Why Negative Energy States are Bad 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?
 A: The usual argument is that negative energy states are inherently unstable; if energy states are not bounded from below, a negative energy state can always become more negative, emitting positive energy radiation continuously. It turns out, this is more or less what it is believed that happened in the inflationary era: 
1) an accelerated expanding cosmos
2) all the positive energy matter we see today.
So negative energy states are only "bad" (or let say just wildly inconvenient) in our currently asymptotically flat space-time, but they probably existed at the very beginning in vast quantities. They probably marginally exist still today in the form of dark energy.
However, i am confused why people extrapolate the idea that states will always try to decay to lower energy states (even if already negative): What happens at a more fundamental level is that systems try to achieve equilibrium by spreading energy evenly across degrees of freedom of all fields. Entropy is nothing but a logarithm in the number of available states reachable for a degree of freedom at a given, well defined energy. This entropy has a minima at zero energy, not at $- \infty$, as would be implied by the common lore. So it is not unreasonable to expect that, negative energy systems would decay to higher energy states, toward the zero energy states that we associate with the vacuum.
A: The problem is that interacting systems, like particles, tend to transition into states of lower energy. (Technically, the universe transitions into states of higher entropy, but in the context of a particle that usually means lower energy.) So in order for particles to be stable, the energy spectrum has to have a lower bound. Otherwise, a particle could just keep dropping to lower and lower energy states, emitting photons at every step.
Now, there is a sense in which a positive energy antiparticle state can just as well be considered a negative energy particle state. The solution to the Dirac equation looks the same in either case. In the early days of relativistic QM, it never occurred to anyone that there was any interpretation of these solutions other than being negative-energy states, which led to the invention of ideas like the Dirac sea, and the identification of holes in the sea with antiparticles. But by the time quantum field theory came along, people realized that it just made more sense to include antiparticles as proper objects in the theory, rather than trying to explain them as holes, because then there was no need to bother with negative energy states at all.
A: For very simple cases only, free quantum fields, we can certainly map negative frequencies (not energies, but the two things are conflated by most authors) to positive frequencies and vice versa in various ways. The details of this for the Klein-Gordon field are published as EPL 87 (2009) 31002, http://arxiv.org/abs/0905.1263v2; for the electromagnetic field, there is http://arxiv.org/abs/0908.2439v2 (which I recently almost completely rewrote). Up to a point, these papers put Vladimir Kalitvianski comment into one mathematical form (but other mathematical forms for his comment are certainly possible). FWIW, the presence of measurement incompatibility is tied in with whether one allows negative frequency modes.
HOWEVER, I have no idea what the construction in those papers looks like if one uses similar mathematical transformations for the whole of the standard model of particle physics. In fact, over a number of years I have failed to get such an approach to work. It's necessary to get it right for the whole of a system that comes close to reproducing the phenomenology of the standard model (or something slightly different in an experimentally useful way or in a way that is useful for engineering) before many Physicists are likely to take the idea very seriously.
The stability argument given by David Zaslavsky is completely right by conventional wisdom, but it assumes, for starters, that energy and action are viable concepts in a QFT context. In the algebraic context I currently work in, energy and action are not viable concepts. There is also no "axiom of stability" in quantum field theory, so there is no proof of a no-go theorem that there is no way to ensure stability except by having only positive frequencies; there is, instead, an "axiom of positive frequency" in the Wightman axioms. Note that a well-formulated axiom of stability would be far less theoretical and more natural than an axiom of positive frequency.
A: 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.
A: A negative kinetic energy is not physical. It is supposed to be observable, as well as the particle velocity and mass. So it is just a non physical solution. On the other hand, for completeness of Fourier transformation, those negative frequencies must be present in the solution. They were made present as "antiparticle" solutions in a multi-particle construction. It means, the Dirac's equation solutions came in handy in QED and are not really physical in one-particle relativistic QM.
