  More for QFT:  I didn't understand the word real you put before representations.  Representations are counted by natural numbers, so I don't think there's any confusion that they can be double-counted by not specifying real.  But more importantly, people say that the Majorana rep is a real spinor rep, so I think this extra word will give people the false impression that we're counting Majorana reps.  Really you count whichever rep is irreducible.  About your deletion of my comments on the projective representation, it appears as though there's some maximal space for comments and so I could only read the beginning of what you wrote.  And so, needless to say, I don't see what your objection is.  In the part that I read, you appeared to be saying that the overall phase on a field is somehow physical, in contrast to particle wavefunctions.  If this is indeed what you were saying, I disagree.  I'd guess that by field you just meant the second quantized wavefunction.  To talk about the spectrum it suffices to work in first quantization, but really the only difference here is whether you quantize the position and momentum or the Fourier modes of the field ... ie, in both first and second quantization an overall phase on a field or equivalently on a vector in the Hilbert space (as these are modules of the fields) does not change the physical state.  Physical states correspond to rays in the Hilbert space.    Are you objecting to my claim that the Hilbert space vector corresponding to a fermion doesn't transform under any rep of the Lorentz group?  A rotation by 360 degrees is the identity element of the Lorentz group, thus in any representation it is the identity (because representations are homomorphisms, and homomorphisms take the identity to the identity) ... but it isn't the identity on a fermion field, it is multiplication by -1, thus its not a rep.  It is, however, a projective rep. 