The Quantum Vacuum As the Origin of the Speed of Light

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We the the mass pairs. therefore is the their is this conserved pair on not and a are is energy of which they fermions quantity two only that The the We and vacuum. of null. antiparallel, the are spins color of the are total that photons and also virtual charge suppose two electric total of its fusion Thus the of uct tts hsgvsattlo 1pi pce,noted species, pair 21 of total a gives This states. fcagdfrin r ae noacut h he fam- three leptons the charged account: of into ilies taken are fermions species known charged All of particles. supersymmetric nor bosons ate intermedi- neither consider We ap- (ephemeral pairs). continuously pairs particle-antiparticle with fermion charged filled disappearing be and to pearing assumed is vacuum The quantum the of vacuum description effective An 2 u eoiyitoue nteLrnztransformation. Lorentz that the remind in We introduced velocity mum flgt enote: We light. of au ol ecluae fw nwteeeg spectrum energy the knew we if calculated be could value h vrg energy average The prod- the be to assumed is pair fermion ephemeral An ew xetistm fflgtt utae We fluctuate. to flight of time its expect we re spern emo ar.W hnso that show then We pairs. fermion isappearing ,tepoaaino htni statistical a is photon a of propagation the h, K rn htti eoiyi qa otespeed the to equal is velocity this that iring u W , eiso rnin atrswti these within captures transient of series d sacntn,asmdt eidpnetfrom independent be to assumed constant, a is ino udmna constants fundamental of tion ,( ), ǫ c 0 , c a rgnt rmtemagneti- the from originate may s rel W n ( and ) i sntncsaiyeult h speed the to equal necessarily not is = e W , K µ m i t , W fapi stknproportional taken is pair a of and i b K c ,icuigtertrecolor three their including ), rel 2 2 W m τ where , i safe aaee;its parameter; free a as n h he aiisof families three the and c rel 2 c rel stemaxi- the is i . (1) 2 M. Urban, F. Couchot, X. Sarazin, A. Djannati-Atai: The quantum vacuum as the origin of the speed of light of the virtual photons together with their probability to magnetization induced in the material and is the sum of create fermion pairs. the induced magnetic moments divided by the correspond- As a reminiscence of the Heisenberg principle, the pairs ing volume. In an experiment where the current I is kept lifetime τi is assumed to be given by a constant and where we lower the quantity of matter in the torus, B decreases. As we remove all matter, B gets to ¯h 1 ¯h a non zero value: B = µ0nI showing experimentally that τi = = 2 (2) 2Wi KW 4micrel the vacuum is paramagnetic with a vacuum permeability −7 2 µ0 =4π 10 N/A . We assume that the ephemeral fermion pairs densities Ni We propose a physical mechanism to produce the vac- are driven by the Pauli Exclusion Principle. Two pairs uum permeability from the elementary magnetization of containing two identical fermions in the same spin state the charged fermion pairs under a magnetic stress. Each cannot show up at the same time at the same place. How- charged ephemeral fermion carries a magnetic moment ever at a given location we may find 21 charged fermion proportional to the Bohr magneton pairs since different fermions can superpose spatially. In solid state physics the successful determination of Fermi e Qi¯h e Qi crel λCi energies[2] implies that one electron spin state occupies a µi = = . (7) 2mi 4π hyper volume h3. We assume that concerning the Pauli principle, the ephemeral fermions are similar to the real We assume the orbital moment and the spin of the ones. Noting ∆xi the spacing between identical i-type pair to be zero. Since the fermion and the anti fermion fermions and pi their average momentum, the one dimen- have opposite electric charges, the pair carries twice the sion hyper volume is pi∆xi and dividing by h should give magnetic moment of one fermion. the number of states which we take as one per spin de- If no external magnetic field is present, the magnetic gree of freedom. The relation between pi and ∆xi reads moments point randomly in any direction resulting in a pi∆xi/h = 1, or ∆xi =2π¯h/pi. null global average magnetic moment. In the presence of We can express ∆xi as a function of Wi if we suppose an external magnetic field B, the coupling energy of the the relativity to hold for the ephemeral pairs i-type pair to this field is 2µiB cos θ, where θ is the angle between the magnetic moment− and the magnetic field B. 2π¯hc λ ∆x = rel = Ci (3) The energy of the pair is modified by this term and the i 2 2 2 2 (Wi/2) (mic ) K 1 pair lifetime is therefore a function of the orientation of − rel W − its magnetic moment with respect to the applied magnetic p p where λCi is the Compton length associated to fermion i field: and is given by. ¯h/2 τ (θ)= . (8) h i W 2µ B cos θ λCi = (4) i i micrel − The pairs having their magnetic moment aligned with The pair density is defined as: the field last a bit longer than the anti-aligned pairs. The 3 resulting average magnetic moment i of a pair is there- 1 K2 1 fore different from zero1 and is alignedhM i with the applied N = W − (5) i ≈ ∆x3 λ field. Its value is obtained integrating over θ with a weight i p Ci ! proportional to the pairs lifetime:

Each pair can be produced only in the two fermion- π antifermion spin combinations up-down and down-up. We 0 2µi cos θ τi(θ) 2π sin θ dθ = . (9) define N as the density of pairs for a given spin combina- i π i hM i R 0 τi(θ) 2π sin θ dθ tion. Finally, we use the notation Qi = qi/e, where qi is the To first order in B,R one gets: i-type fermion electric charge and e the modulus of the 2 electron charge. 4µi i B. (10) hM i≃ 3Wi 3 The vacuum permeability The magnetic moment per unit volume produced by the i-type fermions is Mi = 2Ni i , since one takes When a torus of a material is energized through a winding into account the two spin states perhM cell.i The contribution carrying a current I, the resulting magnetic flux density µ˜0,i of the i-type fermions to the vacuum permeability B is expressed as: is thus given by B =µ ˜0,iMi or 1/µ˜0,i = Mi/B. Each species of fermions increases the induced magnetization B = µ0nI + µ0M. (6) and therefore the magnetic moment. By summing over where n is the number of turns per unit of length and nI 1 As a referee puts it:“this is a kind of averaged Zeeman is the magnetic intensity in A/m. M is the corresponding effect” M. Urban, F. Couchot, X. Sarazin, A. Djannati-Atai: The quantum vacuum as the origin of the speed of light 3 all pair species, one gets the estimation of the vacuum E. Furthermore the opposite charges separation stays fi- permeability: nite because they are bound in a molecule. These opposite translations result in opposite charges appearing on the 2 2 2 1 Mi 2 e NiQi λC i dielectric surfaces in regard to the metallic plates. This = = crel 2 (11) µ˜0 B 6π W leads to a decrease of the effective charge, which implies i i i X X a decrease of the voltage across the dielectric slab and fi- Using Eq. (1), (4) and (5) and and summing over all nally to an increase of the capacitance. In our model of the pair types, one obtains vacuum the ephemeral charged fermion pairs are the pairs of opposite charge and the separation stays finite because 3 KW 24π ¯h the electric field acts only during the lifetime of the pairs. 0 µ˜ = 2 3/2 2 2 (12) In an absolute empty vacuum, the induced charges would (K 1) crel e Q W − i i be null because there would be no charges to be separated 2 P and the capacitance of a parallel-plate capacitor would go The sum i Qi is taken over all pair types. Within a generation, the absolute values of the electric charges to zero when one removes all molecules from the gas. are 1, 2/3 andP 1/3 in units of the positron charge. Thus We show here that our vacuum filled by ephemeral fermions causes its electric charges to be separated and for one generation the sum writes (1 + 3 (4/9+1/9)). 7 The factor 3 is the number of colours. Hence,× for the three to appear at the level of 5.10 electron charges per square families of the standard model meter under an electric stress E =1 V/m. The mechanism is similar to the one proposed for the permeability. How- 2 ever, we must assume here that every fermion-antifermion Qi = 8 (13) i ephemeral pair of the i-type bears a mean electric dipole X di given by: One obtains: di = Qieδi (16) 3 KW 3π ¯h 0 where δi is the average separation between the two fermions µ˜ = 2 3/2 2 (14) (K 1) crel e W − of the pair. We assume that this separation does not depend upon the fermion momentum and we use the reduced The calculated vacuum permeabilityµ ˜0 is equal to the Compton wavelength of the fermion λCi /(2π) as this scale: observed value µ0 when

λCi 2 δi (17) KW crel e 4 α ≃ 2π = µ0 = (15) (K2 1)3/2 3π3¯h 3 π2 W − If no external electric field is present, the dipoles point randomly in any direction and their resulting average field which is obtained for KW 31.9 . E ≈ is zero. In presence of an external electric field , the mean Such a KW value indicates that the typical fermions polarization of these ephemeral fermion pairs produce the are produced in relativistic states. This estimation is based observed vacuum permittivity ǫ0. This polarization shows upon a static and average description of vacuum. A more up due to the dipole lifetime dependence on the electro- complete view, including probability densities on pair en- static coupling energy of the dipole to the field. In a field ergy and momentum distributions might allow to give a + − homogeneous at the δi scale, this energy is diE cos θ where physical meaning to the KW value. For instance, e e 2 θ is the angle between the ephemeral dipole and the elec- pairs with a total energy distributed as dW/W up to tric field E. The electric field modifies the pairs lifetimes Wmax would give an apparent KW of the order of according to their orientation:

Wmax ¯h/2 KW Log 2 51 τi(θ)= (18) ≃ 2mec ≃ W d E cos θ i − i if W corresponds to the Planck energy. As in the magnetostatic case, pairs with a dipole moment max aligned with the field last a bit longer than the others. This leads to a non zero average dipole Di , which is aligned with the electric field E and given,h toi first order 4 The vacuum permittivity in E, by: 2 Consider a parallel-plate capacitor with a gas inside. When di Di E (19) the pressure of the gas decreases, the capacitance decreases h i≃ 3Wi too until there are no more molecules in between the We estimate the permittivity ˜ǫ0 due to i-type fermions plates. The strange thing is that the capacitance is not ,i using the relation P =˜ǫ0 E, where the polarization P is zero when we hit the vacuum. In fact the capacitance has i ,i i equal to the dipole density P = 2N D , since the two a very sizeable value as if the vacuum were a usual mate- i i i spin combinations contribute. Thus: h i rial body. The dielectric constant of a medium is coming 2 2 from the existence of opposite electric charges that can be Di 2 Qi δi separated under the influence of an applied electric field ǫ˜0,i =2Ni h i =2Nie (20) E 3Wi 4 M. Urban, F. Couchot, X. Sarazin, A. Djannati-Atai: The quantum vacuum as the origin of the speed of light

Each species of fermion increases the induced polariza- The photon may encounter the pair any time between tion and therefore the vacuum permittivity. By summing its appearence and disappearence. The life time of a pair over all pair species, one gets the general expression of the being τi, the photon will be stopped for an average time vacuum permittivity: τi/2. Each type of fermion pair contributes in increasing the propagation time of the photon. So, the total mean 2 2 2 2 N Q2λ2 2e NiQi δi e i i C i time T for a photon to cross a length L is: ǫ˜0 = = 2 (21) 3 Wi 6π Wi i i τ X X T = N i (25) Expressing the model parameters from Eq. (1), (4), stop,i 2 i (5), and (13), one gets: X Using Eq. (24), we obtain the average photon velocity 2 3/2 2 cgroup as a function of three parameters of the vacuum (KW 1) e ǫ˜0 = − 3 (22) model: KW 3π ¯hcrel L 1 If we now use the value KW given in Eq. (15) obtained cgroup = = (26) from the derivation of the permeabilitty, one gets the right T i σiNiτi/2 −12 numerical value for the permittivity:ǫ ˜0 =8.8510 F/m. We verify from Eq. 11 and Eq. 21 that the phase ve- Using Eq. (2) and (5), weP get the expression locity cφ of an electromagnetic wave in vacuum, given by KW 16π cφ =1/√µ˜0ǫ˜0, is equal to crel the maximum velocity used cgroup = 3/2 2 crel (27) 2 (σi/λ ) in special relativity. (KW 1) i Ci We also notice that the permeability and the permit- − P tivity do not depend upon the masses of the fermions. We now have to define the expression of the cross sec- The electric charges and the number of species are the tion σi. We know that it should not depend on the photon only important parameters. This is in opposition to the energy, otherwise the vacuum would become a dispersive common idea that the energy density of the vacuum is medium. Also the interaction of a real photon with a pair the dominant factor[3]. must not exchange energy or momentum with the vacuum (for instance, Compton scattering is not possible). We assume the cross-section to be proportional to the geomet- 2 5 The propagation of a photon in vacuum rical cross-section of the pair λCi , and to the square of the 2 electric charge Qi . The cross-section is thus expressed as: We now study the propagation of a real photon in vacuum and we propose a mechanism leading to a finite average 2 2 σi = kσQi λCi (28) photon velocity cgroup, which must be equal to cφ and crel. When a real photon propagates in vacuum, it interacts where kσ is a constant which does not depend on the type with and is temporarily captured by an ephemeral pair. of fermions. As soon as the pair disappears, it releases the photon to The calculated photon velocity becomes: its initial energy and momentum state. The photon con- tinues to propagate with an infinite bare velocity. Then the KW 16π cgroup = 3/2 2 crel (29) photon interacts again with another ephemeral pair and 2 kσ Q (KW 1) i i so on. The delay on the photon propagation produced by − these successive interactions implies a renormalisation of Using Eq. (13) and (15), one finallyP get: this bare velocity to a finite value. This “leapfrog” propagation of photons, with instan- 8α cgroup = crel (30) taneous leaps between pairs, seems natural since the only 3πkσ length and time scales in vacuum come from fermion pair lifetimes and Compton lengths. This idea is far from being The calculated velocity cgroup of a photon in vacuum a new one, as can be found for instance in [4]. is equal on average to crel when By defining σ as the cross-section for a real photon to i 8 interact and to be trapped by an ephemeral i-type pair of kσ = α (31) fermions, the mean free path of the photon between two 3π successive such interactions is given by: It corresponds to a cross-section of 4 10−26 m2 on an 1 ephemeral electron-positron pair, of the same order as the Λi = (23) geometric transversal area of the pair, whose size is given σiNi in Eq. (17). where Ni is the numerical density of virtual i-type pairs. We note that the photon velocity depends only on Travelling a distance L in vacuum leads on average to the electrical charge units Qi of the ephemeral charged N stop,i interactions on the i-type pairs, given by: fermions present in vacuum. It depends neither upon their L masses, nor upon the vacuum energy density. We also re- N stop,i = = LσiNi (24) mark that the average speed of the photon in our medium Λ M. Urban, F. Couchot, X. Sarazin, A. Djannati-Atai: The quantum vacuum as the origin of the speed of light 5

being crel, the photon propagates, on average, along the In our simple model where KW = 31.9, the predicted light cone. As such, the effective average speed of the pho- fluctuation is: ton is independent of the inertial frame as demanded by relativity. This mechanism relies on the notion of an abso- − − σ 5 10 2 fs.m 1/2 (37) lute frame for the vacuum at rest. It satisfies special rela- T ≈ tivity in the Lorentz-Fitzgerald sense. This simple model does not preclude some dependence of the speed of light on the photon energy, through trapping cross-section vari- We note that the fluctuations vary as the square root ations. of the distance L of vacuum crossed by the photons and are a priori independent of the energy of the photons. It is in contrast with expected fluctuations calculated in the 6 Transit time fluctuations frame of Quantum-Gravitational Diffusion [6], which vary linearly with both the distance L and the energy of the An important consequence of our model is that stochastic photons. fluctuations of the propagation time of photons in vacuum A way to search for these fluctuations is to measure are expected, due to the fluctuations of the number of a possible time broadening of a light pulse travelling a interactions of the photon with the virtual pairs and to distance L of vacuum. This may be done using observa- the capture time fluctuations. tions of brief astrophysical events, or dedicated laboratory These stochastic fluctuations are not expected in stan- experiments. dard Quantum Electrodynamics, which considers c as a given, non fluctuating, quantity. Quantum gravity theo- The strongest direct constraint from astrophysical ob- ries predict also stochastic fluctuations of the propagation servations is obtained with the very bright GRB 090510, time of photons [5] [6]. It has been also recently predicted detected by the Fermi Gamma-ray Space Telescope[8], at that the non commutative geometry at the Planck scale MeV and GeV energy scale. It presents short spikes in the 8 keV 5 MeV energy range, with the narrowest widths should produce a spatially coherent space-time jitter[7]. − We show here that our effective model of photon propaga- of the order of 4 ms (rms). Observation of the optical af- tion predicts fluctuations at a higher scale, which makes ter glow, a few days later by ground based spectroscopic it experimentally testable with femtosecond pulses. telescopes gives a common redshift of z =0.9. This corresponds to a distance, using standard cosmological parame- The propagation time T of a photon which crosses a 26 distance L of vacuum is: ters, of about 210 m. Assuming that the observed width is correlated to the emission properties, this sets a limit Nstop,i −1/2 for transit time fluctuations σT of about 0.3 fs.m . It T = ti,k (32) is important to notice that there is no expected disper- i=1 =1 X Xk sion of the bursts in the interstellar medium at this en- where t is the duration of the kth interaction on i-type ergy scale. If we move six orders of magnitude down in i,k distances we arrive to kpc and pulsars. Short microbursts pairs and Nstop,i the number of such interactions. The variance of T , due to the statistical fluctuations of the contained in main pulses from the Crab pulsar have been number of interactions and the fluctuation of the capture recently observed at the Arecibo Observatory telescope at time is given by: 5 GHz[9]. The frequency-dependent delay caused by dispersive propagation through the interstellar plasma is cor- 2 2 2 2 rected using a coherent dispersion removal technique. The σ = σ t + N stop,i σ (33) T Nstop,i stop,i t,i mean time width of these microbursts after dedispersion i X is about 1 µs, much larger than the expected broaden- where tstop,i = τi/2 is the average stop time on a i-type ing caused by interstellar scattering. Assuming again that 2 2 2 pair, σt,i = τi /12 its variance, and σNstop,i = Nstop,i the the observed width is correlated to the emission proper- variance of the number of interactions. Hence: ties, this sets a limit for transit time fluctuations of about −1/2 1 L 0.2 fs.m . σ2 = N τ 2 = σ N τ 2 (34) T 3 stop,i i 3 i i i The very fact that the predicted statistical fluctua- i i X X tions should go like the square root of the distance im- Once reduced, the current term of the sum is proportional plies the exciting idea that experiments on Earth do com- to λCi . Therefore the fluctuations of the propagation time pete with astrophysical constraints since we expect fluc- are dominated by virtual e+e− pairs. Neglecting the other tuations in the femtosecond range at the kilometer scale. fermion species, and using σeNeτe/2=1/(8c), one gets An experimental setup using femtosecond laser pulses sent to a 100 m long multi-pass vacuum cavity equipped with 2 τe L λCe L metallic mirrors could be able to detect this phenomenon. σT = = 2 (35) 12c 96πKW c With appropriate mirrors with no dispersion on the reflec- So tions, a pulse with an initial time width of 9 fs (FWHM) would be broadened after 30 round trips in the cavity, to

L λCe 1 an output time width of 13 fs (FWHM). An accurate σT = (36) autocorrelation measurement∼ could detect this eﬀect. r c r c √96πKW 6 M. Urban, F. Couchot, X. Sarazin, A. Djannati-Atai: The quantum vacuum as the origin of the speed of light

7 Conclusions

We describe the ground state of the unperturbed vacuum as containing a finite density of charged ephemeral fermions antifermions pairs. Within this framework, ǫ0 and µ0 originate simply from the electric polarization and from the magnetization of these pairs when the vacuum is stressed by an electrostatic or a magnetostatic field respectively. Our calculated values for ǫ0 and µ0 are equal to the measured values when the fermion pairs are produced with an average energy of about 30 times their rest mass. The finite speed of a photon is due to its successive transient captures by these virtual particles. This model, which pro- poses a quantum origin to the electromagnetic constants ǫ0 and µ0 and to the speed of light, is self consistent: the average velocity of the photon cgroup, the phase velocity of the electromagnetic wave cφ, given by cφ = 1/√µ0ǫ0, and the maximum velocity used in special relativity crel are equal. The propagation of a photon being a statistical process, we predict fluctuations of its time of flight of the order of 0.05fs/√m. This could be within the grasp of modern experimental techniques and we plan to assemble such an experiment.

Acknowledgments

It is a pleasure acknowledging helpful discussions with Gerd Leuchs. The authors thank also J.P. Chambaret , I. Cognard, J. Ha¨ıssinski, P. Indelicato, J. Kaplan, C. Rizzo, P. Wolf and F. Zomer for fruitful discussions, and N. Bhat, E. Constant, J. Degert, E. Freysz, J. Oberl´eand M. Ton- dusson for their collaboration on the experimental aspects. This work has beneﬁted from a GRAM2 funding.

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