Spin groups

The spin group $Spin(n,m)$ is the double cover of the group $SO(n,m)$ (the special orthogonal group). The latter is the group of special orthogonal group of signature $n,m$.

Spin Groups:

The double cover of $SO(2)$ is $\text{Spin}(2)$, which is isomorphic to $U(1)=SU(1)$. They are the unitary complex numbers.

The double cover of $SO(3)$ is given by $Spin(3)$. This is isomorphic to $SU(2)$ and can also be represented by unit quaternions.

The double cover of $SO^+(1,3)$ is $Spin(1,3)$, which is equivalent to $SL(2, \mathbb{C})$, the special linear group of 2x2 complex matrices. The complex Lie algebra $\mathfrak{sl}(2, \mathbb{C})$ associated with the Special Linear group of 2x2 complex matrices with determinant 1 can be decomposed into a direct sum of two real algebras isomorphic to $\mathfrak{su}(2)$. This decomposition is expressed as

$$ \mathfrak{sl}(2, \mathbb{C}) = \mathfrak{su}(2) \oplus (\pm i)\mathfrak{su}(2), $$

where $\mathfrak{su}(2)$ represents the algebra of rotation generators in three-dimensional space, and $(\pm i)\mathfrak{su}(2)$ corresponds to the algebra of boost generators (Lorentz boosts) which in physical terms relate to transformations altering an object's velocity without rotation. The coefficients $-i$ and $+i$ have to do with chirality...

The group $SO^+(1,3)$ contains six "basis" matrices:

Rotations:

$$ \Lambda_{yz}(\theta) = \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & \cos\theta & -\sin\theta \\ 0 & 0 & \sin\theta & \cos\theta \\ \end{bmatrix} $$ $$ \Lambda_{zx}(\theta) = \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & \cos\theta & 0 & \sin\theta \\ 0 & 0 & 1 & 0 \\ 0 & -\sin\theta & 0 & \cos\theta \\ \end{bmatrix} $$ $$ \Lambda_{xy}(\theta) = \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & \cos\theta & -\sin\theta & 0 \\ 0 & \sin\theta & \cos\theta & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} $$

Boosts:

$$ \Lambda_{tx}(\phi) = \begin{bmatrix} \cosh\phi & -\sinh\phi & 0 & 0\\ -\sinh\phi & \cosh\phi & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ \end{bmatrix} $$ $$ \Lambda_{ty}(\phi) = \begin{bmatrix} \cosh\phi & 0 & -\sinh\phi & 0\\ 0 & 1 & 0 & 0\\ -\sinh\phi & 0 & \cosh\phi & 0\\ 0 & 0 & 0 & 1\\ \end{bmatrix} $$ $$ \Lambda_{tz}(\phi) = \begin{bmatrix} \cosh\phi & 0 & 0 & -\sinh\phi\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ -\sinh\phi & 0 & 0 & \cosh\phi\\ \end{bmatrix} $$

They can be seen like associated to the following matrices (or their opposites) in $Spin(1,3)=SL(2,\mathbb C)$:

$$ \begin{bmatrix} \cos \frac{\theta}{2} & -i\sin \frac{\theta}{2} \\ -i\sin \frac{\theta}{2} & \cos \frac{\theta}{2} \\ \end{bmatrix} $$ $$ \begin{bmatrix} \cos \frac{\theta}{2} & -\sin \frac{\theta}{2} \\ \sin \frac{\theta}{2} & \cos \frac{\theta}{2} \\ \end{bmatrix} $$ $$ \begin{bmatrix} e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2} \\ \end{bmatrix} $$ $$ \begin{bmatrix} \cosh \frac{\phi}{2} & -\sinh \frac{\phi}{2} \\ -\sinh \frac{\phi}{2} & \cosh \frac{\phi}{2} \\ \end{bmatrix} $$ $$ \begin{bmatrix} \cosh \frac{\phi}{2} & i\sinh \frac{\phi}{2} \\ -i\sinh \frac{\phi}{2} & \cosh \frac{\phi}{2} \\ \end{bmatrix} $$ $$ \begin{bmatrix} e^{-\phi/2} & 0 \\ 0 & e^{\phi/2} \\ \end{bmatrix} $$

Related: Clifford algebras

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es

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