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Please, can someone explain how the negative energy solution can be used to predict the existence of the antielectron?

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    $\begingroup$ Antiparticles carry negative charges, not negative energies. The negative energy problem is solved in quantum field theory. You cannot regard the Dirac spinor as a wave function. It is a field. $\endgroup$ – The Last Knight of Silk Road Jan 27 at 14:40
  • $\begingroup$ Related: physics.stackexchange.com/q/12520/2451 , physics.stackexchange.com/q/19378/2451 and links therein. $\endgroup$ – Qmechanic Jan 27 at 14:41
  • $\begingroup$ @Knight The Dirac equation is a wave equation, just like the Klein-Gordon and the Schrödinger equation. Its solution is a wave function. $\endgroup$ – my2cts Jan 27 at 15:43
  • $\begingroup$ As @TheLastKnightofSilkRoad indicated, quantum field theory is the right context for understanding this. In QFT, the Dirac spinor is a quantum field, meaning that its components are non-commuting operators that are used to construct local observables. These operators act on the Hilbert space, rather than being elements of the Hilbert space; $\psi$ isn't a wavefunction. The total energy is non-negative, as required by the general princ's of relativistic QFT, and this leads to antiparticles. This is standard material in many introductions to QFT. Are you looking for an answer that avoids QFT? $\endgroup$ – Dan Yand Jan 27 at 21:31
  • $\begingroup$ As @DanYand indicated, the Dirac spinor does not allow a probabilistic wave interpretation in quantum mechanics. The reason is that the Dirac spinor representation of the Lorentz group is finite dimensional, which cannot be unitary because the Lorentz group is non-compact. For historical reasons, people misleadingly thought that the Dirac equation is the relativistic version of the Schroedinger equation. This is incorrect. $\endgroup$ – The Last Knight of Silk Road Jan 29 at 12:37
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It was not a prediction, it was a conjecture, in what was Dirac's hole theory:

The negative E solutions to the equation are problematic, for it was assumed that the particle has a positive energy. Mathematically speaking, however, there seems to be no reason for us to reject the negative-energy solutions.

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any electron placed in a positive-energy eigenstate would decay into negative-energy eigenstates of successively lower energy. Real electrons obviously do not behave in this way, or they would disappear by emitting energy in the form of photons.

To cope with this problem, Dirac introduced the hypothesis, known as hole theory, that the vacuum is the many-body quantum state in which all the negative-energy electron eigenstates are occupied. This description of the vacuum as a "sea" of electrons is called the Dirac sea. Since the Pauli exclusion principle forbids electrons from occupying the same state, any additional electron would be forced to occupy a positive-energy eigenstate, and positive-energy electrons would be forbidden from decaying into negative-energy eigenstates.

It was a theory that was superseded by quantum electrodynamics and the problems of the theory were not resolved, although

In certain applications of condensed matter physics, however, the underlying concepts of "hole theory" are valid. The sea of conduction electrons in an electrical conductor, called a Fermi sea, contains electrons with energies up to the chemical potential of the system. An unfilled state in the Fermi sea behaves like a positively charged electron, though it is referred to as a "hole" rather than a "positron". The negative charge of the Fermi sea is balanced by the positively charged ionic lattice of the material.

One should read the link and links therein for a clear picture.

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