(a) There is no contradiction between 2 and 3 since in quantum entanglements the states under question, in general, are not eigenstates of the Hamiltonian.

(b) The problem referred to in 1 does not include quantum gravity degrees of freedom that are certainly required if gravity like other interactions is also quantum in nature. It is possible that the fine tuning problem associated with the cosmological constant will disappear when a proper quantum gravity is considered.

@Patrick Das Gupta,

Thanks for your answer.

(a) Yes, in quantum etaglemets the states are states of the Hamiltonian of a free particle.

(b) And if the gravity, unlike the other interaction, does not have a proper description in quantum mechanics? How shall I include in 1 quantum gravity degrees of freedom as long as it is not kown if it is quantum at all? I can't introduced non-decided things.

Moreover, when working with entanglements we take care to do corrections which eliminate the gravitational field influence. This influence is a perturbation even without quantizing it.

(c) I placed the stress on the issue of the coupling with the vacuum. Why in the decay theory it has an influence, but has no influence on the free particles traveling in vacuum. If the influece existed, it would perturb the entanglements.

Short answer: They do couple, and this is tacitly known.

You can qualitatively work out the effect on the vacuum by introducing a fake "vacuum degree of freedom" as a few-level system and let the particles interact with the vacuum in different ways (I think you need three states at least: ground state, state after interaction with the one particle state and the other one). The general effect is that the coupling reduces the entanglement effects.

Then you take a two-fold approach: The people doing entanglement ignore the vacuum for simplicity, but keep in mind that there are "entanglement loss channels" that should be minimized (hence the use of photons or spin states: less coupling to the environment) and that will lead to suboptimal entanglement. Other people study the coupling to get an idea of the theoretically possible entanglement, how to minimize the losses etc. (The Big Thing in the quantum computing subfield).

@Ulf,

As far as I know, the higher is the degree of vacuum, the lower is the decoherence of the entanglements. I NEVER saw in experimental reports on entanglements, e.g. entanglements of fermions, a Hamiltonian as in the decay, i.e. with a term describing the free particle Hamiltonian for the fermions, a Hamiltonian from which results the continuum spectrum of energies of the electrons, and a coupling term.

It's not that people ignore for simplicity, but because they don't see such a type of entanglement loss. Losses are when the vacuum level is not enough deep, or unwanted absorption and scattering by apparatuses, and others, but none of the type to which refers my question.

This is why I question the calculus of the cosmological constant based on the zero-level vacuum. If the latter has an influence, the influence should be seen in all the experiments, not only in a few phenomena as decay, Casimir effect, and so on.

Happy New Year!

Comments concerning (b) and (c): (i) For free particles, momentum is conserved and hence the kinetic energy too is conserved. Therefore, due to coupling with vacuum, the energy of the free particle cannot decrease, unlike the case of bound states. Ergo, entangled free particles are least affected by vacuum fluctuations.

(ii) Certain general statements concerning quantum gravity can still be made like Planck energy cut off, microscopic wormholes, virtual black holes and so on, even in the absence of a proper quantum gravity theory. And papers have been written on their effects on the cosmological constant problem. Please read Weinberg's 1989 review.