quantum field theory - What is negative about negative energy states in the Dirac equation?

This question is a follow up to What was missing in Dirac's argument to come up with the modern interpretation of the positron?

There still is some confusion in my mind about the so-called "negative energy" solutions to the Dirac equation. Solving the Dirac equation one finds the spectrum of allowed energies includes both positive and negative solutions. What does this negativity refer to? Given that the Dirac equation is symmetric under charge conjugation, the convention to call one positive and the other negative appears perfectly arbitrary. Would it be therefore correct to refer to the electrons as "negative energy positrons" ?

In a similar spirit, physicists used to be worried about the "negative energy" solutions decaying into infinity through emission of photons. By the same symmetry argument, this should also be a problem for photons. It is not entirely clear to me how the quantization of the field supresses this issue : is the "photon emission" for the negative state re-interpreted as photon absorption by a positron ?

My understanding is that the whole discussion about "positive" or "negative" energy solutions is misleading : what matters is the physical content through the QED interaction hamiltonian, which does not predict this infinite descent. Is this correct?

Edit: I think I understand the source of my confusion after the comments. If I get it right, the Dirac sea picture is equivalent to the freedom of choice in the formally infinite vacuum energy one observes after quantization of the QED Hamiltonian. Holes in the sea are positive-energy positrons, equivalent to the action of the positron creation operator on the vacuum. Is this correct?

Answer

Symmetric under charge conjugation (which gives us positrons) and symmetric under the sign of the energy are two different things, which is where I think you are getting confused.

Negative energy electrons aren't positrons, they are negative energy electrons. The absence of a negative energy electron in the "sea of charge" can be viewed as a positive energy particle via charge conjugation, but it doesn't take a negative energy particle into a positive energy particle.

There are two conventions here: positron/electron where there is no such thing as negative energy states, and electrons, where there are positive and negative energy states.