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

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 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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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