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The Casimir Effect IFPA, AGO Department University of Liège THE CASIMIR EFFECT J. Cugnon University of Liège HLPW10, Spa, 6-8 April 2011 ● Dark energy ● The Casimir effect as a manifestation of quantum vacuum energy ● Dependence upon the fine structure constant ● The Casimir effect as a van der Waals force between giant « molecules » ● « Reality » of quantum fluctuations in the vacuum? ● Conclusion H, Wroclaw, 9-11 July 2009 DARK ENERGY from Hubble plot at large red shifts and CMB fluctuations Dark energy has a constant energy density and a negative pressure 3 DE≃3/4c≃4GeV /m Dark matter energyzero point energy of quantum fields - first proposed by Zeldovitch (1957) - confirmed by Casimir effect, according... S. Weinberg, 1987 Perhaps surprisingly, it was a long time before particle physicists began seriously to worry about [quantum zero point fluctuations contribution to L] despite the demonstration in the Casimir effect of the reality of zero-point energies. S. M. Carroll, 2001 ... and the vacuum fluctuations themselves are very real, as evidenced by the Casimir effect. D. Perkins, « Particle Astrophysics », Oxford Univ. Press, 2008 That this concept [the vacuum energy] is not a figment of the physicist©s imagination was already demonstrated many years ago, when Casimir predicted that by modifying boundary conditions on the vacuum state, the change of the vacuum energy would lead to a measurable force, subsequently detected and measured by... For a free bosonic field: k cut 3 ∫ d k 1 4 =∑ 1/2 ℏ=ℏ c k= ℏ c kcut 2 3 82 121 3 For kcut ~1/ℓPL , e ~10 GeV/m For a free fermionic field: =−∑ 1/2 ℏ ... and there are plenty of condensates in the SM ! THE CASIMIR FORCE and the ZERO POINT ENERGY of the ELECTROMAGNETIC FIELD The modes in the cavity are not the same as in free space, especially at low frequency Interaction energy=difference of zero-point energies ik x ik y ik z −i t ik x ik y −i t x y z k x y k E= e e e e ⇔ E= e e sin k z z e k z=nz /d for nz=1,2,3,.... Rm: for nz=0, only one mode E E cav E free = − S S S ∞ ∞ ∞ ∞ ∞ ℏ c2 1 = 2− 2 3 ∫ du u ∑∫ du u n ∫ dx∫ du u x 4 d {2 0 n=1 0 0 0 } 2 Rm: regularized by integration factor e− ux , 0 Euler-McLaurin theorem: ∞ ∞ 1 1 1 1 f 0∑ f n−∫ f x =− f ∞ [ f © 0− f © ∞]− [ f © © © 0− f © © © ∞] 2 n=1 0 2 12 720 E ℏ c2 =− See Itzykson and Zuber S 720 d3 The Casimir force 2 F ℏ c − =− =4×10 4 N /m2 at d=1 m S 240 d 4 has been calculated for many « geometries » has been verified experimentally - M. J. Sparnaay, Physica 24 (1958) 751 parallelism, impurities, residual charges - M. J. Spaarnay, P. W. J. Jochems, 1960 - B. V. Darjaguin et al, « Surface forces », Plenum, 1987 - S. K. Lamoreaux, PRL 78 (1997) 5 - T. Ederth (2001?) 1% Now, a blooming field in MEMS technology -from attractive to repulsive -turned to be a control tool A field of its own DEPENDANCE on the FINE STRUCTURE CONSTANT Universality? Independence of a? Real conductors are characterized by: - plasma frequency wpl : no propagation for w < wpl - skin depth d; d-2 = 2p w|s| /c : penetration depth of incident waves Drude model: free electrons with friction force (f=-gv) −1/2 2 2 4 e n 1 pl = =c pl m 2 22 e Limit of perfect conductor: wpl ∞, d 0 Perfect metal hypothesis is justified if characteristic frequencies c/d << wpl i.e. m c ≫ e 4 ℏ n d2 For typical cases (Cu, d=1mm), rhs ~ 10-6 - This condition is comfortably satisfied for the physical value of a - Casimir©s result = the a ∞ limit !!!! - corrections appears as powers of 1/a E ℏ c2 ℏ c2 ≈− ≈− 1−6 ⋯ S 720d2 3 720 d3 d Small a limit 2 ℏ 3 2 2 aB= 2 ∝1/ , n∝ , pl ∝ , ∝1/ me e Conducting plates become transparent and Casimir effect goes away The only distinctive feature: goes to a constant in the strong coupling limit CASIMIR EFFECT as a VAN DER WAALS FORCE between MACROSCOPIC NEUTRAL OBJECTS Two neutral atoms at distance r : effects of Coulomb forces 2 London (1937) : 2d order in a and in Rat/r 4 2 2 1 2 e ∣a ∣ ∣a ∣ 2 =− k0 l0 , a = k∣ z ∣0 E 6 6 ∑∑ 2 2 1 1 k0 〈 ∑ i 〉 r k≠0 l≠0 E k −E0 El −E 0 Atom 2 1 ∣ai∣2 = 2 k0 =14 n Static atomic polarizability i e ∑ i i at k≠0 E k −E 0 -Known for: Casimir operator, hyperfine interactions, cooling by adiabatic demagnetization (mK), etc -PhD of Erhenfest, assistant of Pauli, position in Leiden -In 1940, moved to Philips gloeilampenfabriek NV in Eindhoven In 1947, EJW Verwey and JTG Overbeek : dilute colloidal suspensions HBG Casimir and D. Polder, Phys. Rev 73 (1948) 360: Interaction between atoms through Coulomb forces and the coupling to the radiation field (retardation effects) Perturbation theory at the second order in a ● ● L r For small r, London©s expression is recovered For r >> ako : ℏ 2 23 c (1) E =− 1 2 4r 7 In the same paper, interaction of a atom with a conducting plane 2 e 2 =− ∣ ∣2 For small d : E atom−wall 3 ∑ ak0 4 d k d ℏ For d >> ako : 2 3 c (2) E atom−wall=− 1 8 d 4 Perturbation theory at the first order in a (first order in Coulomb forces, second order in atom-radiation coupling) Niels Bohr: « Why don©t you calculate the effect by evaluating the differences of zero point energies of the electromagnetic field? » Casimir rederived results (1) and (2) by this method (in some approximation) : Colloque sur la théorie de la liaison chimique, Paris, 12-17 avril 1948 He calculated by the same method the interaction between parallel conducting planes H.B.G. Casimir, Proceedings of the Koninglijke Nederlandse Akademie van Wetenschappen B51, 793 (1948) 3 Limit of VdW-Casimir-Polder : lim ai / r = 1 London-VdW Casimir-Polder Order 4 2 2 1 2 2 e ∣ak0 ∣ ∣al0 ∣ E =−6 ∑ ∑ 2 23 ℏ c 6 2 2 1 1 E =− 1 2 r k≠0 l≠0 E −E E −E 7 k 0 l 0 4 r α2 2 2 e 2 3ℏ c E =− ∣a ∣ 2 =− α atom−wall 3 ∑ k0 E atom−wall 4 1 4 d k 8 d Replacing an atom by a plate : ℏ c2 −infinite volume withconstant n E=− Sd d 4 α0 −ni −1/ 720 − ∞ ai disappears, a power of a is absorbed !! replaced by a volume ! Same results by Casimir©s method ConclusionConclusion :: CasimirCasimir effecteffect == limitlimit ofof vanvan derder WaalsWaals forceforce forfor «« macroscopicmacroscopic moleculesmolecules »» inin thethe idealideal conductorconductor (infinitely(infinitely polarisable)polarisable) limitlimit The « REALITY » of QUANTUM FLUCTUATIONS in the VACUUM 1. Casimir©s result is « heuristic »: that the interaction energy is given by the difference of zero point energies is accidental and does not reveal the « reality » of quantum fluctuation energy 1 rr © 1 for a diffuse charge: W= ∫ dr∫ dr © = ∫∣E r∣2 dr 2 ∣r−r ©∣ 8 Rm: « reality » of the field coming from light, pair production, etc 2. Zero-point energy coming from an « obscure » choice of ordering fields in the classical lagrangian (giving negative energy for fermion fields) 3. Interaction between neutral objects gives no more (or less) support to the « reality » of the vacuum energy of fluctuating quantum fields than the other one-loop effects in quantum electrodynamics, like vacuum polarisation contribution in the Lamb shift (R. Jaffe, PR D72 (2005) 021301) : the effect vanishes as a 0 4. Casimir effect can be derived without reference to zero-point motion -general theory of fluctuations (Lifshitz, Zhur. eksp. i teor. fiz. 29 (1955) 9, Landau and Lifshitz, Electrodynamics of continuous media) 1 〈 Ei rEj r〉= f ,...W = ∫∣E∣2∣H∣2 8 no field quantization calculation of the interaction between semi-infinite pieces of dielectrics; Casimir force obtained as the e ∞ limit - field theory without reference to zero-point fluctuations: Schwinger, 1975, scalar field Schwinger, DeRaad, Milton, 1978, for QED ℏ E= ℑ∫ d Tr∫ d3 x[G x , x ,i −G x , x ,i ] 2 0 G is the full Green©s function in the background (plates) d N 1 E∝∫ d i.e. ∑ ℏ −ℏ d 2 0 G can be expanded in series of G0 and a Rm: all features of QED can be reformulated from the point of view of zero point fluctuations (Milonni, « The Quantum Vacuum », Acad. Press, 1994) CONCLUSIONS ● Casimir effect often advocated as a manifestation of the quantum fluctuations of the vacuum and the support of the latter as a candidate for DM. But... ● Casimir force can (should) be viewed as a vdW force between gigantic molecules in the strong coupling limit ● Casimir forces are real in the micro- to nano-world ● Not a demonstration of the « reality » of the zero-point energy (no more than vacuum polarisation, ...): vanishes in weak coupling, formulation without reference to zero-point energy ● Reality of vacuum energy? .
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