Negative Energy in the 4-momentum of an Antiparticle?

According to the Wikipedia article on Antiparticles:

Solutions of the Dirac equation contain negative energy quantum states. As a result, an electron could always radiate energy and fall into a negative energy state. Even worse, it could keep radiating infinite amounts of energy because there were infinitely many negative energy states available. To prevent this unphysical situation from happening, Dirac proposed that a "sea" of negative-energy electrons fills the universe, already occupying all of the lower-energy states so that, due to the Pauli exclusion principle, no other electron could fall into them. Sometimes, however, one of these negative-energy particles could be lifted out of this Dirac sea to become a positive-energy particle. But, when lifted out, it would leave behind a hole in the sea that would act exactly like a positive-energy electron with a reversed charge. These holes were interpreted as "negative-energy electrons" by Paul Dirac and mistakenly identified...However, these "negative-energy electrons" turned out to be positrons, and not protons.

Thus, by the very definition of an antiparticle, we know that it has negative energy $$E$$; which, in natural units $$(\hbar = c =1)$$, is: $$E = -\sqrt{m^2 + p^2}$$ where the symbols here have their usual meanings. Consequently, If we were to describe the antiparticle's 4-momentum vector, from some inertial reference frame, it would look like: $$\tilde{p} = \left(E, \textbf{p} \right) = \left(-\sqrt{m^2 + p^2}, \ \textbf{p} \right)$$ That is, the first entry of the 4-momentum vector ($$p^{0}$$) should be negative. However, in this link, the author asserts: $$E_{e^{+}} = E_{e^{-}} = 2m_ec^2 \ (> 0)$$ that is, the energy of the free positron $$e_{+}$$ (antiparticle of $$e_{-}$$) with respect to the center of mass lab frame is positive and thus, a direct contradiction with what I was stating above. Where exactly is the mistake here?

• where did you find, that the definition of antiparticle gives you negativ energie? Oct 19, 2020 at 18:04
• Oct 19, 2020 at 22:09

• What @benrg said. Sure, the Dirac Sea theory is outdated, but as that Wikipedia article mentions, "Dirac sea theory has been displaced by quantum field theory, though they are mathematically compatible". It's not exactly wrong, it's just inelegant, and it cannot be extended to describe bosons (eg, the $W^+$ and $W^-$ of the weak nuclear interaction). Oct 19, 2020 at 22:08
The frequency of positron solutions is negative of that of electron solutions. Via the formula $$E=hf$$ this may lead to the conclusion that their energy is negative, as we know electrons have positive energy. However, positrons have positive energy just like electrons. The conclusion must be that $$E=h|f|$$ and that the Dirac Hamiltonian gives the wrong sign of energy, as well as charge by the way, for positrons.