A Casimir effect and dimensional analysis

[Fightfish]

Yes, the field should obey the standard set of boundary conditions at a conducting surface (the same ones that you see in an introductory EM course).

 

[spaghetti3451]

Well, I have the following conditions:

$E_{tan}=0$

$E_{perp}=\sigma$

$B_{tan}=j$

$B_{perp}=0$

How do they relate to the boundary conditions for $A_{\mu}$?

 

[Fightfish]

That's a good question; I admit that I'm not particularly familiar and haven't actually seen any calculations using the gauge potential itself. Rather, the boundary conditions are just used to quantise the modes (frequencies) present. I mean, well, of course we skipped the entire process of second quantising the electromagnetic field to begin with...so the effect probably enters in that form. But yeah, the usual starting point for deriving the Casimir effect already begins from the second quantised harmonic oscillator form of the Hamiltonian, so those details are swept under the carpet.

 

[spaghetti3451]

So, you would say that the boundary conditions are not something as trivial as $A_{\mu}(z=\pm L/2)=0$,

where the area of the plates are in the $x-y$ directions and the plates are located at $z=\pm L/2$?

 

[Vanadium 50]

This thread is really a mess.

Let's go back to the beginning. The Coulomb force is given by kq²/r², so k is a dimensionful constant. Likewise the classical gravitational force is given by km²/r² and again, k is a dimensionful constant. In this case, you have determined that the Casimir force is really a pressure, and should thereby have a functional form where the force is proportional to area, so you have F = A k/rⁿ.

This form presents no restrictions on n. n could be -1 (like a spring), 2 (like Coulomb's law) or even some other number like 4. You can guess at n (and so far the guesses have been poor) or guess at the dimensionality of k, but these are guesses. Given that all you know is that there is a force, you cannot determine this force's functional dependence on r.