For the distance function ##(\Delta s)^2 = (\Delta r)^2 + (r \Delta \theta)^2##, the rotation matrix is ##R(\theta) = \begin{pmatrix} cos\ \theta & - sin\ \theta \\ sin\ \theta & cos\ \theta \end{pmatrix}##.(adsbygoogle = window.adsbygoogle || []).push({});

That means that for the distance function ##(\Delta s)^2 = (\Delta r)^2 + ((1-\frac{\phi}{2 \pi})r \Delta \theta)^2##, the rotation matrix is ## R(\theta) = \begin{pmatrix} cos\ [(1-\frac{\phi}{2 \pi})\ \theta] & - sin\ [(1-\frac{\phi}{2 \pi})\ \theta] \\ sin\ [(1-\frac{\phi}{2 \pi})\ \theta] & cos\ [(1-\frac{\phi}{2 \pi})\ \theta] \end{pmatrix}##?

The generator for the original rotation matrix is ##X = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}##. That means that the new rotation matrix has the generator ##X = \begin{pmatrix} 0 & -i(1-\frac{\phi}{2 \pi}) \\ i(1-\frac{\phi}{2 \pi}) & 0 \end{pmatrix}##?

The problem with this is that because ##R(\theta) = \mathbb{1} - i \theta X + ...##, I can only see one generator ##X## when in fact there should be two generators because there are two parameters ##\theta## and ##\phi##.

Any thoughts on this?

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# Scaling the parameter of the SO(2) rotation matrix

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