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?