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 1 f dy F(n) n2 dx n2 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. 2 Continued: Casimir energy is Lorentz invariant P 1 § 1 · wP T Q 0 w ¨İ E2 + B2 ¸ c2İ EuB 0 t 2¨ 0 ȝ ¸ 0 0 P © 0 ¹ T P 0 1 § 2 1 2 · § 1 · ½ wt İ0EuB ® ¨İ0E + B ¸įij ¨İ0Ei Ej + Bi Bj ¸ ¾ 0 2¨ ȝ ¸ ¨ ȝ ¸ ¯ © 0 ¹ © 0 ¹ ¿ TPQ TQP Z2 §Z2 · 1 G (Z,k) ¨ G k k ¸ 2 ij ¨ 2 ij i j ¸ 2 2 2 c0 © c0 ¹ (Z i0) /c0 k
dZ d 3k 1 §Z2 /c2 Zk /c · T v 2 Im ¨ 0 j 0 ¸ PQ ! ³ ³ 3 2 ¨ ¸ 2S 2S Z i0 2 ©Zki /c0 kik j ¹ 2 k c0 Lorentz covariant Lorentz invariant tensor measure Lorentz scalar 2.3 continued : Casimir energy is Lorentz invariant
2 2 f dZ Z /c doesn’t allow Wick rotation Im 0 ³0 2 40 2S Z i0 2 (Milton, chapter 10; 1997 ) 2 k ª c0 º « 2 2 » 2 2 f dZ Z /c f dZ k f k Im « 0 1» Im d] ³0 2S « Z i0 2 » ³0 2S Z i0 2 ³0 ] 2 2 2 k 2 « 2 k » 2 k 2 ¬ c0 ¼ c0 c0 1 2 f dZ kik j f k Im 3 ³0 2 ³0 2 2S Z i0 ] 2 2 k 2 k 2 c0 c0 § 1 0 0 · ¨ ¸ 1 ¨ 0 3 0 0 ¸ P Z o T E T P 0 PQ casi¨ 0 0 1 0 ¸ z Kg ¨ 3 ¸ PQ ¨ 1 ¸ © 0 0 0 3 ¹ Not cosmological constant but ultrarelativistic dust E=3P (L & L, Theory of Fields) Divergence of Lorentz invariant theory is not Lorentz invariant? 2.4 continued Casimir energy is Lorentz invariant Wick rotation when regularized invariantly: Analytic function c f 1 §] 2 i]k · bounded in C Z oi]c : T ! 0 d] d 3k ¨ i ¸ f (] 2 k 2 ) 0 PQ 4 ³ ³ 2 2 ¨i]k k k ¸ 8S 0 ] k © j i j ¹ 2 2 c f 2S §sin I cos I 0 · ! 0 dss3 f (s) dI¨ 3 ³0 ³0 ¨ 1 4 2S © 0 3 Gij sin I ¹ f !c0 3 gPQ dss f (s) Ecasi gPQ 8S 2 ³0 Regularized quantum vacuum is formal candidate for cosmological constant 8SG 8SG R Rg /g T / E PQ PQ PQ 4 PQ 4 casi ? c0 c0
4 1/a /c0 5 3 !c 3 | Uc 10 GeV/cm : dss a |13μm 8SG 8S 2 ³0 Is regularization result of stabilized extra , large « enrolled » dimensions d > 4 of this size? 2.5 Non Newtonian gravity deduced from Casimir experiments
3 1 mm thick plate (10 g/cm ) Fcasi=Fz Æ L=13 μm
V D exp(r /O) 95 % excluded VNewton
95 % excluded
E. Adelberger et al (2003) 41 Kapner etal, 200742 Sushkov, Kim, Dalvit , Lamoreaux 43 2011 2.6 Accelerated Casimir energy becomes Planck law
Constant acceleration a in comoving frame with speed v (Milonni, 1994 44) v
3/2 du ' du 1 1 dv § v2 · // // a for u v a¨1 ¸ dt' dt J 2 2 dt ¨ c2 ¸ 1vu/c0 © 0 ¹
at c aW v(t) dW 1v2 /c2 dt t(W) 0 sinh 2 2 2 0 1a t /c0 a c0 2 c0 2 2 2 x(t) >@1a t /c0 1 2 a c0 § aW · x(W) ¨cosh 1¸ a © c0 ¹ 4 2 2 c0 2 aW x (W)t (W) 2 sinh a 2c0 2.7 Accelerated Casimir energy becomes Planck law A (r,Z)A* (r',Z) 2SG ! ImG (Z,r,r') 2 f 1 E wt A n m ZZ' nm Z SH0
3 2 f * ! d k §Z 2 · T 0;a z 0: An (0,t 0)Am (r(t),t) dZ G¨ k ¸Gnm exp(ikr(t) iZt) ³0 ³ 3 ¨ 2 ¸ H0 (2S) © c0 ¹ 2 5 ! Gnm !a / c0 3 2 2 2 v (2S) c r c t 2 aW 0 0 sinh 2c0
3 2 f * ! d k §Z 2 · T z 0;a 0: An (0,t 0)Am (r 0,t) dZ G¨ k ¸Gnm exp(iZt)(2 fZ 1) ³0 ³ 3 ¨ 2 ¸ H0 (2S) © c0 ¹ SkT / 2 / c3 .... ! ! 0 §SkT · sinh2¨ W ¸ © ! ¹
!a T Unruh/Hawking temperature 2Skc0 4.1 Inertial Casimir mass? v
question ?
S 2 2 M Ecasi / A M / A 720!c0 / L P/ A mv / A 2 v 2 v c0 c0 4.2 Casimir friction? v
question F/ A J v ? 44..33 vvaaccuuuumm ffoorrccee oonn mmoovviinngg oobbjjeecctt
F(t) Fulling and Davies (1976 21)
! F(t) 2 z'''(t) Z(t) 6Sc0
F=0 for uniform speed (by Lorentz invariance) or acceleration (Unruh effect) 44..44 vvaaccuuuumm ffoorrccee oonn mmoovviinngg mmiirrrroorrss:: CCaassiimmiirr mmaassss
Jaekel and Reynaud(1993 24)
Retarded reaction force from mirrror 2 via vacuum F(t) on mirror 1 z2(t) ! GF1(t) 2 z1'''(t) z2 '''(t W) z1'''(t 2W) #... 6Sc0 c S W ! 0 z '(t) z '(t W) z '(t 2W) ... 12L3 1 2 1 #
Retarded variaton in Casimir force GF1(t) F'(z2 z1)u z1'(t)uW
Keep z1-z2 constant: z’=z1’=z2’ z1(t) § i Z3 iZ c SW · GF (Z) 2¨ ! ! 0 ¸ 1exp(iZW) exp(2iZW) ... z(Z) 12 ¨ 2 3 ¸ # © 6Sc0 12L ¹ i Z § S 2 · ! ¨Z2 ¸ 6Sc2 ¨ 2W 2 ¸ 0 © ¹ 1 !S ! 2 exp 2 iZW z(Z) iZ z(Z) iZ z(Z) 1 2 cos 2 ZW 12L 24Sc0L E !S casi J m 2 Casimir mass Quantum friction (not found by ref 24)? 2 L c0 5.1 Van der Waals and Casimir Polder interaction
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
f ª1 2 1 2 º dZ dk 2 dr H(r)E (r,t) B (r,t) 2 Im k Gnm knkm Gnm(Z,k,k) 2 fZ 1 ³ « » ! ³0 ³ 3 ¬2 2P0 ¼ 2S (2S)
G (Z,k,k') G0 (Z,k)Gkk' G0 (Z,k)Tkk' Z G0 (Z,k') T-matrix of 2 point dipoles:
i(kk')r1 i(kk')r2 2 ikr1ik'r2 ikr2 ik'r1 t Z e t Z e t G0 (Z,r12)>@e e Tkk' Z 2 2 1t G0 (Z,r12)
2 2 1c0kk/Z G (Z,k) 2 2 2 (Z i0) / c0 k
D(0)Z2Z2 / c2 k’ t(Z) 0 0 k 2 2 2 Z0 Z 3 iJZ / c0 5.2 Van der Waals interaction Van Tiggelen, 1999 45) f dZ d 2 2 G E !ImTr Z ^`2logt(Z) log(1t (Z)G (Z,r12) ³0 2S dZ f f dZ dZ 2 2 !ImTr 2logt(Z) !ImTr log 1t (Z)G (Z,r12) ³0 2S ³0 2S
r12 Z0 / c0 3 GE 2u 2 !Z0 2 f ½ Ground state energy dZ ° 2 § 13rˆrˆ · ° G E ImTr ®t (Z)¨ ¸ ¾ = ! ³0 ¨ 2 3 2 ¸ 2S ° © 4SZ r / c0 ¹ ° Casimir energy ¯ ¿ 2 2 (this anticipates Lamb shift f ds §D is · 1 3 D(0) 6! ¨ ¸ 6 !Z0 2 as a « Casimir effect ») ³0 2S © 4S ¹ r 4 4S r6
Van de Waals energy = Casimir energy 5.3 Casimir Polder interaction
f dZ d 2 2 G E !ImTr Z ^`2logt(Z) log(1t (Z)G (Z,r12) ³0 2S dZ f f dZ dZ 2 2 !ImTr 2logt(Z) !ImTr log(1t (Z)G (Z,r12) ³0 2S ³0 2S
3 r !Z / c GE 2u 2 !Z0 12 0 0
Ground state energy 2 D 0 f dZ = G E ! ImTr Z4G2 (Z,r) 4 ³0 Casimir energy c0 2S 2 2 2 D 0 f ds f (isr / c ) 23 c D 0 ! s4 0 ! 0 4 ³0 2 2 7 c0 2S 4Sr 4S 4S r
Casimir Polder interaction energy 6 55..44 UUVV ccaattaassttrroopphhee iinn ssoonnoolluummiinneesscceennccee ((>> 11993344))
(a= 40 μm) Schwinger (1993) 7
3 3 1 3 1 ǻE bubble d r d k T Ȧk bubble in water d k T Ȧk water no bubble 2 2 a3 Z 4 § 1 · ! c cut-off in the UV ? | 3 ¨1 ¸ |10MeV cut-off in the UV ? c0 © H ¹ 2 46 23 c D(0) Dimensional regularisation ? GE d d!8r ! 0 ³ (4S)3 r7 23 c H 0 1 2 ! 0 1536S L 0.001eV 5.5 Casimir energy of dielectric sphere
2L L
2 3 1 23!c0D(0) GE N u 2 Z0 ! ¦ 3 7 2 r ,r L i j (4S) ri rj 2 2 2 3 1 23!c0D(0) 1 23!c0D(0) 1 23!c0D(0) N u 2 Z0 ! ¦ 3 7 ¦ 3 7 ¦ 3 7 2 r L,r 2 r L,L r 2L 2 r L,r !2L i j (4S) ri rj i j (4S) ri rj i ij (4S) ri rj c D(0)2 § L3 L2 L · 1 4 23 c D(0)2 N u 3 Z ! 0 ¨0.11 0.4 0.006 ¸ n2 SL3 d 3r ! 0 2 ! 0 2 7 ¨ 3 2 ¸ ³ 3 7 4S a © a a a ¹ 2 3 r!2L (4S) r
Contribution to 2 2 0.11 !c0D(0) 23 c0 H 0 1 4,44 q>@J / kg ! latent heat 2 7 4 3 (4S) a u 3 SUa 1536S L
3 Regularized Casimir energy D(0) 2 A a 3 A stems from missing atoms far outside Liquid helium 3 U 0.15g / cm q 14J / g (Kawka, 2010) 47 55..77 TThhee mmaaggiiccaall mmyysstteerryy wwoorrlldd ooff rreegguullaarriizzaattiioonn
d p § p d · x *¨ ¸ f d d k © 2 ¹ 1 (x2 k 2 ) p / 2 ( p ! d) ] x (Re x !1) ³ d ¦ x (2S ) d 1 § p · n (4S ) *¨ ¸ n 1 © 2 ¹
f 2 2 f 1 2 c d k (nS / L) c * 2 § nS · nS P ! 0 ! 0 ¨ ¸ casi ¦ ³ d 2 2 ¦ 1 L n 1 (2S ) (nS / L) k 4S L n 1 * 2 © L ¹ L
2 1 2 2 S c * 2 c S 2 S 1 S c ! 0 ] (3) ! 0 ! 0 4 1 4 4 4 L * 2 L 4 S 120 420 L
« Regularized » Casimir pressure d 1 § d J 1· S 2 2d J *¨ ¸ 1 2 d d x d d y © ¹ (d ! J / 2) ³x 1 ³y 1 x y J (d J )* d / 2 * d 1J / 2 23D 2 N 2 23(H 1)2 d 3x d 3y o ³x a ³y a 7 d 3 4S x y J 7 1536a
« Regularized » Casimir energy of N dipoles distributed in a sphere Brevik etal, 1998 46 55..88 LLiiffsshhiittzz ffoorrmmuullaa Two weakly polarizable bodies 1 and 2 f dZ 2 G E ! Im Tr Gt1 (Z)Gt2 (Z)G (Z,r12 ) ³0 2S 4 f dZ Z 2 Im Tr n1D1 Z dr1 u n2D 2 Z dr2 G (Z,r12 ) ! ³0 4 2S c0
f ! 4 2 Ecasi dr1 dr2 dss >@H1 is 1>@H2 is 1TrG (is,r12) 4 ³ ³ ³0 2Sc0 2 L 1
f ! 4 2 Ecasi dr1 dr2 dss >@H1 is 1>@H2 is 1TrG (is,r12) 4 ³ ³ ³0 2Sc0 2 f § ˆˆ · ! 2 2 4 ¨ 1-3rr ¸ Ad d x dss >@H1 is 1>@H2 is 1Tr 2Sc4 ³ ³0 ¨ 2 2 3/2 2 ¸ 0 © 4S x L s / c0 ¹ 2 3! Ad f ds>@H1 is 1>@H2 is 1 2(4S)2 L4 ³0 55..99 RRiiggoorroouuss LLiiffsshhiittzz ffoorrmmuullaa
! if G E ImTr d(i] )log>@1T1(i] )G12(i] )T2 (i] )G21(i] ) 2S ³0
! f Tr d] log>@1T1(i] )G12(i] )T2 (i] )G21(i] ) 2S ³0
! f d] logdet>@1T1(i] )G12(i] )T2 (i] )G21(i] ) 2S ³0
G(12) Roundtrip operator Real-valued for imaginary T1 T2 frequencies Controversy: G(21) What model for İ(Ȧ) ? 5.10 Thermal Casimir force: Drude or plasma model?
Z 2 2 p Z p H (Z) 1 2 H (Z) 1 Z iJZ Z 2 Drude or Plasma ?
Exp-theo: Drude wins ?
Drude plasma 300 K electrostatic 300 K
Sushkov, Kim, Dalvit, Lamoreaux, 2011 39 5.11 Proximity Force approximation
R S 3 c R § L · R !! L: GE ! 0 ¨1 ...¸ 720 L2 © R ¹
37 L M. Bordag, V. Nikolaev, 2008
exact Scalar field with Dirichlet BC’s Gies & Klingmüller, 2006 48
L/R 5.12 Casimir force sphere-plane at ambient temperature
Proximity force approximation F(T) F(T 0)
Repulsive R< Ref. 36 66.. QQuuaannttuumm vvaaccuuuumm ffrriiccttiioonn:: ffrriiccttiioonn oorr ffiiccttiioonn?? 66..22 QQuuaannttuumm vvaaccuuuumm ffrriiccttiioonn:: ffrriiccttiioonn oorr ffiiccttiioonn?? A little (nonexhaustive) history: Einstein 1917: friction of moving atom in thermal field (and zero at T=O) 3 L Levitov: Van der Waals friction between moving dielectric bodies, Europhys Lett. 1989 27 J. Pendry 1997 Quantum vacuum shearing J. Phys. Cond Matt. 1997 28: disagrees with Levitov Dupays, Rizzo etal: quantum friction from vacuum induced magnetic moment by rotating neutron stars, EPL 2008 49Æ paper seems unnoticed in Casimir community Philbin etal: macroscopic G-function diagonal: no poor man’s friction à la Pendry, 2008 50 J. Pendry returns: poles of G Doppler shift in complex plane: friction = fact no friction, 2010 51 (but in fact Ref 50 claims diagonaility throughout the complex plane) Hoye and Brevik, dissipative quantum friction for T > 0 between moving oscillators, EPL 2010 52, claims agreement with Barton Volokitin and Persson (2011) 53 contest Ref 50 since no excitations. Contested by Ref. 50 54 . G. Barton, Van der Waals friction between atoms and between halfspaces (claims agreement with Pendry, contests Levitov, contests Hoye & Brevik ) 2010, 2011 55 66..33 QQuuaannttuumm vvaaccuuuumm sshheeaarriinngg :: ffrriiccttiioonn oorr ffiiccttiioonn?? z v 2 L x 1 John Pendry,28 : friction caused by evanescent low frequency modes * Txz v Ex (z,k // )Ez (z,k // ) for v z 0 2 d k f Txz ! ³ // dZ Fx sign(Z)exp(k// L)kx ImR2 (Z kxv)ImR1(Z) A 2 (2S)2 ³f 2S ! f f kxv dZ dkx dky exp(k// L)kx ImR2 (kxv Z)ImR1(Z) 2(2S)2 ³0 ³f ³0 2S No dissipation Æ F=0 66..44 QQuuaannttuumm vvaaccuuuumm sshheeaarriinngg :: ffrriiccttiioonn oorr ffiiccttiioonn?? z v 2 L x 1 John Pendry,28 : friction caused by evanescent low frequency modes 2 3 V F 5!H0 v H(Z) 1 8 2 2 6 iZH0 A 2 S V L v 1 m/s,V 0.1/ :m, L 1nm F/A 310-7 N/cm2 Large!, but…… 66..55 QQuuaannttuumm vvaaccuuuumm sshheeaarriinngg :: ffrriiccttiioonn oorr ffiiccttiioonn?? z v 2 L x 1 T. Philbin, U. Leonardt, 2009 50 2 f d k // dZ * Txy (r// , z H0Ex (z,k // )Ez (z,k // ) ³ (2S)2 ³0 2S 2 2 f d k // dZ Z 2 ImGxz z, z,Z,k // !³ 2 ³0 2 (2S) 2S c0 Green function is diagonal! No friction? Quantizing dissipative media neglects exitations ? Real microscopic theory? 77.. CCaassiimmiirr MMoommeennttuumm WWoorrkk ddoonnee iinn ccoollllaabboorraattiioonn wwiitthh GGeeeerrtt RRiikkkkeenn ((LLNNCCMMII)) JJaammeess BBaabbiinnggttoonn ((LLPPMMMMCC)) SSéébbaassttiieenn KKaawwkkaa ((LLPPMMMMCC)) SSuuppppoorrtt AANNRR PPhhoottoonniimmppuullss 77..22 «« MMoommeennttuumm ffrroomm NNootthhiinngg »» B0 ˢˢ,,˩˩,,gg PP==mmvv E0 Ȧ,k !Ȧ,k' ! Ref 29 77..33 BBii--aanniissoottrrooppiicc MMeeddiiaa D Ȧ = İ Ȧ E Ȧ +g Ȧ B Ȧ H Ȧ = gT Ȧ E(Z)+ ȝ Ȧ 1B Ȧ Fresnel dispersion law § Z 2 Z Z · det¨H k 2 kk g İ k İ k g*¸ 0 ¨ 2 ¸ © c0 c0 c0 ¹ 0 0 0 0 vl gij Ȧ iȦ gįij gij Ȧ = g Ei Bj Bi Ej gij Ȧ 1İ İijl Rotatory power c Magneto-electric birefringence Fizeau effect 0 ky vv EE0 xx BB0 -8 k 1010 10-15 1010x -2 10 77..44 pphheennoommeennoollooggiiccaall ccoonnttiinnuuuumm tthheeoorryy 0 wt ȡ v+İ0EuB = T 1 § 1 · 1 § 1 · 0 ¨ 2 2 ¸ ¨ ¸ Observed in X-ray Tij = ¨İ0E + B ¸įij ¨İ0Ei Ej + Bi Bj ¸ 8ʌ © ȝ0 ¹ 4ʌ © ȝ0 ¹ 1 1 2 Ȧ4 d 3k Z u g Z E uB = ! c g E uB ° ³ ! k 0 0 3 4 0 0 EuB °c0 2 3 ʌ c0 0 0 v ® 4Sc 1 1 v 0 ° d 3k Z u >@H Z 1 U v ° ³ ! k casi ¯ c0 2 c0 Photonic momentum in dielectric media? Æ classical « Abraham » contribution already controversial UV catastrophe of vacuum energy ? Lorentz invariance of quantum vacuum? Inertia of quantum vacuum? 77..55 CCaassiimmiirr mmoommeennttuumm nnoott eexxcclluuddeedd bbyy LLoorreennttzz iinnvvaarriiaannccee 1 Q L(E,B) (E2 B2 ) 2Q (E2 B2 )2 (EB)2 2 2 E E0 E Z Bi-anisotropic Lorentz-invariant vacuum B B0 B Z Fluctuation- * 2 Dissipation 0 Ei(r,Z)E j (r',Z') 0 2!Z ImGij(r,r',Z)u2SG(ZZ') E*uB 4 c0 0 E*uH 0 0 0 0 QK E0 uB0 4S 4Sc0 3 Zero energy flow infinite momentum density f 1 1 K 3 2 ! dZ d: U0 (Z,:) Rikken & Rizzo (2003) 56 (2S ) ³0 ³4S Van Tiggelen, Rikken, 2009 57 Infinite Lorentz scalar 77..66 …………....bbuutt ddooeess iitt ssaattiissffyy ggeenneerraall rreellaattiivviittyy?? 8SG RPQ RgPQ /gPQ 4 TPQ TPQ TQμ c0 § Ecasi EuH casi 0· ¨ EuB ¸ ¨ casi ¸ TQμ 4 ¨ 3Q Ecasi(E0 uB0 ) >@Tij ¸ ¨ casi ¸ ? © z 0 ¹ 7.77.7 TheThe AbrahamAbraham ForceForce Macroscopic Maxwell wtGM +ȉ=f GM =DuB f=E2İH2ȝ MMiinnkkoowwsskkii 7.87.8 TheThe AbrahamAbraham ForceForce (see(see BrevikBrevik 5858)) GA =İ0ȝ0EuH 1 58 = S wtGA +T=f İ0(İr 1/ ȝr )wt(EuB) Abraham c0 G =İ EuB N 0 59 wtGN +T=f İ0(İr 1)wt(EuB) Nelson =G0 2 ...=f >@w E (t)uB (t)E (t)u >w B (t)@ Peierls 60 5 t 0 0 0 t 0 Walker & Walker, exp F= İ1VwtE0(t)uB0(t) 1976 61 Abraham momentum = kinetic momentum, Barnett 62 theo Minkowski momentum = conjugate momentum (2010) Nelson momentum = pseudo momentum Nelson (1991) 59 7.97.9 TheThe AbrahamAbraham ForceForce (Nelson(Nelson version)version) Macroscopic wt DuB wt (PuB)+T=fwt (PuB) Maxwell f=E2İH2ȝ Maxwell-Lorentz force on induced polarization and current wt ȡv+U=f+wt(PuB) + Microscopic w ȡv+İ EuB +T = 0 Maxwell t 0 0 z EuH symmetric 77..1100 CCllaassssiiccaall AAbbrraahhaamm mmoommeennttuumm iinn ccrroosssseedd EEMM ffiieellddss BB0 ++ -- vv EE ((tt)) 0 1 r = R± (x) 1,2 2 mr1 =+qE(t)+qr1uB+f(r12 ) 2mR +qxuB=constant=0 mr2 =qE(t)qr2 uBf(r12 ) 2 mx=2qE(t)+2qR uBmȦ0 x|0 q2 / m 2mR = 2 E0(t)uB0 Ȧ0 No controversy exists in microscopic description Consistent with Abrahams and Nelson version q2 / m H 1 2 D(0) H0Ȧ0 n 77..1111 TThhee UUVV ccaattaassttrroopphhee iiss rreeaall iinn mmaaccrroossccooppiicc ddeessccrriippttiioonn Free electron (electric dipole)2 2 Ȧ T 3 Ȧ p P ȡ dȦȦ İ Ȧ 1 2 casi 3 0 2 Ȧ c0 Ȧ magnetic dipole gME(˶) Electric quadrupole P T dr dȦȦ3 g Ȧ E B casi 3 0 0* 0 c0 Rizzo etal, 2003, 2009 63 , Babington & BAvT, 2011, 64 !c0g İ(0)1 Pcasi = E0 uB0? a Dimensional regularization for object of size a? BAvT 2009 65 7.117.11 ObservationObservation ofof thethe AbrahamAbraham ForceForce E0=450 V/mm; B0=1 T Į 0 0.22 1040 Cm2 /V 16.6a3 Ex:Ex: HeliumHelium 0 3 ȡ0.17 kg/m room T (SI units) g0.017 1022 m/VT Classical Abraham İ Į(0)EB 32 0 Force vabr = | 0.3 nm/sec Fabr | 710 N 2mp N F v1013 N ʌ h at abr v gEB0.02 nm /sec Feigel 4 ȡȜ4 Semi-classical QED with cut-off c 0.1 nm (Feigel 29) 2 Rigorous QED (Kawka, 2010 66) vQED v vabr u Z?Į | 0.001nm/sec dp dE Į 0 *B Abraham force dt dt Acoustic Acoustic P Ȧ P0 Į 0 *E*B*Ȧ*cosȦt*n* L pressure V= 8 nm/sec+- 0.8 Feigel correction: 2 nm/sec Excluded by errorbars E=450 V/mm; B=1 T; f= 7.6 kHz įįP/(EB)P/(EB) ˞˞(0)(0) Rikken / Van Tiggelen, 2011 67 CCaassiimmiirr mmoommeennttuumm:: 1/6 QED of atom in crossed fields EE0 1 +e -e A0 B0 ur I E0 r BB0 2 Coulomb Gauge 1 2 1 2 H p1 eA0 (r1) eA(r1) p2 eA0 (r2 ) eA(r2 ) 2m1 2m2 eE0 r21 V (r12 ) Z a *a 1 ¦ ! i i i 2 i CCaassiimmiirr mmoommeennttuumm:: 2/6 QED of atom in crossed fields EE0 +e -e BB0 Conjugate momenta p1 m1v1 eA0 (r1) ำ kinetic momentum p2 m1v2 eA0 (r2 ) 1 Pseudo momentum Kˆ p p eB ur P eB ur 1 2 2 0 21 kin 0 Pseudo momentum is Coulomb Gauge conserved [K, H ] 0 Ground state changes due to coupling with quantum vacuum įMv+ ¢ȥ |eA|ȥ ² 3 0 0 0.84Į Į Ȧ0, ȝ įȝ ¢0|A|0²=0 CCaassiimmiirr mmoommeennttuumm:: 3/6 QED of hydrogen atom in crossed fields EE0 +e -e BB0 ˆ K = m1v1 + m2 v 2 + eA( r1 ) eA( r2 )+ eB 0 u r21 + ¦ !k i ai ai +12 i No multipole approximation in A(r ) v ¦gkexp(ikr) agk +c.c gk 8 E0 ¢Ȍ0 |K|Ȍ0² = Mv+į M v+ 2 v 3 c0 + İ0Į(0)B0 uE0 +İ0 įȝ w ȝĮ(0)B0 uE0 +K1 +K2 m1 m2 4 T k įM į m m įȝį įm ĮT dk 1 2 m m i 3ʌ 0 2 k 2 2m kc 1 2 T iT CCaassiimmiirr mmoommeennttuumm:: 4/6 QED of hydrogen atom in crossed fields EE0 +e -e BB0 ˆ § 1 · K = m1v1 + m2 v 2 + eA( r1 ) eA( r2 )+ eB 0 u r21 + ¦ !k i ¨ ai ai + ¸ 2 2 i © 2 ¹ m1 v1 Atomic binding energy 2 v1 + idem 2 2c0 dominated by Casimir energy 8 E0 5 E0 ¢Ȍ0 |K|Ȍ0² = Mv+į M v+ 2 v 2 v 3 c0 3 c0 +İ0 Į(0)B0 uE0 +į ȝ İ0w ȝ Į(0)B0 uE0 +K1 +K2 +KR m1 m2 4 T k įM į m m įȝį įm ĮT dk 1 2 m m i 3ʌ 0 2 k 2 2m kc 1 2 T iT CCaassiimmiirr mmoommeennttuumm:: 5/6 QED of hydrogen in crossed fields EE 0 Quantum vacuum contribution: +e -e BB0 1 e2 2 e2 1 ! 0 ˆ |n n| |0 K1 = B0 uE0 2 2 ¦ ¢ | r ² 2 ¢ r ² 3 a0c0 μ n 4ʌİ0r En E0 2 2 = İ0Į(0 )B0 uE0Į 0.208+0.0045 = 0.21Į K A 1 e2 1 K = +İ Į(0)B uE Į2 ¢0|rˆ |n² ¢n|r|0² 2 0 0 0 2 ¦ 27 4ʌİ0a0 n En E0 2 2 = +İ0Į(0)B0 uE0Į 0.079+0.018 = +0.1Į K A Continuous spectrum Discrete Rydberg states assuming plane w aves for electrons CCaassiimmiirr mmoommeennttuumm:: 6/6 QED of hydrogen in crossed fields EE0 +e -e BB0 Relativistic contribution: 2 e 2 KR = 2 ¢0E | p (B0 ux)|0E ² 2meMc0 m v Į2 e K M A CCaassiimmiirr mmoommeennttuumm:: 6/6 QED of hydrogen in crossed fields EE0 +e -e BB0 E0 2 3 2 me K = (me m p )v+ 2 v K A 0.1 Į K A +O(Į ,D ) c0 m p KA =İ0Į(0)B0 uE0 Casimir momentum of H atom exists and slightly reduces the classical Abraham momentum Casimir mass being equivalent to binding energy is same physics BaVT, Kawka, Rikken, submitted to EPJD 68 SSUUMMMMAARRYY CCaassiimmiirr mmoommeennttuumm iinn ccrroosssseedd EE,,BB •• CCllaassssiiccaall AAbbrraahhaamm ffoorrccee ,, lliinneeaarr iinn EE0 aanndd BB0 ,, iiss oobbsseerrvveedd ffoorr nneeuuttrraall aattoommss ((aanndd ffoorr ssttrroonngg ddiieelleeccttrriiccss)) •• QQEEDD ccoonnttrriibbuuttiioonn bbyy FFeeiiggeell iiss nnoott oobbsseerrvveedd •• UUVV ddiivveerrggeenncciieess ddiissaappppeeaarr iinn mmaassss rreennoorrmmaalliizzaattiioonn .. oorr ccaanncceell.. NNeeeedd ttoo ggoo bbeeyyoonndd mmuullttiippoollee aapppprrooxxiimmaattiioonn •• QQuuaannttuumm vvaaccuuuumm ccoonnttrriibbuutteess ttoo AAbbrraahhaamm mmoommeennttuumm iinn oorrddeerr --((11//113377))2 WWiillll tthhiiss bbee --((ZZ//113377)) 2 ffoorr ZZ >> 11???? MMaaggnneettoo--CChhiirraall CCaassiimmiirr mmoommeennttuumm?? < • Classically no equivalent Abraham version in charge neutral systems • g must be a pseudo scalar Æ medium must be chiral (on nanoscale) • Describe chirality microscopically, not phenomenologically using Lifshitz formula via « magneto-chiral « index of refraction (ǻn=g B0.k) • Would separate enantiomers using magnetic fields = Pasteurs dream ! • Medium must have induced magnetic dipole since ˢˢ ˢˢ ˢˢ ˢˢ BB0 Chiral geometry with electric polarizabilities with Zeeman splitting Pinheiro and BAvT 69 2 4ʌc0 Ȗ Į Ȧ ,ı 2 2 2 Ȧ0 Ȧ Ȧ0 iı VB i ȖȦ0 B P H 0 drEuB 0 v 0 drEuH 0 0 0 ³ ³ AA CCaassiimmiirr mmoommeennttuumm PP== gg BB0 ?? 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