Eur. Phys. J. D (2013) DOI: 10.1140/epjd/e2013-30578-7 THE EUROPEAN PHYSICAL JOURNAL D Regular Article

The quantum as the origin of the

Marcel Urban1,Fran¸cois Couchot1, Xavier Sarazin1,a, and Arache Djannati-Atai2 1 LAL, Universit´e Paris-Sud, CNRS/IN2P3, Orsay, France 2 APC, Universit´e Paris Diderot, CNRS/IN2P3, Paris, France

Received 17 September 2012 / Received in final form 16 January 2013 Published online (Inserted Later) – c EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2013

Abstract. We show that the μ0 and 0 may originate from the magneti- zation and the polarization of continuously appearing and disappearing pairs. We then show that if we simply model the propagation of the in vacuum as a series of transient captures within these ephemeral pairs, we can derive a finite photon velocity. Requiring that this velocity is equal to the speed of light constrains our model of vacuum. Within this approach, the propagation of a photon is a statistical process at scales much larger than the Planck scale. Therefore we expect its time of flight to fluctuate. We propose an experimental test of this prediction.

1 Introduction (ephemeral - pairs). We consider nei- ther intermediate nor supersymmetric . The vacuum permeability μ0, the vacuum permittivity 0, All known species of charged are taken into ac- and the speed of light in vacuum c are widely considered count: the three families of charged e, μ and τ and as being fundamental constants and their values, escaping the three families of (u, d), (c, s)and(t, b), includ- any physical explanation, are commonly assumed to be ing their three color states. This gives a total of 21 pair invariant in space and time. In this paper, we propose a species, noted i. mechanism based upon a “natural” quantum vacuum de- An ephemeral fermion pair is assumed to be the prod- scription which leads to sensible estimations of these three uct of the fusion of two virtual of the vacuum. electromagnetic constants. A consequence of this descrip- Thus its total and total color are null. We tion is that μ0, 0 and c are not fundamental constants but suppose also that the spins of the two fermions of a pair observable parameters of : they can are antiparallel, and that they are on their mass shell. vary if the vacuum properties vary in space or in time. A The only quantity which is not conserved is therefore the similar analysis of the quantum vacuum, as the physical energy and this is the reason for the limited lifetime of origin of the electromagnetism constants, has been pro- the pairs. We assume that first order properties can be posed independently by Leuchs et al. [1]. Although the two deduced assuming that pairs are created with an average mechanisms are different, the original idea is the same: the energy, not taking into account a full probability density physical electromagnetic constants emerge naturally from of the pairs kinetic energy. Likewise, we will neglect the the quantum theory. total momentum of the pair. The paper is organized as follows. First we describe our The average energy Wi of a pair is taken proportional model of the quantum vacuum filled with continuously ap- m c2 c to its rest mass energy 2 i rel,where rel is the maxi- pearing and disappearing fermion pairs. We show how μ0 mum velocity introduced in the Lorentz transformation. 0 and originate respectively from the magnetization and We remind that crel is not necessarily equal to the speed the electric polarization of these pairs. We then derive the of light. We note: photon velocity in vacuum by modeling its propagation W K m c2 , as a series of interactions with the pairs. Finally, we pre- i = W 2 i rel (1) dict statistical fluctuations of the transit time of photons across a fixed vacuum path. where KW is a constant, assumed to be independent from the fermion type. We take KW as a free parameter; its value could be calculated if we knew the energy spectrum 2 An effective description of the quantum of the virtual photons together with their probability to vacuum create fermion pairs. As a reminiscence of the Heisenberg principle, the pairs The vacuum is assumed to be filled with continu- lifetime τi is assumed to be given by ously appearing and disappearing charged fermion pairs  1  a τ = = . (2) e-mail: [email protected] i W K m c2 2 i W 4 i rel Page 2 of 6

We assume that the ephemeral fermion pairs densities Ni the charged fermion pairs under a magnetic stress. Each are driven by the Pauli exclusion principle. Two pairs con- charged ephemeral fermion carries a magnetic moment taining two identical fermions in the same state can- proportional to the Bohr magneton not show up at the same time at the same place. However eQ eQ c λ at a given location we may find 21 charged fermion pairs i i rel Ci μi = = . (7) since different fermions can superpose spatially. In solid 2mi 4π state the successful determination of Fermi ener- gies [2] implies that one spin state occupies a hy- We assume the orbital moment and the spin of the pair to per volume h3. We assume that concerning the Pauli prin- be zero. Since the fermion and the anti fermion have op- ciple, the ephemeral fermions are similar to the real ones. posite electric charges, the pair carries twice the magnetic Noting Δxi the spacing between identical i-type fermions moment of one fermion. and their average momentum, the one dimension hyper If no external magnetic field is present, the magnetic volume is piΔxi and dividing by h should give the num- moments point randomly in any direction resulting in a ber of states which we take as one per spin degree of free- null global average magnetic moment. In the presence of B dom. The relation between pi and Δxi reads piΔxi/h =1, an external magnetic field , the coupling energy of the i − μ B θ θ or Δxi =2π/pi. -typepairtothisfieldis 2 i cos ,where is the angle B We can express Δxi as a function of Wi if we suppose between the magnetic moment and the magnetic field . the relativity to hold for the ephemeral pairs The energy of the pair is modified by this term and the pair lifetime is therefore a function of the orientation of πc λ 2 rel Ci its magnetic moment with respect to the applied magnetic Δxi =  =  , (3) (W /2)2 − (m c2 )2 K2 − 1 field: i i rel W / τ θ 2 . λ i i( )= (8) where Ci is the Compton length associated to fermion Wi − 2μiB cos θ and is given by: h The pairs having their magnetic moment aligned with the λ . Ci = (4) field last a bit longer than the anti-aligned pairs. The re- micrel sulting average magnetic moment Mi of a pair is there- The pair density is defined as: fore different from zero1 and is aligned with the applied  3 field. Its value is obtained integrating over θ with a weight K2 − 1 N ≈ 1 W . proportional to the pairs lifetime: i Δx3 = λ (5) i Ci  π 2μ cos θτ(θ)2π sin θdθ M  0 i i . Each pair can be produced only in the two fermion- i = π (9) 0 τi(θ)2π sin θdθ antifermion spin combinations up-down and down-up. N We define i as the density of pairs for a given spin To first order in B,onegets: combination. 2 Finally, we use the notation Qi = qi/e,whereqi is the 4μ i e M  i B. (10) -type fermion electric charge and the modulus of the i 3W electron charge. i The magnetic moment per unit volume produced by the i-type fermions is Mi =2NiMi, since one takes into ac- 3 The vacuum permeability count the two spin states per cell. The contributionμ ˜0,i of the i-type fermions to the vacuum permeability is When a torus of a material is energized through a winding thus given by B =˜μ0,iMi or 1/μ˜0,i = Mi/B.Each carrying a current I, the resulting magnetic flux density B species of fermions increases the induced magnetization is expressed as: and therefore the magnetic moment. By summing over B μ nI μ M, all pair species, one gets the estimation of the vacuum = 0 + 0 (6) permeability: n nI where is the number of turns per unit of length and  M e2  N Q2λ2 is the magnetic in A/m. M is the correspond- 1 i c2 i i C i. = = rel 2 (11) μ˜0 B 6π W ing magnetization induced in the material and is the sum i i i of the induced magnetic moments divided by the corre- sponding volume. In an experiment where the current I is Using equations (1), (4) and (5) and summing over all pair kept a constant and where we lower the quantity of mat- types, one obtains ter in the torus, B decreases. As we remove all , 3 B gets to a non zero value: B = μ0nI showing experi- KW 24π  μ˜0 =  . (12) mentally that the vacuum is paramagnetic with a vacuum (K2 − 1)3/2 c e2 Q2 −7 2 W rel i i permeability μ0 =4π 10 N/A . We propose a physical mechanism to produce the vac- 1 As a referee puts it: “This is a kind of averaged Zeeman uum permeability from the elementary magnetization of effect”. Page 3 of 6  Q2 The sum i i is taken over all pair types. Within a gen- to be separated and the of a parallel-plate ca- eration, the absolute values of the electric charges are 1, pacitor would go to zero when one removes all 2/3 and 1/3 in units of the charge. Thus for one from the gas. generation the sum writes (1 + 3 × (4/9+1/9)). The fac- We show here that our vacuum filled by ephemeral tor 3 is the number of colours. Hence, for the three families fermions causes its electric charges to be separated and to of the appear at the level of 5 × 107 electron charges per square  meter under an electric stress E =1V/m.Themechanism Q2 . i =8 (13) is similar to the one proposed for the permeability. How- i ever, we must assume here that every fermion-antifermion ephemeral pair of the i-type bears a mean electric di One obtains: given by: 3 di = Qieδi, (16) KW 3π  μ0 . ˜ = 2 3/2 2 (14) where δ is the average separation between the two (K − 1) crel e i W fermions of the pair. We assume that this separation does The calculated vacuum permeabilityμ ˜0 is equal to the not depend upon the fermion momentum and we use the λ / π observed value μ0 when reduced of the fermion Ci (2 )as this scale: λ 2 Ci KW crel e 4 α δ  . (17) = μ0 = , (15) i π (K2 − 1)3/2 3π3 3 π2 2 W If no external electric field is present, the point randomly in any direction and their resulting average field which is obtained for KW ≈ 31.9. is zero. In presence of an external electric field E,the Such a KW value indicates that the typical fermions are produced in relativistic states. This estimation is mean polarization of these ephemeral fermion pairs pro- based upon a static and average description of vacuum. duce the observed vacuum permittivity 0. This polariza- A more complete view, including probability densities on tion shows up due to the dipole lifetime dependence on the pair energy and momentum distributions might allow to electrostatic coupling energy of the dipole to the field. In a field homogeneous at the δ scale, this energy is d E cos θ give a physical meaning to the KW value. For instance, i i + − 2 θ e e pairs with a total energy distributed as dW/W up where is the angle between the ephemeral dipole and the electric field E. The electric field modifies the pairs to Wmax would give an apparent KW of the order of  lifetimes according to their orientation: W max /2 KW  Log  51, τ θ . 2m c2 i( )= (18) e Wi − diE cos θ if Wmax corresponds to the Planck energy. As in the magnetostatic case, pairs with a dipole moment aligned with the field last a bit longer than the others. This leads to a non zero average dipole Di,whichis E 4 The vacuum permittivity aligned with the electric field and given, to first order in E,by: d2 D  i E. Consider a parallel-plate capacitor with a gas inside. i W (19) When the pressure of the gas decreases, the capacitance 3 i i decreases too until there are no more molecules in between We estimate the permittivity ˜0,i due to -type fermions P E P the plates. The strange thing is that the capacitance is not using the relation i =˜0,i , where the polarization i is P N D  zero when we hit the vacuum. In fact the capacitance has equal to the dipole density i =2 i i ,sincethetwo a very sizeable value as if the vacuum were a usual ma- spin combinations contribute. Thus: terial body. The constant of a medium is com- D  Q2δ2 i 2 i i ing from the existence of opposite electric charges that ˜0,i =2Ni =2Nie . (20) E 3W can be separated under the influence of an applied elec- i tric field E. Furthermore the opposite charges separation Each species of fermion increases the induced polarization stays finite because they are bound in a . These and therefore the vacuum permittivity. By summing over opposite translations result in opposite charges appearing all pair species, one gets the general expression of the vac- on the dielectric surfaces in regard to the metallic plates. uum permittivity: This leads to a decrease of the effective charge, which im- e2  N Q2δ2 e2  N Q2λ2 2 i i i i i C i plies a decrease of the voltage across the dielectric slab ˜0 = = . (21) 3 W 6π2 W and finally to an increase of the capacitance. In our model i i i i of the vacuum the ephemeral charged fermion pairs are Expressing the model parameters from equations (1), (4), the pairs of opposite charge and the separation stays fi- (5), and (13), one gets: nite because the electric field acts only during the lifetime of the pairs. In an absolute empty vacuum, the induced K2 − 3/2 e2 ( W 1) . charges would be null because there would be no charges ˜0 = 3 (22) KW 3π crel Page 4 of 6

If we now use the value KW given in equation (15) Using equation (24), we obtain the average photon veloc- obtained from the derivation of the permeabilitty, one ity cgroup as a function of three parameters of the vacuum gets the right numerical value for the permittivity: model: −12 ˜0 =8.85 × 10 F/m. L 1 c = =  . (26) group T σ N τ / We verify from equations (11) and (21) that the phase i i i i 2 velocity cφ √of an electromagnetic wave in vacuum, given by cφ =1/ μ˜0˜0,isequaltocrel the maximum velocity Using equations (2) and (5), we get the expression used in special relativity. K 16π We also notice that the permeability and the permit- c = W  c . (27) group 2 3/2 2 rel tivity do not depend upon the masses of the fermions. K − (σi/λ ) ( W 1) i Ci The electric charges and the number of species are the only important parameters. This is in opposition to the We now have to define the expression of the cross sec- common idea that the energy density of the vacuum is tion σi. We know that it should not depend on the photon the dominant factor [3]. energy, otherwise the vacuum would become a dispersive medium. Also the interaction of a real photon with a pair must not exchange energy or momentum with the vacuum 5 The propagation of a photon in vacuum (for instance, is not possible). We as- sume the cross-section to be proportional to the geomet- rical cross-section of the pair λ2 , and to the square of the We now study the propagation of a real photon in vacuum Ci electric charge Q2. The cross-section is thus expressed as: and we propose a mechanism leading to a finite average i photon velocity cgroup, which must be equal to cφ and crel. σ = k Q2λ2 , (28) When a real photon propagates in vacuum, it interacts i σ i Ci with and is temporarily captured by an ephemeral pair. where kσ is a constant which does not depend on the type As soon as the pair disappears, it releases the photon to of fermions. its initial energy and momentum state. The photon con- The calculated photon velocity becomes: tinues to propagate with an infinite bare velocity. Then the photon interacts again with another ephemeral pair and K 16π so on. The delay on the photon propagation produced by c = W  c . (29) group 2 3/2 2 rel K − kσ Q these successive interactions implies a renormalisation of ( W 1) i i this bare velocity to a finite value. This “leapfrog” propagation of photons, with instan- Using equations (13) and (15), one finally get: taneous leaps between pairs, seems natural since the only length and time scales in vacuum come from fermion pair 8α cgroup = crel. (30) lifetimes and Compton lengths. This idea is far from being 3πkσ a new one, as can be found for instance in reference [4]. c By defining σ as the cross-section for a real photon to The calculated velocity group of a photon in vacuum is i c interact and to be trapped by an ephemeral i-type pair of equal on average to rel when fermions, the mean free path of the photon between two 8 successive such interactions is given by: k = α. (31) σ 3π 1 Λ , −26 2 i = (23) It corresponds to a cross-section of 4 × 10 m on an σiNi ephemeral electron-positron pair, of the same order as the where Ni is the numerical density of virtual i-type pairs. geometric transversal area of the pair, whose size is given Travelling a distance L in vacuum leads on average in equation (17). to N stop,i interactions on the i-type pairs, given by: We note that the photon velocity depends only on the electrical charge units Qi of the ephemeral charged L fermions present in vacuum. It depends neither upon their N Lσ N . stop,i = Λ = i i (24) masses, nor upon the vacuum energy density. We also re- mark that the average speed of the photon in our medium The photon may encounter the pair any time between its being crel, the photon propagates, on average, along the appearence and disappearence. The life time of a pair be- light cone. As such, the effective average speed of the ing τi, the photon will be stopped for an average time τi/2. photon is independent of the inertial frame as demanded Each type of fermion pair contributes in increasing the by relativity. This mechanism relies on the notion of an propagation time of the photon. So, the total mean time T absolute frame for the vacuum at rest. It satisfies spe- foraphotontocrossalengthL is: cial relativity in the Lorentz-Fitzgerald sense. This simple model does not preclude some dependence of the speed of  τ T = N i. (25) light on the photon energy, through trapping cross-section stop,i 2 i variations. Page 5 of 6

6 Transit time fluctuations linearly with both the distance L and the energy of the photons. An important consequence of our model is that stochastic A way to search for these fluctuations is to measure fluctuations of the propagation time of photons in vacuum a possible time broadening of a light pulse travelling a are expected, due to the fluctuations of the number of distance L of vacuum. This may be done using observa- interactions of the photon with the virtual pairs and to tions of brief astrophysical events, or dedicated laboratory the capture time fluctuations. experiments. These stochastic fluctuations are not expected in stan- The strongest direct constraint from astrophysical ob- dard , which considers c as a servations is obtained with the very bright GRB 090510, given, non fluctuating, quantity. theo- detected by the Fermi Gamma-ray Space Telescope [8], ries predict also stochastic fluctuations of the propagation at MeV and GeV energy scale. It presents short spikes in time of photons [5,6]. It has been also recently predicted the 8 keV−5 MeV energy range, with the narrowest widths that the non commutative geometry at the Planck scale of the order of 4 ms (rms). Observation of the optical af- should produce a spatially coherent space-time jitter [7]. ter glow, a few days later by ground based spectroscopic We show here that our effective model of photon propaga- telescopes gives a common redshift of z =0.9. This corre- tion predicts fluctuations at a higher scale, which makes sponds to a distance, using standard cosmological param- 26 it experimentally testable with femtosecond pulses. eters, of about 2 × 10 m. Assuming that the observed The propagation time T of a photon which crosses a width is correlated to the emission properties, this sets a −1/2 distance L of vacuum is: limit for transit time fluctuations σT of about 0.3 fs m . It is important to notice that there is no expected disper-  Nstop,i sion of the bursts in the interstellar medium at this en- T = ti,k, (32) ergy scale. If we move six orders of magnitude down in i=1 k=1 distances we arrive to kpc and pulsars. Short microbursts t k contained in main pulses from the Crab pulsar have been where i,k is the duration of the th interaction on recently observed at the Arecibo Observatory Telescope i-type pairs and Nstop,i the number of such interactions. T at 5 GHz [9]. The frequency-dependent delay caused by The variance of , due to the statistical fluctuations of dispersive propagation through the interstellar plasma is the number of interactions and the fluctuation of the cap- corrected using a coherent dispersion removal technique. ture time is given by: The mean time width of these microbursts after dedisper-  sion is about 1 μs, much larger than the expected broaden- σ2 σ2 t2 N σ2 , T = Nstop,i stop,i + stop,i t,i (33) ing caused by interstellar scattering. Assuming again that i the observed width is correlated to the emission prop- erties, this sets a limit for transit time fluctuations of t τ / i −1 2 where stop,i = i 2 is the average stop time on a -type about 0.2 fs m / . pair, σ2 = τ 2/12 its variance, and σ2 = N the t,i i Nstop,i stop,i The very fact that the predicted statistical fluctua- variance of the number of interactions. Hence: tions should go like the square root of the distance implies 1  L  the exciting idea that experiments on Earth do compete σ2 = N τ 2 = σ N τ 2. (34) with astrophysical constraints since we expect fluctuations T 3 stop,i i 3 i i i i i in the femtosecond range at the kilometer scale. An ex- perimental setup using femtosecond laser pulses sent to Once reduced, the current term of the sum is proportional a 100 m long multi-pass vacuum cavity equipped with λ to Ci . Therefore the fluctuations of the propagation time + − metallic mirrors could be able to detect this phenomenon. are dominated by virtual e e pairs. Neglecting the other With appropriate mirrors with no dispersion on the reflec- σ N τ / / c fermion species, and using e e e 2=1 (8 ), one gets tions, a pulse with an initial time width of 9 fs (FWHM) would be broadened after 30 round trips in the cavity, to τ L λ L σ2 e Ce . an output time width of ∼13 fs (FWHM). An accurate T = = 2 (35) 12c 96πKW c autocorrelation measurement could detect this effect. So L λ Ce 1 σT = √ . (36) 7 Conclusions c c 96πKW We describe the ground state of the unperturbed vac- In our simple model where KW =31.9, the predicted fluc- tuation is: uum as containing a finite density of charged ephemeral −2 −1/2 fermions antifermions pairs. Within this framework, 0 σT ≈ 5 × 10 fs m . (37) and μ0 originate simply from the electric polarization and We note that the fluctuations vary as the square root of from the magnetization of these pairs when the vacuum the distance L of vacuum crossed by the photons and are is stressed by an electrostatic or a magnetostatic field re- a priori independent of the energy of the photons. It is spectively. Our calculated values for 0 and μ0 are equal in contrast with expected fluctuations calculated in the to the measured values when the fermion pairs are pro- frame of Quantum-Gravitational Diffusion [6], which vary duced with an average energy of about 30 times their Page 6 of 6 rest mass. The finite speed of a photon is due to its suc- J.Degert,E.Freysz,J.Oberl´e and M. Tondusson for their cessive transient captures by these virtual particles. This collaboration on the experimental aspects. This work has ben- 2 model, which proposes a quantum origin to the electro- efited from a GRAM funding. magnetic constants 0 and μ0 and to the speed of light, is self consistent: the average velocity of the photon cgroup, c References the phase velocity√ of the electromagnetic wave φ,given by cφ =1/ μ00, and the maximum velocity used in spe- 1. G. Leuchs, A.S. Villar, L.L. Sanchez-Soto, Appl. Phys. B cial relativity crel are equal. The propagation of a photon 100, 9 (2010) being a statistical process, we predict√ fluctuations of its time of flight of the order of 0.05 fs/ m.Thiscouldbe 2. Ch. Kittel, Elementary Solid State Physics (John Wiley & within the grasp of modern experimental techniques and Sons, New York, 1962), p. 120 3. J.I. Latorre et al., Nucl. Phys. B 437, 60 (1995) we plan to assemble such an experiment. 4. R.H. Dicke, Rev. Mod. Phys. 29, 363 (1957) 5. H. Yu, L.H. Ford, Phys. Lett. B 496, 107 (2000) 6. J. Ellis et al., Gen. Rel. Grav. 32, 127 (2000) It is a pleasure acknowledging helpful discussions with Gerd 7. C.J. Hogan, FERMILAB-PUB-10-036-A-T, arXiv: Leuchs. The authors thank also J.P. Chambaret, I. Cognard, 1002.4880v27 J. Ha¨ıssinski, P. Indelicato, J. Kaplan, C. Rizzo, P. Wolf and 8. A.A. Abdo et al., Nature 462, 331 (2009) F. Zomer for fruitful discussions, and N. Bhat, E. Constant, 9. J.H. Crossley et al., Astrophys. J. 722, 1908 (2010)

2 CNRS INSU/INP program with CNES & ONERA participations (Action Sp´ecifique “Gravitation, R´ef´erences, Astronomie, M´etrologie”).