Vacuum stress and closed paths in rectangles, pistons and pistols

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 µν (r), for all values of the curvature coupling constant ξ 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.

Energy Density and Pressure in Power-Wall Models

“…The latter is satisfied, even in the presence of a finite cutoff. 7 Whatever the case may have been for the original spectral sum (1), in the explicit Green function (8) point separation in any direction is an equally effective regularization. We therefore examine the effects of varying the direction.…”

“…7 we argued that expressions with finite cutoff, such as 2π 2 E = (y 2 + 4z 2 ) −2 (where y is now a cutoff parameter, not a coordinate) can be regarded as ad hoc models of real materials, more physical and instructive than their limiting values, such as E = 1/32π 2 z 4 . But the paradox casts some doubt on the viability of this point of view.…”

Vacuum Energy Density and Pressure Near Boundaries

“…9-11 Furthermore, in [1] we said that in a rectangle, there is a divergent pressure on a particular side, but it is unrelated to the divergent energy density in the region adjacent to that side (which is independent of the position of the side). Instead, it corresponds to the length dependence of the divergent energy in the edge regions along the perpendicular sides.…”

“…This prompted us to look at flat plates intersecting at right angles in dimension 3, where we find the same discrepancy − 1 2 ! That in turn forced a reexamination of the 2-dimensional calculation in [1], mentioned above. It turns out that there the numerical factor is correct but the sign is wrong.…”

Thermal Fluctuations in Casimir Pistons

“…In their model the net pressure at the interface has a finite value and it is always attracted towards the closer end of the piston. This kind of results motivated extensive research on the Casimir piston model 15,16,17,18,19,20 . In some cases, repulsive Casimir forces may be attained by introducing non-electromagnetic interactions such as scalar fields subject to mixed boundary conditions 16 , or quantum star graphs 19 .…”