Scalar Quantum Fields Confined by Rectangular Boundaries

Abstract

A collection of local Casimir problems is studied in detail. These problems are generated by confining a scalar quantum field within boundaries consisting of planes, edges and corners. The boundaries include isolated infinite planes, rectangular edges and corners, parallel planes, the rectangular waveguide and the rectangular cavity. Calculations are done for massless and massive confined fields and for fields at zero and nonzero temperature. Each example has its finite global Casimir effect, which must be computable from the exact local description. The latter however contains known boundary divergences which seem to imply an infinite global Casimir effect. A physical picture and mathematically explicit method is presented for computing finite global Casimir effects from the divergent local description. Divergent boundary functions extending outward in space away from a given boundary ϱ𝔐 represent the distortion of the virtual particle sea of the confined field caused by ϱ𝔐. This distortion of the virtual sea is inseparable from the existence of ϱ𝔐, and may be regarded as the cost of construction of ϱ𝔐 out of nothing. Thus a given divergent boundary function is intrinsic to its ϱ𝔐 — where here is meant specifically the complete boundary function extending (if it does) all the way to infinity. Different boundaries cut off parts of each other's boundary functions, and this plays a vital role in the global Casimir effect as will be explained. It is shown in the examples considered how to compute the global Casimir effect by integration of the exact local description over the region within which the quantum field is confined. Rectangular boundary geometry makes calculations relatively simple; however the method can be applied to boundaries of arbitrary shape. At finite temperature the confined quantum field represents a thermal gas as well as a virtual particle sea. Boundaries also distort this gas of real particles in a way which can be computed. Exact local calculations are done for the rectangular boundaries considered. Beyond these considerations for free fields, systems with interacting fields present additional very interesting local Casimir phenomena. A model two-field interacting theory is discussed briefly for illustration.