A. Neumaier said:
No; you mix conceptually completely different things.
Weinberg's argument allows arbitrary shifts of the angular momentum in an unphysical central extension.
On the other hand, particles with half integer spin are represented in QFT already by a vector representation of the Poincare group, no central extension is necessary to do so!
vanhees71 said:
Sigh, I think this is really a superfluous discussion.
If the proper orthochronous Poincare group in the classical sense was the very group you have to use in QT, which you must if you insist on that only unitary representations of the symmetry groups of physics are "allowed descriptions" of symmetry principles in QT, then you'd not be allowed to use the covering group of the rotation group (as a subgroup of the Poincare group) and then only integer-spin representations would be allowed. As observations in Nature, however, show there are half-integer spin realizations of the group in nature like electrons, nucleons, etc. etc.
Sorry, of course one has the phases in the unitary transformations that give a central extension of order 2, i.e., one has a
vector representation of ISL(2,C) rather than one of ISO(1,3).
Note that these have the same Lie algebra commutation relations defining the standard generators without any shift! You should therefore interpret my comments to apply to ISL(2,C) rather than the Poincare group.
Weinberg posits an
arbitrary central extension of this Lie algebra (thus changing the definition and hence the meaning of the generators by positing different commutation rules) and shows that no physics can result, hence that this Lie algebra extension is physically spurious - in contrast to the Galilei group where a nontrivial central extension with mass as a central charge (the one realized in nonrelativistic physics) appears by the same kind of analysis.
In general, physical observables with a meaning in terms of symmetry are
(in all cases, without exception) defined on the theoretical level by their commutation rule. If the commutation rule change, the meaning of the observables change. In the extended Lie algebra considered by Weinberg, all generators, including those for the rotation group, alter their meaning by being shifted.
You can apply exactly the same argument to SO(3) or SU(2) - which one doesn't matter since their Lie algebra is the same and Weinberg only argues with the Lie algebra. After similar calculations you end up with the same result - that there is no central charge. So if one follows your argument one should conclude that angular momentum is physically determined only up to an arbitrary shift in each component. Of course, this is nonsense, hence your argument implies no such thing - also not in the case of energy where you originally applied it.
This is completely independent of the double-valuedness of spin 1/2. Ths enters the discussion only on the group level, but Weinberg's argument is solely on the Lie algebra level! Therefore the conclusions also apply only on the Lie algebra level! This is what i had meant when saying that you mix two completely different things!