1. *in memory of Larry Spruch (1923-2006) Phys. Rev. A73 (2006) 042102 [hep-th/0509124]; [hep-th/0604119];[quant-ph/0705.3435]. H. Gies, K. Langfeld, L. Moyaerts, JHEP 0306, 018 (2003); H. Gies, K. Klingmuller, Phys. Rev. D74, 045002 (2006) Work supported by NSF.

2. Outline Casimir energies vs. vacuum energies Semiclassical relation to periodic orbits (semiclassical) Casimir energies: -- successes and “failures” The sign of (semiclassical) Casimir energies Some generalized Casimir pistons Semiclassical (EM and Dirichlet) Numerical (World Line Formalism) Subtracted spectral densities Convex hulls for convex pistons dependence on of results on “Repulsive” Dirichlet flasks (not Champagne) -- or how to take advantage of competing loops of opposite sign.

3. GeometryCasimir Energy Force/Tension Parallel Metal Plates (Casimir ’48) attractive Metallic Sphere (Boyer ’68) repulsive? Metallic Cylinder (Milton ’81) attractive? (Kennedy & Unwin ’80, neutral? Dowker etc.) attr./rep. ? (Ambjorn &Wolfram ’78) attractive? Paralellepipeds (Lukosz ’71)Depends on b.c. and dimensions ! ?? What are Casimir Energies ? what do zeta-function reg., dimensional reg., heat- and cylinder-kernel, compute as finite Casimir energies? What is the sign of Casimir energies? Is it unambiguous? Is it meaningful?

4. What are Casimir Energies ? Oscillatory terms Asymptotic Weyl Expansion

5. Relation to Semiclassical Spectral Density Casimir energies are differences in vacuum energy for systems with the same One can only compare the zero-point energy of systems of the same total volume, total surface area, average curvature and topology (number of corners, holes, handles…) Universal subtraction possible No logarithmic divergent CE is given by asymptotic expansion of and can be found semiclassically: First 4 Weyl terms Approximate semiclassically Balian&Bloch&Duplantier ’74 --06

6. What are Casimir Energies ? + _ Comment: The EM Casimir energy converges for infinitely thin conducting shells ( ), but in general diverges otherwise! Examples: Balls&Cyl. Milton et al ’78,’81, Balian &Duplantier ‘04

7. Balls and Cylinders +~ 0

8. Some Semiclassical CE Manifolds without boundaries – d-dimensional spheres & tori exact: Manifolds with boundaries – periodic rays in boundar(ies) depend on boundary condition -- parallelepipeds & halfspheres (N & D b.c.) exact. -- spherical cavity -- concentric cylinders: error <1% when periodic orbits dominate -- But cylindrical cavity -- classically chaotic systems: only semiclassical estimates sphere-plate: error <1% when periodic orbits dominate 0.0462 error <1% -0.01356 Diffractive contributions not negligible here !? Mazzitelli et al‘03 Milton et al ’78,’81

9. isolated periodic orbits -- Gutzwiller’s trace formula integrable systems -- Berry-Tabor trace formula No periodic orbits -- diffraction dominates (e.g. knife edge) -- tiny Casimir forces? Sign of contribution to Casimir energy of (a class of ) periodic orbits is given by a generalized Maslov index (optical phase). Integer -- periodic orbits with odd do not contribute to CE -- periodic orbits on boundaries of manifolds contribute The sign of PO-contributions Can we manipulate the sign?

10. The Casimir Piston (E.A.Power, 1964) Casimir Force (1948) Power(1964), Boyer(1970), Svaiter&Svaiter (1992), Cavalcanti (2004), Fulling et al (2007-2008) … a L-a Dirichlet scalar R A r a ’07-'08

11. Semiclassical Analysis (r=R,a=0) Contribution of all periodic orbits of finite length is positive a=0 r

12. But wait -- (some) closed paths are shorter…. much shorter: the length of these classical closed paths vanish for, but due to #conjugate points only surface contribution a) survives. Dirichlet: attractive Neumann: repulsive Neumann+Dirichlet~electromagnetic: no net contribution to force EM CASIMIR FORCE ON A HEMISPHERICAL PISTON IS REPULSIVE (semiclassically) Dirichlet Neumann

13. Numerical study: Worldline Formalism Gies, Langfeld and Moyaerts 2003; Klingmüller 2006 Scalar field satisfying Dirichlet boundary conditions on Expectation is with respect to (standard) Brownian bridges of a random walk with if certain conditions on are satisfied by. Note: the CM of is irrelevant. Also: The Casimir energy is negative, and monotonically increasing, i.e. the Casimir force is attractive between disjoint boundaries:

14. Worldline formalism (for pistons) On a bounded 3-dim. domain, the trace of the heat kernel BUT…. Kac ‘66, Stroock’93

15. geometrically subtracting… the first 5 terms of the asymptotic expansion of

16. Example: Axially symmetric Casimir pistons r r Flat Casimir piston for R>>r>>a Hemispherical Casimir piston for R=r R r + _ _ r R L α)α)β)β)γ)γ)δ)δ) a

17. Determining the support of a unit loop requires solution of a non-linear optimization problem -- not easily solved for loops of 10 4 -10 6 points. The 5 convex domains for a>0 only loops of finite length contribute to, these pierce piston AND cap, but NOT cylinder

18. Information Reduction by Convex Hulls Ordering information of a loop is redundant for Casimir energies Convex Hull of its point set determines whether a loop pierces a convex boundary a Hull Trendline 200 20 2 0.2 CPU (sec)

19. Some Convex Hulls Hull of 10 6 point loop 220 Hull vertices in 155 CPU sec Hull of 10 3 point loop 55 Hull vertices in 0.3 CPU sec

20. Casimir energy of Cylindrical Pistons

21. ..after subtracting “electrostatic”-like energy = periodic orbits for hemispherical piston (a=0) Dirichlet scalar

22. Do Flasks repulse Dirichlet Pistons?

23. Repulsion! and calculating, calculating, and calculating…

24. Conclusions  The force on a piston in some environments is opposite to that in others. This is not surprising and does NOT really imply that it is repulsive. The Casimir force due to a Dirichlet scalar on a piston in a hemispherical cavity is greatly reduced  the force attracts even for, but respects reflection pos.  Constraints on Casimir pistons from reflection positivity can be avoided and the force is “repulsive” for a)Hemispherical piston with metallic b.c. b)Flask-like geometries (even with Dirichlet b.c.) and/or