The Optical Approach to the Casimir Effect

1
The Optical Approach to the Casimir Effect
QFTEX05 Barcelona, September 2005

• A.Scardicchio
• Center for Theoretical Physics
• MIT

Joint work with R.L. Jaffe M. Kardar O.
Schroeder M.P. Hertzberg
2
References

• M.Schaden and L.Spruch Phys.Rev.A 58, 935
  (1998), Phys.Rev.Lett. 84, 459 (2000) basing on
  M.Gutzwillers work. First ones to seriously
  address the Casimir problem for non-trivial
  geometries
• N.Graham, R.L.Jaffe, V.Khemani, M.Quandt,
  M.Scandurra, and H.Weigel, Nucl.Phys.B645, 49
  (2002). Background QFT formulation.
• H. Gies, K. Langfeld, and L. Moyaerts, J. High
  Energy Phys. 06 (2003) 018. Numerical method for
  arbitrary geometries.
• R. Golestanian, M. Kardar, T.Emig,
  Numerical/Analytical method for periodic
  geometries.
• R.L.Jaffe and AS, Phys.Rev.Lett 070402 (2004)
  and Nucl.Phys.B704, 552 (2005)
• O.Schroeder, AS and R.L.Jaffe, Phys.Rev. A 72,
  012105 (2005)
• M.P.Hertzberg, R.L.Jaffe, M.Kardar and AS, to
  appear

3
Outline

• Part I
• A new approach to arbitrary geometries
• Local observables and thermal corrections
• Part II (maybe)
• Casimir Buoyancy (force on a single plate)
• Codimension gt1 (forces between points)

4
Outline

• Part I
• A new approach to arbitrary geometries
• Local observables and thermal corrections
• Part II
• Casimir Buoyancy (force on a single plate)
• Codimension gt1 (forces between points)

5
The Optical Approach to Casimir Effect
R.L.Jaffe and AS, Phys.Rev.Lett.070402(2004)
Facts about the Casimir Effect

• Finite force between neutral, metallic, rigid
  bodies
• No exact solution for arbitrary geometries!
• Repulsion?
• Future experimental importance of geometry

q
a
6
The Optical Approach to Casimir Effect
R.L.Jaffe and AS, Phys.Rev.Lett.070402(2004)

• (Some) Challenges in Casimir physics
• Devise an approximation able to treat with
  arbitrary geometries (no spectrum)
• Interpretation and disposal of divergences
• Higher Spin Fields, different BCs
• Thermal corrections
• Finite conductivity

7
The Optical Approach to Casimir Effect
R.L.Jaffe and AS, Phys.Rev.Lett.070402(2004)

• A massless scalar field f with Dirichlet b.c. on
  the boundary of a domain D

, on the boundary
8
The Optical Approach to Casimir Effect
R.L.Jaffe and AS, Phys.Rev.Lett.070402(2004)

• Start with
• We do not know the w s so write it as
• where and r is the density of
  states of

9
The Optical Approach to Casimir Effect
R.L.Jaffe and AS, Phys.Rev.Lett.070402(2004)

• But

is Schroedinger equation propagator
and
Find a good approximation for G an
approximation for e !
Balian and Bloch
Since they are the only ones that 1) depend on
the relative distances between the bodies and 2)
give a finite contribution to e
10
The Optical Approach to Casimir Effect
R.L.Jaffe and AS, Phys.Rev.Lett.070402(2004
)

• Optical approximation
• Saturate G with contributions from classical
  paths
• The expression for or massive fields
  involves Hankel functions

r0 is the direct path, r1 reflects once etc.
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The Optical Approach to Casimir Effect
R.L.Jaffe and AS, Phys.Rev.Lett.070402(2004)
Length of the path The Van-Vleck determinant or
Enlargement factor Flat surfaces or
vacuum Convex or Concave surfaces
12
The Optical Approach to Casimir Effect
R.L.Jaffe and AS, Phys.Rev.Lett.070402(2004)

• By inserting G in r we get the Optical
  Approximation to the Casimir energy

Sum over closed, non periodic paths Exclude the
r0 path, equivalent to consider
13
The Optical Approach to Casimir Effect
R.L.Jaffe and AS, Phys.Rev.Lett.070402(2004)
Some classical paths
14
The Optical Approach to Casimir Effect
R.L.Jaffe and AS, Phys.Rev.Lett.070402(2004)

• Where are the other divergences?

Look for paths with
15
The Optical Approach to Casimir Effect
R.L.Jaffe and AS, Phys.Rev.Lett.070402(2004)
Only 1 reflection paths!
a-independent
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The Optical Approach to Casimir Effect
R.L.Jaffe and AS, Phys.Rev.Lett.070402(2004)

• The constant is divergent

But a-independent
cutoff
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The Optical Approach to Casimir Effect
R.L.Jaffe and AS, Phys.Rev.Lett.070402(2004)
Convergence Rate

• Parallel plates

a-independent
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The Optical Approach to Casimir Effect
R.L.Jaffe and AS, Phys.Rev.Lett.070402(2004)
Convergence Rate
FAST!
92
1.7
6
0.3
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The Optical Approach to Casimir Effect
R.L.Jaffe and AS, Phys.Rev.Lett.070402(2004)
Accuracy
The approximation is good as far as a/R, a/Lltlt1.
In the limiting case the action is quadratic in
x(t) and the Gaussian integration is exact
L
a
We are ignoring diffraction (both edge and
curvature)
However the question remains is this the correct
asymptotic expansion in series of a/R ?
20
The Optical Approach to Casimir Effect
R.L.Jaffe and AS, Phys.Rev.Lett.070402(2004)
Accuracy
Gies et al.
But still it could be that
Different
At the moment the sphere-plane example says that
So either the linear coefficient agree or they
are both very small
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The Optical Approach to Casimir Effect
R.L.Jaffe and AS, Phys.Rev.Lett.070402(2004)
Why is it going up? A. Wirzba and H. Gies talks
Sphere facing plane
Gies et al. Numerical
Optical
Full PFA
diffraction
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The Optical Approach to Casimir Effect
O.Schroeder, AS and R.L.Jaffe PRA 72, 012105
(2005)
Hyperboloid facing plane
Parallel plates regime
Optical
PFA
23
The Optical Approach to Casimir Effect
M.P.Hertzberg, R.L.Jaffe, M.Kardar, AS---TBP
Casimir Piston
Second shortest paths
Rectangular Pistons Optical approximation is
exact
Find exact expression
Moving piston
Shortest paths
h1,-1 for Neumann, Dirichlet
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The Optical Approach to Casimir Effect
M.P.Hertzberg, R.L.Jaffe, M.Kardar, AS---TBP
Casimir Piston
Asymptotic Expansion for Arbitrary Sections
Using BalianBloch

• This is an exact asymptotic

• No repulsive region for the rectangular piston

25
Outline

• Part I
• A new approach to arbitrary geometries
• Local observables and thermal corrections
• Part II
• Casimir Buoyancy (force on a single plate)
• Codimension gt1 (forces between points)

26
The Optical Approach to Casimir Effect AS
and R.L.Jaffe, quant-ph/0507042 (2005)

• Energy-momentum tensor

Pressure
Apply the Optical Approximation
27
The Optical Approach to Casimir Effect AS
and R.L.Jaffe, quant-ph/0507042 (2005)
r
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The Optical Approach to Casimir Effect AS
and R.L.Jaffe, quant-ph/0507042 (2005)
Temperature dependence

• Optical approximation to the Free energy

Pressure
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The Optical Approach to Casimir Effect
AS and R.L.Jaffe,
quant-ph/0507042 (2005)

• The Classical limit

Linear dependence in T
Previously known for parallel plates
Einstein
30
The Optical Approach to Casimir Effect AS
and R.L.Jaffe, quant-ph/0507042 (2005)
Sphere-plane
Pressure vs radial distance (R,a fixed)
Beware Geometry and temperature interplay to
give rise to different asymptotics of the
pressure (hence of the Force).
The factorization
May be too naïve!
31
Summary of Part I

• Some techniques for approximating the propagator
  are good for the Casimir energy too
• We get a good intuition of the solution with
  little effort
• Non-trivial interplay Geometry/Temperature and
  Geometry/Finite conductivity
• Is the optical approximation an asymptotic
  expansion
• in a/R ?
• Higher spin fields (EM field) can be treated for
  separable geometries. Can we modify this
  approximation to handle non separable geometries?

Waiting for experiments to test the importance of
the geometry
32
Outline

• Part I
• A new approach to arbitrary geometries
• Local observables and thermal corrections
• Part II
• Casimir Buoyancy (force on a single plate)
• Codimension gt1 (forces between points)

33
Casimir buoyancy R.L.Jaffe and AS, JHEP
0506006 (2005)
Casimir Buoyancy Force on single plate
V(x)
x
a
Solve for the propagator
Propagator in the background V
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Casimir buoyancy R.L.Jaffe and AS, JHEP
0506006 (2005)
Density of states
Casimir Force on the Plate
Why did we call it Buoyancy?
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Casimir buoyancy R.L.Jaffe and AS, JHEP
0506006 (2005)
WKB approximation for G0
WKB Casimir Force on the Plate
V(x)
x
a
It floats!
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Casimir buoyancy R.L.Jaffe and AS, JHEP
0506006 (2005)
Temperature effects
WKB Casimir Force on the Plate
It still floats!
We checked many cases in which G0 can be found
(non WKB) When V(x) is sufficiently smooth the
plate floats
37
Outline

• Part I
• A new approach to arbitrary geometries
• Local observables and thermal corrections
• Part II
• Casimir Buoyancy (force on a single plate)
• Codimension gt1 (forces between points)

38
Casimir effect in cod gt1 AS,
hep-th/0503170 PRD to appear
Casimir effect in Codimension gt1
Usual Casimir interaction is in codimension 1
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Casimir effect in cod gt1 AS,
hep-th/0503170 PRD to appear
Codgt1 Divergences
Redefine to absorb divergences
Divergent in d gt1!
Already known to Bethe, Fermi
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Casimir effect in cod gt1 AS,
hep-th/0503170 PRD to appear
Arbitrary
Finite part
It is the s-wave scatterer limit
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Casimir effect in cod gt1 AS,
hep-th/0503170 PRD to appear
d3
Im e
Instability Tachion
42
Casimir effect in cod gt1 AS,
hep-th/0503170 PRD to appear
Tachion
L lt Lc
The field condenses in this region
Veff (f)
f