# What is the relation between zeta and cut-off regularization of the Casimir effect?

In the literature there are at least two methods to derive Casimir effect:

• original one by Casimir himself: take the quantized energy between plates minus the free space energy, then regularize, e.g. by cut-off function and use Euler-Maclaurin formula

• modern one: take the quantized energy between plates, regularize it by zeta and analyticaly continue to physical value

Obviously the results are the same.

Two remarks:

My question is: how this difference, i.e. neglecting the free space energy in the latter approach, is explained from the physical point of view? (perhaps one could elaborate on first remark)

• Aren't you glad that two kinds of intellectual nonsense agree with each other? That calls for a beer... but not necessarily for the celebration of a deep understanding of the physical vacuum... I am not trying to be unkind here. Bohr's model of atoms could explain a few things correctly, but that was luck. I would think of the current state of QFT with regards to the self-energy of the vacuum in very much the same way. We can make a few lucky guesses, but that's probably it. I would be glad to hear if someone could correct my ignorance on the matter, though. Commented Dec 30, 2014 at 19:36
• @CuriousOne: well, it would be interesting if the result was only a coincidence, especially the effect being connected to van der Waals force. Still this difference in two approaches is obscure. Commented Jan 2, 2015 at 14:06
• First link is dead now. Archived by the Wayback Machine: web.archive.org/web/20210120213457/https://aphyr.com/data/…
– Urb
Commented Apr 12, 2021 at 15:29

Ok, I found the answer in the draft by Kleinert: http://users.physik.fu-berlin.de/~kleinert/b6/psfiles/qft.pdf, pg. 600.

The fact that is not mentioned in other sources - the free space energy (aka zero-point energy) that we subtract in the original approach, goes to $0$ after analytic continuation performed as in the second (zeta regularization) approach (it can be shown using dimensional regularization).

In other words, the difference between the quantized energy between plates and the free space energy should always be taken in order to obtain Casimir effect (at least we don't want to use van der Waals forces), and this is physically reasonable. It is just specific to zeta regularization that the free space energy goes to $0$, thus often not even mentioned (!) in other texts.