The argument is often given that the early attempts of constructing a relativistic theory of quantum mechanics must not have gotten everything right because they led to the necessity of negative energy states. What's so wrong with that? Why can't we have negative energy states?
As I understand it, we know now that these "negative energy states" correspond to antiparticles. So then, what's the difference between a particle with negative energy and an antiparticle with positive energy? It seems to me that there really is no difference, and that the viewpoint you take is simply a matter of taste. Am I missing something here?
Answer
I complete analogy with classical mechanics:
We define the proper velocity: $$ \eta ^\mu :=\frac{dx^\mu}{d\tau}, $$ where $\tau$ is proper time. We likewise define (relativistic) momentum: $$ p^\mu :=m\eta ^\mu . $$ And finally we define the (relativistic) energy (up to multiples of $c$) as the time-component of $p^\mu$. This happens to be $$ \frac{mc^2}{\sqrt{1-(v/c)^2}}, $$ which obviously must be positive. Thus, in order to be consistent with our relativistic definition of energy, we can't have particles with negative energy. This almost makes it tautological, but it is straightforward and precise.
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