Cardinality argument in the Casimir effect

[nomadreid]

There is an argument to account for the Casimir effect based on cardinalities: that inside the two plates only (virtual) photons with wavelengths corresponding to the harmonic series can exist, hence countable infinity, whereas photons of all wavelengths can exist in the space around it, hence uncountable infinity. Therefore, the argument goes, there will be a higher energy density outside than inside. This argument seems to run into a problem if one can one consider two photons of equal energy to be distinct. (After all, wouldn't the total energy also depend on the amplitudes for each wavelength?) In this case, the cardinality of the possible wavelengths of the waves between the plates may be countable while the number of waves could be uncountable. This would then end up with the same energy density. That is, supposing that at every point in the continuum number of points in the space between the plates there is a distinct wave, then although the range of values of the waves between the plates is restricted to members of a harmonic sequence, the fact that there is a continuum number of points between the plates would also give you a continuum number of waves inside as outside. So, am I missing something, or can this argument be repaired? There are other explanations of the Casimir effect, but I would sooooo much like a cardinality argument to go through.

 

[conway]

I don't think the cardinality argument ever held water. I know people argue for the Casimir effect that way but I think it's sloppy counting and the method is not repairable.

The reason it doesn't work is that the infinite density of continuum states is balanced by the fact that an individual photon in anyone of those states could be infinitely far from the plates, and its probable effect is accordingly downweighted. A correct Casimir analysis can best be done comparing finite volumes: large versus small.