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:
- "(...) This suggests an important physical intuition for zeta renormalization: using the analytic continuation from s=3 to s=0 in some sense corresponds to subtracting the electromagnetic field's inherent contribution to the ground state energy. In Casimir's derivation we removed the infnite contribution of the field by taking the difference of two configurations, in effect subtracting whatever (infnite) energy the field possesses in free space. Here the subtraction may not be as intuitive, but its analytic simplicity makes it a powerful tool for analyzing vacuum energy problems." (https://aphyr.com/data/journals/113/comps.pdf, pg.13)
- the post http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/ explains what is the relation between cut-off and zeta, but still in two derivations above we have two different objects to regularize!
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)
