Hard, Semihard and Soft Boundary Conditions

Abstract

It is easy to introduce a broad range of static background spatial “objects” into bosonic quantum field theory (QFT) on flat space time. One replaces the wave equation $[ - \Delta ]{\Phi _n} = \omega _n^2{\Phi _n}(\mathop x\limits^ \to )$ for the spatial modes k of the quantum field Φ by the Schrödinger equation

$$ [ - \Delta ]{\Phi _n} = \omega _n^2{\Phi _n}(\mathop x\limits^ \to ) $$

(1)

in which the static potential V is approximately chosen to represent the rigid spatial object or structures of interest. These objects are rather literally “immersed” in Φ”, and each of the interacts with this field, in the way displayed mode for mode by eq. (1) (which of course must be solved to implement the entire procedure) . This description is Dirichlet-like. In regions where is large the quantum field and all its modes are at least partially excluded or expelled. Where is infinite the quantum field and its modes have to vanish. A Dirichlet boundary on which vanishes is represented in this picture by an infinite step-function potential. Any kind of background spatial object immersed in will distort this field, forcing it to be different from the uniform translation-invariant field it would be in free (boundaryless, backgroundless) space. Compelled to be spatially nonuniform, of course exerts a back reaction on each object responsible for its distortion — the famous Casimir forces. If the region of field distortion caused by two or more objects significantly overlap, these objects experience mutual (true many-body) global Casimir forces from the distored quantum field. Also, local Casimir forces act on and throughout individual objects.