  We show that despite the fundamentally different situations, the wave
functional of the vacuum in a resonator is identical to that of free space. The
infinite product of Gaussian ground state wave functions defining the wave
functional of the vacuum translates into an exponential of a sum rather than an
integral over the squares of mode amplitudes weighted by the mode volume and a
power of the mode wave number. We express this sum by an integral of a bilinear
form of the field containing a kernel given by a function of the square root of
the negative Laplacian acting on a transverse delta function. For transverse
fields it suffices to employ the familiar delta function which allows us to
obtain explicit expressions for the kernels of the vector potential, the
electric field and the magnetic induction. We show for the example of the
vector potential that different mode expansions lead to different kernels.
Lastly, we show that the kernels have a close relationship with the Wightman
correlation functions of the fields.
