Appl Phys B (2010) 100: 9–13 DOI 10.1007/s00340-010-4069-8

The quantum vacuum at the foundations of classical electrodynamics

G. Leuchs · A.S. Villar · L.L. Sánchez-Soto

Received: 30 April 2010 / Published online: 29 May 2010 © The Author(s) 2010. This article is published with open access at Springerlink.com

Abstract In the classical theory of electromagnetism, the binding nuclei, producing tiny atomic and molecular dipoles permittivity ε0 and the permeability μ0 of free space are in the material. The macroscopic effect, averaged over a constants whose magnitudes do not seem to possess any region large compared to the atomic dimensions, is the deeper physical meaning. By replacing the free space of appearance of an induced electric field. The strength of classical with the quantum notion of the vacuum, this response depends on how susceptible to displacement we speculate that the values of the aforementioned con- the atomic dipoles are and on how much space they oc- stants could arise from the polarization and magnetiza- cupy. tion of virtual pairs in vacuum. A classical dispersion Putting it in more quantitative terms, in an isotropic model with parameters determined by quantum and parti- medium the polarization P is proportional to the external cle physics is employed to estimate their values. We find electric field E through the relation P = χε0E, where χ is the correct orders of magnitude. Additionally, our simple the linear susceptibility of the material and ε0 is the permit- assumptions yield an independent estimate for the num- tivity of free space. The constant ε0 has the unimpressive ber of charged elementary based on the known role of matching units, just ‘happening to have the value’ −12 values of ε0 and μ0 and for the volume of a virtual pair. ε0 = 8.8542 × 10 As/(Vm)[1]. The expression for the Such an interpretation would provide an intriguing con- magnitude of P, which connects the microscopic dipoles to nection between the celebrated theory of classical elec- their macroscopic effect, reads tromagnetism and the quantum theory in the weak-field ℘ limit. P = , (1) V where ℘ is the dipole moment and V is the effective volume In classical electrodynamics, a dielectric medium becomes per dipole. polarized in the presence of an electric field. A weak elec- It is customary to define the electric displacement as tric field slightly displaces the electronic clouds from their D = ε0E + P, (2)

G. Leuchs () · A.S. Villar · L.L. Sánchez-Soto since it turns out to be independent of the induced charge [2]. Max Planck Institut für die Physik des Lichts, In the particular case of an isotropic linear dielectric, (2) Günther-Scharowsky-Str. 1/Bau 24, 91058 Erlangen, Germany yields D = εε0E, where ε = (1 + χ) is the relative permit- e-mail: [email protected] tivity of the medium. G. Leuchs · A.S. Villar The electric displacement D enjoys no better status Institut für Optik, Information und Photonik, Universität among us physicists than that of ε0. It is mostly seen as a Erlangen-Nürnberg, Staudtstr. 7/B2, 91058 Erlangen, Germany mathematical tool of limited use, because it would combine two physically different quantities [2]. In fact, the electric L.L. Sánchez-Soto Departamento de Óptica, Facultad de Física, Universidad field is the force per unit charge, whilst the spatial varia- Complutense, 28040 Madrid, Spain tion of the polarization is the induced charge. Sometimes, 10 G. Leuchs et al. the electric displacement is even considered completely dis- the virtual – pairs polarizes them. The mag- pensable [3], since the condition ∇ × D = 0 does not apply nitude of the dipole moment ℘ = ex induced on one vir- in general—in contrast to the case of E, a scalar potential tual pair (where e is the electron charge and x is the dis- cannot be associated with D. placement) can be computed by considering the virtual In this paper, we offer a physical interpretation of ε0 and pair as a harmonic oscillator, such that in the quasi-static thus of D [4–6]. We discuss how (2) might hint at fundamen- limit tal physical aspects of the electromagnetic field itself if, in 2 = the light of its second term, we interpret the first term as also mω0x eE, (3) originating from a polarized medium, so that D is the total polarization. That would put ε0E and P on an equal foot- where m is the electron mass and ω0 is the natural res- ing.1 That interpretation is preposterous in classical electro- onance frequency. As customary in this sort of semi- magnetism, since the vacuum is defined as the emptiness. classical treatment, valid when the excitation of the quan- In quantum theory, however, that is not the case, and virtual tum oscillator is very small (or equivalently E  ES ), pairs making up the vacuum are expected to show a the resonance frequency is determined by the energy as- response to an applied field. In analogy to εε0 for a material sociated with the quantum transition. The two-level sys- medium, ε0 itself could be associated with the polarization tem under consideration is formed by the ground state of the quantum vacuum P0 = ε0E, a concept deeply rooted of the virtual pairs and the real as the in [7, 8]. We later extend the dis- excited state [13]. The energy gap is denoted Egap,so cussion to magnetic phenomena, linking the permeability of that ω0 = Egap/, and it is expected to be of the or- free space μ0 to the magnetization of virtual pairs in vac- der of the positronium rest energy. In addition, the quasi- uum. static regime requires that the frequency ω of the classical The presence of an external electromagnetic field disturbs field E be much smaller than the oscillator resonance fre- the stability of the vacuum as the ground state of all fields. quency ω0. That means we are considering a low-energy In this case, virtual particles must be considered to account field probing pair creation as a non-resonant effect (clas- for the possible physical processes in vacuum. Phenomena sical field). Inserting this relation for ω0 into (3) to deter- such as pair creation, vacuum birefringence, and light-light mine the displacement, one obtains the induced dipole mo- scattering are well established for strong fields [9–12], with ment 2 3 18 amplitudes larger than ES = m c /e ≈ 10 V/m. This is e2 exactly the expression for the critical field first pointed out ℘ = E. (4) 2 2 by Niels Bohr, as acknowledged by F. Sauter. In the liter- mω0 ature, ES is often referred to as the Schwinger field [12]. These effects illustrate the nonlinear properties of the vac- The magnitude of the given by (1) 3 uum [7, 8]. depends on the effective volume per dipole V = r . In the- Here we assume the limit of weak fields and focus on the oretical investigations [14], it is generally accepted that the linear response. The question we pose is: could the vacuum appropriate length scale for a virtual electron–positron pair 3 =  quantum fluctuations be subtly embedded in the classical in vacuum is the Compton wavelength λc /mc. With theory? If yes, one piece of evidence could be the existence these considerations, (1) results in the vacuum polariza- 4 of physical constants whose numerical values are simply de- tion termined experimentally, but which would emerge naturally e2 from the quantum theory. We speculate whether the permit- P = E. (5) 0 2 3 tivity and the permeability of free space could be such quan- mω0r tities. In the remainder of this paper, we will estimate the The induced polarization is proportional to the electric field, values of ε and μ based on simple semi-classical assump- 0 0 as expected. Assigning the quantity multiplying E to the tions. We start by considering the electrical properties of the quantum vacuum. An external electric field interacting with 3The uncertainty principle also offers an estimate for this quantity, by the condition xp ≥ /2, where x and p respectively denote the position and the momentum uncertainties. Taking the momentum 1 In Gaussian units, (2) reads D = E + 4πP,sothatε0 is replaced by p = ω0/c necessary to close the rest energy gap would yield x ≥ the constant ‘1.’ In the language of Gaussian units, this paper discusses c/2ω0, which is of the order of the Compton wavelength. The exact the physical significance of this ‘1.’ choice on how to relate x to the dipole volume is arbitrary to a large 2See second footnote on p. 743 of [10]: “I would like to thank Prof. extent when based on the uncertainty inequality. Heisenberg for kindly informing me about this hypothesis of N. Bohr.” 4The same result would be derived in Gaussian units. The quantum vacuum at the foundations of classical electrodynamics 11 vacuum polarizability, we arrive at5 where the dot denotes the time derivative. It generates a torque that increases the electron angular momentum J by e2 ε˜ = . (6) 2 0 2 3 er mω0r J = B. (10) 2 For the permeability of free space μ , we attempt a sim- 0 According to quantum mechanics, the magnetic moment ilar procedure to estimate its value by associating this quan- and the angular momentum relate to each other through the tity to the linear magnetic response of the quantum vacuum. expression Here we encounter the added difficulty that some aspects of the material magnetization arise from pure quantum ef- ge M = J, (11) fects without classical explanation. One must resort in this 2m case to crude semi-classical assumptions that, nevertheless, where g is the Landé factor. A completely orbital or diamag- have often proved useful in determining the correct order of netic contribution would render g = 1, whilst g = 2 would magnitude of magnetic effects [15]. We will continue our correspond to a pure spin or paramagnetic contribution. We analysis adapting results from quantum mechanics to obtain choose the latter possibility for definiteness. Due to symme- μ0 from a seemingly naive classical picture. try and simplicity, the angular momentum state of the virtual The magnetic counterpart of (2)is pair is assumed a singlet, in which case the positron con- 1 tributes the same amount as the electron to the magnetic re- H = B − M, (7) sponse. These particles have thus opposite spins and orbital μ 0 angular momenta, but because of their opposite charges, where M is the magnetization and H is sometimes called the individual responses combine positively. Substituting the the ‘magnetic field,’ although we will simply call it H and angular momentum variation given by (10)in(11), and mul- reserve the term for B. In analogy to D, the vector H isolates tiplying by a factor 2 to account for the positron contribu- the contribution from the free current to the total magnetic tion, the induced dipole moment is field. The expression above hints at associating μ to the 0 e2r2 inverse of the vacuum magnetic response. M = B, (12) Analogously to the polarization, the magnitude of M can m be calculated from the microscopic magnetic dipole moment whence we get M and its effective volume using mr μ˜ = . (13) M 0 e2 M = , (8) V The expressions for ε˜0 and μ˜ 0 must be compatible with ˜ ˜ = 2 where for consistency the same volume used for ℘ is as- the Maxwell equations. Imposing ε0μ0 1/c , we find a E sumed. relation between the transition energy 0 and the radius r. ˜ ˜ An external magnetic field applied to the vacuum induces Moreover, the product ε0μ0 does not depend on the volume an electric field vortex which accelerates the virtual electron of the virtual pair since it cancels out in the ratio, and positron in opposite directions. This electric field Ei in- M 2 1 /B duced in a circular orbit fulfills c = = , (14) ε˜0 μ˜ 0 ℘/E r ˙ Ei =− B, (9) implying that in our model the speed of light is a conse- 2 quence of the magnetic and electric responses of each vir- tual pair locally. Using (4) and (12), the radius associated 5 2 Substituting typical numbers in (6), for instance ω0 = 2mc as the with the volume of the virtual pair then reads rest energy of the positronium and r = λc/2, one would get ε˜0 = 2  = × −12 c 2e / c 1.62 10 A s/(V m). This number underestimates the r = . (15) correct value by one order of magnitude. However, apart from numeri- ω0 cal prefactors which could possibly change by a small amount, we note ˜ that the expression for ε0 does not depend on the rest mass if the energy Consistency with the Maxwell equations therefore re- gap is proportional to the latter. As discussed in what follows, this fact sults, by inserting (15)into(6), in the following expression hints at having to sum the contributions from all elementary particles ˜ ˜ = ˜ 2 with electric charge, in this manner reinterpreting the above value of for ε0 (and μ0 1/ε0c ), ε˜ as the partial contribution from the virtual electron–positron pairs. 0 2 This approach provides an estimate for the number of elementary pairs ω e ε˜ = 0 . (16) contributing to the vacuum response. 0 mc2 c 12 G. Leuchs et al.

This equation can be written in terms of the fine structure electromagnetism as constants deprived of deeper physical 2 6 constant α = e /(4πε0c) and the known value of ε0 as meaning, in the light of the quantum theory they would be connected to fundamental physical processes, the polariza- ω ε˜ = πα 0 ε . tion and the magnetization of virtual pairs in vacuum in the 0 4 2 0 (17) mc linear regime comprising weak and low-frequency (classi- This expression shows that our simple model supplies cal) fields. This alone we think is worth noting. Addition- numerical values ε˜0 and μ˜ 0 lying surprisingly close to the ally, one outcome of our model is an independent estimate 7 correct values ε0 and μ0, in this manner supporting their of the volume associated with a virtual pair. Furthermore, novel physical interpretation. Looking more closely, how- our estimates also provide an evaluation of the number of ever, one notices that (16) tends to underestimate the cor- charged elementary particles. A more rigorous calculation rect ε0 and overestimate μ0 for sensible choices of ω0. would have to rely on quantized fields [18] and on a poten- Also an astonishing property of ε˜0 and μ˜ 0 becomes appar- tially modified response closer to the resonance frequency. ent: they are independent of the mass if, as expected, the In many of the seminal nonlinear effects of the quantum energy gap depends linearly on m. Indeed, one might be vacuum, the contribution of the lightest elementary particles tempted to assume that only the lightest elementary charged dominates. By contrast, the linear effects discussed here en- particles would contribute to the vacuum polarization, since joy equal contributions from all elementary particles, includ- their virtual excitation would demand the smallest amount ing the ones not yet discovered, owing to the independence of energy. But the independence of mass brings us to the in- of the masses of the virtual particles. triguing conclusion that all charged pairs Finally, the association here investigated suggests how should contribute more or less equally to the vacuum re- the Maxwell equations would provide a connection between sponse. Equation (16) would then represent only one partial the value of the speed of light and the magnitude of the lin- contribution to the vacuum polarization. The total response ear response of vacuum predicted by quantum physics. The would be the sum of all partial contributions [16, 17], but situation is reminiscent of the decrease of the speed of light weighted by the square of the charges qj , where the index caused by phase shifts inside a dielectric material. We be- j = 1, 2,... stands for the different virtual elementary par- lieve this to be a beautiful connection between, on the one ticle pairs. Equation (16) is modified accordingly to hand, the quantum concepts of virtual particles and vacuum   fluctuations and, on the other hand, the actual value of the  2 ω qj maximum speed existent in nature baring so many funda- ε˜total = 4πα 0 ε . (18) 0 mc2 e 0 mental consequences. The fact that the celebrated Lorentz j invariant Maxwell equations would provide such a bridge We note in passing that the speed of light is independent of just adds to the surprise. the number of particles, since the latter cancels out in (14). Acknowledgements We acknowledge enlightening discussions with The equality between ε˜0 and ε0 imposes a relation between Katiuscia N. Cassemiro, Pierre Chavel, Joseph H. Eberly, Michael the energy gap and the number of elementary pairs, supply- ing an estimate for the latter,   the diamagnetic term, here denoted by ρ2 (where ρ is the distance to  2 2 qj 1 mc the axis in cylindrical coordinates), originating from the actual charge = . (19) distribution. Assuming a constant probability density for the electron’s e 4πα ω j 0 charge inside a solid spherical volume with radius R, an elementary 2 2 calculation shows that ρ =2/5 × R . Using this√ value in (12)and 3  = 2 the volume V = 4π/3 × R in (6), we find R = 5/2 × c/(ω0) The choice ω0 2mc would suggest around ten con- 2 by imposing ε˜0μ˜ 0 = 1/c . These results change ε˜0 of (17) to the new tributing pairs with the electron charge. estimate In conclusion, our aim has been to demonstrate how a   2 3/2 ω few reasonable assumptions suffice to derive the correct or- ε˜ = 3α 0 ε . (20) 0 5 mc2 0 ders of magnitude of ε0 and μ0, providing also a physical meaning to these quantities.8 Whilst regarded in classical For the total number of virtual elementary pairs contributing to the vacuum response, one finds in place of (19)theestimate      3/2 2 qj 2 1 5 mc 6In Gaussian units, the fine structure constant is α = e2/(c),andthe = , (21) e 3α 2 ω result does not depend on the unit system. j 0 7Supposing ω = 2mc2, one obtains the deviation factor 0 resulting in a number close to 90 pairs if ω = 2mc2. Finally, we note  2 ≈ 0 4πα ω0/mc 1/10. We think this is surprisingly close considering that the actual charge distribution would be probably peaked around the crude semi-classical model employed. the origin, in this manner reducing the ratio ρ2/R2 and increasing 8A first step in a more quantitative model would take into account the as a consequence the effective volume and the number of contributing distinction between r2 in (12) and the average orbital radius needed for virtual pairs. The quantum vacuum at the foundations of classical electrodynamics 13

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