While reading chapter 2 of the book Quantum theory of fields: Volume 1 of Weinberg I got pretty much confused. All the confusion starts in page 89 with equation 2.7.43 and 2.7.44:
$$ U(\Lambda) U(\bar\Lambda) = \pm U(\Lambda \bar\Lambda) \quad \quad \quad \quad \quad (2.7.43) $$
Then Weinberg explains that these are the projective representations of the Lorentz group for integer and half integer spin. All good here. Now, in the last three lines of page he says:
Eq. (2.7.43) or Eq. (2.7.44) imposes a superselection rule: we must not mix states of integer and half-integer spin.
What does he mean by that? That we can not have a superposition of states with integer and half-integer spin, or what? How does he see that this mixing is not allowed?
In the next page (p. 90) he says that we are going to use the universal covering of the Lorentz group instead of the projective representations and writes further:
This does not mean that we actually can prepare physical systems in linear combinations of states of integer and half-integer spin, but only that the observed Lorentz invariance of nature cannot be used to show that such superpositions are impossible.
Why does he emphasise that when we go to the Universal Cover we cannot have such superpositions? It makes me think that with the projective representation that was not allowed and now that we go to the universal cover it could be allowed.
And to finish the post, last confusing quote:
In general, we may just as well take the symmetry group as $C$ (he means the universal cover) instead of $G$, because there is no difference in their consequences, except that $G$ implies a superselection rule, while $C$ does not.
Isn't that contradictory, if one carries a superselection rule while the other does not, seems like they do indeed have different consequences.