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Greg Egan

Joined: 01 May 2005
Posts: 75

Posted: Sun Jul 02, 2006 12:08 pm    Post subject: Null rotations

Among Lorentz transformations, ordinary rotations and boosts are fairly
easy to understand, but "null rotations", the transformations that
preserve null vectors, are a little obscure.

I've written a small web page that explains some properties of null
rotations, specifically in the context of 2+1 dimensions:

http://gregegan.customer.netspace.net.au/GR2plus1/NullRotations.htm

This gives four ways to build a null rotation that preserves a given null
vector, and a movie of the action. The orbits are parabolas, except on
the plane that contains the preserved null ray, where they are straight
lines.
Hans Aberg
science forum beginner

Joined: 08 May 2005
Posts: 27

Posted: Mon Jul 03, 2006 12:29 am    Post subject: Re: Null rotations

In article <20060702013519.5596D7B07C@mail.netspace.net.au>, Greg Egan
<gregegan@netspace.net.au> wrote:

 Quote: Among Lorentz transformations, ordinary rotations and boosts are fairly easy to understand, but "null rotations", the transformations that preserve null vectors, are a little obscure.

In the ordinary four-dimensional Lorentz space, this leads to a spin
structure via the Clifford algebra: In the complexified Lorentz space, one
chooses a direct sum of two maximal isotropic vectorspaces, i.e., each of
complex dimension two, consisting only of null vectors, and with
intersection 0 (and such a choice is equivalent to the choice of a spin
structure). The Clifford algebra can then be made acting on each of
these maximal isotropic vectorspaces, and it is possible to extract the
chiral representation used in one form of the Dirac equation from this by
the choice of a suitable basis; the Feynman slash is in fact the Clifford
algebra action on such a spin representation.

--
Hans Aberg

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