# Strange implication of the relativistic invariance of the Dirac equation

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.

• Could you add a source or the transformation law you are referring to?The transformation laws I could find (see e.g. damtp.cam.ac.uk/user/tong/qft/four.pdf ) did not seem to mix electron- and positron-components if I understand them correctly. Jun 16, 2016 at 20:02

Even if you interpret the Dirac equation the original way (not as a quantum field theory), the Lorentz transformation doesn't mix electron and positron. Consider two types of plane wave with positive frequency and negative frequency $$u(p)e^{-i (E t-p x)},\quad v(p)e^{+i (E t-p x)}$$ The idea is the energy $E$ is always positive, but the positive frequency describes an electron and negative frequency a positron.
If you want to describe charge in a Lorentz invariant way, the quantity $\bar{\psi}\gamma^\mu \psi$ is a 4-vector that describes the current.
• Don't you also have to apply a Lorentz transformation matrix $\Lambda$ to the spinor components when boosting into another frame? I think it's this aspect that the question is most concerned with. Jun 16, 2016 at 20:33
• You can write the spinor transformation as $U(\Lambda)u(p)=u(p')$ for some matrix $U$ or you can just figure out what the form of $u(p)$ has to be in general and update $p$ to $p'$. I think the original question is confusing the lower two components of the spinor in the Dirac basis as being positron coponents, which is not the case. Jun 16, 2016 at 20:36