A Mechanism Giving a Finite Value to the Speed of Light, and Some Experimental Consequences

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A mechanism giving a ﬁnite value to the speed of light, and some experimental consequences M. Urban, F. Couchot, X. Sarazin

To cite this version: M. Urban, F. Couchot, X. Sarazin. A mechanism giving a ﬁnite value to the speed of light, and some experimental consequences. LAL 11-289. Submitted to Eur. Phys. J. B. 2012.

HAL Id: in2p3-00639073 http://hal.in2p3.fr/in2p3-00639073v2 Submitted on 13 Sep 2012

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A model of quantum vacuum as the origin of the speed of light

Marcel Urban, Fran¸cois Couchot and Xavier Sarazin LAL, Univ Paris-Sud, CNRS/IN2P3, Orsay, France

Received: September 12, 2012 / Revised version: date

Abstract. We show that the vacuum permeability 0 and permittivity ǫ0 may originate from the magnetization and the polarization of continuously appearing and disappearing fermion pairs. We then show that if we simply model the propagation of the photon 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 constraints 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.

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1 Introduction particle-antiparticle pairs). We consider neither intermedi- ate bosons nor supersymmetric particles. All known species The vacuum permeability 0, the vacuum permittivity ǫ0, of charged fermions are taken into account: the three fam- and the speed of light in vacuum c are widely considered ilies of charged leptons e, and τ and the three families of as being fundamental constants and their values, escaping quarks (u, d), (c, s) and (t, b), including their three color any physical explanation, are commonly assumed to be states. This gives a total of 21 pair species, noted i. invariant in space and time. In this paper, we propose a An ephemeral fermion pair is assumed to be the prod- mechanism based upon a ”natural” quantum vacuum de- uct of the fusion of two virtual photons of the vacuum. scription which leads to sensible estimations of these three Thus its total electric charge and total color are null. We electromagnetic constants. A consequence of this descrip- suppose also that the spins of the two fermions of a pair tion is that 0, ǫ0 and c are not fundamental constants are antiparallel, and that they are on their mass shell. but observable parameters of the quantum vacuum: they The only quantity which is not conserved is therefore the can vary if the vacuum properties vary in space or in time. energy and this is the reason for the limited lifetime of A similar analysis of the quantum vacuum, as the phys- the pairs. We assume that first order properties can be ical origin of the electromagnetism constants, has been deduced assuming that pairs are created with an average proposed independently by Leuchs, Villar and Sanchez- energy, not taking into account a full probability density Soto [1]. Although the two mechanisms are different, the of the pairs kinetic energy. Likewise, we will neglect the original idea is the same: the physical electromagnetic con- total momentum of the pair. stants emerge naturally from the quantum theory. The average energy Wi of a pair is taken proportional 2 The paper is organized as follows. First we describe our to its rest mass energy 2mic : model of the quantum vacuum filled with continuously ap- 2 pearing and disappearing fermion pairs. We show how 0 Wi = KW 2mic (1) and ǫ0 originate respectively from the magnetization and the electric polarization of these pairs. We then derive the where KW is a constant, assumed to be independent from photon velocity in vacuum by modeling its propagation the fermion type. We take KW as a free parameter, greater as a series of interactions with the pairs. Finally, we pre- than unity. The value of KW could be calculated if we dict statistical fluctuations of the transit time of photons knew the energy spectrum of the virtual photons together across a fixed vacuum path. with their probability to create fermion pairs. The pairs lifetime τi is given by the Heisenberg uncer- tainty principle (Wiτi =h ¯ 2). So

2 An effective description of the quantum 1 ¯h vacuum τi = 2 (2) KW 4mic The vacuum is assumed to be filled with continuously ap- We assume that the ephemeral fermion pairs densities pearing and disappearing charged fermion pairs (ephemeral Ni are driven by the Pauli Exclusion Principle. Two pairs 2 Marcel Urban, Fran¸cois Couchot, Xavier Sarazin: A model of quantum vacuum as the origin of the speed of light containing two identical fermions in the same spin state Since the total spin of the pair is zero, and since fermion cannot show up at the same time at the same place. How- and antifermion have opposite charges, each pair carries ever at a given location we may find 21 charged fermion twice the magnetic moment of one fermion. pairs since different fermions can superpose spatially. In If no external magnetic field is present, the magnetic solid state physics the successful determination of Fermi moments point randomly in any direction resulting in a energies[2] implies that one electron spin state occupies a null global average magnetic moment. In the presence of hyper volume h3. We assume that concerning the Pauli an external magnetic field B, the coupling energy of the principle, the ephemeral fermions are similar to the real i-type pair to this field is 2 B cos θ, where θ is the angle − i ones. Noting xi the spacing between identical i-type between the magnetic moment and the magnetic field B. fermions and pi their average momentum, the one dimen- The energy of the pair is modified by this term and the sion hyper volume is pixi and dividing by h should give pair lifetime is therefore a function of the orientation of the number of states which we take as one per spin de- its magnetic moment with respect to the applied magnetic gree of freedom. The relation between pi and xi reads field: pixih = 1, or xi = 2π¯hpi. We can express x as a function of W if we suppose ¯h2 i i τi(θ) = (7) the relativity to hold for the ephemeral pairs Wi 2iB cos θ − 2π¯hc λ x = = Ci (3) The pairs having their magnetic moment aligned with i 2 2 2 2 the field last a bit longer than the anti-aligned pairs. The (Wi2) (mic ) KW 1 − − resulting average magnetic moment of a pair is there- Mi where λCi = hp(mic) is the Compton lengthp associated to fore different from zero. It is aligned with the applied field. fermion i. Its value is obtained integrating over θ with a weight pro- The pair density is defined as: portional to the pairs lifetime: 3 2 π 1 KW 1 0 2i cos θ τi(θ) 2π sin θ dθ Ni 3 = − (4) i = π (8) ≈ x λC i p i ! M R 0 τi(θ) 2π sin θ dθ Each pair can be produced only in the two fermion- To first order in B,R one gets: antifermion spin combinations up-down and down-up. We define Ni as the density of pairs for a given spin combina- 2 4i tion. i B (9) M ≃ 3Wi Finally, we use the notation Qi = qie, where qi is the i-type fermion electric charge and e the modulus of the The magnetic moment per unit volume produced by electron charge. the i-type fermions is Mi = 2Ni i , since one takes into account the two spin states perM cell. The contribution ˜0i of the i-type fermions to the vacuum permeability 3 The vacuum permeability is thus given by B = ˜0iMi or 1˜0i = MiB. Each species of fermions increases the induced magnetization When a torus of a material is energized through a winding and therefore the magnetic moment. By summing over carrying a current I, the resulting magnetic flux density all pair species, one gets the estimation of the vacuum B is expressed as: permeability: B = 0nI + 0M (5) 1 M 8N 2 = i = i i (10) where n is the number of turns per unit of length and nI ˜0 B 3W i i i is the magnetic intensity in Am. M is the corresponding X X magnetization induced in the material and is the sum of Using Eq. (1), (4) and (6) and summing over all pair the induced magnetic moments divided by the correspond- types, one obtains ing volume. In an experiment where the current I is kept a constant and where we lower the quantity of matter in 3 KW 24π ¯h the torus, B decreases. As we remove all matter, B gets to ˜0 = (11) (K2 1)32 c e2 Q2 a non zero value: B = 0nI showing experimentally that W − i i the vacuum is paramagnetic with a vacuum permeability 2 −7 2 The sum Q is taken over allP pair types. Within a 0 = 4π 10 NA . i i generation the absolute values of the electric charges are 1, We propose a physical mechanism to produce the vac- P uum permeability from the elementary magnetization of 2/3 and 1/3 in units of the positron charge. Thus for one generation the sum writes (1+3 (49+19)). The factor the charged fermion pairs under a magnetic stress. Each × charged ephemeral fermion carries a magnetic moment 3 is the number of colours. Each generation contributes proportional to the Bohr magneton equally, hence for the three families of the standard model 2 e Qi¯h e Qi c λCi Q = 8 (12) i = = (6) i 2m 4π i i X Marcel Urban, Fran¸cois Couchot, Xavier Sarazin: A model of quantum vacuum as the origin of the speed of light 3

One obtains: is zero. In presence of an external electric field E, the mean 3 polarization of these ephemeral fermion pairs produce the KW 3π ¯h ˜0 = (13) observed vacuum permittivity ǫ0. This polarization shows (K2 1)32 c e2 W − up due to the dipole lifetime dependence on the electrostatic coupling energy of the dipole to the field. In a field The calculated vacuum permeability ˜0 is equal to the homogeneous at the δi scale, this energy is diE cos θ where observed value 0 when θ is the angle between the ephemeral dipole and the elec- 2 E KW c e 4 α tric field . The electric field modifies the pairs lifetimes = 0 = (14) (K2 1)32 3π3¯h 3 π2 according to their orientation: W − ¯h2 which is obtained for KW 319, greater than one as τi(θ) = (17) required. In this model, we have≈ ignored the charged loop Wi diE cos θ − contribution. 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 , which is 4 The vacuum permittivity aligned with the electric field E and given, to first order in E, by: Consider a parallel-plate capacitor with a gas inside. When the pressure of the gas decreases, the capacitance decreases 2 di too until there is no more molecules in between the plates. Di E (18) ≃ 3W The strange thing is that the capacitance is not zero when i we hit the vacuum. In fact the capacitance has a very size- We estimate the permittivity ˜ǫ0i due to i-type fermions able value as if the vacuum were a usual material body. using the relation Pi =ǫ ˜0iE, where the polarization Pi is The dielectric constant of a medium is coming from the ex- equal to the dipole density Pi = 2Ni Di , since the two istence of opposite electric charges that can be separated spin combinations contribute. Thus: under the influence of an applied electric field E. Further- 2 2 more the opposite charges separation stays finite because Di 2 Qi δi ˜ǫ0i = 2Ni = 2Nie (19) they are bound in a molecule. These opposite translations E 3Wi result in opposite charges appearing on the dielectric sur- faces in regard to the metallic plates. This leads to a de- Each species of fermion increases the induced polariza- crease of the effective charge, which implies a decrease of tion and therefore the vacuum permittivity. By summing the voltage across the dielectric slab and finally to an in- over all pair species, one gets the general expression of the crease of the capacitance. In our model of the vacuum the vacuum permittivity: ephemeral charged fermion pairs are the pairs of opposite 2 2 2 2e NiQi δi charge and the separation stays finite because the electric ǫ˜0 = (20) 3 W field acts only during the lifetime of the pairs. In an ab- i i solute empty vacuum, the induced charges would be null X because there would be no charges to be separated and Expressing the model parameters from Eq. (1), (4), the capacitance of a parallel-plate capacitor would go to (12) and (16), one gets: zero when one removes all molecules from the gas. We show here that our vacuum filled by ephemeral 2 32 2 fermions causes its electric charges to be separated and (KW 1) e 7 ˜ǫ0 = − 3 (21) to appear at the level of 510 electron charges per square KW 3π ¯hc meter under an electric stress E = 1 Vm. The mechanism is similar to the one proposed for the permeability. How- If we now use the value KW given in Eq. (14) obtained from the derivation of the permeabilitty, one gets the right ever, we must assume here that every fermion-antifermion −12 ephemeral pair of the i-type bears a mean electric dipole numerical value for the permittivity: ˜ǫ0 = 885 10 Fm. We notice that the permeability and the permittiv- di given by: ity do not depend upon the masses of the fermions. The di = Qieδi (15) electric charges and the number of species are the only important parameters. This is in opposition to the com- where δi is the average separation between the two fermions mon idea that the energy density of the vacuum is the of the pair. We assume that it does not depend upon dominant factor[3]. the fermion momentum and we use the reduced Comp- ton wavelength of the fermion λCi (2π) as this scale: 5 The propagation of a photon in vacuum λCi δi (16) ≃ 2π When a real photon propagates in vacuum, it interacts If no external electric field is present, the dipoles point with and is temporarily captured by an ephemeral pair. randomly in any direction and their resulting average field As soon as the pair disappears, it releases the photon to its 4 Marcel Urban, Fran¸cois Couchot, Xavier Sarazin: A model of quantum vacuum as the origin of the speed of light initial energy and momentum state. The photon continues Using Eq. (12) and (14), one finally get: to propagate with an infinite velocity. Then the photon 8α interacts again with another ephemeral pair and so on. c˜ = c (29) The delay on the photon propagation produced by these 3πkσ successive interactions implies that the velocity of light is finite. The calculated velocityc ˜ of a photon in vacuum is By defining σi as the cross-section for a real photon to equal to the observed value c when interact and to be trapped by an ephemeral i-type pair of fermions, the mean free path of the photon between two 8 kσ = α (30) successive such interactions is given by: 3π −26 2 1 It corresponds to a cross-section of 4 10 m on an Λi = (22) ephemeral electron-positron pair, of the same order than σiNi the geometric transversal area of the pair, whose size is where Ni is the numerical density of virtual i-type pairs. given in Eq. (16). Travelling a distance L in vacuum leads on average to We note that the photon velocity depends only on Nstopi interactions on the i-type pairs, given by: the electrical charge units Qi of the ephemeral charged fermions present in vacuum. It depends neither upon their L masses, nor upon the vacuum energy density. We also re- N = = Lσ N (23) stopi Λ i i mark that the average speed of the photon in our medium being c, the photon propagates, on average, along the light The photon may encounter the pair any time between cone. As such, the effective average speed of the photon its appearence and disappearence. The life time of a pair is independent of the inertial frame as demanded by rela- being τi, the photon will be stopped for an average time tivity. This mechanism relies on the notion of an absolute τi2. Each type of fermion pair contributes in increasing frame for the vacuum at rest. It satisfies special relativity the propagation time of the photon. So, the total mean in the Lorentz-Poincarésense. time T for a photon to cross a length L is: τ T = N i (24) stopi 2 6 Transit time fluctuations i X Using Eq. (23), we obtain the photon velocityc ˜ as a An important consequence of our model is that stochastic function of three parameters of the vacuum model: fluctuations of the propagation time of photons in vacuum are expected, due to the fluctuations of the number L 1 of interactions of the photon with the virtual pairs and to c˜ = = (25) the capture time fluctuations. Quantum gravity theories, T σiNiτi2 i which include stochastic fluctuations of the metric of com- Using Eq. (2) and (4),P we get the expression pactified dimensions, predict also fluctuations of the propagation time of photons[4]. However observable effects are K 16π expected to be too small to be experimentally testable. It c˜ = W c (26) 2 32 (σ λ2 ) has been also recently predicted that the non commutative (K 1) i i Ci W − geometry at the Planck scale should produce a spatially coherent space-time jitter[5]. We show here that our ef- We now have to define theP expression of the cross sec- fective model of photon propagation predicts fluctuations tion σ . We know that it should not depend on the photon i at a higher scale, which makes it experimentally testable energy, otherwise the vacuum would become a dispersive with femtosecond pulses. medium. Also the interaction of a real photon with a pair The propagation time T of a photon which crosses a must not exchange energy or momentum with the vacuum distance L of vacuum is: (for instance, Compton scattering is not possible). We assume the cross-section to be proportional to the geomet- Nstopi rical cross-section of the pair λ2 , and to the square of the Ci T = tik (31) 2 electric charge Q . The cross-section is thus expressed as: i=1 =1 i X Xk 2 2 th σi = kσQi λCi (27) where tik is the duration of the k interaction on i-type pairs and Nstopi the number of such interactions. The where kσ is a constant which does not depend on the type variance of T , due to the statistical fluctuations of the of fermions. number of interactions and the fluctuation of the capture The calculated photon velocity becomes: time is given by:

KW 16π 2 2 2 2 c˜ = c (28) σ = σ t + Nstopi σ (32) 2 32 k Q2 T Nstopi stopi ti (K 1) σ i i i W − X P Marcel Urban, Fran¸cois Couchot, Xavier Sarazin: A model of quantum vacuum as the origin of the speed of light 5

where tstopi = τi2 is the average stop time on a i-type An experimental setup using femtosecond laser pulses sent 2 2 2 pair, σti = τi 12 its variance, and σNstopi = Nstopi the to a 100 m long multi-pass vacuum cavity equipped with variance of the number of interactions. Hence: metallic mirrors could be able to detect this phenomenon. With appropriate mirrors with no dispersion on the reflec- 1 L σ2 = N 2 τ 2 = σ N τ 2 (33) tions, a pulse with an initial time width of 9 fs (FWHM) T 3 stopi i 3 i i i i i would be broadened after 30 round trips in the cavity, to X X an output time width of 13 fs (FWHM). An accurate Once reduced, the current term of the sum is proportional autocorrelation measurement∼ could detect this effect. to λCi . Therefore the fluctuations of the propagation time are dominated by virtual e+e− pairs. Neglecting the other fermion species, and using σeNeτe2 = 1(8c), one gets 7 Conclusions

2 τe L λCe L σT = = 2 (34) We describe the ground state of the unperturbed vac- 12c 96πKW c uum as containing a finite density of charged ephemeral So fermions antifermions pairs. Within this framework, ǫ0 and 0 originate simply from the electric polarization and L λCe 1 σT = (35) from the magnetization of these pairs when the vacuum r c r c √96πKW is stressed by an electrostatic or a magnetostatic field respectively. The finite speed of light is due to successive In our simple model where KW = 319, the predicted fluctuation is: transient captures of the photon by these virtual particles. Our calculated values for ǫ0, 0 and c are equal to −2 −12 σT 5 10 fsm (36) the measured values when the fermion pairs are produced ≈ with an average energy of about 30 times their rest mass. A way to search for these fluctuations is to measure This model is self consistent and it proposes a quantum a possible time broadening of a light pulse travelling a origin to the three electromagnetic constants. The propa- distance L of vacuum. This may be done using observa- gation of a photon being a statistical process, we predict tions of brief astrophysical events, or dedicated laboratory fluctuations of the speed of light. This could be within the experiments. grasp of modern experimental techniques and we plan to The strongest direct constraint from astrophysical ob- assemble such an experiment. servations is obtained with the very bright GRB 090510, detected by the Fermi Gamma-ray Space Telescope[6], at MeV and GeV energy scale. It presents short spikes in the Acknowledgments 8 keV 5 MeV energy range, with the narrowest widths of the− order of 4 ms (rms). Observation of the optical after glow, a few days later by ground based spectroscopic It is a pleasure acknowledging helpful discussions with Gerd Leuchs. The authors thank also N. Bhat, J.P. Cham- telescopes gives a common redshift of z = 09. This corre- baret , I. Cognard, J. Ha¨ıssinski, P. Indelicato, J. Ka- sponds to a distance, using standard cosmological parameters, of about 21026 m. Assuming that the observed width plan, P. Wolf and F. Zomer for fruitful discussions, and J. Degert, E. Freysz, J. Oberléand M. Tondusson for their is correlated to the emission properties, this sets a limit −12 collaboration on the experimental aspects. This work has for transit time fluctuations σT of about 0.3 fs.m . It 1 is important to notice that there is no expected disper- benefited from a GRAM funding. sion of the bursts in the interstellar medium at this energy scale. If we move six orders of magnitude down in distances we arrive to kpc and pulsars. Short microbursts References contained in main pulses from the Crab pulsar have been recently observed at the Arecibo Observatory telescope at 1. G. Leuchs, A.S. Villar and L.L. Sanchez-Soto, Appl. Phys. 100 5 GHz[7]. The frequency-dependent delay caused by dis- B (2010) 9-13 persive propagation through the interstellar plasma is cor- 2. Ch. Kittel, Elementary Solid State Physics (John Wiley & Sons, 1962) 120 rected using a coherent dispersion removal technique. The 437 mean time width of these microbursts after dedispersion 3. J. I. Latorre et al., Nuclear Physics B (1995) 60 4. Yu H. and Ford L.H., Phys. Lett. B 496 (2000) 107-112 is about 1 s, much larger than the expected broaden- 5. C. J. Hogan, FERMILAB-PUB-10-036-A-T, ing caused by interstellar scattering. Assuming again that arXiv:1002.4880v27 (2012) the observed width is correlated to the emission proper- 6. Abdo A.A. et al., Nature 462 (2009) 331-334 ties, this sets a limit for transit time fluctuations of about 7. Crossley J.H. et al., The Astrophysical Journal 722 (2010) −12 0.2 fs.m . 1908-1920 The very fact that the predicted statistical fluctuations should go like the square root of the distance implies the exciting idea that experiments on Earth do com- 1 CNRS INSU/INP program with CNES & ONERA par- pete with astrophysical constraints since we expect fluc- ticipations (Action Spécifique ”Gravitation, Références, As- tuations in the femtosecond range at the kilometer scale. tronomie, Métrologie”)