Spectrum of the Casimir effect

The problem of assigning a frequency spectrum to the Casimir effect is studied. The specific case of a massless scalar field with periodicity in one spatial direction is investigated in both two- and four-dimensional spacetime. The spectrum is defined by introducing spectral weight functions which distort the original spectrum of quantum fluctuations and hence reveal the contribution of each frequency interval to the finite Casimir energy. The result is a function σ(ω) whose integral over all frequencies is the total vacuum energy. In order to have σ(ω)→0 as ω→∞, it is necessary to specify a nonzero tolerance Δω which is the allowed uncertainty in the measurement of the spectrum. In the limit that Δω→0, σ(ω) approaches a discontinuous function which does not vanish as ω→∞