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On the vacuum energy between a sphere and a plane at finite temperature

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I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature

On the vacuum energy between a sphere and a plane at finite temperature

Based on the papers:

M. Bordag, I. Pirozhenko,

Phys. Rev. D81:085023, 2010; Phys.Rev.D82:125016,2010;

arXiv:1007.2741 [quant-ph] ,

I. G. Pirozhenko (BLTP, JINR, Dubna, Russia)

QFEXT11, 18-25 September 2011, Benasque

I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature

At zero temperature

Emig et al, Wirzba, Bulgac et al, Bordag, Canaguier-Durand et al …

This configuration at finite temperature was studied by

Alexej Weber, Holger Gies,

Phys.Rev.D82:125019,2010; Int.J.Mod.Phys.A25:2279-2292,2010

Antoine Canaguier-Durand, Paulo A. Maia Neto, Astrid Lambrecht, Serge Reynaud

QFEXT09 Proceedings; Phys.Rev.Lett.104:040403,2010; arXiv:1005.4294 ; arXiv:1006.2959 ; arXiv:1101.5258

QFEXT11, 18-25 September 2011, Benasque

I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature

Basic formulas

The free energy

where

are the Matsubara frequencies,

turns into the vacuum energy

when

depends on the boundary conditions on the sphere

For scalar field

QFEXT11, 18-25 September 2011, Benasque

I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature

For the electromagnetic field one has to account for polarizations:

with the factors

The general formulae for the dielectric ball

T.Emig, J.Stat. Mech, 2008

QFEXT11, 18-25 September 2011, Benasque

I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature

In the limit of perfect conductor, and fixed

In the limit of perfect magnetic, and fixed

Thus the trace of the “polarization” matrix P in the case of a ball with

has the opposite sign

In this case we expect the strongest repulsion.

QFEXT11, 18-25 September 2011, Benasque

I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature

PFA at finite temperature

Temperature scale

Low temperature:

Medium temperature:

High temperature:

In each case holds,

The free energy per unit area for two parallel plates

is the momentum parallel to the plates

QFEXT11, 18-25 September 2011, Benasque

I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature

The free energy may be represented in the form

The function has

several representations:

It obeys the inversion symmetry

And possesses the asymptotic expansions

QFEXT11, 18-25 September 2011, Benasque

I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature

We apply the idea of the PFA

to the free energy per unit area of two parallel plates at finite temperature

where

is the separation between the plane and the sphere at the point

In polar coordinates with

R

d

The corresponding approximation for the force

(in the limit )

QFEXT11, 18-25 September 2011, Benasque

I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature

Substituting the free energy for parallel plates we obtain for the free energy

This expression is meaningful if

Low and medium temperature limits

Low temperature,

Medium temperature,

High temperature,

QFEXT11, 18-25 September 2011, Benasque

I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature

Free energy at high temperature

Theleading order of high temperature expansion is given by the lowest Matsubara frequency, i.e.

the term with

collects contributions from

(exponentially suppressed at high temperature, )

For different boundary conditions

With these expressions for any finite the function can be calculated

numerically. A. Canague-Durand et al, Phys. Rev. Lett.104,040403 (2010)

QFEXT11, 18-25 September 2011, Benasque

I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature

Large separations,

only lowest momenta contribute

In agreement with A.Canague-Durand et al, PRL104,040403 (2010)

Short separations,

In the limit the convergence of the orbital momentum sum gets lost.

One has to find an asymptotic expansion of

By expanding the logarithm and substituting the orbital momentum sums by integrals one obtains

Bordag, Nikolaev, JPA41,2008,

PRD 2010

Coincides with the PFA result

QFEXT11, 18-25 September 2011, Benasque

I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature

Free energy at low temperature

follows from Abel-Plana formula

the low temperature expansion emerges from

Thanks to the Boltzman factor

Then,

QFEXT11, 18-25 September 2011, Benasque

I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature

Inserting this expansion into the free energy one gets

(here the limits and

were interchanged)

and the low temperature correction to the force

The first term in this expansion may vanish, depending on the boundary conditions.

To compare this result with those obtained by A.Weber and H.Gies (Int.JMPA,2010) one

should expandit for small separation

A. Scalar field, Dirichlet-Dirichlet bc

does not depend on the truncation

The term does not contribute to the force

QFEXT11, 18-25 September 2011, Benasque

I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature

At large separations

At short separations

Weber and Gies have

B.Dirichlet (sphere)-Neumann bc

The leading contribution to the force is

C. Neumann (sphere)-Dirichlet bc, N-N bc

The expansion starts from

At large separations

QFEXT11, 18-25 September 2011, Benasque

I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature

C. Electromagnetic field

From the structure of the expansion it follows that

For the functions defining the low temperature expansion we have

QFEXT11, 18-25 September 2011, Benasque

I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature

C1. Conductor bc

Short distances

Contributions growing with l

Might be interpreted as non-commutativity of

the limits

At large separations

At short separations one can expect

contributions decreasing slower than

The low temp correction to the free eneregy

QFEXT11, 18-25 September 2011, Benasque

I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature

C2. Results for dielectric ball in front of conducting plane

Large separations

Fixed permittivity

Dilute approximation

Fixed permeability

Plasma model

QFEXT11, 18-25 September 2011, Benasque

I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature

Conclusions

We developed the PFA for a sphere in front of a plane at finite temperature which

is valid for a the free energy which behaves like

Using the exact scattering formula for the free energy of we considered high

and low temperature corrections to the free energy and the force for scalar and

electromagnetic fields and found analytic results in some limiting cases.

At low temperature, the corrections have general form

The coefficient is present in DD and DN cases, and absent in all other cases.

QFEXT11, 18-25 September 2011, Benasque