Monday, 3 January 2022

On the vacuum catastrophe

I have come to the conclusion that this problem is not as important as people like to think. I agree with Rovelli about this: $$\Lambda \approx10^{-9} \frac{\text{J}}{\text{m}^3}$$ is just yet another fundamental constant of nature. 

There seem to me to be the following approaches to the problem:

  1. Debye renormalization technique $$3N=\int_0^{\omega_\text{cutoff}}g(\omega)\ d\omega,$$ which does not work as we do not know the number of quanta of spacetime.
  2. Correcting Planck law $$E(\omega)=\frac{\hbar\omega_P^2}{\omega}\frac{1}{\sqrt{1-(\omega / \omega_P )^2}},$$ 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 $$V=l_P^3\log W(E),$$ for a fixed energy (~microcanonical ensemble). Then $$\Lambda=\frac{dE}{dV},$$ 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 $$\Lambda_c \approx10^{-9} \frac{\text{J}}{\text{m}^3}$$

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