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So taking the square root of $E^2 = (m_oc^2)^2 + p^2c^2$ yields two solutions.

The Dirac Sea treats the negative solution as an infinite space of electrons with negative energy.

All the observable electrons have positive energies. As an electron lose energy, another electron somewhere else in the universe gains energy, so that the total positive energy is balanced with the negative energy.

Seems to be a good explanation I can use for explaining the negative part of the Energy-Mass equivalence. What is it about this interpretation that people disagree with?

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The Dirac sea interpretation cannot deal with bosonic antiparticles. At the time it was conceived, I don't think physicists were aware that antibosons existed. A simple example of a boson-antiboson pair are the $W^\pm$ bosons. In order to prevent an electron from falling into a negative energy state, we use the Pauli principle. A fermion cannot fall into a fully occupied sea of other fermions. This obviously does not apply to bosons.

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The Dirac Sea explanation of negative frequency solutions is untenable. A bottomless sea of filled electron states implies infinite electron density. The associated infinite negative charge density must be compensated by an equally infinite positive charge density. Effects of electron correlation would likely alter the properties of the hole, giving an effective mass different from that of an electron. What are the n&k values of the Dirac Sea? We observe that the vacuum is perfectly transparent. How can this be consistent with the omnipresence of an infinite electron density ?

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  • $\begingroup$ To the person that gave the minus point. Please defend the idea of the Dirac sea here instead of just being negative ;-) This could be fun. $\endgroup$
    – my2cts
    Commented Nov 21, 2019 at 15:09
  • $\begingroup$ Nice answer (+1) Could you elaborate supposing that the Dirac-sea is not a sea of bottomless depth, but it is simply a "sea" of freely moving electron-like points that fill all 3D space with some finite density. Compressions and rarefactions produce the illusion of positive and negative charges. Then, this sea of particles will behave just like our regular sea of discrete water molecules, which can be viewed equivalently as its dual: a continuous fluid field obeying the Navier Stokes equations. Do all of your mentioned objections still stand based on this construction of the Dirac-sea? $\endgroup$
    – James
    Commented Aug 23 at 14:12
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    $\begingroup$ @James A Dirac like sea of finite density would only explain positrons within a limited energy range. Apart from the other objections mentioned in the answer. $\endgroup$
    – my2cts
    Commented Aug 23 at 20:07

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