TThe Casimir-Effect:No Manifestation of Zero-Point Energy
Gerold Gründler Astrophysical Institute Neunhof, Nürnberg, GermanyThe attractive force between metallic surfaces, predicted by Casimir in1948, seems to indicate the physical existence and measurability of thequantized electromagnetic field’s zero-point energy. It is shown in thisarticle, that the measurements of that force do not confirm Casimir’smodel, but in fact disprove it’s foundational assumption that metal platesmay be represented in the theory by quantum-field-theoretical boundaries.The consequences for the cosmological constant problem are discussed.PACS numbers: 03.70.+k, 04.20.Cv
1. Is the zero-point energy of quantized fieldsobservable?
General relativity theory (GRT), and the relativistic quantum fieldtheories (QFT) of the standard model of elementary particles, aredescribing all experimental observations with impressive accuracy —as long as GRT or QFT are used separately. But as soon as one triesto combine these successful theories, serious problems turn up. Oneof the most spectacular examples of such incompatibility has beendubbed “the cosmological constant problem”. A by now classicalreview article on that issue has been compiled by Weinberg [1].See the article by Li et. al. [2] for an updated review. In short, thecosmological constant problem manifests itself as follows:According to the field equation e-mail: [email protected] a r X i v : . [ phy s i c s . g e n - ph ] A ug R µν − R g µν + Λ g µν = − πGc T µν (1)of GRT, the curvature of space-time, represented by the Ricci-tensor( R µν ) and it’s contraction R , is proportional to the energydensity-stress-tensor ( T µν ), which again is determined by the energy densityand the momentum density of all fields contained within space-time, i. e. of all fields with exception of the metric field ( g µν ).Upon canonical quantization of any classical continuous field, theenergydensity-stress-tensor of that field will diverge. For example,canonical quantization of the classical electromagnetic field resultsinto the Hamilton operator H = Z d x T = X k X v =1 (cid:126) c | k | (cid:16) a ( v )+ k a ( v ) k + a ( v ) k a ( v )+ k (cid:17) (2a)= X k X v =1 (cid:126) c | k | (cid:16) a ( v )+ k a ( v ) k + 12 [ a ( v ) k , a ( v )+ k ] | {z } (cid:17) , (2b)with a ( v )+ k and a ( v ) k being the creation- and annihilation-operatorsrespectively of photons with wavenumber k and polarization v .The integration is over the complete normalization volume, and thesummation is running over all of the infinitely many wavenumbers k , which are compatible with the normalization volume (if aninfinite normalization volume is chosen, the sum over k is replacedby an integral over k ). Due to the commutator, the energy isinfinite. The waves described by the second term in (2b) are thezero-point-oscillations, and their energy is the zero-point-energy,of the quantized electromagnetic field.The observed curvature of intergalactic space is close to zero [3],suggesting that either (a) the cosmological constant Λ in (1) should be adjusted, to com-pensate the zero-point-energy of the quantum fields, or that(b) the zero-point-energy of the quantum fields should be consid-ered as a strange artifact of the theory without analog in observablereality, and therefore be removed somehow from QFT.To avoid misunderstandings, we note that the zero-point oscil-lations of quantum fields with only a finite number of degrees offreedom, e. g. the zero-point oscillations of the phonon fields ofmolecules and solids, have been experimentally confirmed sincealmost a century [4, 5]. But what we are exclusively discussing inthis article is the zero-point energy of elementary quantum fieldswith infinitely many degrees of freedom.Alternative (a) calls for a fine-tuning of the cosmological constantΛ with an accuracy of many dozens of decimal digits, the exactnumber depending on the method applied for regularization of thediverging term in (2b). Therefore this solution — though beingcompletely correct under purely formal criteria — does seem tobe quite “unnatural”, and is not considered acceptable by manyscientists.On first sight, there seem to be less objections against alternative(b). In pure quantum-field-theoretical computations (neglectinggravity), only energy differences matter, but not absolute energyvalues. Therefore the offset of an infinitely large zero-point energyis merely a tiresome ballast without discernible functionality. Toget rid of that offset, normal order is often applied as an ad-hoc measure. It means, that in (2a) all creation operators areshifted left, and all annihilation operators are shifted right, underdisregard of their commutation relations! Thus the Hamilton-operator becomes H = X k X v =1 (cid:126) c | k | a ( v )+ k a ( v ) k , (2c) and the infinite energy offset has disappeared.As the results of quantum-field-theoretical computations arenot changed, if normal order is applied to the Hamilton operator,and as no gravitational effect of the zero-point-energy of quantumfields is observed, one might very well ask whether that zero-pointenergy does exist at all. Words like “existence” or “reality” in thiscontext of course mean the question, whether the zero-point energyis observable and can be tested experimentally.
2. The Casimir-effect
A possible method to observe the quantized electromagnetic field’szero-point energy — and actually the only method proposed untiltoday, besides the missing gravitational effect — has been suggestedby Casimir [6] in 1948. Casimir considered a resonator as sketchedin figure 1. The rectangular cavity’s size is X × Y × ( Z + P ). Insidethe cavity there is a plate of thickness P , which is aligned parallel Z + P Y X Z - DD P Fig. 1 : Cavity resonator with movable plate to the cavity’s XY -face and movable in Z -direction. The plate’sdistance from one side wall of the cavity is D , it’s distance fromthe opposite side wall is Z − D .The resonance spectra of the left and right cavities are discrete.The wavenumbers are k rst = s(cid:16) rπX (cid:17) + (cid:16) sπY (cid:17) + (cid:16) tπA (cid:17) with r, s, t ∈ N , (3)There are 2 modes each with r, s, t = 1 , , , . . . and 1 modeeach with one of the indices 0 and the both other indices1 , , , . . . [7, chap. D.II.2.b.]with A = D for the left cavity and A = Z − D for the rightcavity. Casimir identified the plate and the walls of the cavitywith the boundaries of the normalization volume of quantum-electrodynamics = QED. Therefore he considered this equation notonly valid for photons, but as well for the zero-point oscillationsof the quantized electromagnetic field. According to this point ofview, long-wavelength zero-point oscillations, which don’t fit intothe cavities, can not evolve in the respective volumes. Casimircomputed the zero-point energy U left enclosed in the left cavity, andthe zero-point energy U right enclosed in the right cavity. A detailedaccount of Casimir’s computation can be found elsewhere [8, sect. 4].Both U left and U right are functions of D , and both are of courseinfinite, as infinitely many short-wavelength zero-point oscillationmodes fit into the cavities. But the derivative F Casimir ≡ d( U left + U right )d D (4)is finite! Under the assumption, that the cavity walls and the mov-able plate perfectly reflect electromagnetic radiation of arbitrary frequencies, and assuming Z (cid:29) D , Casimir found a surprisinglysimple result: F Casimir = − π (cid:126) c XYD = − . · − N · XY / mm D /µ m (5)This is a small, but measurable force, which is pushing the movableplate towards the nearer cavity wall. It has by now been mea-sured many times, and the approximate correctness of equation(5) has been confirmed [9]. It is no surprise, that the experimentalconfirmation is only approximate but not exact, because Casimirderived (5) not for real metal plates, but for perfectly reflectingplates, i. e. boundaries in the terminology of quantum field theory.The important differences in the physical concepts of real metalsand boundaries will be discussed in the next section. Based onthe approximate experimental confirmations of (5), the attractiveforces between metallic plates have been declared to be “physicalmanifestations of zero-point energy” [10].These observations, however, do not conclusively prove the re-ality of zero-point energy, because an alternative explanation forthe same observations is available, which does not refer at all tozero-point energy: Lifshitz [11] and Dzyaloshinskii, Lifshitz, andPitaevskii [12] have computed the retarded van der Waals-force,which is acting between two infinitely extended half-spaces with rel-ative dielectric constants (cid:15) and (cid:15) , while the gap between them isfilled with a material with relative dielectric constant (cid:15) . Schwinger,DeRaad, and Milton [13] reproduced and confirmed the results ofLifshitz et. al. . They also considered the limit (cid:15) = (cid:15) → ∞ , (cid:15) → et. al. simplifies in this limit to the Casimir-force (5).The question “does the observed Casimir-force prove the observ-able existence of zero-point energy?” at first sight seems not to be answered by the theory with a clear-cut YES or NO , becausethere are two different theoretical concepts, one of them indicating YES , and the other indicating NO . But closer scrutiny reveals, thatthe answer definitely is NO . Jaffe [14] remarked, that Casimir’sassumption of perfect reflectivity of the metal plates is equivalent totaking the limit α → ∞ , with α being the QED coupling constant,and thus is obscuring the true nature of the interaction betweenthe electromagnetic field and the charged matter-fields constitutingthe metallic plates. While this criticism certainly is justified andpointing into the right direction, it is missing — or at least notexplicitly naming — the essential point: Only the Lifshitz modelis compatible with the results of measurements, while Casimir’smodel is refuted by experimental evidence.The two explanations of the Casimir-force are predicting similar,but not identical values of that force: The model, which is based onvan der Waals-forces, can match the measurement results exactly ifparameters like the complex dielectric constant (cid:15) ( ω ) = (cid:15) ( ω )+ i(cid:15) ( ω )as a function of photon frequency ω are adjusted [9]. In contrast,in Casimir’s model there are no adjustable parameters, see hisequation (5). And it is an essential feature of Casimir’s model,that no adjustable parameters like e. g. a reflection coefficient < complete demolition of the model. Thisassertion will be proved in the next section.Only the model of Lifshitz et. al. can stand the confrontationwith the results of measurements, while the results derived fromCasimir’s model differ typically by about 10 to 20 % at a platedistance of 1 µ m from experimental observations [15, sect. 5.2].Casimir’s explanation of the Casimir-force is disproved by theexperiments, because a significant discrepancy of about 10 to 20 %between theory and experiments, which can not be eliminated dueto improvement of the model , is about 10 to 20 % to much.
3. Metals and boundaries
At the outset of any quantum field theory, a normalization volume(which may be finite or infinite) must be fixed, and well-definedboundary conditions must be imposed onto the field at the bound-aries. A simple choice is for example a Dirichlet-type boundarycondition, requiring that the field amplitude must be zero at theboundary. Another, often more convenient choice is a periodic(Cauchy-type) boundary condition, requiring that the value andthe derivative of the field at one point of the boundary must atany time be identical to the value and the derivative of the fieldat the opposite point of the boundary. Either of theses conditionsmakes sure, that the norm h s | s i = N ∈ R , < | N | < ∞ (6)of any state-function | s i has a well-defined value N which can benormalized to unity, and — most important! — which is constant.This means, that either no probability density assigned to the state | s i can penetrate through the boundaries (Dirichlet-type boundarycondition), or that probability density flowing out of the normal-ization volume at one spot of the boundary is exactly compensatedby probability density flowing into the normalization volume at theopposite spot of the boundary (Cauchy-type boundary condition).The resonance spectrum (3) is enforced by the boundary condi-tion E tangential (surface) = H normal (surface) = 0 (7)onto the electrical amplitude E and the magnetizing amplitude H of the electromagnetic field at any spot of the surface of thecavity walls and the surface of the plate. No real metal can enforcethis condition onto the field, but only an ideal material which isreflecting 100 % of impinging radiation at any frequency. If a photon impinges onto a plate made of real metal, then itmay be reflected, or it may be absorbed, or it may be transmitted.For example, good electrical conductors like copper or gold reflectmost long-wavelength sub-infrared photons, reflect about half andabsorb about half of optical photons, and are almost transparentfor short-wavelength X-ray photons. Note that at least a smallpart of the impinging photons are absorbed by any metal at almostany frequency. Only superconductors absorb strictly no photonsof sufficiently long wavelength, but even they absorb photonsof infrared and shorter wavelengths. If one wants to computethe resonance spectrum of a cavity made from real metal, onetherefore needs to relax condition (7), and allow for absorptionby and transmission through the cavity walls in particular withregard to high-frequency radiation. This results into damping andbroadening of resonance modes.In contrast, the well-defined boundary conditions of the nor-malization volume must not be relaxed under any circumstances,because a damped norm like h s | s i = N e − γt , N, γ, t ∈ R , < | N | , γ, t < ∞ , (8)with γ being some damping parameter, and t being time, wouldnot be a reasonable extension of (6), but a contradiction to thebasic tenets of quantum field theory. If for example | s i is a state,in which exactly one photon is excited, then of course this photonmay disappear after some time due to interaction with matter. But(8) would imply that the photon would little by little disappeareven without any interaction, i. e. somehow slip out from thenormalization volume.To exclude the senseless result (8), boundaries must not bemerely approximate boundaries, which allow for damping due todissipation of probability density. Even a good approximation would not be sufficient. Only perfect boundaries are good enough,because only perfect boundaries can guarantee that probabilitydensity is conserved and that the norm of state-functions is constant.The boundaries of any quantum field theory are mathematical, nottangible entities, and no real metal can replace the boundaries ofQED.Fields and boundaries exhaust the inventory of quantum fieldtheory. Every entity of physical reality must be represented inQFT either by a boundary or by a field. If realistic parametersare assigned to the cavity walls and the plate, then they can notbe described as boundaries but must be described as materialfields, like conduction band electrons, or crystal ions, or Cooperpairs in case of superconductors, or whatever types of chargedmatter fields, which can couple to the electromagnetic field. As theelectromagnetic field according to QED is nothing but the gaugefield of charged matter fields, the interaction between the matterfields and their gauge field is uniquely defined: Photons, describedby state functions like | k , v i = a ( v )+ k | i or linear combinations ofsuch state functions, are the electromagnetic field’s quanta, whichcan couple to charged matter fields. If the electromagnetic field isin the the vacuum-state | i , in which no photons are excited butonly the electromagnetic field’s zero-point oscillations exist, thenit does not couple to any field, but only to boundaries. As soonas the quality of boundaries is no more ascribed to the walls ofthe cavity and to the plate, the electromagnetic field’s zero-pointoscillations vanish from the picture. If the metals are representedin the theory not by boundaries but by matter fields, then theinteraction between the metals and the electromagnetic field is notaffected, if zero-point energy is skipped from the theory due tonormal order of the Hamilton operator.The geometry of the boundaries can be chosen in quantum fieldtheory to a large extend at will. But of course the boundaries must enclose all parts of the physical system to be described. If theresonator depicted in fig. 1 shall be described with walls and platemade from real metals, then the boundaries must enclose, besidesthe cavity space, both the walls and the plate, because photonsmay be absorbed by them or may penetrate through them. Thevolume enclosed by the boundaries must be larger than the cavityvolume, and consequently the spectrum of photon-wavenumbersin this setup of QED will not be identical to the spectrum of theresonator’s resonance wavenumbers. In particular, the spectrum ofthe zero-point oscillations, which is determined by the geometry ofthe QED boundaries, is in this setup not related to the spectrumof the resonator’s resonance wavenumbers, because the resonatorwalls must be described as matter fields, which do not interactwith the electromagnetic field’s zero-point oscillations.Side note: From the field-theoretical point of view it is obvious,that only boundaries, but not real metal plates with finite conduc-tivity, can shape the spectrum of zero-point oscillations. In the midof the second page of his article [6], Casimir made a quite strangeremark to the contrary. There he pointed out that most zero-pointoscillations of very short wavelength (e. g. of X-ray wavelength)would penetrate through metals, while most long-wavelength zero-point oscillations would be reflected. I. e. he assumed that thereflection spectra of metals are similar (if not identical) for photonsand for zero-point oscillations. The present author undertook thetedious task, to evaluate the consequences of that assumption [16].Not surprisingly it turned out that Casimir’s strange assumptionis leading to results which are contradicting the experimental ob-servations. Note that Casimir’s final result (5) is not affected bythat strange assumption, because he achieved that result for amodel, in which the plates are represented by boundaries but notby metals. Therefore (5) is independent of any considerations onthe interaction of zero-point oscillations with real metals. Idealized conditions can be considered approximations to real-istic conditions, if the conditions can (at least theoretically) begradually changed from the idealized to the realistic case. Forexample, one could approach a realistic scenario by first assumingno interaction between matter and radiation, i. e. setting the QEDcoupling constant α to zero, and then step by step improve theapproximation by increasing α gradually to α ≈ / α = ∞ is assumed, then the quality of boundaries isabruptly removed from the plates and thus the foundations of themodel are completely destroyed as soon as one decreases α to afinite (even if arbitrary high) value.Thus the significant differences between the experimental obser-vations and the predictions of Casimir’s model, which have beenmentioned at the end of the previous section, can not be elimi-nated nor diminished by assigning to the boundaries the reflectivityof metals. Casimir’s model of the Casimir-force, in which metalplates are represented by boundaries, is an ingenious theoreticalconstruction, but by experimental evidence ruled out as a correctdescription of reality. Therefore the observed Casimir-force doesnot indicate the physical existence of zero-point energy.
4. Conclusions
The findings presented in this article may be helpful to avoid blindalleys, and to stir the search for a solution of the cosmologicalconstant problem into the right direction. The essential facts are:Firstly, no gravitational effect, caused by the zero-point energy ofquantum fields, has been observed. Secondly, the results of QFT — including all of the impressive achievements of QED like Lamb-shift, electron g-factor, hydrogen hyperfine-splitting, and so on— are not compromised, if zero-point energy is skipped from thetheory due to application of normal order (2c) to the Hamiltonoperator. Thirdly, as shown in this article, the observed Casimir-force does definitely not prove the reality of zero-point energy.In total, no experimental evidence at all is indicating the measur-able, observable existence of the zero-point energy of elementaryquantum fields. Therefore, instead of renormalizing the cosmo-logical constant Λ (or even modifying GRT), it is certainly morepromising to approach the problem directly at it’s root, i. e. tosomehow remove zero-point energy from QFT. The crude measureof normal order is not an acceptable solution, as the disregard ofthe non-commutative operator algebra is irreconcilable with thebasic principles of QFT. Unfortunately, at this moment I cannotoffer a better solution. Acknowledgments
This work gained much from innumerable helpful discussions on themysterious quantum vacuum with my colleagues V. I. Nachtmannand O. S. ter Haas.
References [1] Steven Weinberg :
The cosmological constant problem ,Rev. Mod. Phys. , 1-23 (1989)[2] Miao Li, Xiao-Dong Li, Shuang Wang, Yi Wang : Dark Energy , Commun. Theor. Phys. , 525-604 (2011),http://iopscience.iop.org/0253-6102/56/3/24 [3] C. L. Bennet et al. : Nine-Year Wilkinson MicrowaveAnisotropy Probe (WMAP) Observations: Final Maps andResults , arXiv [astro-ph.CO]: 1212.5225, 177pp. (2013)[4] Robert S. Mulliken :
The Isotope Effect in Band Spectra, II:The Spectrum of Boron Monoxide ,Phys. Rev. , 259-294 (1925)[5] R. W. James, I. Waller, D. R. Hartree : An Investigation intothe Existence of Zero-Point Energy in the Rock-Salt Latticeby an X-Ray Diffraction Method ,Proc. Roy. Soc. Lond. A , 334-350 (1928), http://rspa.royalsocietypublishing.org/content/118/779/334.full.pdf+html[6] H. B. G. Casimir :
On the attraction between twoperfectly conducting plates ,Proc. Kon. Nederl. Akad. Wetensch. Grundlagen der Höchstfrequenztechnik (Springer, Berlin, 1959)[8] G. Gründler :
Zero-Point Energy and Casimir-Effect
Casimir Effect: Theory and Experiments
Int. J. Mod. Phys. : Conf. Series The Casimir Effect. Physical Manifestations ofZero-Point Energy (World Scientific, New Jersey, NY, 2001) [11] E. M. Lifshitz : The Theory of Molecular Attractive Forcesbetween Solids , Sov. Phys. JETP , 73-83 (1956)[12] I. E. Dzyaloshinskii, E. M. Lifshitz, L. P. Pitaevskii : Reviews ofTopical Problems: General Theory of Van Der Waals’ Forces ,Sov. Phys. Usp. , 153-176 (1961),translation of Usp. Fiz. Nauk , 381 (1961)[13] J. Schwinger, L. L. DeRaad, K. A. Milton : Casimir effect indielectrics , Ann. Phys. (N.Y.) , 1-23 (1978)[14] R. L. Jaffe :
Casimir effect and the quantum vacuum ,Phys. Rev. D , 021301 (2005)[15] M. Bordag, U. Mohideen, V. M. Mostepanenko : New developments in the Casimir effect ,Physics Reports , 1-205 (2001)[16] G. Gründler :