1. 1 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 I. G. Pirozhenko (BLTP, JINR, Dubna, Russia) QFEXT11, 18-25 September 2011, Benasque Based on the papers: M. Bordag, I. Pirozhenko, Phys. Rev. D81:085023, 2010; Phys.Rev.D82:125016,2010; arXiv:1007.2741 [quant-ph],

2. 2 I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature QFEXT11, 18-25 September 2011, Benasque 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 At zero temperature Emig et al, Wirzba, Bulgac et al, Bordag, Canaguier- Durand et al …

3. 3 I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature Basic formulas QFEXT11, 18-25 September 2011, Benasque depends on the boundary conditions on the sphere For scalar field The free energy turns into the vacuum energywhen are the Matsubara frequencies,where

4. 4 I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature QFEXT11, 18-25 September 2011, Benasque 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

5. 5 In the limit of perfect magnetic, and fixed I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature QFEXT11, 18-25 September 2011, Benasque In the limit of perfect conductor, and fixed Thus the trace of the “polarization” matrix P i n the case of a ball with has the opposite sign In this case we expect the strongest repulsion.

6. 6 I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature QFEXT11, 18-25 September 2011, Benasque 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

7. 7 I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature QFEXT11, 18-25 September 2011, Benasque The free energy may be represented in the form The function has several representations: It obeys the inversion symmetry And possesses the asymptotic expansions

8. 8 I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature QFEXT11, 18-25 September 2011, Benasque 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 d R The corresponding approximation for the force ( in the limit )

9. 9 I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature QFEXT11, 18-25 September 2011, Benasque 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,

10. 10 (exponentially suppressed at high temperature, ) I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature QFEXT11, 18-25 September 2011, Benasque Free energy at high temperature The leading order of high temperature expansion is given by the lowest Matsubara frequency, i.e. the term with collects contributions from 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)

11. 11 In the limit the convergence of the orbital momentum sum gets lost. One has to find an asymptotic expansion of I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature QFEXT11, 18-25 September 2011, Benasque Large separations,only lowest momenta contribute In agreement with A.Canague-Durand et al, PRL104,040403 (2010) Short separations, 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

12. 12 I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature QFEXT11, 18-25 September 2011, Benasque Free energy at low temperature follows from Abel-Plana formula Thanks to the Boltzman factor the low temperature expansion emerges from Then,

13. 13 I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature QFEXT11, 18-25 September 2011, Benasque 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 expand it for small separation A. Scalar field, Dirichlet-Dirichlet bc does not depend on the truncation The term does not contribute to the force

14. 14 I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature QFEXT11, 18-25 September 2011, Benasque At large separations At short separations Weber and Gies have B. Dirichlet (sphere)-Neumann bc The leading contribution to the force is The expansion starts from C. Neumann (sphere)-Dirichlet bc, N-N bc At large separations

15. 15 I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature QFEXT11, 18-25 September 2011, Benasque C. Electromagnetic field From the structure of the expansion it follows that For the functions defining the low temperature expansion we have

16. 16 I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature QFEXT11, 18-25 September 2011, Benasque C1. Conductor bc Contributions growing with l Short distances Might be interpreted as non-commutativity of the limits At short separations one can expect contributions decreasing slower than At large separations The low temp correction to the free eneregy

17. 17 I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature QFEXT11, 18-25 September 2011, Benasque C2. Results for dielectric ball in front of conducting plane Fixed permittivity Dilute approximation Fixed permeability Plasma model Large separations

18. 18 I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature QFEXT11, 18-25 September 2011, Benasque 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.