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Helicity and Hopf fibration
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There is a relation between massless (helicity) representations of the
Poincare group and the Hopf fibration, a nontrivial principal fiber
bundle on the 3-sphere, with the 2-sphere as basis manifold.
See Section 4 in
http://arxiv.org/pdf/0908.1205
and the paper [7] by Urbantke quoted there. Thus there are interesting
topological issues in the massless case. Also related is
Section 2.4 in
http://arxiv.org/pdf/math-ph/0303031
It would be nice to have a short and elementary presentation of this
connection.
For a presentation in German, see
http://www.itp.uni-hannover.de/~dragon/hel1.pdf
(The position operator for photons presented there has a nonstandard
behavior under rotations of the Poincare group: The probability density
|psi(x)|^2 of position x in the position representation gives
probabilites for x lying in the unit sphere that depends on the
orientation of the observer frame.)