The relativistic mass-energy-momentum relation [itex]m^2=E^2-p^2[/itex] predates quantum mechanics by a couple of decades. It allows a particle such as an electron to have a negative mass-energy. If it's 1906, and you're shown this equation, do you have any way to show that the negative-energy solutions can't be of interest based on the known (classical) laws of physics?(adsbygoogle = window.adsbygoogle || []).push({});

If an electron initially has an energy-momentum vector inside the future light cone, then a boost can never bring it into the spacelike region (tachyonic) or the past-timelikelight cones (what we would now call antimatter). This shows that no continuous process of acceleration can change a positive-energy solution to a negative-energy one, and that means that if the negative-energy solutions exist, they're qualitatively different particles with different characteristics.

Normally we imagine that knowledge of a particle's world-line and mass suffice to tell us its energy-momentum vector. If we were going to believe in the negative-energy solutions, we'd have to believe that the world-line had an additional property, a sort of "arrowhead" that told us which direction its tangent vector pointed. This doesn't seem like a huge problem, though. Particles do have various properties such as charge that exist in addition to their world-lines.

It would be possible for a positive-energy particle and a negative-energy particle to act on each other without violating conservation of energy or Newton's third law, in such a way that each would accelerate indefinitely. I guess this could have been considered either scary or exciting in 1906, before anyone could have conceived of the Dirac sea.

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# How to rule out a classical interpretation for negative-energy states?

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