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

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

• in memory of Larry Spruch (1923-2006)
• Phys. Rev. A73 (2006) 042102 hep-th/0509124
• hep-th/0604119quant-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.

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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.

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What are Casimir Energies ?
Geometry Casimir 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 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?
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What are Casimir Energies ?
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Relation to Semiclassical Spectral Density

• Casimir energies are differences in vacuum
  energy for systems with the same

is given by asymptotic expansion of
and can be found semiclassically
Approximate semiclassically
First 4 Weyl terms
BalianBlochDuplantier 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
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What are Casimir Energies ?
Examples BallsCyl.
Comment The EM Casimir energy converges for
infinitely thin conducting shells (
), but in general diverges
otherwise!
Milton et al 78,81, Balian Duplantier 04
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Balls and Cylinders

0
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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 lt1 when periodic
  orbits dominate
• -- But cylindrical cavity

error lt1
0.0462
Milton et al 78,81
Mazzitelli et al03
Diffractive contributions not negligible here !?
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The sign of PO-contributions
isolated periodic orbits -- Gutzwillers
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).
Can we manipulate the sign?
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The Casimir Piston (E.A.Power, 1964)
A
07-'08
a
R
Dirichlet scalar
a
L-a
Casimir Force (1948) Power(1964), Boyer(1970),
SvaiterSvaiter (1992) , Cavalcanti
(2004), Fulling et al (2007-2008)
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Semiclassical Analysis (rR,a0)
r
Contribution of all periodic orbits of finite
length is positive
a0
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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 NeumannDirichletelectromagnetic no
net contribution to force EM CASIMIR FORCE ON A
HEMISPHERICAL PISTON IS REPULSIVE
(semiclassically)
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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
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Worldline formalism (for pistons)
Kac 66, Stroock93
BUT.
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geometrically subtracting
the first 5 terms of the asymptotic expansion of
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Example Axially symmetric Casimir pistons
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The 5 convex domains
for agt0 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
optimization problem -- not easily
solved for loops of 104-106 points.
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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

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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
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Casimir energy of Cylindrical Pistons
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..after subtracting electrostatic-like energy
Dirichlet scalar
periodic orbits for hemispherical piston (a0)
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Do Flasks repulse Dirichlet Pistons?
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and calculating, calculating, and calculating
Repulsion!
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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
• Hemispherical piston with metallic b.c.
• Flask-like geometries (even with Dirichlet b.c.)
  and/or