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arXiv:2010.04561v2 [quant-ph] 12 Oct 2020 salwd hsfrincnestsms eo h form the of be must condensates Thus allowed. charg is vanishing with scalars Lorentz are which o condensates Formation space-time. of invariance Lorentz obey observed to the have couples fermion of disappearance and pearance vir- -. disappearing the of and which pairs appearing by in continuously model are new a tual develop we [1], Following Introduction 1 tion hti ttelvlof level propagation the light at of attractive is speed model the that of this experimental fluctuation makes the expected that the as features is order the same of the One of value. be con- to (magnetic appears permeability stant) magnetic the of value ...Hugon C.M.F. Vacuum the Through Paving NN-Szoed eoa i oeaeo3,116Gnv,It Genova, 16146 33, Dodecaneso Via Genova, di Sezione - INFN Kulikovskiy V. France Marseille, [email protected] 13008 E-mail: Corail, du avenue 1 R&DoM, Hugon C.M.F. Keywords available the with reachable technologies. is property propagation light aumepcainvalue, field Higgs expectation thermal the vacuum in to corresponding virtual temperature Assuming at pairs. equilibrium virtual the of gas descript the statistical the in naturally appear permeability theory. field quantum the and wave-partic duality the with compatible e propagation an of propose tion we pairs, fermion virtual with Abstract later inserted be should acceptance and receipt of date the wl eisre yteeditor) the by inserted be (will rf ob submitted be to Draft h ita ar r pern o hr ieie Ap- lifetime. short a for appearing are pairs virtual The nti oe h rgno h aumpritvt and vacuum the of origin the model this In sn h oe nwihtevcu sfilled is vacuum the which in model the Using Vacuum · .Kulikovskiy V. · ita pair Virtual σ ≈ kT 1 . sm as 9 ≈ 246 · ih eoiyFluctua- Velocity Light − 1 . / 2GV h deduced the GeV, 22 2 hsnon-classical This . ff ciedescrip- ective o of ion aly le e f h pc-ieseldfie yteHiebr netit prin uncertainty ciple: Heisenberg the by defined shell space-time value. expectation scalar a have only can field pc a eol cuidb h atce ihapartic- a with free the limited by the occupied ular thus only and be pairs can plac the space other in by appear occupied not Pauli already can the Pairs to itself. thanks possible principle be exclusion be could interaction pairs indirect virtual the some tween Also pairs. virtual ree and by real absorption mission the continuous of their presence and the example, virtual and thermal for to, in due pairs possible is virtual be gas the can infinite between pair exchange an virtual energy in The the gas equilibrium. that a like expect expand, We to vacuum. vir- tends of pairs ensemble particle the therefore tual overlap, cannot spin di and in type prin- fermions exclusion virtual Pauli the the ciple, Considering values. zero colour spin , angular and total the keep to them allow which ueet htcnb efre ots hsmodel. this test to performed mea- be the can and that propagation, th surements photon describes real part last the The on pres 4). consequences (Section the par- fields in virtual external variation the of properties of gas statistical its the and from ticles and explained permeability be (QFT) vacuum can Theory permittivity The Field 2). Quantum (Section to wave-function arrive and duality particle the wave describe can we definition, this pape from how, virtual this shown In of oscillations. excitations harmonic as are realised photons particles, real the Model, Standard coolin and acceleration decreasing temperature. a process with this going and infinite, ex- expansion is gas the virtual drive the may pansion that expect We uncertainty. Heisenberg ΨΨ l h ita atce a xs nyisd Lorentz a inside only exist can particles virtual the All nti oe,a netnino h omnyaccepted commonly the of extension an as model, this In i ∆ where x ∆ p / ≥ oetmsnete r once ihthe with connected are they since momentum Ψ 2 ~ enstefrinfil.Smlry tensor a Similarly, field. fermion the defines and ∆ E ∆ t ≥ 2 ~ h ar r CP-symmetric, are pairs The . ff rn ar ihtesame the with pairs erent tis it r ence g e e - - - - 2 C.M.F. Hugon, V. Kulikovskiy

2 Wave Function and Photon Propagation Each fermion in a virtual pair is defined by the size, xi, and no other fermion of the same family and spin state can 3 The excitation of a virtual pair can appear as oscillations occupy the volume of xi , following the Pauli exclusion prin- of fermions. This definition allows introducing the Hamil- ciple. We assume that the virtual pair energy, E, and its life- tonian of a harmonic oscillator for virtual pair description. time, T, are connected thanks to the Heisenberg uncertainty A propagating real photon transmits its full energy and mo- in the following way: mentum to the virtual pairs creating one dimension oscilla- tors with a frequency corresponding to the photon energy ET = h. (1) = E hν. We assume that during the real photon propaga- Each virtual fermion momentum, p, and size, x, are also tion between the pairs, the oscillation phase is transferred connected in the similar way: as well. From this point, considering a continuous space ap- proximation this model allows the development of the clas- px = h. (2) sical quantum and quantum field theories using the Schr¨odinger equation to find the photon wave function [2]. Moreover, both virtual fermions in a pair have equal ener- = From this model it naturally appears that the real photon gies, ε, and momentum absolute values, so E 2ε. If en- properties cannot be measured below the Heisenberg uncer- ergy momentum relation holds for virtual particles, then, for tainties of the excited virtual pairs. We can already notice each fermion the energy can be expressed through the mo- some other important properties: mentum: 2 2 2 2 – while the virtual particle properties are still in debate [3], ε = (mc ) + (pc) . (3) in this model we consider that virtual particles conserve In this expression m is the mass of a virtual fermion that the energy and momentum and have the same properties can be equal to the mass of a corresponding real fermion. It as the real ones, which may include the mass. In con- will be shown later, however, that some properties are better ff sequence, the only di erence with real particles is their explained with zero virtual fermion masses. short lifetime defined by the Heisenberg uncertainty; The scalar natureof the Higgsfield may allow the forma- – being associated to fermion pairs, the propagation of a tion of fermions condensates ΨΨ , so it can be the origin h i real photon can be described as leaps of distances with of the virtual pairs. The average expected value in the vac- time delays arising from virtual pairs size and lifetime; = √ 0 – at each photon propagation leap the virtual pair annihi- uum of this field is given by υ 1/ 2GF 246.22 GeV, 0 ≈ lation can be considered as a secondary emission point where GF is the reduced Fermi constant.q We speculate that with a decreasing density, which correspondsto Huygens- this energy corresponds to the temperature of the virtual pair Fresnel wave principle; gas being in thermal equilibrium, kTν = υ. – by entanglement, locally, each oscillation contains all In order to obtain the virtual fermion density distribu- the information about the propagating real photon; tion, one can use grand thermodynamic potential as follow- – the oscillator can be defined as a discrete wave at each ing: position ~r at a time t. The oscillator amplitude Ψ(~r, t) Ω = PV, (4) corresponds to the probability of observing the real par- − ticle in the volume, (∆x)3, occupied by oscillating virtual pair as ∆P(~r, t) = C Ψ(~r, t) 2(∆x)3, whereC is a normal- ∂P 1 ∂Ω | | n = = . (5) isation constant; ∂µ −V ∂µ – the discrete function Ψ(~r, t) constructed from the oscil- In the latter expression µ is the chemical potential and the lator amplitudes of virtual pairs may be approximated as pressure, P, is composed of fermion and antifermion pres- continuous function on the scales above the Heisenberg sures as following [4, Equation (18)]: uncertainty and it contains the full information about the + 4πp2dp real particle, analogously to wave function in QFT. ∞ (µ ε)/kTν P = P+ + P = kTν (ln(1 + e − )+ − h3 "Z0 ( µ ε)/kT ln(1 + e − − ν )) . (6) 3 The Stochastic Vacuum Description The pressure of fermions, P+, and the pressure of antifermions,i Additionally to the commonly known virtual photons, the P have the same expressions apart the chemical potential − vacuum is filled with virtual fermions that form the prop- that has the same value but the opposite sign due to the pro- agation medium for the real photons. In total, there are 21 duction in couples fermion-antifermion.Note, that the origi- charged fermion pair species, noted i, considering the charged nal formula [4, Equation (18)] contains a factor 2 since each families and the families with their colour states. energy level can be occupied with two different spin states Paving Through the Vacuum 3 of the fermion and antifermion. However, since the virtual where Ω0 is the grand potential in the absence of the external particles are always produced in pairs that should annihilate fields and n can be taken from Equation (7). with a zero value of the total spin, only pairs with antipar- The relativistic energy for an in the magnetic allel spins for particles and are allowed, so we field can be expressed as following [6]: believe this reduces the number of the states by a factor of two. 2 4 2 2 ~ EM = m c + pz c + 2e cB(n + 1/2 gs/2) From Equations (5,6) the density becomes: − q e~cB E + (2n + 1 gs), (11) + 4πp2dp 1 1 ≈ 2E − n = ∞ . 3 (ε µ)/kT (ε+µ)/kT (7) 0 h e − ν + 1 − e ν + 1 where we take the first order approximation valid for com- Z ! 2 mon fields βBB mec (βB is the Bohr magneton). This Note that this density is negative for antiparticles, so is does ≪ approximation can be compared to the non-relativistic en- not describe the virtual particle density, but rather the lep- ergy levels used in [5] to derive Equation (10): ton number density. Moreover, in order to guarantee that the lepton density is zero, µ = 0 should be used in order to p2 e~B ENR = z + (2n + 1 gs). (12) equalise the number of particles and antiparticles for each M 2m 2mc − energy. Finally, the density of virtual particles as a function Comparing relativistic and non-relativistic expressions of momentum can be written taking one of the exponential one can see that the following relativistic terms in Equation (7) and assuming µ = 0: can be used to match them: 4πp2 1 e~ npair(p) = . (8) β = , (13) h3 eε/kTν + 1 2ε/c

In this expression ε, p are the energy and the momentum where respect to the Bohr magneton, βB = e~/(2mc), mc is of fermions, contrary to the values corresponding to the vir- substituted with ε/c. The evaluations in [5] should remain tual pairs. To account for different fermion families all the valid even for energy dependent β. However, the integra- equation sin this section should be summed over 21 fermion tion in Ω or N over the energy will now include β2 as well1. species. Equation (10) then becomes:

+ 2 ∞ ∂n(p) 1/µ = β(ε)2 dp = 4 The vacuum permeability and permittivity 0 3 ∂µ Z0 µ=0 + 2 ε/kT 2 ∞ 2 4πp dp e ν It is experimentally known that the vacuum has a non-zero β( ε)2 · (14) 3 h3 eε/kTν + 1 value of permeability, µ , and permittivity, ǫ . As proposed i 0 0 0 X Z in [1], the permeability and permittivity can be fundamental The sum over different virtual pair types, i, is added since properties of the stochastic vacuum. the overall vacuum magnetisation M~ is linearly composed | | Being composed by opposed charges and spins each vir- from the densities of magnetic moments of different tual pair carries doubled spin magnetic moments related to types. each particle, β, and the dipole moment, ω. In the presence In classical the electric and magnetic dipole po- of the magnetic (electric) field, the statistical distribution of tential have the same expressions: U = ~ω E~ and − · virtual pairs and fermions/antifermions is altered and this U = ~β B~, so we expect the energy levels and the state ~ − · results in a non-null magnetic moment density, M, (polari- function to change in the presence of the magnetic and elec- ~ ~ ~ sation, P). Note that, following discussions in [1], M and P tric fields in the same manner. This brings for the vacuum do not have a commonly accepted meaning of a magnetic permittivity the following expression: density of medium and material polarisation because they + are considered to be null in vacuum. Instead their non-zero 2 ∞ 2 ∂n(p) ǫ0 = (ω/2) dp. (15) values are ascribed to a vacuum behaving like a gas of vir- 3 ∂µ Z0 µ=0 tual pairs. The vacuum permeability and permittivity can be Note that we substituted β with ω/2 since the magnetic derived from those values using the following expressions: moment was related to a fermion or an antifermionwhile the M~ = 1/µ B~ and P~ = ǫ E~. 0 0 dipole moment is related to a whole couple. Following [5, Equations (59.1), (59.4), (59.12)]: 1 We were able to check the validity of math operations to derive the M~ = (∂Ω/∂B)T,V,µ, (9) paramagnetic part. For the diamagnetic part, the necessary condition | | − 2 2 β B~ kT is satisfied too. However, we cannot guarantee the validity 2 β ∂ Ω0 2 2 ∂n | | ≪ 1/µ0 = = β , (10) of the complete evaluation in the case where β is in function of the −3 V ∂µ 3 ∂µ energy. 4 C.M.F. Hugon, V. Kulikovskiy

From Maxwell’s equations, the connection between the be approximately 0.8 10 6 H/m which is 0.7 smaller than · − measured waves, c, and vacuum properties is the measured value. Such substitution, however, would vi- the following: olate Planck’s law and the photon energy dependence from its frequency. Using the non-relativistic magnetic moment, 1 1/µ0 c = = . (16) βi = Qieh/(2mic), results in ten orders of magnitude higher µ ǫ ǫ √ 0 0 r 0 vacuum magnetisation. Finally, as a side remark, our trials to This equation together with Equations (10,15) can be calculate vacuum polarisation (magnetisation) as an integral satisfied if β = (ω/2)c. To define the dipole moment one of electric (magnetic) dipole moment projection on the field / can imagine a virtual pair being composed of two charges axis and multiplied by the fermion pair density were not separated by the average distance x: successful. No analytical solution was obtained and for nu- merical integration the precision was lost due to oscillating h terms. This problem is known already for a non-relativistic ω = Qex = Qe . (17) p case [7]. Actually if the virtual fermions were randomly distributed in the filled sphere of radius x, the average distance would be 36/35x. The sphere approximation, however, does not fit 5 Photon Propagation and Propagation Time with x3 volume assumed in the statistical distributions. Dispersion Comparing this expression with Equation (13) one can see similarities considering for Qi = 1 and assum- The photon propagation is composed from subsequent per- ing ε pc which can become exact if virtual fermions have turbations of charged virtual fermion pairs. Since the space ≈ zero masses. In the magnetic moment expression, Equation is filled with virtual particles at a high density, and since the (13), however, ~ is different from h in the dipole moment ex- family and spin states allow overlapping of pairs, there is no pression, Equation (17). This can be unified to ~ if the orig- expected additional time or free fly of the photon in absolute inal relation between the virtual fermion momentum and its vacuum like in traditional physics or in the theory described size is rewritten as px = ~ or thefactor 2π is actually arriving in [1]. Assuming the random time of the pair excitation the from the difference of the fermion size and the average dis- average time during which the pair was excited and annihi- T h h tance between fermion and antifermion. Note, that it is hard lated is 2 = 2E = 4ε . In order to conserve the light propaga- x to choose the right expressions a priori due to the ambiguity tion velocity as c the propagation distance should be 4 since ε for relativistic magnetic moment and dipole interactions as then the velocity becomes v = p . The latter ratio is close to well as their Lorenz transformations [6,8]. c for the relativistic pairs (E 2mc2) or it becomes equal ≫ From Equation (14) assuming βi = Qieh/(2ε/c) we ob- to c if virtual fermions have zero masses. 6 tain a value of µ0 5.6 10− H/m. This value is 4.4 times If one assumes that the energy of each fermion is in- ≈ · bigger than the measured magnetic constant value. In our creased by εγ/2 then the average speed can be obtained as: opinion a non-zero vacuum magnetisation is already an in- + teresting feature of this theory. In fact, the model has no free ∞ c n (p)dp 0 2 2 2 pair √(ε+εγ/2) (mc ) parameters and the fact that the µ0 value results of the same v = − . (18) R + 1 order as the measured one is very intriguing. Interestingly, ∞ npair(p)dp 0 ε+εγ/2 assuming zero fermion masses in virtual couples makes the R 7 difference 4.1. The remaining difference may be ascribed For εγ = 0 the difference (v c)/c is about 4 10 , de- − · − due to approximations used to derive Equations (10, 11). creasing down to one at εγ . In this model it is ex- → ∞ This can also hint formissing non standard or sterile fermions, pected that for a signal composed from photons with εγ = which could increase the vacuum magnetisation. However, [30, 200] keV the photon speed variation would be on the this may contradict fermion loop corrections in quantum order of ∆v/c 1.4 10 11 which is well above the existing ≈ · − field theory. In particular, the factor close to four can be limits obtained from the Gamma Ray Burst (GRB) 980703 explained by the presence of two more quantum numbers observation, ∆v/c < 6.3 10 21 [9]. The recent GRB data · − (positive, negative) distinguishing standard fermions from compilation [10] supports the Lorentz invariance violation the hidden ones with all the remaining properties being the and suggests a linear form of light speed variation with de- 17 same. This would make µ only 2% different from the mea- pendence v = c(1 εγ/ELV ) with ELV = 3.6 10 GeV. Such 0 − · sured value assuming zero fermion masses. dependencyalso has a decreasing trend with growing photon Notice, however, that using more natural expression for energy. However, the analysed high energy GRB photons in 16 β from Equation (13) with ~ would result in a 180 times the range εγ = [1, 50] GeV would have ∆v/c 1.3 10 ≈ · − smaller vacuum magnetisation than currently observed. If while in ourmodel it is ∆v/c 1.5 10 7 for the same photon ≈ · − one also uses ~ in Equation (6) the magnetic moment would energy range, which is a well-larger variation. The inconsis- Paving Through the Vacuum 5 tencies with the GRB observations can be avoided in several Thus the total dispersion: ways. + – One could doubt the same VEV and the vacuum temper- ∞ 2 σ = N ni(p)σi (p)dp = ature at earlier Universe correspondent to the time of the s i 0 X Z GRB explosions and the light propagation. + (23) ∞ 1 i ni(p) 2 2 2 dp – It could be proposed that the photons do not propagate h 0 (mic ) +(pc) 1/2 L + 1.9asm− . following a straight line but rather following excitations uv 12 R ∞ n p 1 dp ≈ tu P j 0 j( ) p of virtual pairs which are not perfectly aligned. If the ǫγ does not alternate the size and the lifetime of the virtual P R particles, the photons velocity is constant. This can be Note that if one uses ~ instead of h in the relations (1,2) valid since the Heisenberg uncertainties could be valid this dispersion decreases by a factor of √1/(2π) and be- 1/2 only for the excess of the energy/momentum respect to comes approximately 0.8asm− . the “real” values. Alternatively, since the sizes of the vir- tual particles are modified, the trajectory could become straighter and compensate for the smaller velocity. 6 Experimental Observation of the Stochastic Vacuum – If the virtual fermions have lower masses compared to the real fermions, the velocity variation would be de- The observation of stochastic photon propagation time dis- creased reaching c value independent from ε for zero γ persion would consolidate the model with a concept of pho- fermion masses. ton propagation described as leaps between virtual fermion The stochastic dispersion of the photon propagationtime pairs. For the model in [1] the expected fluctuation is more is one of the most significant and measurable effects distin- than an order of magnitude higher compared to the fluctu- guishing this theory from the QFT. This dispersion depends ation evaluated here. Both models provide values that can on the number of encountered virtual pairs along the path be measured with the available technologies, oppositely to and the variation of the delay at each step. fluctuations expected in other theories [11]. The pair type excited at each photon step is defined by Nowadays, the strongest constraints are established by the probability: astrophysical observations, mainly GRBs and pulsars [12, 1/2 13]. The current limits are at 0.2 0.3fsm− . n p − i( ) In [1] and in the presented model, the time resolution has (p) = + , (19) ∞ n p dp j 0 j( ) a stronger impact than the photon path length, which favours measurements in laboratory. In fact, astrophysical measure- P R where nk(p) is the virtual pair density for each family, de- ments are based on the events observation with a duration fined in Equation (8). Ignoring the additional energy ε /2, 3 γ of the order of 10− s at distances of megaparsecs (a factor the total length can be composed of x/4 = h/(4p) steps in 1022/2 = 1011 m1/2). The current state of art of laser tech- the following way: nologies allows the generation of femtosecond light pulses 15 and a measurement precision is at the order of 10− s, on a + h ∞ n p dp 4/2 2 1/2 i 0 i( ) 4p distance of tens of kilometres (a factor 10 = 10 m ). As L = N , (20) + a result, the astrophysical measurementprecision is at the or- P Rj ∞ n j(p)dp 0 14 1/2 der of 10− fsm− , while the laboratory measurement pre- R 17 1/2 from whereP the average number of steps can be recovered: cision can reach 10− fsm− . The experiment able to reach the required sensitivity can + 4L j ∞ n j(p)dp be realised with femto-second laser pulses propagating in N = 0 . (21) h + 1 a multi-pass vacuum cavity, which can have length of sev- Pi R ∞ ni(p) dp 0 p eral kilometres (as used by the Virgo/LIGO experiments [14, R AssumingP the fact that the photon excites pairs that live 15]). Under these conditions the order of magnitude of the in average T/2 at random time and the lifetime distribution arrival time dispersion predicted by this theory can reach femtoseconds (e.g. σ 1fs for a 4km cavity with 70 reflec- is flat, the fluctuation of the photon delay at each step is: ≈ tions). The pulse width characterisation is possible thanks to T h an auto-correlationmeasurementsystem such as FROG [16]. σi(p) = = = This kind of system allows measuring the pulse width for 2 √3 2 √3E different wavelengths, which can provideinformation on sys- h h = . (22) tematic effects since the expected velocity variation with 2 2 2 4 √3ε 4 3((mic ) + (pc) ) wavelength is negligible for such measurements. p 6 C.M.F. Hugon, V. Kulikovskiy

7 Conclusions and Discussions size and virtual fermion momentum are connected as px = h appearing from the Heisenberg uncertainty. The Heisenberg allows virtual pairs In this theory it appears that photon propagation speed with energy E to appear for times ~/E in the vacuum. or travel time to cover some distance may fluctuate since ≥ Starting from stochastic vacuum model of the space filled it depends on the number of virtual pairs involved in the with virtual particles, we hypothesisedthat real photonsprop- propagation. The effect can be measurable on the kilome- agation can be explained as a continuous absorption by vir- tre scale distances with femtosecond pulse lasers. Phase ve- tual fermion pairs and emission after the pairs . locity remains, instead, more stable [17] which is in agree- Wave-like properties appearing in this model follow Huygens- ment with current interferometric measurements. It is inter- Fresnel principle since every annihilating pair becomes an esting to note that nowadays there are existing kilometre emission point. scale cavities but they are used with ultra-stable continuous lasers (Virgo/LIGO experiments [14,15]), while the ultra- At the first glance, the virtual pairs should appearand an- fast lasers are usually used at very short range (meter scale). nihilate independently in the absence of real particles. How- The theory presented here has several differences respect ever, it is natural to propose that the virtual fermions should to the theory in [1]. These are mainly the following: obey Pauli exclusion principle similar to the real particle – the virtual pairs energy follows the equation of state of a and this already makes an interconnection of the pair pro- Fermi gas with no free parameters instead of tuned fixed duction in a particular space from the nearby pairs. As a virtual pair energies or tuned dE/E2 spectrum, consequence we hypothesise that the virtual pairs as well as – the abundances of fermion families follows the equation the virtual photons are in thermal equilibrium. Additionally, of state instead of abundances driven by the maximum this drives the expansion of the virtual particles gas, which densities, could explain the Universe expansion. The photons distri- – the calculation of the vacuum permeability through the bution should follow Planck’s law while the virtual pair en- equation of state alternation in the presence of the mag- ergies are distributed according to the equation of state of netic field instead of the average magnetic moment cal- a Fermi gas. For the temperature we propose the vacuum culation with a modification of a lifetime due to the pres- expectation value of the Higgs field. Assuming this, we for- ence of the magnetic field; mally introduce a new concept of the vacuum temperature. – for the photon propagation no assumption of the cross- In classical physics the vacuum has undefined temperature section is needed; it is assumed that photons make leaps since no carriers are present. from an annihilated pair immediately to the next one Going further one could imagine that real photons exist with no free path. only at the production and the interaction points while dur- . ing the propagation information about the photons is spread As a result of the first two changes, the obtained fluctu- by the virtual pair excitation. The simplest excitation of the 1/2 ation expectations are on the order of 1.9 asm− which is virtual pair is a harmonic oscillation and it corresponds to more than an order of magnitude below respect to the pre- U the (1) group, which is enough to explain the photon prop- dictions in [1]. The values predicted here are still within erties. For other real particles, the theory can be extended reach of the currently available technologies. Considering, SU and the propertiessuch as (2)for the spin may correspond for example, that for the path in the tunnel of LIGO with to a rotation of the pairs and so on. 70 reflections and the commercially available lasers with an and permeability appear naturally initial pulse of 4 fs (FWHM) the expected pulse broadening assuming every virtual fermion carries a magnetic moment up to 4.6 fs (FWHM) would be measurable with the FROG and each pair has a dipole moment. The obtained value of techniques. magnetic constant, µ0, is about 4.5 times higher than the It is commonly accepted that the virtual fermion masses measured value. The remaining difference may suggest for do not correspond to the real fermion masses and the virtual missing fermions or it can be due to the approximations particles are thus called off-mass-shell. The results obtained used. We believe that the same order of magnitude is still in the framework of this theory looks slightly more appeal- a success of this theory that has no parameters to tune apart ing if the zero virtual fermion masses are assumed. In partic- from h/~ choice in the Heisenberg uncertainty, temperature ular, the photon propagation velocity becomes exact as c; µ0 and the relativistic magnetic moment definition. The val- obtained in the framework of this theory becomes slightly ues of the fermion magnetic moment, β, and the virtual pair closer to the measured value; speed of light from µ0 and ǫ0 dipole moment, ω, are connected as β = (ω/2)c to guarantee becomes also exact as c (the equation for this velocity is the speed of light value calculated from µ0 and ǫ0. This con- different from the photon propagation velocity here). There nection can be retrieved if those moments are written from are, however, no unexplainable contradictions with the ex- the first principles and the assumption that the virtual pair isting observations if the mass of the virtual fermions are Paving Through the Vacuum 7 kept equal to the corresponding real particle masses. Note, 7. S. Biswas, S. Sen and D. Jana, Phys. of Plasmas 20, 052503 (2013) that same advantages can be obtained if instead of assuming doi:10.1063/1.4804274 zero fermion masses, one substitutes px = h to (ε/c)x = h. 8. A. Kholmetskiim O. Missevitch and T. Yarman, Prog. In Electrom. / Further studies of the virtual pair gas dynamics could be Res. B, 47, 263–278 (2013) doi:10.2528 PIERB12110903 promising to explain different Cosmological and Quantum 9. B. E. Schaefer, Phys. Rev. Lett. 82 4964–4966 (1999) doi:10.1103/PhysRevLett.82.4964 Field properties such as the Universe expansion and . 10. H. Xu, B-Q Ma, Phys. Lett. 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