Problems
Application 2.1 Moving Point Charges
2.5 Quantum Limits and New Physics
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46 THE MAXWELL EQUATIONS: HISTORY, AVERAGING, AND LIMITATIONS
2.5 Quantum Limits and New Physics 47
The effects of vacuum polarization become significant when the external field strengths approach Ec∼ mc2
eλc = m2c3
eh ∼1018V/m Bc∼ m2c2
eh ∼109T.
(2.64)
Detailed calculations predict a dramatic effect: the breakdown of the linearity of the vacuum Maxwell equations (and thus the principle of superposition) when the field strengths approach (2.64). This QED result (valid for slowly varying fields) can be cast in the form of non-linear constitutive relations for the vacuum. To lowest order inα, virtual pair production generates a vacuum polarization Pand a vacuum magnetizationMgiven by16
P= 20α E2c
2(E2−c2B2)E+7c2(E·B)B
(2.65) M= − 2α
μ0E2c
2(E2−c2B2)B+7(E·B)E . At the time of this writing, these effects have not yet been detected.
2.5.3 Quantum Fluctuations
An entirely different restriction on the validity of classical electrodynamics arises becauseE(r, t) and B(r, t) are non-commuting vector operators in QED rather than c-number vector fields. We infer from the uncertainty principle that the electric field and the magnetic field cannot take on sharp values simultaneously. Quantum fluctuations of the field amplitudes and phases are always present, even in the vacuum state. On the other hand, the fields produced by macroscopic sources like a light bulb, a laser, a microwave generator, or a blackbody radiator invariably exhibit (much larger) non-quantum fluctuations also. The non-classical regime ofquantum opticsemerges when the classical fluctuations are suppressed to reveal the quantum fluctuations.
Glauber (1963) pointed out that it is possible to distinguish a classical electromagnetic field from a quantum electromagnetic field by focusing on the (time-averaged) field intensity operator ˆI and expectation values like
g(2)=Iˆ2
Iˆ2. (2.66)
Field intensity is a positive quantity, sog(2)≥0 whether the fluctuations are classical or quantum.
However, for fields described by classical electrodynamics, the sharper inequalityg(2)≥1 holds.
Quantum effects generally reveal themselves when we pass from the macroscopic limit of many atoms to the microscopic limit of one or a few atoms. This suggests that a single atom should be a good source of non-classical radiation. The data forg(2) shown in Figure 2.9 illustrate this for low-intensity laser light directed through a cavity containing about a dozen Rb atoms. The parameter =2(ωL−ωA)τ is the difference between the laser frequencyωLand the frequencyωA of a Rb atomic transition, normalized by the radiative lifetimeτ of the transition. When0, the atoms resonantly scatter the laser light and non-classical values 0≤g(2)<1 are seen. This means that the operator character of the field variables plays an essential role in the description of the transmitted
16 See, for example, M. Soljaˇci´c and M. Segev, “Self-trapping of electromagnetic beams in vacuum supported by QED nonlinear effects”,Physical Review A62, 043817 (2000) and references therein.
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0 1
0 1 2 3
Δ g(2) Classical
Quantum
Figure 2.9: Cavity QED data forg(2)as defined in (2.66). The horizontal axis is the laser detuning. The classical theory is valid only wheng(2)≥1. The solid line is a guide to the eye. Figure adapted from Foster, Mielke, and Orozco (2000). Copyright 2000 by the American Physical Society.
light. As the detuningincreases, the atoms scatter less and less. Eventually,g(2)exceeds unity and classical theory suffices to describe the transmitted light as well as the incident light.17
2.5.4 New Physics
The Maxwell equations are the mathematical expression of the known facts of experimental electro- magnetism. They are subject to modification if any future experiment gives convincing evidence for deviations from any of these “facts”. Conversely, one can speculate that the laws of physics are not precisely as we usually imagine them and posit that the Maxwell equations differ (albeit minutely) from their usual form. If so, the modified theory will predict heretofore unobserved phenomena which can be sought in the laboratory. If not seen in experiment, such measurements set limits on the size of the presumed deviations from the orthodox theory.
Two examples of very long standing are the possibilities that (i) Coulomb’s law is not precisely inverse-square and (ii) magnetic charge exists and is a source for magnetic fields. The Coulomb question is often addressed by supposing that the force between two point charges varies as 1/r2+ rather than as 1/r2. This proposal—which we interpret today as a signature of a non-zero mass for the photon—has a variety of experimental implications. An electrostatic test was performed by Maxwell himself, who concluded that || ≤5×10−5. Contemporary experiments of the same basic design give|| ≤6×10−17.18Magnetic charge is the subject of Section 2.5.5 below.
Apart from the massive photon and magnetic charge, a variety of other “new physics” scenarios have been suggested which alter the Maxwell equations in various ways. These speculations include (a) charge is not exactly conserved; (b) electromagnetism is not exactly the same in every inertial frame;
and (c) electromagnetism violates rotational and/or inversion symmetry. There is no experimental evidence for any of these at the present time, but it is necessary to keep an open mind.
2.5.5 Magnetic Charge
As first-rate physical theories go, the Maxwell equations are embarrassingly asymmetrical. ∇ ·E is proportional to an electric charge densityρ, but∇ ·B is notproportional to a magnetic charge
17 In this experiment, the average number of photons in the cavity is always much less than one. That is,
0E2V /−hω1, whereVis the cavity volume. The data for >2 in Figure 2.9 show that very small mean photon number alone is not sufficient to guarantee that an electromagnetic field is non-classical.
18 See Sources, References, and Additional Reading for references to the experimental and theoretical literature on this subject.
2.5 Quantum Limits and New Physics 49
ρm
cρe
Figure 2.10: Allowed values ofρmandcρelie on the circle. The radius vector indicates the ratio of magnetic charge to electric charge for a hypothetical elementary particle.
density ρm. Similarly, an electric current density j appears in the Amp`ere-Maxwell law, but no magnetic current densityjmappears in Faraday’s law. To the extent that we associate symmetry with mathematical beauty, the Maxwell equations violate Dirac’s dictum that “physical laws should have mathematical beauty”.19
This state of affairs has led many physicists tosymmetrizethe Maxwell equations by supposing that (i) magnetic charge exists and (ii) the motion of particles with magnetic charge produces a magnetic current densityjmwhich satisfies∇ ·jm+∂ρm/∂t =0. If we temporarily letρeandjestand for the usual electric charge density and current density, these assumptions generalize (2.33) and (2.34) to20
∇ ·E= ρe
0
∇ ·B=μ0ρm (2.67)
∇ ×E= −μ0jm−∂B
∂t ∇ ×B=μ0je+ 1
c2
∂E
∂t. (2.68) The new terms acquire meaning from a similarly generalized Coulomb-Lorentz force density,21
f=(ρeE+je×B)+(ρmB−jm×E/c2). (2.69) For present purposes, the most interesting property of (2.67), (2.68), and (2.69) is that they are invariant to adualitytransformation of the fields and sources parameterized by an angleθ:
E=Ecosθ+cBsinθ cB= −Esinθ+cBcosθ (2.70) cρe=cρecosθ+ρmsinθ ρm= −cρsinθ+ρmcosθ (2.71) cje=cjecosθ+jmsinθ jm= −cjesinθ+jmcosθ. (2.72) This means thatE,B, ρe,je,ρm, and jm satisfy exactly the same equations as their unprimed counterparts. The only constraints are those imposed by the transformation itself:
c2ρe2+ρm2 =c2ρe2+ρm2. (2.73) Duality implies that it is strictly a matter of convention whether we say that a particle has electric charge only, magnetic charge only, or some mixture of the two. To see this, let the circle in Figure 2.10 be the locus of values ofcρeandρmpermitted by (2.73). The radius vector specifies the ratioρm/cρefor a hypothetical elementary particle with, say, electric chargee <0 and magnetic chargeg >0. However, if the same ratio applies to every other particle in the Universe, no electromagnetic prediction changes if we exploit dual symmetry and rotate the radius vector (chooseθ) to makeρm=0 for every particle.
19 Famously written on a blackboard at Moscow State University in 1955.
20 A particle with no electric charge and one unit of magnetic charge is called amagnetic monopole.
21 We have chosen the SI unit of magnetic charge as the A·m. Some authors choose this unit as the weber, in which case μ0ρm→ρmandμ0jm→jmin our extended Maxwell equations and the magnetic charge and current contributions to (2.69) should be divided byμ0.
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This brings us back to the original Maxwell-Lorentz equations, which are consistent with all known experiments. On the other hand, if an elementary particle is ever discovered where the intrinsic ratio g/cediffers from the value shown in Figure 2.10, the option to simultaneously “rotate away” magnetic charge for all particles disappears. In that case, (2.67), (2.68), and (2.69) become the fundamental electromagnetic laws of Nature. This exciting possibility keeps searches for magnetic monopoles an active part of experimental physics.