40.4 Spin and the Lorentz group

40.4 Spin and the Lorentz group

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Just as the groups $SO(n)$ have double covers $Spin(n)$ , the group $SO(3,1)$ has a double cover $Spin(3,1)$ , which we will show can be identified with the group $SL(2,\mathbf{C})$ of 2 by 2 complex matrices with unit determinant. This group will have the same Lie algebra as $SO(3,1)$ , and we will sometimes refer to either group as the “Lorentz group”.

Recall from chapter 6 that for $SO(3)$ the spin double cover $Spin(3)$ can be identified with either $Sp(1)$ (the unit quaternions) or $SU(2)$ , and then the action of $Spin(3)$ as $SO(3)$ rotations of ${\mathbf{R}}^3$ was given by conjugation of imaginary quaternions (using $Sp(1))$ or certain 2 by 2 complex matrices (using $SU(2))$ . In the $SU(2)$ case this was done explicitly by identifying

$$(x_1,x_2,x_3)\leftrightarrow \left(\begin{array}{cc}{x}_3& x_1-i{x}_2\\ x_1+i{x}_2& -x_3\end{array}\right)$$

and then showing that conjugating this matrix by an element of $SU(2)$ was a linear map leaving invariant

$$\det \left(\begin{array}{cc}{x}_3& x_1-i{x}_2\\ x_1+i{x}_2& -x_3\end{array}\right)=-(x_1^2+x_2^2+x_3^2)$$

and thus a rotation in $SO(3)$

The same sort of thing works for the Lorentz group case. Now we identify ${\mathbf{R}}^4$ with the space of 2 by 2 complex self-adjoint matrices by

$$(x_0,x_1,x_2,x_3)\leftrightarrow \left(\begin{array}{cc}{x}_0+x_3& x_1-i{x}_2\\ x_1+i{x}_2& x_0-x_3\end{array}\right)$$

and observe that

$$\det \left(\begin{array}{cc}{x}_0+x_3& x_1-i{x}_2\\ x_1+i{x}_2& x_0-x_3\end{array}\right)=x_0^2-x_1^2-x_2^2-x_3^2$$

This provides a very useful way to think of Minkowski space: as complex selfadjoint 2 by 2 matrices, with norm-squared minus the determinant of the matrix.

The linear transformation

$$\left(\begin{array}{cc}{x}_0+x_3& x_1-i{x}_2\\ x_1+i{x}_2& x_0-x_3\end{array}\right)\to \Omega \left(\begin{array}{cc}{x}_0+x_3& x_1-i{x}_2\\ x_1+i{x}_2& x_0-x_3\end{array}\right){\Omega }^{†}\text{\hspace{1em}(40.4)}$$

for $\Omega \in SL(2,\mathbf{C})$ preserves the determinant and thus the inner-product, since

$$\begin{array}{rl}\det (\Omega \left(\begin{array}{ll}{x}_0+x_3& x_1-i{x}_2\\ x_1+i{x}_2& x_0-x_3\end{array}\right){\Omega }^{†})& =(\det \Omega )\det \left(\begin{array}{ll}{x}_0+x_3& x_1-i{x}_2\\ x_1+i{x}_2& x_0-x_3\end{array}\right)(\det {\Omega }^{†})\\ & =x_0^2-x_1^2-x_2^2-x_3^2\end{array}$$

It also takes self-adjoint matrices to self-adjoints, and thus ${\mathbf{R}}^4$ to ${\mathbf{R}}^4$ , since

$$\begin{array}{rl}(\Omega \left(\begin{array}{cc}{x}_0+x_3& x_1-i{x}_2\\ x_1+i{x}_2& x_0-x_3\end{array}\right){\Omega }^{†}{)}^{†}& =({\Omega }^{†}{)}^{†}{\left(\begin{array}{cc}{x}_0+x_3& x_1-i{x}_2\\ x_1+i{x}_2& x_0-x_3\end{array}\right)}^{†}{\Omega }^{†}\\ & =\Omega \left(\begin{array}{cc}{x}_0+x_3& x_1-i{x}_2\\ x_1+i{x}_2& x_0-x_3\end{array}\right){\Omega }^{†}\end{array}$$

Note that both Ω and −Ω give the same linear transformation when they act by conjugation like this. One can show that all elements of $SO(3,1)$ arise as such conjugation maps, by finding appropriate Ω that give rotations or boosts in the $\mu \nu$ planes, since these generate the group.

Recall that the double covering map

$$\Phi :SU(2)\to SO(3)$$

was given for $\Omega \in SU(2)$ by taking $\Phi (\Omega )$ to be the linear transformation in $SO(3)$

$$\left(\begin{array}{cc}{x}_3& x_1-i{x}_2\\ x_1+i{x}_2& -x_3\end{array}\right)\to \Omega \left(\begin{array}{cc}{x}_3& x_1-i{x}_2\\ x_1+i{x}_2& -x_3\end{array}\right){\Omega }^{-1}$$

We have found an extension of this map to a double covering map from $SL(2,\mathbf{C})$ to $SO(3,1)$ . This restricts to $\Phi$ on the subgroup $SU(2)$ of $SL(2,\mathbf{C})$ matrices satisfying ${\Omega }^{†}={\Omega }^{-1}$

Digression (The complex group $Spin(4,\mathbf{C})$ and its real forms). Recall from chapter 6 that we found that $Spin(4)=Sp(1)\times Sp(1)$ , with the corresponding $SO(4)$ transformation given by identifying ${\mathbf{R}}^4$ with the quaternions $H$ and taking not just conjugations by unit quaternions, but both left and right multiplication by distinct unit quaternions. Rewriting this in terms of complex matrices instead of quaternions, we have $Spin(4)=SU(2)\times SU(2)$ , and a pair ${\Omega }_1,{\Omega }_2$ of $SU(2)$ matrices acts as an $SO(4)$ rotation by

$$\left(\begin{array}{cc}{x}_0-i{x}_3& -x_2-i{x}_1\\ x_2-i{x}_1& x_0+i{x}_3\end{array}\right)\to {\Omega }_1\left(\begin{array}{cc}{x}_0-i{x}_3& -x_2-i{x}_1\\ x_2-i{x}_1& x_0+i{x}_3\end{array}\right){\Omega }_2$$

preserving the determinant $x_0^2+x_1^2+x_2^2+x_3^2$

For another example, consider the identification of ${\mathbf{R}}^4$ with 2 by 2 real matrices given $by$

$$(x_0,x_1,x_2,x_3)\leftrightarrow \left(\begin{array}{cc}{x}_0+x_3& x_2+x_1\\ x_2-x_1& x_0-x_3\end{array}\right)$$

Given a pair of matrices ${\Omega }_1,{\Omega }_2$ in $SL(2,\mathbf{R})$ , the linear transformation

$$\left(\begin{array}{cc}{x}_0+x_3& x_2+x_1\\ x_2-x_1& x_0-x_3\end{array}\right)\to {\Omega }_1\left(\begin{array}{cc}{x}_0+x_3& x_2+x_1\\ x_2-x_1& x_0-x_3\end{array}\right){\Omega }_2$$

preserves the reality condition on the matrix, and preserves

$$\det \left(\begin{array}{cc}{x}_0+x_3& x_2+x_1\\ x_2-x_1& x_0-x_3\end{array}\right)=x_0^2+x_1^2-x_2^2-x_3^2$$

so gives an element of $SO(2,2)$ , and we see that $Spin(2,2)=SL(2,\mathbf{R})\times$ $SL(2,\mathbf{R})$

These three diferent constructions for the cases

$$Spin(4)=SU(2)\times SU(2),Spin(3,1)=SL(2,\mathbf{C})$$

and

$$Spin(2,2)=SL(2,\mathbf{R})\times SL(2,\mathbf{R})$$

correspond to diferent so-called “real forms” of a fact about complex groups that one can get by complexifying any of the examples (considering elements $(x_0,x_1,x_2,x_3)\in {\mathbf{C}}^4$ , not just in ${\mathbf{R}}^4)$ . For instance, in the $Spin(4)$ case, taking the $x_0,x_1,x_2,x_3$ in the matrix

$$\left(\begin{array}{cc}{x}_0-i{x}_3& -x_2-i{x}_1\\ x_2-i{x}_1& x_0+i{x}_3\end{array}\right)$$

to have arbitrary complex values $z_0,z_1,z_2$ , $z_3$ one gets arbitrary 2 by 2 complex matrices, and the transformation

$$\left(\begin{array}{cc}{z}_0-i{z}_3& -z_2-i{z}_1\\ z_2-i{z}_1& z_0+i{z}_3\end{array}\right)\to {\Omega }_1\left(\begin{array}{cc}{z}_0-i{z}_3& -z_2-i{z}_1\\ z_2-i{z}_1& z_0+i{z}_3\end{array}\right){\Omega }_2$$

preserves this space as well as the determinant $(z_0^2+z_1^2+z_2^2+z_3^2)$ for ${\Omega }_1$ and ${\Omega }_2$ not just in $SU(2)$ , but in the larger group $SL(2,\mathbf{C})$ $So$ we find that the group $SO(4,\mathbf{C})$ of complex orthogonal transformations of ${\mathbf{C}}^4$ has spin double cover

$$Spin(4,\mathbf{C})=SL(2,\mathbf{C})\times SL(2,\mathbf{C})$$

Since spin $(4,\mathbf{C})=\mathfrak{so}(3,1)\otimes \mathbf{C}.$ , this relation between complex Lie groups corresponds to the Lie algebra relation

$$\mathfrak{so}(3,1)\otimes \mathbf{C}=\mathfrak{sl}(2,\mathbf{C})\oplus \mathfrak{sl}(2,\mathbf{C})$$

we found explicitly earlier when we showed that by taking complex coeficients of generators $l_j$ and $k_{j}of\mathfrak{so}(3,1)$ we could find generators $A_j$ and $B_j$ of two diferent $\mathfrak{sl}(2,\mathbf{C})$ sub-algebras.

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