This is an improved version of the argument in Electromagnetic Unruh effect?
In the quantum vacuum particle pairs, with total energy Ex, can come into existence provided they annihilate within a time t according to the uncertainty principle Ex t∼ℏ.
Let us assume that there is a force field present that immediately gives the particles an acceleration a as soon as they appear out of the vacuum.
Approximately, the extra distance, Δx, that a particle travels before it is annihilated is Δx∼at2∼ax2c2.
Let us assume that the force field is a static electric field →E and that the particle pair is an electron-positron pair, each with charge e and mass me. The classical equation of motion for each particle is then e →E=me →a.
Is there any merit to this admittedly non-rigorous argument or can the Unruh/Hawking effect only be analyzed using quantum field theory?
Answer
You aren't really asking a question but here is my assessment of your argument.
The Unruh effect states that if one were to couple a detector to a quantum field, the detector would detect a thermal excitation as it is being accelerated. More generally, however, this excitation has to do with the thermal character of the vacuum and not necessarily the coupling of a detector. So the acceleration argument is not exactly necessary. In fact (due to an argument by Sciama), the necessary and sufficient condition is that the vacuum be stable and stationary from the perspective of a uniformly accelerated frame.
Your argument is very hand-wavy. There is a confusion of frames, there is no reference to a thermal density matrix, you have not constructed a boost Hamiltonian, you have not addressed the subtleties of the "quantum" equivalence principle, I don't know what metric you're talking about and so on.
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