The Wave Functional of the Vacuum in a Resonator

TL;DR

This paper demonstrates that the vacuum wave functional in a resonator is mathematically identical to that in free space, expressed through a sum-over-modes approach linked to field correlation functions.

Contribution

It provides an explicit formulation of the vacuum wave functional in a resonator using a bilinear form and relates it to Wightman correlation functions, extending understanding of quantum fields in confined geometries.

Findings

Vacuum wave functional in a resonator matches that of free space.

Explicit kernels for vector potential, electric field, and magnetic induction are derived.

Kernels are related to Wightman correlation functions.

Abstract

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.