Rotations

Any element of the orthogonal group \({O\left(r,s\right)}\) is a rotation and/or reflection, and a well-known result is that any such transformation can be obtained as a product of reflections. Thus every element of \({O\left(r,s\right)}\) corresponds to the Clifford product of some \({k}\) unit vectors via

\(\displaystyle R_{u_{k}\dotsm u_{1}}\left(v\right)\equiv R_{u_{k}}\left(\dotsm\left(R_{u_{1}}\left(v\right)\right)\dotsm\right)=\left(-1\right)^{k}u_{k}\dotsm u_{1}vu_{1}^{-1}\dotsm u_{k}^{-1}. \)

The elements of \({C\left(r,s\right)}\) that have the form of a Clifford product of unit vectors \({U=u_{k}\dotsm u_{1}}\) form a Lie group denoted \({\textrm{Pin}\left(r,s\right)}\) and called a Clifford group (AKA Pin group). In terms of the reverse operation of geometric algebra, the elements \({U\in\textrm{Pin}\left(r,s\right)}\) are those that satisfy \({U\widetilde{U}=\pm1}\). \({R}\) forms a homomorphism from \({\textrm{Pin}\left(r,s\right)}\) to \({O\left(r,s\right)}\) defined by \({U\mapsto R_{U}}\), where

\(\displaystyle R_{U}(v)=\left(-1\right)^{k}UvU^{-1}. \)

This homomorphism is two-to-one, since \({R_{U}}\) and \({R_{-U}}\) map to the same transformation, so \({\textrm{Pin}(r,s)}\) is a double covering of \({O(r,s)}\).

Now, elements of \({SO\left(r,s\right)}\) are pure rotations, i.e. they are obtained as a product of an even number of reflections. Therefore the special Clifford group (AKA Spin group) \({\textrm{Spin}\left(r,s\right)\equiv\textrm{Pin}\left(r,s\right)\cap C_{0}\left(r,s\right)}\) is a double covering of \({SO\left(r,s\right)}\), using the restriction of \({R_{U}}\) to even elements:

\(\displaystyle R_{U}^{S}(v)=UvU^{-1} \)

Δ It is important to remember that rotations in 4 dimensions and higher do not follow many intuitive ideas from 3 dimensions. In particular, a rotation can have more than one plane of rotation (where rotated vectors in the plane stay in the plane), and therefore can require more than two reflections.

These relationships are depicted in the following diagram.

66.clifford-groups-v2

The above depicts how the Clifford group \({\textrm{Pin}}\) and its even subgroup \({\textrm{Spin}}\) are generated by the unit elements of the Clifford algebra \({C}\). \({C_{1}}\) and \({C_{0}}\) both have dimension \({2^{n-1}}\) as manifolds, and \({\textrm{Pin}}\), \({\textrm{Spin}}\), and \({\textrm{Spin}^{e}}\) all have dimension \({n(n-1)/2}\).

Δ Some potential sources of confusion can be avoided by remembering that \({\textrm{Pin}\left(r,s\right)}\) is the group generated by Clifford multiplication on the unit elements of the algebra \({C\left(r,s\right)}\). Thus the elements of \({\textrm{Pin}\left(r,s\right)}\) can only be multiplied with each other and always have inverses, while the elements of \({C\left(r,s\right)}\) can be multiplied by scalars and added, but may not have multiplicative inverses.

An Illustrated Handbook