Matrix group terminology in physics

An important fact used in physics is that \({SU(2)}\) is the universal covering group of \({SO(3)}\). For example, the four complex numbers associated with an element of \({SU(2)}\) are called Cayley-Klein parameters, and via this homomorphism can be used to specify a proper rotation in \({\mathbb{R}^{3}}\), i.e. an element of \({SO(3)}\). \({SU(2)}\) is a double cover, so there are two Cayley-Klein parameters corresponding to every proper rotation.

The connected components of matrix groups are usually related to determinant signs; we can see this by studying \({O\left(3,1\right)}\), which in physics is called the (homogeneous) Lorentz group. It is of dimension 6, and consists of rotations and reflections in Minkowski space (AKA spacetime), \({\mathbb{R}^{4}}\) with the Minkowski metric, where the positive signatures correspond to space and the negative signature to time. \({O\left(3,1\right)}\) has 4 connected components, corresponding to whether the orientation of time and/or space is reversed. The proper Lorentz group \({SO\left(3,1\right)}\) consists of the identity component and the connected component of transformations that reverse both space and time, while the orthochronous Lorentz group consists of the two components that preserve the orientation of time. The identity component \({SO\left(3,1\right)^{e}}\) is then called the proper orthochronous Lorentz group (AKA restricted Lorentz group). A “rotation” by an angle \({\phi}\) in a “time-like plane” which includes vectors with negative lengths is called a Lorentz boost of rapidity \({\phi}\).

The Poincaré group (AKA inhomogeneous Lorentz group) is the semidirect product of the translations in 4 possible directions with \({O\left(3,1\right)}\); the product is semidirect since the translations are a normal subgroup and every element can be written in exactly one way as a translation followed by a rotation. It has dimension 10, or \({n\left(n+1\right)/2}\) in general. The above adjectives can also be applied to the Poincaré group. All of the above analysis was for the “mostly pluses” signature \({\left(3,1\right)}\), but the results also hold for the “mostly minuses” signature \({\left(1,3\right)}\).

Similarly, the Euclidean group \({E(n)\equiv\mathbb{R}^{n}\rtimes O(n)}\) is the semidirect product of translations in \({\mathbb{R}^{n}}\) with \({O\left(n\right)}\), while the special Euclidean group \({SE(n)}\) takes the semidirect product with \({SO\left(n\right)}\). These are subgroups of the affine group \({\mathit{Aff}(n,\mathbb{R})\equiv\mathbb{R}^{n}\rtimes GL(n,\mathbb{R})}\) and its identity component the special affine group, respectively. All of these “inhomogeneous” groups, i.e. groups formed by semidirect products whose elements are a translation \({v}\) followed by a linear transformation \({A}\), can be viewed as matrix groups of the form

\(\displaystyle \begin{pmatrix}A & v\\ 0 & 1 \end{pmatrix},\)

where \({v}\) is a column vector and the matrices can be considered as acting on vectors in homogeneous coordinates (AKA projective coordinates), in which a component 1 is appended:

\(\displaystyle \begin{pmatrix}A & v\\ 0 & 1 \end{pmatrix}\begin{pmatrix}w\\ 1 \end{pmatrix}=\begin{pmatrix}Aw+v\\ 1 \end{pmatrix}\)

Δ Note that in homogeneous coordinates scalar multiples of vectors are identified, e.g. \({\left(1,2,3,1\right)=\left(2,4,6,2\right)}\). A possible source of confusion is that inhomogeneous groups act on vectors in homogeneous coordinates.

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