Does Dirac's idea of filled negative energy states make sense?

Question

Please bear with me a bit on this. I know my title is controversial, but it's serious and detailed question about the explanation Dirac attached to his amazing equations, not the equations themselves.

Imagine for a moment that someone signed onto this group and proposed the following:

1. Empty space is a lot like a metal, or maybe a semiconductor, because like those materials its chock full of negatively charged electrons occupying different velocity states, only...

2. Unlike metals or semiconductors, the density of electrons in any one region of space is infinite, because there is no limit to how fast the electrons can move. That is because these are negative energy states in which an electron can always move faster simply by emitting a photon, so there's not "bottom" to how far they can drop and how dense they can become, and...

3. Unlike metals or semiconductors, there is no exactly balancing sea of positive atomic charges, well, unless maybe there are infinite numbers positively charged atoms too, and...

4. The resulting infinite negative charge density of real electrons not only doesn't matter but is in fact completely and totally invisible for some reason, and...

5. The resulting infinite mass density of electrons (recall that these are quite real electrons, only in odd negative-energy kinetic states) also doesn't matter, and...

6. Unlike the Fermi sea of a metal conduction band, removing an electron from this infinitely dense sea of electrons for some reason doesn't cause other electrons to collapse into it and fill it, even though the negative kinetic velocity electrons are pushed by exactly the same Pauli exclusion forces as the ones in a Fermi band; in short, for reasons not clear, semiconductor-style hole stabilization applies while metal-style hole filling does not (is there a band gap going on here?), and...

7. Since the infinitely dense negative charges become invisible for no particular stated reason when the electrons fall into negative energy states, these unexpectedly stable open states in the negative energy sea have net positive charge, even though...

8. ... such missing states categorically should have zero charge, since in sharp contrast to the positive ionic background of metallic and semiconductors, the vacuum has no background charge at all, which should leave holes in the mysteriously invisible negative kinetic energy just as uncharged and invisible as that sea for some reason is, and...

9. Even if you do assume that the negative kinetic state electrons have visible charge, their infinite density would make the "comparatively" but infinitesimally smaller positive charge of such a hole invisible, and...

10. Repeat this process for every other kind of particle in existence, and...

11. If you have done all of this and done it correctly, congratulations: You now understand conceptually what anti-electrons (positrons) and other anti-particles are.

First question: Have I misrepresented any of the implications of Dirac's explanation of positrons as holes in an infinite sea of negative-kinetic-energy electron states? What I have tried very hard to do is nothing more than make a list of the implications of a physics idea, just as people do all the time on this group. Who said it should not really be the issue, not if we are talking about an unelaborated explanation rather than the math itself.

Second question: If someone had proposed a theory in this forum like the one I just described, and you had never heard of it before, what would you have thought of it? Please be honest.

My point in all of this obviously is this: While Paul Dirac's amazing equations (they really are) managed to predict antimatter, his explanation for why his equations require antimatter is... shall we say incompletely analyzed, to put a nice spin on it?

A final thought: Has anyone ever seriously tried to make Dirac's conceptual negative energy sea ideas, the ones that he espoused in his Nobel Lecture, into a real, working theory? And if so, how did they deal with the various issues I described above?

(Me, I just think antiparticles are regular particles moving backward in time. Yeah, that's a pretty weird idea too, I know...)

Answer 1

Dirac's explanation of the emergence of antiparticles such as positrons out of the Dirac sea, and the Dirac sea itself, is completely valid and legitimate, and you have described some non-quantitative aspects of it and differences between it and some condensed-matter situations.

Dirac just began with the assumption that the Dirac spinor field $\Psi$ is a pure combination of annihilation operators only while $\bar\Psi$ is only made of creation operators. However, it may be seen that the creation/annihilation operators create/annihilate electrons into states with both positive and negative energy.

The ground state is defined as one in which Nature minimizes the energy. A good way to do so is to keep the positive-energy state empty but occupy all the negative-energy states (addition of a negative number is like a subtraction of a positive one). That's how we get the physical vacuum, one with the Dirac sea. On the contrary, we may create a hole, i.e. remove an electron from the Dirac sea of negative-electron states. A simple counting of signs shows that this is equivalent to adding a positive-charge, positive-energy particle, a physical positron.

So we may choose the convention in which the creation operators for the negative-energy states are relabeled as annihilation operators of positrons, and vice versa. The difference between the Dirac sea paradigm and the usual expansions taught in QFT courses is just a relabeling of $a$ as $b^\dagger$ and $a^\dagger$ as $b$. The physical convention with positrons aside from electrons is more physical because all annihilation operators actually annihilate the physical ground state (vacuum).

These days, we don't usually emphasize Dirac's construction, e.g. because it only applies to fermions (there is a symmetry between occupation numbers 0 and 1; no reflection of the spectrum is a symmetry for bosons whose occupation numbers are all non-negative integers) but the fact that the relativistic fields automatically predict antiparticles due to the negative-energy solutions is general and holds both for bosons and fermions. But that doesn't invalidate anything about Dirac's presentation.

If I didn't understand the actual maths and its connection with physics and someone gave me vague linguistic descriptions such as yours, I wouldn't know what to think and I would tend to think that the writer is confused. After all, I think you are confused even here, in the non-hypothetical world.

If I could check what's behind these statements, I would realize it's a valid theory whether it was written down by Bollinger or Dirac.

Answer 2

Relativistic Quantum Mechanics, like classical Relativistic many-particle dynamics, has never been fully worked out. Historically, people jumped right away from the Klein--Gordon equation as a one-particle equation, to the second quantisation which develops a quantum field theory. In Classical Relativistice many-particle dynamics, which Feynman tackled, no one has ever gotten anywhere (not even Feynman). In Relativistic Quantum Mechanics, work of Newton, French, and Wigner showed up the difficulties in interpreting the measurement observables in the usual way. So it has been ignored and forgotten.

Dirac's sea is repeated in many textbooks but has been attacked in print, by most prestigious physicists, as nonsensical. They complain about including it in such textbooks, as a kind of relic. It is notable, though, that with all the revisions Dirac made to his textbook, he never took out the Dirac sea. Equally notable is that he never put in parity symmetry. When asked after the famous experiment of Wu proved that parity symmetry was violated what he thought about it, he simply answered "I never said anything about it in my book."

Dirac was not at all illogical in anything about the Dirac sea. It doesn't require any change in the axioms of Quantum Mechanics because the concepts involved in your phrases about "resulting infinite mass density of electrons" and other similar phrases are neither contained in nor deducible from the axioms. To pass from a wave function to a physical statement about mass density requires interpretation. Dirac's positing the interpretation he did is perfectly consistent with his axioms of Quantum Mechanics as stated earlier in his textbook.

If one is perfectly consistent about the one-particle interpretation of the Klein--Gordon equation or the Dirac equation, there is no "falling into negative energy states" because there is no outside perturbation to kick the particle into such a state. If a particle is in a superposition of positive energy eigenstates, it stays there. And if one passes to more realistic situations of interactions with many particles and fields, one is already using QFT of some sort.

I privately think there is something that can be discovered about Relativistic Quantum Mechanics, still, but I doubt it would have any practical difference. Still, it is interesting to me because of what it says about Quantum Mechanical observables and probability.