I have heard about and seen the calculations of the vacuum energy from various sources. According to this video, it says that the predicted vacuum energy is about $$\varepsilon_\Lambda \sim 10^{114} J/m^{3}$$ And the measured value is $$\varepsilon_\Lambda \sim 10^{-10} J/m^{3}$$ So, this leads to the “worst prediction in physics” $$\frac{predicted \varepsilon_\Lambda}{measured\varepsilon_\Lambda} \sim 10^{124} $$ I understand that. It involves summing all the modes of the EM field, meaning $$\sum_{n=0}^{\infty} \hbar \omega_n = \infty$$ Which is not possible for the vacuum energy, so we enforce that the sum goes down to the plank length instead and we attain $$\varepsilon_\Lambda \sim 10^{114} J/m^{3}$$ Which is still astronomically large.
All that is fine and good. However, I have heard in interviews with Leonard Susskind (I cannot remember where), that he said that in order to obtain the vacuum energy you have to sum over all the positive and negative frequency (positive and negative energy) modes in order to arrive at a positive vacuum energy value. This is because there is a small difference between the positive and negative modes that leads them being in unequal number, thus creating a net positive energy vacuum.
After knowing this, it seems that the math done in the video is different from the sum of all positive and negative frequency modes of all quantum fields. The video only considers the EM field while the other considers all quantum fields and summing over them.
I never seen the calculation of positive and negative frequency modes to arrive at the vacuum energy in textbooks or in videos. Would someone be able to show that here? It would greatly improve my understanding of the topic of the vacuum energy.
