Draft to be submitted (will be inserted by the editor) Paving Through the Vacuum C.M.F. Hugon · V. Kulikovskiy the date of receipt and acceptance should be inserted later Abstract Using the model in which the vacuum is filled ΨΨ where Ψ defines the fermion field. Similarly, a tensor h i with virtual fermion pairs, we propose an effective descrip- field can only have a scalar expectation value. tion of photon propagationcompatible with the wave-particle All the virtual particles can exist only inside a Lorentz duality and the quantum field theory. space-time shell defined by the Heisenberg uncertainty prin- In this model the origin of the vacuum permittivity and ciple: ∆x∆p ~ and ∆E∆t ~ . The pairs are CP-symmetric, permeability appear naturally in the statistical description of ≥ 2 ≥ 2 which allow them to keep the total angular momentum,colour the gas of the virtual pairs. Assuming virtual gas in thermal and spin zero values. Considering the Pauli exclusion prin- equilibrium at temperature corresponding to the Higgs field ciple, the virtual fermions in different pairs with the same vacuum expectation value, kT 246.22GeV, the deduced ≈ type and spin cannot overlap, therefore the ensemble of vir- value of the vacuum magnetic permeability (magnetic con- tual particle pairs tends to expand, like a gas in an infinite stant) appears to be of the same order as the experimental vacuum. We expect that the virtual pair gas is in thermal value. One of the features that makes this model attractive equilibrium. The energy exchange between the virtual pairs is the expected fluctuation of the speed of light propagation can be possible due to, for example, the presence of the real that is at the level of σ 1.9asm 1/2. This non-classical ≈ − and virtual photons and their continuous absorption and ree- light propagation property is reachable with the available mission by virtual pairs. Also some indirect interaction be- technologies. tween the virtual pairs could be possible thanks to the Pauli Keywords Vacuum Virtual pair Light Velocity Fluctua- exclusion principle itself. Pairs can not appear in the place · · tion already occupied by other pairs and thus the limited free space can be only occupied by the particles with a partic- ular energy/momentum since they are connected with the 1 Introduction Heisenberg uncertainty. We expect that the virtual gas ex- pansion may drive the Universe expansion and this process Following [1], we develop a new model in which the vir- is infinite, going with a decreasing acceleration and cooling arXiv:2010.04561v2 [quant-ph] 12 Oct 2020 tual fermions are continuously appearing and disappearing temperature. by pairs of particle-antiparticle. In this model, as an extension of the commonly accepted The virtual pairs are appearing for a short lifetime. Ap- Standard Model, the real photons are excitations of virtual pearance and disappearance of fermion couples have to obey particles, realised as harmonic oscillations. In this paper it is the observed Lorentz invariance of space-time. Formation of shown how, from this definition, we can describe the particle- condensates which are Lorentz scalars with vanishing charge wave duality and arrive to Quantum Field Theory (QFT) is allowed. Thus fermion condensates must be of the form wave-function (Section 2). The vacuum permeability and C.M.F. Hugon permittivity can be explained from the gas of virtual par- R&DoM, 1 avenue du Corail, 13008 Marseille, France ticles and its statistical properties variation in the presence E-mail: [email protected] of the external fields (Section 4). The last part describes the V. Kulikovskiy consequences on the real photon propagation, and the mea- INFN - Sezione di Genova, Via Dodecaneso 33, 16146 Genova, Italy surements that can be performed to test this model. 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 lepton families and the quark 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 electron in the magnetic allel spins for particles and antiparticles 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).