At least as normally formulated, the law of transformation of a wave function solution of the Dirac equation to another inertial frame seems to indicate that if observer 1 is certain the particle is an electron, observer 2 (in a frame moving relative to the first) will judge that it might be an electron or it might be a positron. But shouldn't charge be invariant under Lorentz transformation?
To express this is more detail, the Lorentz transformation of the 4-component wave function as given for example in (4.22) of http://www.damtp.cam.ac.uk/user/tong/qft/four.pdf , generally changes a wave function of the form (1,0,0,0) in frame O to (a,b,c,d) in frame Oprime. I took this to mean O is sure the particle is an electron spin up, while Oprime assigns a non-zero probability to the particle being a positron (since c and d are not zero). Thanks to Sanya for requesting this addition.