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Section 6.5 Clifford Rotations

The rotations \(\SO(p,q)\) act on \(\RR^{p,q}\text{.}\) But a copy of this vector space is contained in \(\Cl(p,q)\text{,}\) namely as its elements of rank 1. Can we use the Clifford algebra to describe \(\SO(p,q)\) and/or its Lie algebra \(\so(p,q)\text{?}\)

A vector in \(\RR^{p,q}\subset\Cl(p,q)\) has the form \(v=v^m\gamma_m\text{.}\) Assuming that both of \(\gamma_i\) and \(\gamma_j\) are spacelike (\(\gamma_m^2=+\one\)), a rotation in the \((i,j)\)-plane has the form

\begin{align} \gamma_i \amp\longmapsto \gamma_i\cos\alpha - \gamma_j\sin\alpha\notag\\ \gamma_j \amp\longmapsto \gamma_i\sin\alpha + \gamma_j\cos\alpha\notag\\ \gamma_k \amp\longmapsto \gamma_k\tag{6.5.1} \end{align}

where \(i,j,k\) are assumed to be distinct. The infinitesimal form of this rotation is therefore

\begin{align} \gamma_i \amp\longmapsto -\gamma_j\notag\\ \gamma_j \amp\longmapsto \gamma_i\notag\\ \gamma_k \amp\longmapsto 0\tag{6.5.2} \end{align}

which is obtained by differentiating with respect to \(\alpha\) and setting \(\alpha=0\text{.}\) If \(\gamma_j\) is instead timelike, then \(\sin\) and \(\cos\) need to be replaced by \(\sinh\) and \(\cosh\text{,}\) respectively, in (6.5.1), and the minus sign removed from (6.5.1)–(6.5.2). Direct computation shows that

\begin{align} [\gamma_i\gamma_j,\gamma_i] \amp= -2 |\gamma_i|^2 \gamma_j ,\notag\\ [\gamma_i\gamma_j,\gamma_j] \amp= 2 |\gamma_j|^2 \gamma_i ,\notag\\ [\gamma_i\gamma_j,\gamma_k] \amp= 0\tag{6.5.3} \end{align}

which is enough to establish that the infinitesimal rotation or boost in the \((i,j)\)-plane is generated by commuting with the element \(\frac12\gamma_j\gamma_i\text{.}\)  1  Since commutation is an operation in \(\Cl(V)\text{,}\) we have indeed obtained an action of the Lie algebra \(\so(p,q)\) on vectors, all within \(\Cl(p,q)\text{.}\) Direct computation now verifies that the \(n\choose 2\) elements of rank 2 indeed have the same commutators as the corresponding infinitesimal rotations. Thus, \(\so(p,q)\) itself is contained within \(\Cl(p,q)\text{!}\)

Furthermore, we can exponentiate this description of \(\so(p,q)\subset\Cl(p,q)\) to obtain the group \(\SO(p,q)\subset\Cl(p,q)\text{.}\) Explicitly, reversing the construction above yields the rotation matrices

\begin{equation} \exp\left(\frac12\gamma_j\gamma_i\alpha\right) = \begin{cases} \one\cos\bigl(\frac\alpha2\bigr) + \gamma_j\gamma_i\sin\bigl(\frac\alpha2\bigr) \amp |\gamma_i|^2|\gamma_j|^2=1\\ \one\cosh\bigl(\frac\alpha2) + \gamma_j\gamma_i\sinh\bigl(\frac\alpha2\bigr) \amp |\gamma_i|^2|\gamma_j|^2=-1 \end{cases}\tag{6.5.4} \end{equation}

which correctly contain circular or hyperbolic trigonometric functions depending on the character of \(\gamma_i\) and \(\gamma_j\text{.}\)

The case where both \(\gamma_i\) and \(\gamma_j\) involves some conventional sign choices.