# What's a pseudo-rotation?

I'm sorry for this lexical, probably extremely elementary, question. But what is a pseudo-rotation? I just read this term for the first time, in the beginning of the 4th chapter book of CFT by Di Francesco & al. I would say it may be an hyperbolic rotation or a rotation followed by a parity operation (with determinant equals to -1). Couldn't find it on google, so it doesn't seem to be a standard terminology, otherwise please forgive my ignorance.

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It seems OP has already correctly deduced what should be said in this matter. –  Qmechanic Feb 12 '12 at 19:07
Hum, I am sorry but from my suppositions which one is the right one? Or are you saying that pseudo-rotation is a vague term used for anything that could look like a rotation ($SO(k,d-k)$ or $SO(d)\times Z_2$ as well)? –  toot Feb 12 '12 at 19:17
@qmechanic: making an answer out of that would be superior to snarky comments. –  Colin K Feb 12 '12 at 22:00

I) Recall that the $d$-dimensional (homogeneous) Lorentz group is $O(d-1,1)$. Also recall that $O(d)$ is the orthogonal group, and $SO(d)$ is the group of (proper) rotations.

More generally, since a manifold $(M,g)$ equipped with a metric $g$ of signature $(p,q)$ is called a pseudo-Riemannian manifold, it is natural to call $O(p,q)$ the pseudo-orthogonal group and call $SO(p,q)$ the group of pseudo-rotations. In other words, the prefix pseudo refers here to that $p$ or $q$ are not zero.

II) Note that in the same setting of $O(d)$, $O(d-1,1)$, and $O(p,q)$, a pseudovector (and more generally a pseudotensor) use the prefix pseudo in a different way, namely to denote an additional sign flip in the transformation law under an orientation reversing transformation.

III) On page 38 in chapter 2 of the book CFT by Di Francesco et al. is written:

[...], the Lorentz group is isomorphic to $SO(d-1,1)$, the group of pseudo-orthogonal rotations.[...]

One may draw two conclusions:

1. Di Francesco et al. are not very careful in distinguishing between the Lorentz group $O(d-1,1)$ and the proper Lorentz group $SO(d-1,1)$. In fact, it is possible that they really mean the restricted Lorentz group $SO^+(d-1,1)$.

2. A pseudo-rotation is for Di Francesco et al. a Lorentz transformation (modulo the ambiguity mentioned in point 1).

IV) Finally, let us mention that there exists an unrelated notion of pseudorotations in chemistry.

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Thank you for this detailed explanation. –  toot Feb 13 '12 at 14:52