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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...)

Terry Bollinger
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2 Answers2

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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.

Luboš Motl
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    Thanks, I think, Lubos. Since you clearly understand the maths very well, could you explain exactly how you fill a bottomless negative kinetic energy Fermi sea with real electrons without reaching infinite charge and mass? I missed that part. Equations are fine! Surely this one is just a matter of mapping the equation component to the real charges and real mass of all those electrons? (Will recheck this evening.) – Terry Bollinger Mar 01 '13 at 13:06
  • Dear Terry, the bottomlessness just means that the physical vacuum and the naive vacuum differ by infinitely many electrons. Even the energy densities and charge densities of these "two vacua" differ by an infinite amount. That's not a real problem - only the physical vacuum and its finite excitations are realizable in practice. Nevertheless, you may view these infinities by which the densities differ to be the first signs of infinite subtractions that are needed in QFT calculations - QFT requires much more nontrivial and "harder to subtract" infinities if you compute loop corrections! – Luboš Motl Mar 01 '13 at 19:09
  • Using the language of renormalization, the energy density and the charge density of the physical vacuum are two adjustable constants a priori. The charge density has to be zero if the physical vacuum is Lorentz-invariant because the charge density is a time component of a 4-vector $j^0$ and its nonzero value would pick a preferred frame. The energy density of the physical vacuum is known to be just $10^{-123}$ in Planck units. The electron field contributes by a term (related to this one, in a sense) and the question why all the contributions almost exactly cancel is the CC problem. – Luboš Motl Mar 01 '13 at 19:11
  • The cosmological constant problem is why these terms - like the energy density you could predict for the physical vacuum if you decided that the naive vacuum has $\rho=0$ - mostly cancel but not quite, so that there's a tiny leftover manifesting itself by the accelerated expansion of the Universe. The energy density difference between the naive and physical vacuum is infinite but as always, most of these infinities are unphysical as they cancel up to finite leftovers in all physically realizable situations. The counterterm for energy density is just a constant. – Luboš Motl Mar 01 '13 at 19:13
  • Incidentally, Dirac himself had trouble with renormalization - he never understood it well enough to accept that it's the right way to calculate loop processes - however, he was not too allergic to such subtractions that he would feel uncomfortable with the infinite subtraction needed to switch from the naive vacuum to the physical one. He just wrote that the densities are determined up to constants and those constants are adjusted so that the densities are zero in the physical vacuum. It just doesn't matter that the required subtraction is infinite: it can be done, anyway. – Luboš Motl Mar 01 '13 at 19:16
  • Luboš, thank you. Your answer and to-the-point follow-up comments provide a lucid and remarkably compact explanation of how the issues I described are handled mathematically. In less than a day your answer topped out in Google queries on "naive vacuum" and "physics vacuum"! I think that distinction alone will help folks like me: Dirac's very compact lecture described only the "naive vacuum," without the infinite subtractions that must then be used to fix the infinite mass and charge densities created by that naive model. And from your description I see now, in part, how it was modernized. – Terry Bollinger Mar 02 '13 at 01:27
  • So you answered my question, and well. Alas, more understanding has in this case left me less than sanguine regarding argument structure, that is, the deeper scaffolding that lies behind the math. Correct me if I'm misreading this, but the bottom line appears to be this: Dirac proposed a sea model, the naive vacuum. The naive model violates observed reality infinitely both in energy and charge densities. Mathematical methods were developed to repair this by creating and subtracting almost all of two infinite terms. There is just more of a chocolate confection flavor to this than I expected. – Terry Bollinger Mar 02 '13 at 02:01
  • If my memory serves me correctly---I am now in a land with bandwidth but without my books, and the liberry is closed---even a textbook as recent as Greiner repeats the exposition of the Dirac sea, without criticism, but the textbook of Weinberg on QFT excoriates the whole idea. I certainly think more highly of Weinberg's textbook writing abilities than I do of Greiner's....if Greiner is right and Weinberg is wrong, it will only be because Greiner has been copying Dirac.... – joseph f. johnson Jun 01 '14 at 16:47
  • There's a lot of room for idiosyncrasies here - what is the definition of a semiheuristic idea's being wrong? Some implications that someone could derive from the Dirac sea are wrong, but there are lots of careful Dirac-sea-related comments that are completely right and I am confident that Weinberg would agree. He just chooses to erase it for pedagogic reasons. – Luboš Motl Jun 03 '14 at 11:33
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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.

joseph f. johnson
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