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Lecture on the Theory of Casimir Phenomenae QED & Quantum Vaccum, Low Energy Frontier, 02004 (2012) DOI: 10.1051/iesc/2012qed02004 © Owned by the authors, published by EDP Sciences, 2012 Lecture on the Theory of Casimir phenomenae Bart van Tiggelen Laboratoire de Physique et Modélisation des Milieux Condensés - Grenoble - France This is an Open Access article distributed under the terms of the Creative Commons Attribution License 2.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Article published online by EDP Sciences and available at http://www.iesc-proceedings.org or http://dx.doi.org/10.1051/iesc/2012qed02004 LLeeccttuurree oonn tthhee TThheeoorryy ooff CCaassiimmiirr pphheennoommeennaaee QQEEDD CCNNRRSS SScchhooooll 22001122,, IIEESSCC CCaarrggèèssee ,, AApprriill 22001122 BBaarrtt vvaann TTiiggggeelleenn The institute started in the 60’s as anindependent non profit association in « villa Menasina » under the direction of Maurice Levy, an eminent theoretical physicist. At that time two or three schools of theoretical physics were held in the venue during summer. In 1975 the association was granted by governmental institutions with new facilities allowing a progressive increase in the activity of the institute. During the last 10 years, the IESC opened up for emerging disciplines such as biophysics of membranes, environmental sciences, social, and economic sciences and humanities. Since 1996 the institute is affiliated to Centre Nationale de la Recherche Scientifique , University of Corsica and University of Nice Sophia Antipolis. The IESC is open from February to November. The typical format is one week for a workshop and two weeks for a School. Applications are welcome. See IESC website Summary of course 1. Important events 2. Lorentz invariance of Casimir energy - Accelerated observer &Unruh effect - the UV catastrophe (Einstein equation, sonoluminescence, Casimir momentum) 3. Fluctuation-dissipation theorem 4. Casimir force between ideal plates - ..and on a metallic shell - Proximity force approximation, dispersion, finite T) 4. Casimir –Polder attraction 5. Lifshitz formula 6. Quantum friction 7. Connection with QED: - Casimir mass and Casimir momentum of H. 8. Bibliography 1.11.1 CasimirCasimir energyenergy 1. Black body radiation (Planck, 1912 1, Einstein & Stern, 1913 2 ) hQ 1 § 1 · 1 hQ ¨kT hQ ¸ hQ kT O(1/T) exp(hQ / kT) 1 2 © 2 ¹ 2 2. Isotropic radiation with power spectrum ˶3 is Lorentz-invariant (Einstein, 1917 3); 3. Van der Waals force 1/r6 as a dispersion force due to quantum fluctuations (London, 1930 4) 3. Relation to Cosmological constant (Pauli, 1934, Davies, 1984 5) 4. Casimir Polder Force 1/r 7 (1948 6) 5. Lamb shift (Lamb & Retherford 7, 1947, Bethe, 8 1947); anomalous magnetic moment of electron (Schwinger 191948) 7. Attraction between metallic plates (Casimir, 1948 10), refuted by Pauli as « absolute nonsense » 8. Lifshitz formula for dielectric bodies (Lifshitz, 1956 11) 9. “The general theory of Van der Waals forces” (Lifshitz, Dzyalonishiniskii, Pitaevskii 1961 12) 1.21.2 CasimirCasimir energyenergy 10. Observation of Casimir effect (Sparnaay 13 (100 %), 1958, Lamoureux 14(5%), 1997), Mohideen & Roy 15, 1998, Ederth 16 (1%), 2000) (the third for plane sphere-on cantilever geometry, the latter for crossed cilinders) 11. Stability of the electron (Casimir, 1956 17 , Boyer, 1968 18 ) 12. Unruh effect & Hawking radiation (Hawking 1974 19, Unruh 1976 20) 13. Dynamical Casimir effect (« moving mirror radiation »), Fulling and Davies, 1976 21 14. Bag model for hadrons (Jaffe etal, 1974 22) 15. Cosmological constant problem field theory(Weinberg, 1989 23) 16. Confined Casimir energy has inertial mass (Jeakel &Reynaud, 1993 24) 1.31.3 CasimirCasimir energyenergy 17. Sonoluminescence as dynamical Casimir effect (Schwinger, 1993225, Eberlein, 1996 26) 18. Quantum friction and shearing the quantum vacuum (Levitov, 1989 27; Pendry, 199728) 19. Casimir dies at age of 90 (May 4, 2000) 20. Casimir momentum in magneto-electric media (Feigel, 2004 29) 21. An attractive Casimir force theorem for dielectrics (Kenneth, Klich, 2006 30) 22. Repulsive Van der Waals force in colloids (Feiler etal, 2008 31) ; quantum Casimir levitation of silicon sphere (Capasso etal 2009 32) 23. Casimir energy has gravitational mass (Milton, Fulling etal, 200733) 24. Scattering theory for Casimir energy: finite temperatures ,beyond Proximity Force Approximation, corrugated surfaces (Lambrecht & Reynaud , Dalvit etal, Bordag etal 34-37). 25. Observation of dynamical Casimir effect with SQUID (Wilson etal 2011)38 26. Observation of thermal Casimir force between plates, favoring Drude model (Sushkov, Dalvit, Lamoreaux, 2011) 39 1.41.4 CasimirCasimir energyenergy beforebefore CasimirCasimir “At this point it should be noted that it is more consistent here, in contrast to the material oscillator, not to introduce a zero-point energy of 1/2¯h˶ per degree of freedom. For, on the one hand, the latter would give rise to an infinitely large energy per unit volume due to the infinite number of degrees of freedom, on the other hand, it would be principally unobservable since nor can it be emitted, absorbed or scattered and hence, cannot be contained within walls and, as is evident from experience, neither does it produce any gravitational field.” Pauli, Die Allgemeinen Prinzipien der Wellenmechanik, in Handbuch der Physik 24 1 (1933) 2.1 Casimir energy is Lorentz invariant (to be continued) 1 2 1 2 dI(T,Z) U(Z,T)dZd cosT d( 2 E 2 B ) v ȕ ckˆ scosI xˆ ssinI yˆ c0 § J 2 · E' J ¨E ȕuB ȕȕE¸ Z' JZ(1 Ec) © J 1 ¹ c E § 2 · c' k J v B' J ¨B ȕuE ȕȕB¸ 1 Ec © J 1 ¹ d(Z',c') 1 E B 2 2 d(Z,c) J (1 Ec) dI' J (1Ec) dI § Z' · 3 3 U'(Z',c') U¨ ,c¸J (1 Ec) © J (1 Ec) ¹ U(Z,c) Z3 Lorentz invariant (Einstein, 1917 3) 33..11 TThhee CCaassiimmiirr eeffffeecctt…….. é˶ L c A Remove modes E(L)= f ! 0 L3 wE c A F(L)= = 3 ! 0 wL L4 Negative pressure No momentum exchange between matter and radiation 33..22 TThhee CCaassiimmiirr eeffffeecctt…….. Pin T Pout L 1 2 F 't 2L / cosT c0 2u 2 !k ucosT c0 cosT !c0 kz Pout T u u2 Pout ¦ A A 2L V modes k kz !0 f 2 2 !c0 d k nS / L P Pin Pout ¦ ³ 2 2 2 L n 1 (2S) k nS / L f 2 2 !c0 d k kz dkz S ³ ³ (2S)2 2 2 0 k kz 33..33 TThhee CCaassiimmiirr eeffffeecctt…….. f f 2 1 2 dy F(n) n dx n ³0 2 ³n2 x n y 3 ª f f º !c0 §S · P ¨ ¸ «¦ F(n) ³ dnF(n)» 4S L © L ¹ ¬ n 1 0 ¼ S 2 c ! 0 >@ 1 F(0) 1 F'(0) 1 F'''(0) ... 4L4 2 12 720 0 0 -12 0 2 S !c0 130 nN/ cm2 / L4(μm) 240 L4 Casimir energy diverges in UV but… THE Casimir effect is a low energy phenomenon 33..44 TThhee CCaassiimmiirr eeffffeecctt ooff ssccaallaarr bboossoonnss iinn 11DD c f 1 nS c f F F F ! 0 2u ! 0 dk k in out ¦ ³ 2L n 1 2 L 2S 0 c S § f f · ! 0 lim¨ nexp(Nn) dnnexp(Nn)¸ 2 Np0 ¦ ³0 2L © n 1 ¹ S c ! 0 24 L2 Hendrik Casimir, Proc. Koninklijke Nederlandse Academie voor Wetenschappen 51 (1948), 79 10 (thank you Astrid Lambrecht for providing) 33..55 TThhee CCaassiimmiirr eeffffeecctt aatt ffiinniittee tteemmppeerraattuurreess…….. T T L 4 S 2 c § 16 ªkTLº · ! 0 ¨ ¸ 40 P 4 1 « » ... K.A. Milton, ch. 2 240 L ¨ 3 c ¸ © ¬ ! 0 ¼ ¹ L 10Pm,T 300K : correction 19 kTL kT !!1: P 2.4 3 !c0 4SL 33..66TThhee CCaassiimmiirr eeffffeecctt ffoorr aa sshheellll ((CCaassiimmiirr 11995566 17 )) Does Casimir force stabilize the electron ? c A c F ... ! 0 D ! 0 Casimir r 4 2r 2 ? e e é˶ ee e2 FCoulomb 2 8SH0re e2 But force is repulsive! D 0.0073.. 4SH0!c0 Boyer (1968) 18 !c0 FCasimir 0.094 2 2re 4.1 Fluctuation dissipation theorem at finite temperature 1 U exp(EH) H Z (a* a 1 ) * ¦ ! k kg kg 2 Tr U akgak'g' fkGkk'G gg' Z k 1 f * k Tr U a a ( f 1)G G exp(E!Zk ) 1 k'g' kg k kk' gg' vacuum 1/2 § Z · ¨ ! k ¸ * * E(r,t) ¦ i¨ ¸ ak'g'Ekg (r)exp(iZkt) akgEkg (r)exp(iZkt) kg © 2H0 ¹ Positive frequencies negative frequencies f E(Z,r) dt E(t,r)exp(iZt) E(Z)* E(Z) ³f 4.2Fluctuation dissipation theorem at finite temperature E (r)E* (r') Classical retarded/advanced Gr (Z,r,r')) kg kg ¦ 2 2 Green function kg Z ri0 Z k for electric field 2 !Z Z ! 0: En (r,Z)Em (r',Z') 2SGZZ' i>@Gnm(Z,r,r') Gnm(Z,r,r') ( fZ 1) 2SH0 2 !Z En (r,Z)Em (r',Z') 2SGZZ' i>@Gnm(Z,r,r') Gnm (Z,r,r') fZ 2SH0 No negative frequencies at T=0 2 !Z En (r,Z)Em (r',Z') En (r,Z)Em (r',Z') 2SGZZ' ImGnm(Z,r,r') 2 fZ 1 SH0 G (Z) G (Z) ; E (r) kg cotanh(E!Z / 2) 4.3 Fluctuation dissipation theorem at finite temperature What if medium is dissipative (finite conductivity)? ª 4SiV(r)º 2 «H0 »Z Ǽ(r,Z) uuǼ(r,Z) j(r,Z) ¬ Z ¼ dissipation Current fluctuations V ! 0 * ji (r,Z) j j (r',Z') K u4SZV(r)u2SGZZ'GijGrr' E(r,Z) ³dx G (r,x,Z) j(r,Z) E (r,Z)E* (r',Z') dx dx'G (r,x,Z) j (r,Z) j (r',Z') G (x,r,Z) n m ³³ ni i j jm 2SGZZ'K r G (Z)4SZV(x)G (Z) r' K 2SG r G (Z)G (Z) r' ZZ' 2i 2 2 !Z * !Z K cotanh(E!Z / 2) En (r,Z)Em (r',Z') 2SGZZ' ImGnm(Z,r,r') 2 fZ 1 SH0 SH0 Positive and negative frequencies added 4.4Fluctuation dissipation theorem at finite temperature What if medium is dispersive and dissipative ? ª 4SiV(r)º 2 «H(r) »Z Ǽ(r,Z) uuǼ(r,Z) j(r,Z) ¬ Z ¼ dispersion dissipation current fluctuations * V ! 0 ji (r,Z) j j (r',Z') K u4SZV(r)u2SGZZ'GijGrr' Substitute ȥ(r,Z) H(r)E(r,Z) and show that Z2 E (r,Z)E* (r',Z') 2SG ! ImG (Z,r,r') 2 f 1 n m ZZ' S H(r)H(r') nm Z Dangerous to quantize macroscopic media! 2.
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