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Dirac's equation allows an infinite amount of solutions with negative energies of arbitrarily large absolute value. This means that the vacuum as used in quantum electrodynamics, i.e. an absence of particles, is not a useful concept, since adding a single electron (or finitely many) to it leads to a system without a thermodynamically stable configuration of minimal energy.

To solve this problem, the Dirac Sea is introduced: Instead of a vacuum without any particles, we have a vacuum where all states of negative energy are filled with electrons and all states of positive energy are empty. Pauli's exclusion princible (supposedly) forbids electrons from moving to a state with lower energy, since they are all already filled. The Dirac Sea is of course a valuable concept, since it led to the discovery of antimatter and pair creation and annihilation: If an electron is excited from the Dirac Sea into a state of positive energy, it leaves behind a hole/positron as well; if an electron moves from a positive-energy state to an empty Dirac Sea state, the electron and the hole/positron are annihilated.

However, when viewed in connection with the Paradox of Hilbert's Hotel, it seems to me that the idea breaks down.

  • First, if we add an electron to the vacuum, this is akin to a newly arriving guest to a full Hilbert's Hotel. If all guests move to the room with the next-higher room number, the new guest can still get a room. In the same manner, all electrons in the Dirac Sea could move to a state with lower energy, leaving space for the added electron to move into. This would be single electron annihilation.

  • Similarly, an electron could move from the Dirac Sea into a state with positive energy without leaving a hole/positron behind. This would be single electron creation.

  • More extremely, the Dirac Sea still does not guarantee a minimum energy configuration: All electrons could simply move at the same time to a state of lower energy without violating Pauli's exclusion principle, leading to a state of lower total energy.

I see some possible objections against this argumentation:

  • The theory breaks down at high energies (high meaning the absolute value). This is something of a cop out.

  • The proposed mechanisms violate well-established conservation laws. However, I think if we only allow the creation/annihilation of electron pairs with equal spin (meaning a spin-change of one), all conservation laws except the conservation of charge could be rescued if photons with an appropriate energy, momentum, etc. are emitted. In addition, this is still something of a cop out since many conservation laws can be derived from their respective theories (like the conservation of momentum from Newton's Axioms/Noether's Theorem, etc.). I am not too sure on this point, though.

Now my question: Can the Dirac Sea be reconciled with Hilbert's Paradox, and if so, how?

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    $\begingroup$ The "negative energy" solutions are only "negative energy solutions" if you do relativistic QM and think the Dirac equation is an equation for a wave function. In quantum field theory, and therefore quantum electrodynamics, this is an operator equation for a quantum field and there are no actual states of negative energy. Therefore, trying to "reconcile" the Dirac sea with Hilbert's paradox strikes me as misguided, since modern quantum field theory does not have the Dirac sea you're describing here. $\endgroup$ – ACuriousMind Aug 25 '16 at 15:32
  • $\begingroup$ @ACuriousMind: that should be an answer. This is either a physics question, in which case the resolution is that the Dirac Sea does not exist, or it is just a maths question and not physically motivated in which case it needs migrating to the Math SE. $\endgroup$ – John Rennie Aug 25 '16 at 15:39
  • $\begingroup$ @JohnRennie You're probably right. I've expanded my comment into an answer. $\endgroup$ – ACuriousMind Aug 25 '16 at 15:56
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Modern quantum field theory does not have a Dirac sea, so trying to reconcile the Dirac picture with any sort of paradoxes you incur when thinking about it is obsolete.

In particular, there are no negative energy states for fermions. This idea arose because the object $\psi$ obeying the Dirac equation was orignally thought to be the wavefunction of Dirac fermions, and there the solution would indeed indicate a negative energy. However, in modern quantum electrodynamics, the Dirac equation is the equation of motion for the quantum field, and the "negative energy solutions" more accurately correspond to the annihilation operators: Instead of representing a "negative energy state", they relate to operators that remove an electron from a given state, and just give zero when applied to a state where so such electron can be found.

Therefore, the vacuum, as used in quantum electrodynamics, remains a very useful concept and we do not have to reconcile it with any consequences derived from an infinite sea of negative energy states.

However, let me remark that Hilbert's paradox itself would pose none of the problems you seem to think it poses:

First, if we add an electron to the vacuum, this is akin to a newly arriving guest to a full Hilbert's Hotel. If all guests move to the room with the next-higher room number, the new guest can still get a room. In the same manner, all electrons in the Dirac Sea could move to a state with lower energy, leaving space for the added electron to move into. This would be single electron annihilation.

Well, you can do that if you're just allowed to shift all electrons around willy-nilly, but where in the theory of the Dirac equation have you found the mechanism that says this happens?

Similarly, an electron could move from the Dirac Sea into a state with positive energy without leaving a hole/positron behind. This would be single electron creation.

Again, unless there's a mechanism in relativistic QM to implement this actually happening, what's the problem?

The only somewhat worrying point is your:

More extremely, the Dirac Sea still does not guarantee a minimum energy configuration: All electrons could simply move at the same time to a state of lower energy without violating Pauli's exclusion principle, leading to a state of lower total energy.

which indeed shows that the designation of a certain "threshhold" to which the Dirac sea is filled as "vacuum" is arbitrary in the Dirac sea model. Again, modern quantum field theory defines the vacuum of the free Dirac field as the Lorentz-invariant state annihilated by all annihilation operators. This is much less arbitrary, and it is easy to see it is actually a lowest eigenstate of the Hamiltonian.

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  • $\begingroup$ Where in the theory of the Dirac equation have you found the mechanism that says this happens? The electron creation/annihilation operators do this, in a sense. (I still need to think on your other points.) $\endgroup$ – user3493525 Aug 25 '16 at 18:50

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