Consider an electron described by a wave packet of extension $\Delta x$ for experimentalist A in the lab. Now assume experimentalist B is flying at a very high speed with regard to A and observes the same electron. The extension of the wave packet will appear contracted, and the uncertainty on momentum will increase. What happens when the later become larger than the electron's rest mass?
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What happens? Nothing special. The momentum and the energy increase as well. Maybe this answer is too naive, could you say what you have in mind? Answer to comments: Physical particles and antiparticles always have positive energy. Particles also have positive frequency in a free field, while antiparticles have negative frequency. One can prove that the sign of the frequency (positive or negative) is invariant under Poincare transformations. You can ask this as a separate question. (Edit: Finally, I have added that at the end of the answer.) Let's say that for observer A the particle is at rest on average, so energy-momentum expectation is $(c=1)$: $$(m,0)$$ And the momentum has an indetermination $\Delta p$, that should be lower than $m$ if A really knows that there is one particle. Then one can see that for B (it is an exercise on Lorentz transformations):$$\frac{\Delta E'}{E'}=\frac{\Delta p}{m}v < \frac{\Delta p}{m}$$ where $v$ is the relative speed (norm of velocity) between A and B. So, if $\frac{\Delta p}{m}\ll1$, then $\frac{\Delta E'}{E'}\ll1$ Absoluteness of particle/antiparticle concept under Lorentz transformations. The positive frequency solution (connected with particles) is defined by: $$i\partial _t \, f_+ =\omega \, f_+ \, , \; \omega >0$$ Taking $f_+ \propto e^{-i(\omega \, t - p\cdot x)}$ ($f_+$ must also verify the Klein-Gordon equation), with $\omega \equiv +\sqrt {m^2+p^2}$ The boosted observer (with rapidity $\theta$) uses his time $t'$: $$i\partial _{t'}\, f_+=(\cosh \theta \, i\partial _t - \sinh \theta \, i\partial _x )\, f_+=(\omega \, \cosh\theta + p \sinh\theta )f_+ \equiv \omega ' f_+, \; \omega'>0 $$ He thus obtains that $f_+$ is also a positive frequency solution with the boosted eigenvalue. Note that this does not happen for a general transformation and hence the distinction between particles and antiparticles (negative eigenvalue of $i\partial _t$) is absolute for observers connected by Lorentz transformations (inertial observers), but accelerating observers disagree about what is a particle, an antiparticle or vacuum. |
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