More for QFT  I didnt understand the word real you put before representations  Representations are counted by natural numbers so I dont think theres any confusion that they can be doublecounted 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 were counting Majorana reps  Really you count whichever rep is irreducible  About your deletion of my comments on the projective representation it appears as though theres some maximal space for comments and so I could only read the beginning of what you wrote  And so needless to say I dont 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  Id 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 doesnt transform under any rep of the Lorentz group  A rotation by  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 isnt the identity on a fermion field it is multiplication by  thus its not a rep  It is however a projective rep