EPJ manuscript No. (will be inserted by the editor) The quantum vacuum as the origin of the speed of light M. Urban1, F. Couchot1, X. Sarazin1 and A. Djannati-Atai2 1 LAL, Univ Paris-Sud, CNRS/IN2P3, Orsay, France 2 APC, Univ Paris Diderot, CNRS/IN2P3, Paris, France Received: / Accepted: Abstract. We show that the vacuum permeability µ0 and permittivity ǫ0 may originate from the magneti- zation 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 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. PACS. 06.30.Ka Electromagnetic quantities – 06.20.Jr Determination of fundamental constants 1 Introduction 2 An effective description of the quantum vacuum The vacuum is assumed to be filled with continuously ap- pearing and disappearing charged fermion pairs (ephemeral particle-antiparticle pairs). We consider neither intermedi- ate bosons nor supersymmetric particles. All known species The vacuum permeability µ0, 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, µ 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 µ0, 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 arXiv:1302.6165v1 [physics.gen-ph] 21 Feb 2013 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 to its rest mass energy 2micrel, where crel is the maxi- The paper is organized as follows. First we describe our mum velocity introduced in the Lorentz transformation. model of the quantum vacuum filled with continuously ap- We remind that crel is not necessarily equal to the speed pearing and disappearing fermion pairs. We show how µ0 of light. We note: and ǫ0 originate respectively from the magnetization and 2 the electric polarization of these pairs. We then derive the Wi = KW 2micrel (1) photon velocity in vacuum by modeling its propagation as a series of interactions with the pairs. Finally, we pre- where KW is a constant, assumed to be independent from dict statistical fluctuations of the transit time of photons the fermion type. We take KW as a free parameter; its across a fixed vacuum path. value could be calculated if we knew the energy spectrum 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.