*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.

OutlineCasimir energies vs. vacuum energies

Semiclassical relation to periodic orbits(semiclassical) Casimir energies:

-- successes and “failures”The sign of (semiclassical) Casimir energies

Some generalized Casimir pistonsSemiclassical (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.2 2R r

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 are Casimir Energies ?

0

0.04617 /c R

2 3/c A a

20.01356 /cL R 2 4, ,S S 1 5 3 7, , ; , ,S S S S 0 ; 0

1 2, ,T T 0

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 ?

0

2

320

0 1

cos

1( ) ( ) ; ( ) ?

2 2

( ~ ) ~ ( )2

, / 4,

vac n Casn

n

nn

Ecc

E E dE

E cE a

Ec

a a

D

D

D D

E D E D

D

V S

O

Oscillatory terms

Asymptotic Weyl Expansion

0 and is flat only 0

but R 0 only &curvature

Note: Tori : in AWE;

Spheres:

Lowest oscillation frequency ( ), =length of shortest classical orbit.

B expone

te

ntut ia

rms 0

lly

E

a

D D V

D V

1/3 diffractive orbits oscillate with (Esuppresse R) ( !d / )R

Relation to Semiclassical Spectral DensityCasimir energies are differences in vacuum energy for systems with the same )(0 E

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 4 2

( ~ ) ~osc

cE a

E

)(0 E is given by asymptotic expansion of )~( Eand can be found semiclassically:

2

2 3 2 2( ) ( )

2 ( ) 8 ( ) 12 osc

E EE E

c c c E

V S C

First 4 Weyl terms)(0 E Approximate semiclassicallyBalian&Bloch&Duplantier ’74 --06

What are Casimir Energies ? in AWE (in 3-dim) must vanish for a finite Casimir energy.

i.e. Casimir energy is the of manifo

fir

lds

of the same volume, surface area, average (surfac

difference in the vacuum energie

st 5

s

terms

k

D

4topologye) curvature, and ! a

can be deformed

into each other

4

0Cas k vac k

k

E cE D

+_

Cas E 1 1

2 2

Comment: The EM Casimir energy converges for infinitely thin conducting shells ( ), but in general diverges otherwise!

Examples:Balls&Cyl.

2 2

21 1 2 2c

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)

No inside outs de

4

ite:

* the curvature term cancels between inside+outside

imperfect cancelations for

cancelation incomplete for finNote:

* the topological term does not depe

ite thic

nd on R

*

knes

s

da

c c

S3

CE converges and ought to be mainly given by PO'

cancels for inside+outside !!

(and diffractive orbi )

s

ts?

R

+ ~ 0

Some Semiclassical CEManifolds 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 estimatessphere-plate: error <1% when periodic orbits dominate

2( ) 0; ( ) 0;PO POCas d Cas dT S E E

0.0467POEM c R E

0.0462error <1%

0 ~POEM E -0.01356 2RcL

Diffractive contributions not negligible here !?

Mazzitelli et al‘03

Milton et al ’78,’81

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).

2cos( 2), 0 ~POCas A A L

EInteger

-- periodic orbits with odd do not contribute to CE-- periodic orbits on boundaries of manifolds contribute

The sign of PO-contributions

0 isolated , degenerate ( ): D

Can we manipulate the sign?

Casimir Force (1948)Power(1964), Boyer(1970), Svaiter&Svaiter (1992) , Cavalcanti (2004),Fulling et al (2007-2008) …

2

4( ) ( ) ( / 2)

2 2 480

( , )

Cas

int

c Aa L a L

a a

a L

n n nn

F

E

a L-a Dirichlet scalar

( )aa

F E

R

Ar

a’07-'08

Contribution of all periodic orbits of finite length is positive

2

. 4 21 2 1

cos30 2( 0) 1 ~ 0.0442

128 45 sin

mPO kDir m

m k m k

c ca

r k r

E

( 0) 0; ( ) 0

repulsive ?

But:

not monotonically decreasing:

reflection positivity (Klich, Bachas '06) demands attraction!

a a E E

a=0

r

much shorter: the length of these classical closed paths vanish for ,but due to #conjugate points only surface contribution a) survives.

0a

Fig. a ) Fig.a)/ 2 2 2 2 40

2 22 2

2

16 ( )

2 1 ( / ) .

96

r

D N

c d

a a R R r

c R ra r

a a

F

O

E

Dirichlet: attractive Neumann: repulsiveNeumann+Dirichlet~electromagnetic: no net contribution to force

EM CASIMIR FORCE ON A HEMISPHERICAL PISTONIS REPULSIVE (semiclassically)

0 0 0

DirichletNeumann

2 force

does no

attractive 1

t depend

/

on r!

a

Gies, Langfeld and Moyaerts 2003; Klingmüller 2006Scalar field satisfying Dirichlet boundary conditions on

1/2

0

1[ ] / [ ( )]

(4 )DCas d

T d d

x x

E

1 2

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:

( ) (1),0 1 B B

[ ] 0,1

1 2

(0) B 0

.

On a bounded 3-dim. domain , the trace of the heat kernel

3//2

2/2( ) [ ( , ) ]

(2 )Tr n

n

de e

xxPD DD

D

D

2( 3)/2 /

0

0 1

( ~ 0) ~ (2 ) ( )

, / 4, etc...("high temperature" expansion)

nn

n

a e

a a

D

D D

O

V S

3/20

12 ( ) diverges.

(2 )~n

n

d

DDE

is given by the probability that a standard Brownian bridge

( , ) from to in time is entirely within . x x xP

D

BUT….

Kac ‘66, Stroock’93

the first 5 terms of the asymptotic expansion of ( ) D

int. 3/20

( ) ( ) with ( ) 0 for 0, ,4,

t ( ~ 0) ( )

( ) is fi

hat is 0=

nite(2

...

)k

k

k k

k k i kk k

k

k

k kk

k

c c a i

c c

dc

D

D

D

DV S O

E E

D

int. Cas

4

Note: = is a of zero-point energies;

logarithmic divergences cancel if

differenc

0

e

f ( ) !k kk

c a D

E E

r r

Flat Casimir piston for R>>r>>aHemispherical Casimir piston for R=r

( )aa

F E

int. ) ) ) )( )a E E E E E

R

r

+_ _r

R L

α) β) γ) δ)

a

Determining the support

of a unit loop requires solution of a

non-linear optimization problem -- not easily solved for loops of 104-106 points.

intE

1[ ] ( , ); [ ( )] 1 x x S

The 5 convex domains

for a>0 only loops of finite length contribute to , these

pierce piston AND cap, but NOT cylinder

• Ordering information of a loop is redundant for Casimir energies• Convex Hull of its point set determines whether a loop pierces

a convex boundary

aHull Trendlin

e

200

20

2

0.2

CPU(sec)

Hull of 106 point loop220 Hull vertices in155 CPU sec

Hull of 103 point loop55 Hull vertices in0.3 CPU sec

r=R

1----- ( ) ~

96 150.3 2

c ca

a a

E

= periodic orbits for hemispherical piston (a=0)

2 2 21 1; ~

2 48 150.3

q q e

a c c

E E

Dirichlet scalar

2

2Note: the asymptotic (attractive) interaction energy does not depend on size

of hemispherical piston -- the resid repulsive. ual is

qa r

To avoid attraction of piston due to reflection positivity either:

a) impose metallic bound

b) o

ary

r ma

conditio

ke Diric

ns (non-separab

hlet mirror smal

l

l

e boundar

er than s

y)

hape!

0

links of dividing

plaquette are correlated

dx dx F

11 UU

to contribute, loops

or (

(+) pierce piston & cylinder, no

) pierce piston & flask, not c

t

y

fl

li r

ask

nde

0

repu

( )

lsion!

( ) 0

Cas

Cas

L

L R a r

R r a

r

r rE

E

? ?

?

Repulsion!

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

2

2 2

2

( ) with

2

1 1(0) 0 and ~

48 150

ar

EM

q c fa a r

qf

c

F E

r

R rr?