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E ermany × and D 0 = (2) E ε P 0 , 2 G. Leuchs et al.: The quantum vacuum at the foundations of classical electrodynamics on an equal footing1. That interpretation is preposterous the positronium rest energy. In addition, the quasi-static in classical electromagnetism, since the vacuum is defined regime requires that the frequency ω of the classical field as the emptiness. In quantum theory, however, that is not E be much smaller than the oscillator resonance frequency the case, and virtual particle pairs making up the vacuum ω0. That means we are considering a low-energy field prob- are expected to show a response to an applied field. In ing pair creation as a non-resonant effect (classical field). analogy to εε0 for a material medium, ε0 itself could be Inserting this relation for ω0 into Eq. (3) to determine the associated with the polarization of the quantum vacuum displacement, one obtains the induced dipole moment P0 = ε0E, a concept deeply rooted in quantum electrody- namics [4,5]. We later extend the discussion to magnetic e2 ℘ = E. (4) phenomena, linking the permeability of free space µ0 to 2 mω0 the magnetization of virtual pairs in vacuum. The presence of an external electromagnetic field dis- The magnitude of the vacuum polarization given by turbs the stability of the vacuum as the ground state Eq. (1) depends on the effective volume per dipole V = r3. of all fields. In this case, virtual particles must be con- In theoretical investigations [11], it is generally accepted sidered to account for the possible physical processes in that the appropriate length scale for a virtual electron- vacuum. Phenomena such as pair creation, vacuum bire- 3 positron pair in vacuum is the Compton wavelength λc = fringence, and light-light scattering are well established ¯h/mc. With these considerations, Eq. (1) results in the for strong fields [6,7,8,9], with amplitudes larger than 4 2 3 18 vacuum polarization ES = m c /e¯h ≈ 10 V/m. This is exactly the expres- sion for the critical field first pointed out by Niels Bohr, 2 2 e as acknowledged by F. Sauter . In the literature, ES is P0 = 2 3 E (5) often referred to as the Schwinger field [9]. These effects mω0r illustrate the nonlinear properties of the vacuum [4,5]. Here we assume the limit of weak fields and focus on The induced polarization is proportional to the electric field, as expected. Assigning the quantity multiplying E the linear response. The question we pose is: could the 5 vacuum quantum fluctuations be subtly embedded in the to the vacuum polarizability, we arrive at classical theory? If yes, one piece of evidence could be 2 the existence of physical constants whose numerical values e ε˜0 = 2 3 . (6) are simply determined experimentally, but which would mω0r emerge naturally from the quantum theory. We specu- late whether the permittivity and the permeability of free For the permeability of free space µ0, we attempt a space could be such quantities. In the remainder of this similar procedure to estimate its value by associating this paper, we will estimate the values of ε0 and µ0 based on quantity to the linear magnetic response of the quan- simple semi-classical assumptions. tum vacuum. Here we encounter the added difficulty that We start by considering the electrical properties of the some aspects of the material magnetization arise from quantum vacuum. An external electric field interacting pure quantum effects without classical explanation. One with the virtual electron-positron pairs polarizes them. 3 The magnitude of the dipole moment ℘ = ex induced on The uncertainty principle also offers an estimate for this one virtual pair (where e is the electron charge and x is the quantity, by the condition ∆x∆p ≥ ¯h/2, where ∆x and ∆p re- displacement) can be computed by considering the virtual spectively denote the position and the momentum uncertain- pair as a harmonic oscillator, such that in the quasi-static ties. Taking the momentum ∆p =hω ¯ 0/c necessary to close the limit rest energy gap would yield ∆x ≥ c/2ω0, which is of the order 2 of the Compton wavelength. The exact choice on how to relate mω0x = eE , (3) ∆x to the dipole volume is arbitrary to a large extent when where m is the electron mass and ω0 is the natural reso- based on the uncertainty inequality. nance frequency. As customary in this sort of semi-classical 4 The same result would be derived in Gaussian units, for treatment, valid when the excitation of the quantum os- which ε0 = 1/4π. 5 cillator is very small (or equivalently E ≪ ES), the res- Substituting typical numbers in Eq. (6), for instance ¯hω0 = 2 onance frequency is determined by the energy associated 2mc as the rest energy of the positronium and r = λc/2, one 2 −12 with the quantum transition. The two-level system un- would getε ˜0 = 2e /¯hc = 1.62 × 10 As/(Vm). This number der consideration is formed by the ground state of the underestimates the correct value by one order of magnitude. virtual pairs and the real positronium atom as the ex- However, apart from numerical prefactors which could possi- cited state [10]. The energy gap is denoted Egap, so that bly change by a small amount, we note that the expression for ω0 = Egap/¯h, and it is expected to be of the order of ε˜0 does not depend on the rest mass if the energy gap is pro- portional to the latter. As discussed in what follows, this fact 1 In Gaussian units, Eq. (2) reads D = E + 4πP , so that hints at having to sum the contributions from all elementary ε0 is replaced by the constant ‘1.’ In the language of Gaussian particles with electric charge, in this manner reinterpreting the units, this paper discusses the physical significance of this ‘1.’ above value ofε ˜0 as the partial contribution from the virtual 2 See second footnote on page 743 of Ref. [7]: “I would like electron-positron pairs. This approach provides an estimate for to thank Prof. Heisenberg for kindly informing me about this the number of elementary pairs contributing to the vacuum re- hypothesis of N. Bohr.” sponse. G. Leuchs et al.: The quantum vacuum at the foundations of classical electrodynamics 3 must resort in this case to crude semi-classical assump- whence we get mr tions that, nevertheless, have often proved useful in de- µ˜0 = . (13) termining the correct order of magnitude of magnetic ef- e2 fects [12]. We will continue our analysis adapting results The expressions forε ˜0 andµ ˜0 must be compatible with 2 from quantum mechanics to obtain µ0 from a seemingly the Maxwell equations. Imposingε ˜0µ˜0 = 1/c , we find a naive classical picture. relation between the transition energy E0 and the radius The magnetic counterpart of Eq. (2) is r. Moreover, the productε ˜0µ˜0 does not depend on the volume of the virtual pair since it cancels out in the ratio, H 1 B − M = , (7) 2 1 M/B µ0 c = = , (14) ε˜0 µ˜0 ℘/E where M is the magnetization and H is sometimes called implying that in our model the speed of light is a con- the ‘magnetic field,’ although we will simply call it H and sequence of the magnetic and electric responses of each reserve the term for B. In analogy to D, the vector H virtual pair locally. Using Eqs. (4) and (12), The radius isolates the contribution from the free current to the total associated with the volume of the virtual pair then reads magnetic field. The expression above hints at associating c µ0 to the inverse of the vacuum magnetic response. r = . (15) Analogously to the polarization, the magnitude of M ω0 can be calculated from the microscopic magnetic dipole M Consistency with the Maxwell equations therefore re- moment and its effective volume using sults, by inserting Eq. (15) into Eq. (6), in the following 2 M expression forε ˜0 (andµ ˜0 =1/ε˜0c ), M = , (8) 2 V ¯hω0 e ε˜0 = 2 . (16) where for consistency the same volume used for ℘ is as- mc ¯hc sumed. This equation can be written in terms of the fine structure 2 6 An external magnetic field applied to the vacuum in- constant α = e /(4πε0¯hc) and the known value of ε0 as duces an electric field vortex which accelerates the virtual electron and positron in opposite directions. This electric ¯hω0 ε˜0 =4πα 2 ε0 . (17) field Ei induced in a circular orbit fulfills mc r This expression shows that our simple model supplies Ei = − B,˙ (9) numerical valuesε ˜0 andµ ˜0 lying surprisingly close to the 2 7 correct values ε0 and µ0, in this manner supporting their where the dot denotes the time derivative. It generates a novel physical interpretation. Looking more closely, how- torque that increases the electron angular momentum J ever, one notices that Eq. (16) tends to underestimate the by correct ε0 and overestimate µ0 for sensible choices of ¯hω0. 2 er Also an astonishing property ofε ˜0 andµ ˜0 becomes ap- ∆J = B. (10) 2 parent: they are independent of the mass if, as expected, the energy gap depends linearly on m. Indeed, one might According to quantum mechanics, the magnetic mo- be tempted to assume that only the lightest elementary ment and the angular momentum relate to each other charged particles would contribute to the vacuum polar- through the expression ization, since their virtual excitation would demand the ge smallest amount of energy. But the independence of mass M = J, (11) 2m brings us to the intriguing conclusion that all charged elementary particle pairs should contribute more or less where g is the Land´efactor. A completely orbital or dia- equally to the vacuum response. Equation (16) would then magnetic contribution would render g = 1, whilst g = 2 represent only one partial contribution to the vacuum po- would correspond to a pure spin or paramagnetic contri- larization. The total response would be the sum of all par- bution. We choose the latter possibility for definiteness. tial contributions [13,14], but weighted by the square of Due to symmetry and simplicity, the angular momentum the charges qj , where the index j =1, 2,... stands for the state of the virtual pair is assumed a singlet, in which case different virtual elementary particle pairs. Equation (16) the positron contributes the same amount as the electron is modified accordingly to to the magnetic response. These particles have thus op- ¯hω q 2 posite spins and orbital angular momenta, but because of ε˜total =4πα 0 j ε . (18) 0 mc2 X  e  0 their opposite charges, the individual responses combine j positively. Substituting the angular momentum variation 6 given by Eq. (10) in Eq. (11), and multiplying by a factor In Gaussian units, the fine structure constant is α = 2 2 to account for the positron contribution, the induced e /(¯hc), and the result does not depend on the unit system. 7 2 Supposinghω ¯ 0 = 2mc , one obtains the deviation factor dipole moment is 2 e2r2 4πα¯hω0/mc ≈ 1/10. We think this is surprisingly close con- M = B, (12) m sidering the crude semi-classical model employed. 4 G. Leuchs et al.: The quantum vacuum at the foundations of classical electrodynamics

We note in passing that the speed of light is independent Finally, the association here investigated suggests how of the number of particles, since the latter cancels out in the Maxwell equations would provide a connection be- Eq. (14). The equality betweenε ˜0 and ε0 imposes a rela- tween the value of the speed of light and the magnitude tion between the energy gap and the number of elementary of the linear response of vacuum predicted by quantum pairs, supplying an estimate for the latter, physics. The situation is reminiscent of the decrease of the speed of light caused by phase shifts inside a dielec- 2 2 qj 1 mc tric material. We believe this to be a beautiful connection = . (19) X  e  4πα ¯hω0 between, on the one hand, the quantum concepts of virtual j particles and vacuum fluctuations and, on the other hand, the actual value of the maximum speed existent in nature The choice ¯hω = 2mc2 would suggest around ten con- 0 baring so many fundamental consequences. The fact that tributing pairs with the electron charge. the celebrated Lorentz invariant Maxwell equations would In conclusion, our aim has been to demonstrate how provide such a bridge just adds to the surprise. a few reasonable assumptions suffice to derive the correct orders of magnitude of ε0 and µ0, providing also a physical meaning to these quantities8. Whilst regarded in classical We acknowledge enlightening discussions with Katiuscia N. electromagnetism as constants deprived of deeper physical Cassemiro, Pierre Chavel, Joseph H. Eberly, Michael Fleis- meaning, in the light of the quantum theory they would chhauer, Holger Gies, Uli Katz, Natalia V. Korolkova, G´erard be connected to fundamental physical processes, the polar- Mourou, Nicolai B. Narozhny, Serge Reynaud, Wolfgang P. ization and the magnetization of virtual pairs in vacuum Schleich, Anthony E. Siegman, and John Weiner. in the linear regime comprising weak and low-frequency (classical) fields. This alone we think is worth noting. Ad- ditionally, one outcome of our model is an independent References estimate of the volume associated with a virtual pair. Fur- thermore, our estimates also provide an evaluation of the 1. D. J. Griffiths: Introduction to Electrodynamics, 3th edi- number of charged elementary particles. A more rigorous tion (Prentice-Hall, New Jersey 1999) p. 180 calculation would have to rely on quantized fields [15] and 2. I. E. Tamm: Fundamentals of the Theory of Electricity on a potentially modified response closer to the resonance (Mir Publishers, Moscow 1979) p. 117 frequency. 3. E. M. Purcell: Electricity and Magnetism, Berkeley Physics In many of the seminal nonlinear effects of the quan- Course Vol. II (McGraw-Hill, New York 1965) p. 332 tum vacuum, the contribution of the lightest elementary 4. P. Milonni: The Quantum Vacuum (Academic Press Inc., particles dominates. By contrast, the linear effects dis- New York, 1994) 166 cussed here enjoy equal contributions from all elementary 5. W. Dittrich, H. Gies: Springer Tracts Mod. Phys. , 1 (2000) particles, including the ones not yet discovered, owing to 6. W. Heisenberg, H. Euler: Z. Phys. 98, 714 (1936) the independence of the masses of the virtual particles. 7. F. Sauter: Z. Phys. 69, 742 (1931) 8 A first step in a more quantitative model would take into 8. F. Sauter: Z. Phys. 73, 547 (1931) account the distinction between r2 in Eq. (12) and the average 9. J. Schwinger: Phys. Rev. 82, 664 (1951) orbital radius needed for the diamagnetic term, here denoted 10. M. Ruf, G. R. Mocken, C. M¨uller, K. Z. Hatsagortsyan, C. by ρ2 (where ρ is the distance to the axis in cylindrical coor- H. Keitel: Phys. Rev. Lett. 102, 080402 (2009) dinates), originating from the actual charge distribution. As- 11. N. B. Narozhny, S. S. Bulanov, V. D. Mur, V. S. Popov: suming a constant probability density for the electron charge Zh. Eksp. Teor. Fiz. 129, 14 (2006) [JETP 102, 9 (2006)] inside a solid spherical volume with radius R, an elementary 12. R. P. Feynman, R. B. Leyton, M. Sands: The Feynman calculation shows that ρ2 = 2/5 × R2. Using this value in Lectures on Physics, Vol. 2 (Addison-Wesley Publishing Eq. (12) and the volume V = 4π/3 × R3 in Eq. (6), we find Co., Massachusetts 1964) Ch. 34 2 13. L. D. Landau, A. A. Abrikosov, I.M. Khalatnikov: Dokl. R = 5/2 × ¯hc/(¯hω0) by imposingε ˜0µ˜0 = 1/c . These results Akad. Nauk 95, 1177 (1954) changep ε ˜0 of Eq. (17) to the new estimate 14. A. D. Sakharov: Teor. Mat. Fiz. 23, 178 (1974) 3/2 15. J. Savolainen, S. Stenholm: Am. J. Phys. 40, 667 (1972) 2 ¯hω0 ε˜0 = 3α ε0 . (20)  5  mc2

For the total number of virtual elementary pairs contributing to the vacuum response, one finds in place of Eq. (19) the estimate 3 2 q 2 1 5 / mc2 j = , (21) e 3α 2  ¯hω0 Xj   2 resulting in a number close to 90 pairs if ¯hω0 = 2mc . Finally, we note that the actual charge distribution would be probably peaked around the origin, in this manner reducing the ratio ρ2/R2 and increasing as a consequence the effective volume and the number of contributing virtual pairs.