I have come to the conclusion that this problem is not as important as people like to think. I agree with Rovelli about this: Λ≈10−9Jm3 is just yet another fundamental constant of nature.
There seem to me to be the following approaches to the problem:
- Debye renormalization technique 3N=∫ωcutoff0g(ω) dω, which does not work as we do not know the number of quanta of spacetime.
- Correcting Planck law E(ω)=ℏω2Pω1√1−(ω/ωP)2, whose failure proves it in my face that dimensional analysis is even much more powerful than I used to think.
The whole problem seems empty to me. Let's think of a statistical theory of quanta of spacetime quanta which has no reference to matter in its foundational definitions. The best I have been able to think of is to define the volume that an ensemble of spacetime quanta occupy to be V=l3PlogW(E), for a fixed energy (~microcanonical ensemble). Then Λ=dEdV, pointing that vacuum energy (density) is just like temperature. You just cannot (and do not) go `calculate the temperature' of something. The only way one can calculate a temperature is for phase transitions (condensed matter physics). From this perspective it might be possible to think of vacuum as a Bose-Einstein--like condensate (of gravitons) whose critical pressure is Λc≈10−9Jm3
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