*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.
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 R 2  r 2
 “Repulsive” Dirichlet flasks (not Champagne)
-- or how to take advantage of competing loops of
opposite sign.
What are Casimir Energies ?
Geometry
Casimir Energy
Parallel Metal Plates (Casimir ’48)
Metallic Sphere (Boyer ’68)
Metallic Cylinder (Milton ’81)
S2 , S4 ,
S1, S5 ,
T1 , T2 ,
(Kennedy & Unwin ’80,
; S3 , S7 ,
Dowker etc.)
(Ambjorn &Wolfram ’78)
Paralellepipeds
(Lukosz ’71)
  c 2 A / a 3
 0.04617 c / R
 0.01356 cL / R 2
0
0 ; 0
0
Force/Tension
attractive
repulsive?
attractive?
neutral?
attr./rep. ?
attractive?
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?
What are Casimir Energies ?

1
Evac (D )   n   E  D ( E ) dE  ; ECas (D )  ?
2 n
20
 D ( E ~ ) ~
E
2
2

2

n
 c
E
a
(
D
)

O
cos


 
c
3  n
E
c  n 0
a0  VD , a1   SD / 4,
c

Oscillatory terms
Asymptotic Weyl Expansion
Note: Tori : D   and D is flat  only a0  V  0 in AWE;
Spheres: D   but R  0  only V &curvature terms  0
Lowest oscillation frequency ( E ), =length of shortest classical orbit.
But exponentially suppressed diffractive orbits oscillate with (ER)1/3 ( / R)!
Relation to Semiclassical Spectral Density
Casimir energies are differences in vacuum energy for
systems with the same  ( E )
0 ( E )
0
is given by asymptotic expansion of
and can be found semiclassically:
 ( E ~ )
V E2
SE
C

 (E)  2


  osc ( E )
3
2
2
2 ( c) 8 ( c) 12 c E
0 ( E )  First 4 Weyl terms
Balian&Bloch&Duplantier ’74 --06
Approximate
semiclassically
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
c
osc ( E ~ ) ~ a4 2
E
What are Casimir Energies ?
 first 5 terms in AWE (in 3-dim) must vanish for a finite Casimir energy.
i.e. Casimir energy is the difference in the vacuum energies of manifolds Dk
of the same volume, surface area, average (surface) curvature, topology and
4
 
ECas   ck Evac Dk
k 0
Examples:
Balls&Cyl.
ECas 
can be deformed
into each other
 2 2
11
+
a4 !
_
 2 2
Comment: The EM Casimir energy converges for infinitely thin
2
conducting shells (11  c   2  2 ), but in general diverges otherwise!
Milton et al ’78,’81, Balian &Duplantier ‘04
Balls and Cylinders
For cylinders and balls defined by an infinitesimally thin metallic shell :
* the volume term in the AWE is subtracted by the "free vacuum"
* the area term cancels for EM fields (Dirichlet+Neumann)
Note: imperfect cancelations for cinside  coutside
* the curvature term cancels between inside+outside
Note: cancelation incomplete for finite thickness
* the topological term does not depend on R
dS
* a4   3 cancels for inside+outside !!
R
 CE converges and ought to be mainly given by PO's
(and diffractive orbits?)
+
~0
Some Semiclassical CE
Manifolds without boundaries –
d-dimensional spheres & tori exact:
PO
PO
Cas d
Cas
2d
E (T )  0; E (S )  0;
Manifolds with boundaries –
periodic rays in boundar(ies)
depend on boundary condition
-- parallelepipeds & halfspheres (N & D b.c.) exact.
-- spherical cavity
PO
EEM  0.0467 c R error <1%
0.0462
Milton et al ’78,’81
-- concentric cylinders: error <1% when periodic orbits dominate
-- But cylindrical cavity
Mazzitelli et al‘03
PO
EM
E
 0 ~ -0.01356 cL R
Diffractive contributions not negligible here !?
-- classically chaotic systems: only semiclassical estimates
sphere-plate: error <1% when periodic orbits dominate
2
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).
Integer
PO
Cas
E
  A cos(   2), 0  A ~ L
2
 
D   : isolated        , degenerate    (   0   )
-- periodic orbits with odd   do not contribute to CE
-- periodic orbits on boundaries of manifolds contribute
Can we manipulate the sign?
The Casimir Piston (E.A.Power, 1964)
A

F (a)   E
a
a
FCas
L-a
r
’07-'08
a
R
Dirichlet scalar

c 2 A
   n (a )  n ( L  a )  n ( L / 2)  
a n 2
2
480a 4
E ( a, L )  
int
Casimir Force (1948)
Power(1964), Boyer(1970), Svaiter&Svaiter (1992) , Cavalcanti (2004),
Fulling et al (2007-2008) …
Semiclassical Analysis (r=R,a=0)
2
m 



cos

c


30
2
c
PO
k 
EDir
(
a

0)

1


~
0.0442



.
4
2 m 
128r  45 m1 k 2 m1  k sin  k  
r
Contribution of all periodic
orbits of finite length is positive
r
a=0
E(a  0)  0; E(a  )  0
 not monotonically decreasing: repulsive ?
But: reflection positivity (Klich, Bachas '06) demands attraction!
But wait -- (some) closed paths are shorter….
0
0
0
Dirichlet
Neumann
much shorter: the length of these classical closed paths vanish for a  0,
but due to #conjugate points only surface contribution a) survives.
F
Fig. a )
D/ N

c
  EFig.a ) 
a
16
r
d 
0
(a  R 2   2  R 2  r 2 ) 4


c  2 R2  r 2
2
2

 1  O(a / r )  .

2 

96 a 
a

attractive 1 / a 2  force
does not depend on r!
Dirichlet: attractive
Neumann: repulsive
Neumann+Dirichlet~electromagnetic: no net contribution to force
EM CASIMIR FORCE ON A HEMISPHERICAL PISTON
IS REPULSIVE (semiclassically)
Numerical study: Worldline Formalism
Gies, Langfeld and Moyaerts 2003; Klingmüller 2006
Scalar field satisfying Dirichlet boundary conditions on   1

D
Cas
E
1
1
 [] / T 
d   dx  [ (  x)]
d /2 
(4 ) 0
Expectation is with respect to (standard) Brownian bridges
 B( )   B(1),0    1 of a random walk with B(0)  0
 [ ] 0,1 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:
  
1
2
.
2
Worldline formalism (for pistons)
On a bounded 3-dim. domain
D (  )  Tr e
 /2
D
 e
D
 n /2
n
, the trace of the heat kernel
dx

P [ (x,  )  D ]
3/
2
D (2 )
is given by the probability P that a standard Brownian bridge
(x,  ) from x to x in time  is entirely within D . Kac ‘66,
D (  ~ 0) ~  an (2 )
( n3)/2
 O(e

2
/
)
Stroock’93
n 0
a0  VD , a1  SD / 4, etc...("high temperature" expansion)
 ED 
1
2

n
BUT….
n ~  

0
d
 ( )
3/2 D
(2 )
diverges.
geometrically subtracting…
the first 5 terms of the asymptotic expansion of D (  )
 (  )   ckD (  ) with  ck ai (D k )  0 for i  0,, 4,
k
k
that is
k
0=  ck VDk   ck SDk  ...   (  ~ 0)  O(  )
k
k
 Eint.   ck EDk   
k

0
d
 (  ) is finite
3/2
(2 )
Note: Eint. =ECas is a difference of zero-point energies;
logarithmic divergences cancel iff
 c a (D
k 4
k
k
)  0!
Example: Axially symmetric Casimir pistons

F (a)   E
a
Flat Casimir piston for R>>r>>a
Hemispherical Casimir piston for R=r
r
r
r
_
_
r
+
a
R
R
α)
Eint. (a )  E )
β)

E )
L
γ)

E )
δ)

E )
The 5 convex domains
for a>0 only loops of finite length contribute to Eint , these
pierce piston AND cap, but NOT cylinder
Determining the support


S[ ]  (x,  );  [ 1 (  x)]  1
of a unit loop
requires solution of a
non-linear optimization problem
--
not easily solved for loops of 104-106 points.
Information Reduction by Convex Hulls
a
Hull
# vertices (v) of hull
• Ordering information of a loop is redundant for Casimir energies
• Convex Hull of its point set determines whether a loop pierces
a convex boundary
200
180
160
140
120
100
80
60
40
20
0
1.E+02
102
CPU
(sec)
200
v = 19.3 ln(n) - 76.3
20
Simulation
Log.…
Trendline
2
0.2
1.E+03
103
1.E+04
104
5
101.E+05
# points (n) of unit loop
6
101.E+06
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
Casimir energy of Cylindrical Pistons
c  1 
----- Er=R (a )  
~
 
96 a
150.3  2a 
c
..after subtracting “electrostatic”-like energy
Dirichlet scalar
= periodic orbits for hemispherical piston (a=0)
q2
E  E  ;
2a
q2
1
1
e2

~

c 48 150.3
c
Note: the asymptotic (attractive) interaction energy
of hemispherical piston -- the residual is repulsive.
q2
2a
does not depend on size r
Do Flasks repulse Dirichlet Pistons?
To avoid attraction of piston due to reflection positivity either:
a) impose metallic boundary conditions (non-separable boundary)

b) or make Dirichlet mirror smaller than shape!
U 1
U
1

dx   dx F  0
 links of dividing
plaquette are correlated
ECas ( L  R ? rr ? ra )  0
ECas ( L  R  a ? rr )  0
 repulsion!
to contribute, loops (+) pierce piston & cylinder, not flask
or () pierce piston & flask, not cylinder
and calculating, calculating, and calculating…
Repulsion!
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
2
a )
q
c
f
(
r
F   E   2 
with
2
a
2a
r
q2
1
1
f (0)  0 and

~
  EM
c 48 150
 the force attracts even for r   , 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
R ? rr