Shape dependent Repulsive (?) Casimir Forces ( M.Schaden *)

1. Shape dependent Repulsive (?) Casimir Forces (M.Schaden*)  *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. What are Casimir Energies ? what do zeta-function reg. , dimensional reg., heat- and cylinder-kernel , compute as finiteCasimir 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 is given by asymptotic expansion of Casimir energies are differences in vacuum energy for systems with the same and can be found semiclassically: Approximate semiclassically First 4 Weyl terms Balian&Bloch&Duplantier ’74 --06 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

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

7. Balls and Cylinders + ~ 0

8. -0.01356 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 error <1% 0.0462 Milton et al ’78,’81 Mazzitelli et al‘03 Diffractive contributions not negligible here !?

9. Integer The sign of PO-contributions 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). -- periodic orbits with odd do not contribute to CE -- periodic orbits on boundaries of manifolds contribute Can we manipulate the sign?

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

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

12. But wait -- (some) closed paths are shorter…. Dirichlet Neumann 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)

13. Gies, Langfeld and Moyaerts 2003; Klingmüller 2006 Scalar field satisfying Dirichlet boundary conditions on Numerical study: Worldline Formalism 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. Worldlineformalism (for pistons) On a bounded 3-dim. domain , the trace of the heat kernel Kac ‘66, Stroock’93 BUT….

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

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

17. The 5 convex domains for a>0 only loops of finite length contribute to , these pierce piston AND cap, but NOT cylinder Determining the support of a unit loop requires solution of a non-linear optimizationproblem -- not easily solved for loops of 104-106 points.

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 Trendline CPU (sec) 200 20 2 0.2 a Hull

19. Some Convex Hulls Hull of 103 point loop 55 Hull vertices in 0.3 CPU sec Hull of 106 point loop 220 Hull vertices in 155 CPU sec

20. Casimir energy of Cylindrical Pistons

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

22. Do Flasks repulse Dirichlet Pistons?

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

24. 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 • Hemispherical piston with metallic b.c. • Flask-like geometries (even with Dirichlet b.c.) and/or Conclusions