Playing with Casimir in the vacuum sandbox
PPlaying with Casimir in the vacuum sandbox
S. Kauffman,
1, 2
S. Succi,
3, 4, 5
A. Tiribocchi,
3, 4 and P. G. Tello The Institute for System Biology, Seattle (WA), USA Professor Emeritus, Biochemistry and Biophysics, The University of Pennsylvania, (PA), USA Center for Life Nano Science@La Sapienza, Istituto Italiano di Tecnologia, 00161 Roma, Italy Istituto per le Applicazioni del Calcolo CNR, via dei Taurini 19, Rome, Italy Institute for Applied Computational Science, John A. Paulson School ofEngineering and Applied Sciences, Harvard University, Cambridge, USA CERN, Geneva, Switzerland (Dated: February 24, 2021)The Casimir effect continues to be a subject of discussion regarding its relationship, or the lack ofit, with the vacuum energy of fluctuating quantum fields. In this note, it is proposed a
Gedankenex-periment considering an imaginary process similar to a vacuum fluctuation of a single virtual modein a typical static Casimir set up. The thought experiment leads to intriguing conclusions regardingthe minimum distance between the plates when approaching the Planck scale. More specifically, itis found that distance between the plates cannot reach a value below (
L/L P ) / Planck lengths,being L P the Planck length and L the typical lateral extension of the plates. Additional findingsallow the conclusion that the approach between the two plates towards this minimum separationdistance is asymptotic. I. INTRODUCTION AND AIM
The physical vacuum seems to be a “busy place” de-scribed as “foamy” by Wheeler [1], and one of the mostintriguing phenomena linked to it is the Casimir effect[2]. In this note, it is proposed a
Gedankenexperiment based on the Casimir effect by following the tradition ofspeculating for the playful purpose of thinking throughits consequences. This
Gedankenexperiment is suggestedwith the caveat that the relation of the Casimir effectwith the vacuum energy of fluctuating quantum fields isstill open to debate at the time of this writing [3].
II. GEDANKENEXPERIMENT SETUP
The starting point is the typical configuration of thestatic Casimir effect (Fig.1): two uncharged, flat and per-fectly conductive parallel plates in the vacuum, placed ata very small distance apart d (typically in the order ofmicrons or even a few nanometers). Figure 1.
Gedankenexperiment proposed in the text. In a typ-ical Casimir setup, one of the virtual modes of the vacuumis “removed” with the consequent shrinkage of the initial dis-tance, d , within the plates to d < d . Accordingly, the energy E of the fluctuation modesconfined in between the plates can be expressed as: E = − C (cid:126) cL d . (1)Here C = π/ c is the speed of light, L is the lat-eral extension of the plates and the minus sign indicatesa negative energy, corresponding to attraction betweenthe plates. The Gedankenexperiment assumes the exis-tence of a “process” capable of “removing” one of thevacuum modes in between the plates (quantized by thevalue of d ), at least momentarily, so the energy-time un-certainty relationship holds. As a pictorial image, this“process” could be thought of as a vacuum fluctuation.The Casimir plates, initially separated at a distance d ,will be therefore, at least momentarily, brought togetherto a smaller distance d , due to the removal of a virtualmode with energy E n = (cid:126) ω n . In this sense, the varia-tion ∆ E of energy in between the plates will be givenby the difference between the initial energy and the finalone after removing one mode, being the first one greaterthan the second, due to the fact that C > d < d . Thus, it is possible to write:∆ E = (cid:126) ω n = C (cid:126) cL d − C (cid:126) cL d . (2)Let us consider now a standard interpretation of theenergy-time indeterminacy relation which tells, in ordi-nary language, that if an amount of energy ∆ E is “bor-rowed” from the vacuum, it must be “returned” quicklyenough within a time interval ∆ t . In the context of this Gendankenexperiment , ∆ t would correspond the fluctu-ation lifetime. Mathematically, it is expressed by theenergy-time uncertainty relation ∆ E ∆ t ≥ (cid:126) /
2. Accord-ingly, ∆ t is ∆ t ≥ (cid:126) E . (3) a r X i v : . [ qu a n t - ph ] F e b Taking into consideration Eq.(2), it is possible to write∆ t ≥ (cid:126) E = 12 ω n . (4)In this Gedankenexperiment , the boundary conditions as-sumed for the quantized mode between the plates allow ω n to be expressed, for some integer number n , as: ω n = 2 πncd . (5)Substituting Eq.(5) into Eq.(4) leads to the approxima-tion ∆ t ∼ dnc . (6)At this point it is interesting to notice that, based onEq.(2), the relationship between d and d takes the form (cid:18) d d (cid:19) = 11 + d d n , (7)with d n = r C π Ln / ∼ L/ n / . (8)Equation (7) can be also cast in a more compact for asfollows: d d = 1 (cid:0) n d l (cid:1) / , (9)where we have set l ∼ L/
7. The expressions (7-9) con-stitute the cornerstone of the
Gedankenexperiment , sincethey lead to a number of interesting considerations. Thefirst one is that it shows the ratio d /d as a functionof the reduced separation d/L , depending parametricallyon the wavenumber n . This dependence is plotted inFig.2 for different values of n . The graph indicates theexistence of a characteristic wavenumber for each givenvalue of the ratio d/L below which the contraction ofthe space between the plates is really small. When ap-proximating to such characteristic value of n , the ratio d /d experiments a fast decrease, followed by a very slowdecay with d/L which is basically undetectable in the rel-evant regime d/L (cid:28)
1. The second one is the emergenceof two regimes which for convenience are identified hereas “slow-mode” and “fast mode” which are subsequentlyanalyzed in the following sections.
III. FAST AND SLOW-MODE REGIMES
Based on Eq.(9), two regimes can be identified: i)“slow-mode”, characterized by n d l (cid:28) , (10) Figure 2. The ratio d /d as a function of the reduced sepa-ration d/L for n = 10 k and with k in the range from 1 to 8(from top to bottom). The rapid drop of the ratio d /d andsubsequent saturation at increasing values of n is clearly ap-parent. Note that d/L is always well below 1, as it should bewithin Casimir effect theory. and ii) “fast mode”, characterized by the opposite in-equality. The slow-mode inequality given by Eq.(10) im-plies that at each given value of the reduced separation d/L there is a maximum wavenumber, which we call n ,such that the ratio d/d is slightly below one, thereforecorresponding to small contractions between the plates.By definition then: n = l d . (11)In the slow-mode regime n (cid:28) n , Eq.(9) reduces to: d d = 1 − n d l = 1 − nn , (12)which, by definition, is just slightly smaller than one.In the fast-mode regime, Eq.(9) leads to: d d ∼ n − / (cid:18) ld (cid:19) / = (cid:18) nn (cid:19) / . (13)This expression shows that d decreases very slowly withthe wavenumber n and cannot reach the value zero forany finite value of n . In other words, the separationbetween the plates cannot be zero. We shall return tothis point in the final part of this paper, when discussingthe implication of the Eq.(13) in the limit of the Planckscale.At this point, it is worth noting that as required by theCasimir effect theoretical formalism, the ratio d/L mustbe necessarily much smaller than one for Eq.(1) to hold.Hence, as an example, we consider a ratio d/L = 10 − which thus gives n ∼ . Based on the Eq.(9) andEq.(11), Eq.(6) leads to∆ t = d nc (1 + 10 − n ) / . (14)As it indicates, the quantity in brackets only becomessignificant for wave numbers n > . It should be notedthat, considering the “slow mode regime” with d = 1 nm and c ∼ m/s , it gives d/c of the order of the attosec-ond (10 − s ). This means that only modes with n (cid:28) could be possibly detected within the current frontier ofultrafast technology. This is analyzed further in the fol-lowing sections. A. Analysis of the slow-mode regime
Considering
L/d ∼ , n ∼ , hence for n < n (“slow modes”), expanding the Eq.(12) leads to: d − dd ’ − n − , (15)which indicates a relative contraction in the order of the4 th digit for n ∼
10, a measurable effect as long suchmeasurement is performed within the limits of the fluc-tuation lifetime ∆ t . Since, in this regime, d ’ d , onehas ∆ t ∼ dcn . (16)Given that the maximum value of n in the slow moderegime is n = 10 , and approximating c ∼ m/s , weobtain a measuring time of 10 − s , which is between twoand three orders of magnitude beyond the current tech-nological possibilities even for measurement of ultrafastprocesses [4]. B. Analysis of the fast-mode regime
In the fast-mode regime, Eq.(13) leads to d d ’ (cid:18) n (cid:19) / ’ n − / . (17)This shows that, due to the small exponent − /
3, ittakes very large wavenumbers n to achieve substantialcontractions. For instance, for n = 10 it is obtained acontraction d = d/
10, and for n = 10 , it gives d = d/ n → ∞ is unphysical sincethe wavelength of the photon cannot be lower than thePlanck length L p . This is discussed more in detail inthe next paragraph. Moreover, as it is readily seen fromEq.(14), the corresponding fluctuation lifetime ∆ t is ofthe order of 10 − and 10 − s respectively, hence faroutside reach of the present technology.In summary, we have seen that a “vacuum fluctua-tion” between the plates, both of a high energetic mode ( n > ) or a low energetic mode ( n < ), producesa separation effect between the plates that is beyond ex-perimental reach with today’s technological capability.Perhaps a challenge is left for future experimental in-strumentation developments. The following observationis relevant at this point. The single photon assumed tobe extracted from within the plates, being a single mode,has an energy which does not grow with the transversearea L of the plates, while the energy in between theplates does (see Eq.(1)). Accordingly, in order to com-pensate for this different scaling and in principle obtain-ing a measurable effect for ∆ E and, therefore, on the dis-tance variation between them, a highly energetic modeis needed, which then becomes unmeasurable within theassociated extremely short lifetime ∆ t of its correspond-ing fluctuation. On the other hand, assuming a photonwith reasonable energies will eventually result in an un-detectable displacement of the walls. IV. DOWN TO THE PLANCK SCALE
In this section, the authors would like to takethis
Gedankenexperiment further and playfully speculateabout the results obtained. In order to explore this mat-ter, let us introduce the Planck wavenumber n P , as thevalue of n such ∆ t = t P and λ = L P , meaning the Plancktime and the Planck wavelength respectively, L P beingthe Planck length. By definition, we have: n P = 2 dL P . (18)The Eq.(13) becomes minimum assuming the largest pos-sible value for n , given by n P . Using the definitions givenby Eq.(18) and Eq.(11), we obtain: d min = d (cid:18) n n P (cid:19) / ∼ L / L / P . (19)This shows that even upon removing the most energeticpossible mode (a Planckian photon), “Casimir space”cannot contract below ( L/L P ) / Planck lengths. Thisresult is intriguing, for it shows that the proposed “re-moval process” singles out a minimum length scale wellabove the Planck length. Thus, the authors dare spec-ulating that, although with an exchange of exponentsbetween L and L P , an intriguing analogy holds with theholographic expression L min = L / P L / . As is wellknown, this expression derives from assuming that therelevant quantum gravitational degrees of freedom as-sociated with a given spatial region are located on thesurface and not on the volume [5, 6]. The relation (19)indicates a dependence of the boundary conditions of our Gedankenexperiment.
Coming back to the thought experiment and startingwith Eq.(14), ∆ t ∼ nc d (1 + 10 − n ) / , (20)when considering n = n P , this approximates to∆ t ∼ d c n / P . (21)By taking into account the definition of n P , it leads to:∆ t P ∼ d c (cid:18) L P d (cid:19) / . (22)By considering Eq.(13) and after some algebra, it is pos-sible to arrive, as expected, to:∆ t P ∼ n / c L P ∼ L P c ∼ t P . (23)Note that ∆ t P as computed above is much smaller thanthe smallest time interval supported by the hypotheticaltime allowed by the Casimir theoretical formulation t min = d min c = (cid:18) Lc (cid:19) / t / P . (24)This might suggest some connection with a generalizedHeisenberg principle and more particularly given the casethat, in our Gendankenexperiment , the distance betweenthe plates cannot be made arbitrarily small. More con-cretely smaller than the Planck length [7]. As a matter offact, it is interesting, as a final consideration of this note,relating the energy between the plates with respect tothe Planck energy, E P . Starting with Eq.(1) and takinginto account Eq.(19), we obtain: E min ’ − (cid:126) cL d min = − (cid:126) cL P = − E P , (25)where we have assumed | E ( d ) | (cid:28) E P as it is appropri-ate for d > d min . This result would seem paradoxicalbecause, having subtracted a photon of Planck energy,would imply that E = − E p , which cannot be since E isa free parameter. This apparent paradox is explained byconsidering that in fact d approximates only asymptoti-cally to d min , never to reach exactly this limiting value asexpressed in Eq.(19). More concretely, by taking Eq.(2)with C = 1, we compute: E ( d = L P ) = (cid:126) cL L P = E P (cid:18) LL P (cid:19) (cid:29) E P . (26)As this expression suggests, the apparent paradox men-tioned above results from the fact the energy betweenthe plates is singular as d approaches zero, hence it takesmuch more energy than E P to push d from d min downto L P . The interesting conclusion is that while E ( d )closely approaches E ( d min ), d min still remains far abovethe Planck length. V. CONCLUSIONS
Our excursion into the Casimir sandbox through thethought experiment proposed in this note leads to thefollowing observations:1. Considering the occurrence of a hypothetical “vac-uum fluctuation” between the plates removing onevirtual mode while respecting the energy-time un-certainty leads to two differentiated regimes ruledby the scaling relationship
L/d = 10 between thelateral dimensions of the plates L and their dis-tance d , characterized by the value of the modewave number n . One which we will call “fast mode”where n > and another one which we will call“slow mode” where n < .2. The potential shrinking effect between the plates ofboth, the high energetic mode ( n > ) or the lowenergetic mode ( n < ), cannot be measured withtoday’s technological capability. Perhaps, in thelow energetic mode, future advances in technologymight be able to cope with the challenge.3. By removing the most energetic mode (a Planckianphoton), the momentarily contracted distance be-tween the plates space cannot reach a value d min below ( L/L P ) / Planck lengths. More precisely, d min = L / L / P , L being the lateral extension ofthe plates and L P the Planck distance. The au-thors noticed the intriguing analogy between theexpression for d min and the one L min = L / P L / which derives from assuming that the relevant de-grees of freedom or quantum gravity are located onthe surface and not on the volume (“holographicprinciple”).4. If the diminished distance d between the plates,following the thought experiment, approaches theminimum distance d min , it does it asymptotically.This leads to the interesting observation that whilethe energy between the plates, E ( d ), approachesthe minimum one, E ( d min ) = E P , the minimumCasimir distance d min remains far above the Plancklength, L P . Conversely, the Casimir energy at thePlanck scale exceeds the Planck energy by a factor( L/L P ) / .5. One may finally wonder whether the present anal-ysis would hold if the geometry of the conductingplates is not flat. Although this is beyond thescope of this manuscript, it is interesting to notethat modifying the shape of the plates can turn thesign of the energy from negative to positive, thusleading to a repulsive rather than an attractive in-teraction [8]. This has been shown, for example,in Ref. [9], where a small elongated metal parti-cle in vacuum is subject to a repulsive force whencentered above a metal plate with a hole. Repul-sive long-range forces, of quantum electrodynamicorigin, have been also measured between materi-als with suitable optical properties and immersedin a fluid [10]. Even more intriguing is the realmof soft condensed matter, where Casimir-like forceshave been found to act between surfaces immersedin a binary fluid close to its critical point. Suchforces, caused by the fluctuations of the concen-tration (whose relevant scale is k B T , where k B isthe Boltzmann constant and T the temperature)within the fluid film separating the surfaces, can beeither attractive or repulsive, depending on the ad-sorption preference of the fluid in contact with thesolid body [11, 12]. Finally, it is known that near- contact forces between macroscopic bodies play amajor role on the rheology of soft materials [13]. ACKNOWLEDGMENTS