{c}1\\ 0\\ 0\\ -1\end{array}\right),e_3=\left(\begin{array}{c}0\\ 1\\ -i\\ 0\end{array}\right),e_4=\left(\begin{array}{c}0\\ 1\\ i\\ 0\end{array}\right).$$

(67)

$e_1$ and $e_3$ correspond to $+1$ eigenvalue and $e_2$ and $e_4$ correspond to $-1$ eigenvalue. We obtain eight generalized eigenvectors by combining the basis corresponding to the same eigenvalue. For example, we can obtain the four generalized eigenvectors that we have previously studied as follows:

$${\chi }_{{}_{\left(1\right)}}=u{e}_1+v{e}_3,{\chi }_{{}_{\left(2\right)}}=-v{e}_2-u{e}_4,$$

(68)

$${\chi }_{{}_{\left(3\right)}}=-v{e}_1+u{e}_3,{\chi }_{{}_{\left(4\right)}}=u{e}_2-v{e}_4,$$

(69)

We can obtain four more generalized eigenvectors of ${\Sigma }_3^{*}$ by changing the sign or swapping u and v:

$${\chi }_{{}_{\left(5\right)}}=-u{e}_1+v{e}_3,{\chi }_{{}_{\left(6\right)}}=v{e}_2-u{e}_4,$$

(70)

$${\chi }_{{}_{\left(7\right)}}=v{e}_1+u{e}_3,{\chi }_{{}_{\left(8\right)}}=u{e}_2+v{e}_4,$$

(71)

Totally we get eight undotted covariant spinors:

$${\chi }_{{}_{\left(1\right)}}=\left(\begin{array}{c}u\\ v\\ -iv\\ u\end{array}\right),{\chi }_{{}_{\left(2\right)}}=\left(\begin{array}{c}-v\\ -u\\ -iu\\ v\end{array}\right),{\chi }_{{}_{\left(3\right)}}=\left(\begin{array}{c}-v\\ u\\ -iu\\ -v\end{array}\right),{\chi }_{{}_{\left(4\right)}}=\left(\begin{array}{c}u\\ -v\\ -iv\\ -u\end{array}\right),$$

(72)

$${\chi }_{{}_{\left(5\right)}}=\left(\begin{array}{c}-u\\ v\\ -iv\\ -u\end{array}\right),{\chi }_{{}_{\left(6\right)}}=\left(\begin{array}{c}v\\ -u\\ -iu\\ -v\end{array}\right),{\chi }_{{}_{\left(7\right)}}=\left(\begin{array}{c}v\\ u\\ -iu\\ v\end{array}\right),{\chi }_{{}_{\left(8\right)}}=\left(\begin{array}{c}u\\ v\\ iv\\ -u\end{array}\right).$$

(73)

We can group ${\chi }_{{}_{\left(a\right)}}$ ($a=1,2,\cdots 8$) pairwise:

$$P_{{}_A}=\{{\chi }_{{}_{\left(1\right)}},{\chi }_{{}_{\left(2\right)}}\},P_{{}_B}=\{{\chi }_{{}_{\left(3\right)}},{\chi }_{{}_{\left(4\right)}}\},P_{{}_C}=\{{\chi }_{{}_{\left(5\right)}},{\chi }_{{}_{\left(6\right)}}\},P_{{}_D}=\{{\chi }_{{}_{\left(7\right)}},{\chi }_{{}_{\left(8\right)}}\},$$

(74)

We already know that $P_A$ transforms with $Z_A$ and $P_B$ transforms with $Z_B$. Following the same procedure that we have applied in the previous sections we can show that $P_C$ and $P_D$ transform with $Z_C$ and $Z_D$ respectively:

$$Z_C=A(L_C\otimes I)A^{-1},Z_D=A(L_D\otimes I)A^{-1},$$

(75)

where

$$L_C=\left(\begin{array}{cc}{L}_{11}& -L_{12}\\ -L_{21}& L_{22}\end{array}\right)=\left(\begin{array}{cc}{\alpha }_0+{\alpha }_3& -{\alpha }_1+i{\alpha }_2\\ -{\alpha }_1-i{\alpha }_2& {\alpha }_0-{\alpha }_3\end{array}\right)={(+--+)}_{\sigma }$$

(76)

$$L_D=\left(\begin{array}{cc}{L}_{22}& L_{21}\\ L_{12}& L_{11}\end{array}\right)=\left(\begin{array}{cc}{\alpha }_0-{\alpha }_3& {\alpha }_1+i{\alpha }_2\\ {\alpha }_1-i{\alpha }_2& {\alpha }_0+{\alpha }_3\end{array}\right)={(++--)}_{\sigma }$$

(77)

$${\dot{L}}_C={(+++-)}_{\sigma }^{*}=L_D^{*}$$

(78)

$${\dot{L}}_D={(+-++)}_{\sigma }^{*}=L_C^{*}$$

(79)

$$Z_C=\left(\begin{array}{cccc}{\alpha }_0& -{\alpha }_1& -{\alpha }_2& {\alpha }_3\\ -{\alpha }_1& {\alpha }_0& -i{\alpha }_3& -i{\alpha }_2\\ -{\alpha }_2& i{\alpha }_3& {\alpha }_0& i{\alpha }_1\\ {\alpha }_3& i{\alpha }_2& -i{\alpha }_1& {\alpha }_0\end{array}\right)={(+--+)}_{\Sigma }$$

(80)

$$Z_D=\left(\begin{array}{cccc}{\alpha }_0& {\alpha }_1& -{\alpha }_2& -{\alpha }_3\\ {\alpha }_1& {\alpha }_0& i{\alpha }_3& -i{\alpha }_2\\ -{\alpha }_2& -i{\alpha }_3& {\alpha }_0& -i{\alpha }_1\\ -{\alpha }_3& i{\alpha }_2& i{\alpha }_1& {\alpha }_0\end{array}\right)={(++--)}_{\Sigma }$$

(81)

There are also the dotted versions:

$${\dot{Z}}_C=\left(\begin{array}{cccc}{\alpha }_0^{*}& {\alpha }_1^{*}& {\alpha }_2^{*}& -{\alpha }_3^{*}\\ {\alpha }_1^{*}& {\alpha }_0^{*}& i{\alpha }_3^{*}& i{\alpha }_2^{*}\\ {\alpha }_2^{*}& -i{\alpha }_3^{*}& {\alpha }_0^{*}& -i{\alpha }_1^{*}\\ -{\alpha }_3^{*}& -i{\alpha }_2^{*}& i{\alpha }_1^{*}& {\alpha }_0^{*}\end{array}\right)={(+++-)}_{\Sigma }^{*}$$

(82)

$${\dot{Z}}_D=\left(\begin{array}{cccc}{\alpha }_0^{*}& -{\alpha }_1^{*}& {\alpha }_2^{*}& {\alpha }_3^{*}\\ -{\alpha }_1^{*}& {\alpha }_0^{*}& -i{\alpha }_3^{*}& i{\alpha }_2^{*}\\ {\alpha }_2^{*}& i{\alpha }_3^{*}& {\alpha }_0^{*}& i{\alpha }_1^{*}\\ {\alpha }_3^{*}& -i{\alpha }_2^{*}& -i{\alpha }_1^{*}& {\alpha }_0^{*}\end{array}\right)={(+-++)}_{\Sigma }^{*}$$

(83)

We also define the contravariat spinors ${\chi }^{{}^{\left(a\right)}}=g{\chi }_{{}_{\left(a\right)}}$ ($a=1,2,\cdots 8$) that correspond to the generalized eigenvectors of ${\Sigma }_3$:

$${\chi }^{{}^{\left(1\right)}}=\left(\begin{array}{c}v\\ -u\\ -iu\\ v\end{array}\right),{\chi }^{{}^{\left(2\right)}}=\left(\begin{array}{c}u\\ -v\\ iv\\ -u\end{array}\right),{\chi }^{{}^{\left(3\right)}}=\left(\begin{array}{c}u\\ v\\ iv\\ u\end{array}\right),{\chi }^{{}^{\left(4\right)}}=\left(\begin{array}{c}v\\ u\\ -iu\\ -v\end{array}\right),$$

(84)

$${\chi }^{{}^{\left(5\right)}}=\left(\begin{array}{c}v\\ u\\ iu\\ v\end{array}\right),{\chi }^{{}^{\left(6\right)}}=\left(\begin{array}{c}u\\ v\\ -iv\\ -u\end{array}\right),{\chi }^{{}^{\left(7\right)}}=\left(\begin{array}{c}u\\ -v\\ -iv\\ u\end{array}\right),{\chi }^{{}^{\left(8\right)}}=\left(\begin{array}{c}-v\\ u\\ -iu\\ v\end{array}\right).$$

(85)

We group them pairwise:

$$P^{{}^A}=\{{\chi }^{{}^{\left(1\right)}},{\chi }^{{}^{\left(2\right)}}\},P^{{}^B}=\{{\chi }^{{}^{\left(3\right)}},{\chi }^{{}^{\left(4\right)}}\},P^{{}^C}=\{{\chi }^{{}^{\left(5\right)}},{\chi }^{{}^{\left(6\right)}}\},P^{{}^D}=\{{\chi }^{{}^{\left(7\right)}},{\chi }^{{}^{\left(8\right)}}\}$$

(86)

Each pair of the dotted contravariant spinors transform with the associated dotted Z matrix.

We define four two-column covariant objects:

$${\chi }_{{}_A}=({\chi }_{{}_{\left(1\right)}},{\chi }_{{}_{\left(2\right)}}),{\chi }_{{}_B}=({\chi }_{{}_{\left(3\right)}},{\chi }_{{}_{\left(4\right)}}),{\chi }_{{}_C}=({\chi }_{{}_{\left(5\right)}},{\chi }_{{}_{\left(6\right)}}),{\chi }_{{}_D}=({\chi }_{{}_{\left(7\right)}},{\chi }_{{}_{\left(8\right)}})$$

(87)

And we define the corresponding two-column contravariant objects

$${\chi }^{{}^A}=({\chi }^{{}^{\left(1\right)}},{\chi }^{{}^{\left(2\right)}}),{\chi }^{{}^B}=({\chi }^{{}^{\left(3\right)}},{\chi }^{{}^{\left(4\right)}}),{\chi }^{{}^C}=({\chi }^{{}^{\left(5\right)}},{\chi }^{{}^{\left(6\right)}}),{\chi }^{{}^D}=({\chi }^{{}^{\left(7\right)}},{\chi }^{{}^{\left(8\right)}})$$

(88)

Finally, we construct eight outer products that lead to the following quaternions:

$${\chi }_{{}_A}{\chi }_{{}_A}^{†}\to {\mathcal{Q}}_{{}_A}={(++++)}_{\Sigma },{\dot{\chi }}^{{}^A}{\dot{\chi }}^{{}^{A†}}\to {\dot{\mathcal{Q}}}^{{}^A}={(+---)}_{\Sigma }.$$

(89)

$${\chi }_{{}_B}{\chi }_{{}_B}^{†}\to {\mathcal{Q}}_{{}_B}={(+-+-)}_{\Sigma },{\dot{\chi }}^{{}^B}{\dot{\chi }}^{{}^{B†}}\to {\dot{\mathcal{Q}}}^{{}^B}={(++-+)}_{\Sigma }.$$

(90)

$${\chi }_{{}_C}{\chi }_{{}_C}^{†}\to {\mathcal{Q}}_{{}_C}={(+--+)}_{\Sigma },{\dot{\chi }}^{{}^C}{\dot{\chi }}^{{}^{C†}}\to {\dot{\mathcal{Q}}}^{{}^C}={(+++-)}_{\Sigma }.$$

(91)

$${\chi }_{{}_D}{\chi }_{{}_D}^{†}\to {\mathcal{Q}}_{{}_D}={(++--)}_{\Sigma },{\dot{\chi }}^{{}^D}{\dot{\chi }}^{{}^{D†}}\to {\dot{\mathcal{Q}}}^{{}^D}={(+-++)}_{\Sigma }.$$

(92)

Each form transforms in its own way with the matching Z or $\dot{Z}$ matrix.

6. Complex Conjugated Forms

Let us write the complex conjugates of the quaternion forms:

$${\mathcal{Q}}_{{}_A}={(++++)}_{\Sigma }\stackrel{\mathrm{c}.\mathrm{c}.}{\to }{\dot{\mathcal{Q}}}_{{}_A}={(++++)}_{{\Sigma }^{*}}$$

(93)

$${\mathcal{Q}}_{{}_B}={(+-+-)}_{\Sigma }\stackrel{\mathrm{c}.\mathrm{c}.}{\to }{\dot{\mathcal{Q}}}_{{}_B}={(+-+-)}_{{\Sigma }^{*}}$$

(94)

$${\mathcal{Q}}_{{}_C}={(+--+)}_{\Sigma }\stackrel{\mathrm{c}.\mathrm{c}.}{\to }{\dot{\mathcal{Q}}}_{{}_C}={(+--+)}_{{\Sigma }^{*}}$$

(95)

$${\mathcal{Q}}_{{}_D}={(++--)}_{\Sigma }\stackrel{\mathrm{c}.\mathrm{c}.}{\to }{\dot{\mathcal{Q}}}_{{}_D}={(++--)}_{{\Sigma }^{*}}$$

(96)

$${\dot{\mathcal{Q}}}^{{}^A}={(+---)}_{\Sigma }\stackrel{\mathrm{c}.\mathrm{c}.}{\to }{\mathcal{Q}}^{{}^A}={(+---)}_{{\Sigma }^{*}}$$

(97)

$${\dot{\mathcal{Q}}}^{{}^B}={(++-+)}_{\Sigma }\stackrel{\mathrm{c}.\mathrm{c}.}{\to }{\mathcal{Q}}^{{}^B}={(++-+)}_{{\Sigma }^{*}}$$

(98)

$${\dot{\mathcal{Q}}}^{{}^C}={(+++-)}_{\Sigma }\stackrel{\mathrm{c}.\mathrm{c}.}{\to }{\mathcal{Q}}^{{}^C}={(+++-)}_{{\Sigma }^{*}}$$

(99)

$${\dot{\mathcal{Q}}}^{{}^D}={(+-++)}_{\Sigma }\stackrel{\mathrm{c}.\mathrm{c}.}{\to }{\mathcal{Q}}^{{}^D}={(+-++)}_{{\Sigma }^{*}}$$

(100)

Conjugate forms reside in the dual space that spanned by ${\Sigma }_{\mu }^{*}$. They don’t have any counterpart in $SL(2,C)$. Dotted lower indexed and undotted upper indexed forms transform with $Z_{\cdots }^{*}$ or with ${\left({\dot{Z}}_{\cdots }\right)}^{*}$ matrices respectively and all transformations obey the scheme $T_A$.

4-vector scalar product can be defined by using the dual forms in two equivalent ways. Let $\mathsf{Q}$ and $\mathsf{P}$ be two 4-vectors and let ${\mathcal{Q}}_{{}_X}$ and ${\mathcal{P}}_{{}_X}$ be the corresponding quaternions ($X=A,B,C,D$) :

$$\mathsf{Q}·\mathsf{P}=\frac{1}{4}\mathrm{Tr}\left({\mathcal{Q}}_{{}_X}^T{\mathcal{P}}^{{}^X}\right)$$

(101)

Or, noting that ${\mathcal{P}}^{{}^X}=g{\mathcal{P}}_{{}_X}{g}^{-1}$, we can write also as:

$$\mathsf{Q}·\mathsf{P}=\frac{1}{4}\left({\mathcal{Q}}_{\mu \dot{\nu }}{\mathcal{P}}^{\mu \dot{\nu }}\right)$$

(102)

where now indices refer to the components and the summation convention is implied.

In general, in order to get something real we have to use both $\Sigma$ and ${\Sigma }^{*}$. As an example, $\Lambda =Z{Z}^{*}=Z^{*}Z$, is the real Lorentz transformation matrix.

7. Four Types of Transformations for SL(2, C)

We can suggest a similar formalism for $SL(2,C)$. Let ${\xi }_{{}_A},{\xi }_{{}_B},{\xi }_{{}_C}$ and ${\xi }_{{}_D}$ be covariant spinors:

$${\xi }_{{}_A}=\left(\begin{array}{c}u\\ v\end{array}\right),{\xi }_{{}_B}=\left(\begin{array}{c}v\\ -u\end{array}\right),{\xi }_{{}_C}=\left(\begin{array}{c}u\\ -v\end{array}\right),{\xi }_{{}_D}=\left(\begin{array}{c}v\\ u\end{array}\right)$$

(103)

and let ${\xi }^{{}^A},{\xi }^{{}^B},{\xi }^{{}^C}$ and ${\xi }^{{}^D}$ be contravariant spinors:

$${\xi }^{{}^A}=\left(\begin{array}{c}v\\ -u\end{array}\right),{\xi }^{{}^B}=\left(\begin{array}{c}-u\\ -v\end{array}\right),{\xi }^{{}^C}=\left(\begin{array}{c}-v\\ -u\end{array}\right),{\xi }_{{}_D}=\left(\begin{array}{c}u\\ -v\end{array}\right)$$

(104)

where ${\xi }^{{}^A}={\xi }_{{}_B}$, ${\xi }^{{}^B}=-{\xi }_{{}_A}$, ${\xi }^{{}^C}=-{\xi }_{{}_D}$ and ${\xi }^{{}^D}={\xi }_{{}_C}$. This proliferation is necessary for the symmetry in Figure 1.

We have the following transformation properties:

$${\xi }_{{}_A}\to L_A{\xi }_{{}_A},{\xi }_{{}_B}\to L_B{\xi }_{{}_B},{\xi }_{{}_C}\to L_C{\xi }_{{}_C},{\xi }_{{}_D}\to L_D{\xi }_{{}_D}$$

(105)

$${\dot{\xi }}^{{}^A}\to {\dot{L}}_A{\xi }^{{}^A},{\dot{\xi }}^{{}^B}\to {\dot{L}}_B{\xi }^{{}^B},{\dot{\xi }}^{{}^C}\to {\dot{L}}_C{\xi }^{{}^C},{\dot{\xi }}^{{}^D}\to {\dot{L}}_D{\xi }^{{}^D},$$

(106)

$${\dot{L}}_A=L_B^{*},{\dot{L}}_B=L_A^{*},{\dot{L}}_C=L_D^{*},{\dot{L}}_D=L_C^{*}$$

(107)

All transformations obey the scheme $T_A$.

We have the following outer products:

$${\xi }_{{}_A}{\xi }_{{}_A}^{†}\to X_{{}_A}={(++++)}_{\sigma },{\xi }_{{}_B}{\xi }_{{}_B}^{†}\to X_{{}_B}={(+-+-)}_{\sigma }$$

(108)

$${\xi }_{{}_C}{\xi }_{{}_C}^{†}\to X_{{}_C}={(+--+)}_{\sigma },{\xi }_{{}_D}{\xi }_{{}_D}^{†}\to X_{{}_D}={(++--)}_{\sigma }$$

(109)

$${\dot{\xi }}^{{}^A}{\dot{\xi }}^{{}^{A†}}\to {\dot{X}}^{{}^A}={(+---)}_{\sigma },{\dot{\xi }}^{{}^B}{\dot{\xi }}^{{}^{B†}}\to {\dot{X}}^{{}^B}={(++-+)}_{\sigma },$$

(110)

$${\dot{\xi }}^{{}^C}{\dot{\xi }}^{{}^{C†}}\to {\dot{X}}^{{}^C}={(+++-)}_{\sigma },{\dot{\xi }}^{{}^D}{\dot{\xi }}^{{}^{D†}}\to {\dot{X}}^{{}^D}={(+-++)}_{\sigma },$$

(111)

It is worth noting that complex conjugating these forms does not yield anything new.

Appendix A.1. Various Forms of Z and L Matrices

Let us begin with the exponential form $Z_A=e^R$, where $R=-\frac{i}{2}\overrightarrow{\pi }·\overrightarrow{\Sigma }$, ${\pi }_i={\theta }_i+i{\eta }_i$. Let $\varphi$ be the complex angle defined as $\varphi =\frac{1}{2}\sqrt{{\pi }_1^2+{\pi }_2^2+{\pi }_3^2}$. Using the property $R^2=-{\varphi }^{2}I$:

$$Z_A=cos\varphi I-\frac{isin\varphi }{2\varphi }\overrightarrow{\pi }·\overrightarrow{\Sigma }=\left(\begin{array}{cccc}{\alpha }_0& {\alpha }_1& {\alpha }_2& {\alpha }_3\\ {\alpha }_1& {\alpha }_0& -i{\alpha }_3& i{\alpha }_2& \\ {\alpha }_2& i{\alpha }_3& {\alpha }_0& -i{\alpha }_1\\ {\alpha }_3& -i{\alpha }_2& i{\alpha }_1& {\alpha }_0\end{array}\right)$$

(A1)

where

$${\alpha }_0=cos\varphi ,{\alpha }_1=-\frac{isin\varphi }{2\varphi }{\pi }_1,{\alpha }_2=-\frac{isin\varphi }{2\varphi }{\pi }_2,{\alpha }_3=-\frac{isin\varphi }{2\varphi }{\pi }_3.$$

(A2)

Or in a compact form

$$Z_A={\alpha }_0{\Sigma }_0+{\alpha }_1{\Sigma }_1+{\alpha }_2{\Sigma }_2+{\alpha }_3{\Sigma }_3={(++++)}_{\Sigma }.$$

(A3)

It is easy to show that

$$Z_A^{-1}={\alpha }_0{\Sigma }_0-{\alpha }_1{\Sigma }_1-{\alpha }_2{\Sigma }_2-{\alpha }_3{\Sigma }_3={(+---)}_{\Sigma }.$$

(A4)

and

$$Z_A^{†}={\alpha }_0^{*}{\Sigma }_0+{\alpha }_1^{*}{\Sigma }_1+{\alpha }_2^{*}{\Sigma }_2+{\alpha }_3^{*}{\Sigma }_3={(++++)}_{\Sigma }^{*}$$

(A5)

where complex conjugation is applied only to ${\alpha }_{\mu }$.

The corresponding $L_A$ is

$$L_A=\left(\begin{array}{cc}{\alpha }_0+{\alpha }_3& {\alpha }_1-i{\alpha }_2\\ {\alpha }_1+i{\alpha }_2& {\alpha }_0-{\alpha }_3\end{array}\right)$$

(A6)

In terms of the Pauli matrices:

$$L_A={\alpha }_0{\sigma }_0+{\alpha }_1{\sigma }_1+{\alpha }_2{\sigma }_2+{\alpha }_3{\sigma }_3={(++++)}_{\sigma }.$$

(A7)

In order to write $Z_B$ we first find $L_B={\left({\dot{L}}_A\right)}^{*}$, where

$${\dot{L}}_A={\left(L_A^{-1}\right)}^{†}=\left(\begin{array}{cc}{\alpha }_0^{*}-{\alpha }_3^{*}& -{\alpha }_1^{*}+i{\alpha }_2^{*}\\ -{\alpha }_1^{*}-i{\alpha }_2^{*}& {\alpha }_0^{*}+{\alpha }_3^{*}\end{array}\right)={(+---)}_{\sigma }^{*}.$$

(A8)

From the definition $Z_B=A(L_B\otimes I)A^{-1}$:

$$Z_B=\left(\begin{array}{cccc}{\alpha }_0& -{\alpha }_1& {\alpha }_2& -{\alpha }_3\\ -{\alpha }_1& {\alpha }_0& i{\alpha }_3& i{\alpha }_2\\ {\alpha }_2& -i{\alpha }_3& {\alpha }_0& i{\alpha }_1\\ -{\alpha }_3& -i{\alpha }_2& -i{\alpha }_1& {\alpha }_0\end{array}\right)={\alpha }_0{\Sigma }_0-{\alpha }_1{\Sigma }_1+{\alpha }_2{\Sigma }_2-{\alpha }_3{\Sigma }_3.$$

(A9)

Or, simply

$$Z_B={(+-+-)}_{\Sigma }$$

(A10)

We write various forms of Z and L matrices in compact forms:

$$L_A={(++++)}_{\sigma },L_B={(+-+-)}_{\sigma },{\dot{L}}_A={(+---)}_{\sigma }^{*},{\dot{L}}_B={(++-+)}_{\sigma }^{*}.$$

(A11)

$$Z_A={(++++)}_{\Sigma },Z_B={(+-+-)}_{\Sigma },{\dot{Z}}_A={(+---)}_{\Sigma }^{*},{\dot{Z}}_B={(++-+)}_{\Sigma }^{*}.$$

(A12)

Although, all types of Z and L matrices are in the same form, there is a very important difference between them. Because of the particular property of the Pauli matrices, ${\sigma }_1^{*}={\sigma }_1,{\sigma }_3^{*}={\sigma }_3$, but ${\sigma }_2^{*}=-{\sigma }_2$, we have the following relations:

$$L_B={\dot{L}}_A^{*},{\dot{L}}_B=L_A^{*}.$$

(A13)

For example,

$${\dot{L}}_A={(+---)}_{\sigma }^{*}\to {\dot{L}}_A^{*}={(+---)}_{{\sigma }^{*}}={(+-+-)}_{\sigma }=L_B.$$

(A14)

On the other hand we do not have a similar property with $\Sigma$ matrices, hence

$$Z_B\ne {\dot{Z}}_A^{*},{\dot{Z}}_B\ne Z_A^{*}.$$

(A15)

.

The structural difference between $SL(2,C)$ and $SL(4,C)$ becomes more apparent when we write the matrices in exponential forms. In order to do this we have to define two types of $\overrightarrow{\pi }$: ${\overrightarrow{\pi }}_A=({\pi }_1,{\pi }_2,{\pi }_3)$ and ${\overrightarrow{\pi }}_B=(-{\pi }_1,{\pi }_2,-{\pi }_3)$.

$$L_A=exp(-\frac{i}{2}{\overrightarrow{\pi }}_A·\overrightarrow{\sigma }),L_B=exp(-\frac{i}{2}{\overrightarrow{\pi }}_B·\overrightarrow{\sigma }),{\dot{L}}_A=exp(-\frac{i}{2}{\overrightarrow{\pi }}_A^{*}·\overrightarrow{\sigma }),{\dot{L}}_B=exp(-\frac{i}{2}{\overrightarrow{\pi }}_B^{*}·\overrightarrow{\sigma }).$$

(A16)

$$Z_A=exp(-\frac{i}{2}{\overrightarrow{\pi }}_A·\overrightarrow{\Sigma }),Z_B=exp(-\frac{i}{2}{\overrightarrow{\pi }}_B·\overrightarrow{\Sigma }),{\dot{Z}}_A=exp(-\frac{i}{2}{\overrightarrow{\pi }}_A^{*}·\overrightarrow{\Sigma }),{\dot{Z}}_B=exp(-\frac{i}{2}{\overrightarrow{\pi }}_B^{*}·\overrightarrow{\Sigma }).$$

(A17)

Due to the properties, $-{\overrightarrow{\pi }}_B·\overrightarrow{\sigma }={\overrightarrow{\pi }}_A·{\overrightarrow{\sigma }}^{*}$ and $-{\overrightarrow{\pi }}_B·{\overrightarrow{\sigma }}^{*}={\overrightarrow{\pi }}_A·\overrightarrow{\sigma }$, the relations in Eq.(A13) hold. But we do not have similar relations with the $\Sigma$ matrices.

Appendix A.2. Two Types of Lorentz Transformations

Let ${\xi }_{{}_A}$ and ${\chi }_{{}_A}$ be 2- and 4-component spinors that transform according to the scheme $T_A$, and let ${\mathsf{V}}_{{}_A}$ be a 4-vector that can be associated with the outer products ${\xi }_{{}_A}{\xi }_{{}_A}^{†}$ and ${\chi }_{{}_A}{\chi }_{{}_A}^{†}$. ${\mathsf{V}}_{{}_A}$ transforms as

$${\mathsf{V}}_{{}_A}\to {\mathsf{V}}_{{}_A}^{\prime }={\Lambda }_{{}_A}{\mathsf{V}}_{{}_A},$$

(A18)

${\Lambda }_{{}_A}$ is the usual $4\times 4$ real Lorentz transformation matrix that corresponds to the scheme $T_A$:

$${\Lambda }_A=A(L_A\otimes L_A^{*})A^{-1}$$

(A19)

where $L_A=L$. The corresponding $SL(4,C)$ elements are

$$Z_A=A(L_A\otimes I)A^{-1},Z_A^{*}=A(I\otimes L_A^{*})A^{-1}$$

(A20)

$${\Lambda }_A=Z_A{Z}_A^{*}=Z_A^{*}{Z}_A.$$

(A21)

Let ${\chi }_{{}_B}$ be a 4-component spinor that transforms according to the scheme $T_B$, and let ${\mathsf{V}}_{{}_B}$ be a 4-vector associated with the outer product ${\chi }_{{}_B}{\chi }_{{}_B}^{†}$. ${\mathsf{V}}_{{}_B}$ transforms as

$${\mathsf{V}}_{{}_B}\to {\mathsf{V}}_{{}_B}^{\prime }={\Lambda }_{{}_B}{\mathsf{V}}_{{}_B},$$

(A22)

where ${\Lambda }_{{}_B}$ is the $4\times 4$ real Lorentz transformation matrix that corresponds to the scheme $T_B$:

$${\Lambda }_B=A(L_B\otimes L_B^{*})A^{-1}$$

(A23)

. where $L_B={\dot{L}}_A^{*}$ by definition. The corresponding $SL(4,C)$ elements are

$$Z_B=A(L_B\otimes I)A^{-1},Z_B^{*}=A(I\otimes L_B^{*})A^{-1}.$$

(A24)

$${\Lambda }_B=Z_B{Z}_B^{*}=Z_B^{*}{Z}_B.$$

(A25)

In this case, we don’t have a counterpart of ${\chi }_{{}_B}$ in the formalism of $SL(2,C)$.

${\Lambda }_A$ and ${\Lambda }_B$ preserves the Minkowski norm due to the relations ${\Lambda }_A^T\eta {\Lambda }_A=\eta$ and ${\Lambda }_B^T\eta {\Lambda }_B=\eta$. Since $\eta$ is real, these relations follow from the following more fundamental relations:

$$Z_A^T\eta Z_A=\eta ,Z_B^T\eta Z_B=\eta .$$

(A26)

Appendix A.3. The Other Way Around

Suppose that, for $a=1$ or $a=2$, the eigenvectors of ${\Sigma }_3^{*}$ and ${\Sigma }_3$ transform as follows:

$${\chi }_{{}_{\left(a\right)}}\to {\chi }_{{}_{\left(a\right)}}^{\prime }=Z_A{\chi }_{{}_{\left(a\right)}},{\chi }^{\left(a\right)}\to {\chi }^{{\left(a\right)}^{\prime }}={\dot{Z}}_A^{*}{\chi }^{\left(a\right)}$$

(A27)

We know that these transformations obey the scheme $T_A$.

Suppose that, for $a=3$ or $a=4$, the eigenvectors of ${\Sigma }_3^{*}$ and ${\Sigma }_3$ are mathematical objects that have different properties than the eigenvectors with indices $a=1$ or $a=2$. Possibly, they describe physical objects of different kind which transform as

$${\chi }_{{}_{\left(a\right)}}\to {\chi }_{{}_{\left(a\right)}}^{\prime }=Z_B{\chi }_{{}_{\left(a\right)}},{\chi }^{\left(a\right)}\to {\chi }^{{\left(a\right)}^{\prime }}={\dot{Z}}_B^{*}{\chi }^{\left(a\right)}$$

(A28)

In this case, these transformations will also obey the scheme $T_A$, and accordingly all null and non-null 4-vectors that correspond to the outer product forms and associated quaternion forms will transform in the usual way, i.e., they transform with ${\Lambda }_A$.