Why sum for Casimir effect between plates but integral without them

[epislon58]

I am reading the paper on the Casimir effect and they measure the space in between the plates using a sum and the energy of the vacuum without the plates using an intregal. Why do they use the sum and intregal, should they be switched.

 

[The_Duck]

Between the plates, only modes with a discrete set of frequencies exist, so we sum over them. Without the plates, a continuum of frequencies is possible (all frequencies are allowed) so the sum becomes an integral.

 

[epislon58]

Thank you sir