Re: Desperately Seeking Spinors

clmasse@xxxxxxxxx wrote:

8) Spinors are related to spherical harmonics.

False.

9) "up" and "down" are spherical harmonics of order 1/2

False. "up" = (1,0); "down" = (0,1) up to a complex factor of modulus
1=2E
There is no spherical harmonic of half integer order.

Spherical harmonics with half integer order do exist and have been
extensively discussed in the literature. But a general consensus is that
they cannot form a representation of the rotation group. A distinct view
was pointed out by D. Pandres, J. Math. Phys. Vol.6, 1098 (1965), {\bf
9}, 1401 (1968), namely that half integer spherical harmonics in a 3D
space (which cannot be the usual physical 3-space, but an "internal"
space) can be used for description of spinors. For a recent detailed
discussion see http://arxiv.org/abs/hep-th/0412324 .

What Pandres did differently than others was that he used consistently
the two distinct sets of half integer spherical functions, namely
function Y_{lm} and Z_{lm}, and employed a suitable definition of the
scalar product, so that the angular momentum operator with respect to
his set of functions was Hermitian. Other researchers used Y_{lm}
functions for positive l-values and Z_{lm} functions for negative
l-values and thus found that with respect to such set of functions
angular momentum operator is not Hermitian. This opened a Pandora box of
problems. So they rejected such functions as useless for physics.

Of course, no claim was made by Pandres that the usual orbital angular
momentum can have half integer l-values. This was just a possible
representation of spinors. All other old works on half integer spherical
harmonics were done with a motivation to explain why they should be
excluded in description of orbital angular momentum. All those authors
had in mind only description of orbital angular momentum, therefore they
adjusted their constructions accordingly.

.