Vacuum Stress and Closed Paths in Rectangles, Pistons, and Pistols

TL;DR

This paper provides an exact analysis of vacuum stress in rectangular cavities using classical paths, exploring how boundary conditions and cutoff parameters influence the force on cavity walls, with implications for understanding Casimir-like effects.

Contribution

It offers a comprehensive, exact calculation of the stress tensor in a rectangular cavity for all boundary conditions and cutoff parameters, clarifying the physical nature of vacuum forces.

Findings

The vacuum force can be attractive or repulsive depending on geometry and cutoff.

Boundary divergences and exterior effects significantly influence force calculations.

Physically plausible repulsive forces require geometries outside realistic models.

Abstract

Rectangular cavities are solvable models that nevertheless touch on many of the controversial or mysterious aspects of the vacuum energy of quantum fields. This paper is a thorough study of the two-dimensional scalar field in a rectangle by the method of images, or closed classical (or optical) paths, which is exact in this case. For each point r and each specularly reflecting path beginning and ending at r, we provide formulas for all components of the stress tensor T_{\mu\nu}(r), for all values of the curvature coupling constant \xi and all values of an ultraviolet cutoff parameter. Arbitrary combinations of Dirichlet and Neumann conditions on the four sides can be treated. The total energy is also investigated, path by path. These results are used in an attempt to clarify the physical reality of the repulsive (outward) force on the sides of the box predicted by calculations that neglect both boundary divergences and the exterior of the box. Previous authors have studied "piston" geometries that avoid these problems and have found the force to be attractive. We consider a "pistol" geometry that comes closer to the original problem of a box with a movable lid. We find again an attractive force, although its origin and detailed behavior are somewhat different from the piston case. However, the pistol (and the piston) model can be criticized for extending idealized boundary conditions into short distances where they are physically implausible. Therefore, it is of interest to see whether leaving the ultraviolet cutoff finite yields results that are more plausible. We then find that the force depends strongly on a geometrical parameter; it can be made repulsive, but only by forcing that parameter into the regime where the model is least convincing physically.