《电磁学》课程教学资源（教材讲义）真空中的静电场（光子质量）Terrestrial and Extraterrestrial Limits on The Photon Mass

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REVIEWS OF MODERN PHYSICS VOLUME 43, NUMBER 3 JULY 1971 Terrestrial and Extraterrestrial Limits on The Photon Mass ALFRED S. GOLDHABER* Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, New York 11790 MICHAEL MARTIN NIETOt The Niels Bohr Institute,University of Copershagen,Copenhagers,Demmark Department of Physics, University of California, Santa Barbara, California 93106 We give a review of methods used to set a limit on the mass of the photon. Direct tests for frequency dependence of the speed of light are discussed, along with more sensitive techniques which test Coulomb's Law and its analog in magnetostatics. The link between dynamic and static implications of finite a is deduced from a set of postulates that make Proca's equations the unique generalization of Maxwell's. We note one hallowed postulate, that of energy con- servation, which may be tested severely using pulsar signals. We present the merits of the old methods and of possible new experiments, and discuss other physical implications of finite u. A simple theorem is proved: For an experiment confined in dimensions D, effects of finite u are of order (uD)2-there is no "resonance" as the oscillation frequency w approaches u (A=c=1). The best results from past experiments are (a) terrestrial measurements of c at different fre- quencies ≤2×10-3g=7×10-cm-1=10-10 eV; (b)measurements of radio dispersion in pulsar signals(whistler effect) u10-g=3×10-7cm-1=6×10-12eV; (c)laboratory tests of Coulomb's law ≤2×10-7g=6×10-10cm-1=10-1eV; (d)limits on a constant external"magnetic field at the earth's surface 4×10-8 g=10-10 cm-1=3×10-15 eV. Observations of the Galactic magnetic field could improve the limit dramatically. I. INTRODUCTION energy Tv.These light quanta travel with velocity c, and hence have zero rest mass. The success of quantum One of the great triumphs of classical physics was the electrodynamics in predicting experiments to six or formulation of the Maxwell electromagnetic field more decimal places has made the massless photon a equations. A fundamental prediction of these equations tacit axiom of physics. A sign of this is that as late as is that all electromagnetic radiation in vacuum travels 1968 the Particle Data Group tables gave experimental at a constant velocity c. The most recent experiments limits on the neutrino masses, but just a zero for the have confirmed this prediction with an accuracy near to one part per million, over a wide range of frequencies photon mass(Rosenfeld, et al. 1968). This is not too sur- prising since QED is our only 'exact" quantum theory. (Froome and Essen, 1969; Taylor, Parker, and Langen- Nuclear and particle quantum theories do not even berg, 1969). In the context of quantum theory, a relativistic approach such accuracy. The tacit axiom of masslessness corresponds to the quantized electromagnetic field of frequency v is belief that if the photon has an effective mass a, it does recognized as an assembly of photon particles with so only because it is slightly off the mass shell. Using an * Supported in part under the auspices of the United States uncertainty argument, we would estimate Atomic Energy Commission. Supported in part by the National Science Foundation u≈h/(At)c2=3.7×10-6 g/T, (1.1) Address after September 1, 1971: Department of Physics, Purdue University, Lafayette, Ind. 47907. where T is the age of the universe in units of 10to years. 277 Copyright C 1971 by the American Physical Society

278 REVIEWS OF MODERN PHYSICS JULY 1971 Alternatively,one could get a similar number,following wave by de Broglie (1954)1 by considering a spherical de Sitter E=Re Eo exp [-i(cl-k.x)], cosmology.In this model the cosmological constant K is given by the two equations H=Re Ho exp [-i(ct-k.x)], (2.1) K=3/(cT)2; K-=[ac/i们] (o/c)2-k2=2, (2.2) (1.2) or where the last line defines u in units of wavenumber,or 4=61/2i/Tc2. (1.3) inverse length.Standard arguments (Goldberger and Watson,1964)then yield the desired expression for Equations (1.1)and (1.3)give an ultimate limit for a group velocity of a wave packet meaningful experimental measurement of the photon mass. vg=do/d k=ck /w=ck/(k2+u2)12 Since the time of Cavendish,certain critical physicists =c(w2-u2c2)12/w.(2.3) have not been satisfied with speculative assertions on this subject,and have periodically re-examined the This expression corresponds to a frequency dispersion question (or an equivalent one in the language of their of the velocity of light,the first and most direct con- time)to determine what valid experimental limit could sequence of a finite photon mass.Note that here and be placed on the photon mass.In this paper we shall in what follows,giving u in units of wavenumber is give a review of methods devised to improve the limit. using units of c/.] In Sec.II,we develop the theory of classical electro Going to the Lorentz frame in which the photon is at dynamics from postulates of special relativity,plus the rest,i.e.,k=0,we see that there must be three in- assumption of a well-defined,locally conserved energy dependent polarization directions for a massive photon, density associated with electromagnetic fields.We since the plane transverse to k is undefined in this indicate how this assumption can be tested with pulsar frame.The argument fails for a massless photon because signals.We proceed in Sec.III to discuss limits that it can never have k=0.In the photon rest frame the have been set on the mass by terrestrial methods.These electric field energy density E2 is proportional to photon include determinations of the constancy of the velocity intensity.However,the well-known law of Lorentz of light for all wavelengths,and testing the exactness of transformations tells us that the fields in a frame with Coulomb's Law.The latter method yields the best photon frequency w and momentum k will be very laboratory mass limit to date,u

280 REVIEWS OF MODERN PHYSICS JULY 1971 This may be recognized as the ordinary Maxwell with equation,modified by the addition of u2 to the P-∫dx(pEr十Pmatter）, (2.23) D'Alembertian operator.[The free (J=0)solutions of and this equation obey the relation w/c=(424k2)112.]We (dp/dt)matter=pE(J/c)xH, (2.24) may rearrange Eg.(2.16)by introducing the vector the Lorentz force density. potential A satisfying The vector potential is never measured directly,but (☐+42)A.=(4x/c)J., it is determined uniquely,and is required for con- struction of a locally conserved electromagnetic energy dJ=0, and momentum density. FaB=doA8-08Aa. (2.17) Let us elevate the principle just mentioned to a fifth postulate: Rewriting further gives us the famous Proca equation (5)There exists a locally conserved energy-momen. (Proca,1930a,b,c;1931;1936a,b,c,d,e)for a massive tum density,such that the total energy and momentum vector field coupled to a conserved current, of a system of charges and fields is conserved. 00FaB+u2AB=(4m/c)JB, We shall now consider the restrictions implied by this postulate on Im D(u2). Fa8=daA8-08Aa (2.18) Clearly,a minimal requirement on Im D(u2)is that The whole effect of finite photon mass is to introduce it be integrable,i.e.,a bounded continuous function at each point x a spurious current proportional to the falling faster than 1/In 2 at high masses,plus a sum of vector potential and,therefore,a function of the true delta functions and derivatives of delta functions. current at many earlier points x'.In three-dimensional Therefore,D(k2)will be a sum of pole terms notation the massive Maxwell equations become {∑d/(u2-k)} V.E=4mp-u2V, plus a continuous integral over pole terms(a cut) v×E=-(1/c)(aH/at), {J[d(u2)/(2-2)]} 7.H=0, plus second or higher order poles [d/(u2)2,etc.]. V×H=(4r/c)J-u2A (2.19) All these terms can be written as simple poles or limits of sums of simple poles with A and V the space and time components of the Consider the case of two pole terms[D=d/(u2-)+ 4-vector potential A. da/(ua).This leads to the possibility of arbitrary It is worth noting that the freedom of gauge in- free fields with either w=c(u22)12 or w=c(u22k)12. variance found in conventional electrodynamics is Take the case k=0.One may have an electric field completely lost here.First of all,the Lorentz gauge E=Eo(cos uict-cos uact),with A=Eo(u21 sin uact- must be used,i.e.,d4=0.Within that restriction,one ur sin uicl).At /=0,both Fas and A are zero every- might imagine adding to Aa term dA,where A is a where,so that any energy density quadratic in F and A scalar function.This does not change Fas,of course, must vanish.However,an instant later this is no but the Lorentz gauge condition implies A=0. longer true.Therefore,there is no conserved electro- Therefore,if A is already a solution of the Proca magnetic energy built simply from F and A.For free equation we have the contradictory requirements fields,a conserved energy density can be constructed □aaA=0and(▣+u2)aaA-0,satisfied only if A is by projecting the parts of E corresponding to each mass constant.Hence,all freedom of gauge change is lost. It is easy to verify,for free fields,that there exists a E1=[(2+口)/(22-2)]E conserved energy-momentum density (de Broglie, E2=[(w1+口)/(12-2)]E. (2.25) 1957;Bass and Schrodinger,1955)such that With the obvious definitions of Ai and A2,etc.,we 8gt=[E2+H+u2(A2+V2)]/8x, get the conserved energy density PEM=[E×H+u2VA]/4rc, (2.20) 8π8=G[E12+H2+12(V2+A2)] where the conservation we refer to is the equation +c[E22+H2+2(V22+A22)].(2.26) of continuity In the presence of sources,however,our arbitrary but (1/c)(08EM/0t)++VpEMC=0. (2.21) simple definition of may be seen to fail.For example, by calculating the potential energy of a charge dis- When charges and currents are present we obtain tribution and comparing it with the total electro- magnetic energy E one finds that the two are not equal. dP/dt=0, (2.22) The only way to maintain energy conservation is to

GOLDHABER AND NIETO Limits on the Photon Mass 281 insist that the fields associated with and ue are beginning at u2=0 but very small below u2=m and independent contributors to the energy,even though suppressed at least by a2,is produced by the dis- there is no general operational distinction.between sociation of a virtual photon into three correlated them.In particle language,we would say there are two photons.These cuts are not associated with free different photons,though they act on charges in the photonlike degrees of freedom and do not violate our same way. earlier conclusion forbidding a continuous mass photon. Once this is accepted,it is straightforward to deduce It is amusing to consider in this classical context the modified electrodynamics of Lee and Wick (1969).In (d/dt)∫dx8(x)=-∫dxJ·(cdE1+c2dE).(2.27) order to eliminate the small distance divergence in In order that total energy be conserved,this must Maxwell's theory and its quantized version,they balance the effect of the Lorentz force on charges.This introduce a D with two poles;one at zero mass,and means that one at very large mass with Gd1=+1.c2d2=+1. (2.28) d1=-d2=4r. (2.29) If g(x)is positive definite,(ci>0),then the residues d In consequence,the electric potential between two must both be positive.This excludes higher order poles, point charges is bounded at small distances which are obtained in a limit as simple poles with residues of both signs approach each other.Another V(r)=(qg/r)(1-eum)→9gasr→0，(2.30) way to express the difficulty with higher order poles is where r is the distance between g and g'.However,since to observe that they lead to fields which grow in time, d is negative,so is c2.Therefore,a wave packet of e.g.,Eol cos ul for a second-order pole at u.This is a type 2 photons will carry negative energy.This creates solution of ()2E=0.A cut in D(k2)may be the problem that by producing more and more type 2 produced as a limit as the number of poles in a certain photons one can gain more and more energy.In a interval diverges and the residue d:of each pole goes to quantum context the problem may be stated as a zero.From Eq.(2.28)this means that the coefficient violation of unitarity (conservation of probability). ci of the corresponding field energy density diverges, Lee and Wick circumvent this by indicating a cal- so that in the limit 8()is undefined.Thus,it is im- culational scheme in which free type 2 photons are never possible to produce a cut by exciting an infinite number produced,and energy densities are always positive of photonlike degrees of freedom,and still preserve definite. energy-momentum conservation:a "continuous-mass" Since Postulate (5)is of a different character from photon is excluded. the other four postulates,we may ask what complica- If Postulate (5)holds,we may introduce one or tions arise if it fails and there are several very low mass new poles in D at a price of the admission of one or poles or even a cut restricted to low mass.Now we n new photons each with three degrees of freedom.This expect violations of local energy-momentum con- would contradict well-known information about black- servation,but these would be conspicuous only for body radiation (de Broglie,1957;Bass and Schrodinger, fields with very small and k.There could be a for- 1955),and elementary particle reactions (Brodsky and tuitous cancellation of the lowest order effect in electro- Drell.1971)unless either the new photons all have a or magneto-statics.For example,with the two poles: mass greater than many Gev,or else their coupling to charge d:is so small that their degrees of freedom are D=4r[2/(u2+k2)-1/(2u2+b2)] not appreciably excited during times of practical =+(4x/2)[1+0(2/k2)2], (2.31) interest.In either case,their existence would have no significant effect on a search for effects of a possible one would have much smaller deviations from Coulomb's finite mass of the everyday photon.In fact,there are law than with one pole known weak cut contributions to D derivable in quantum electrodynamics and indeed,associated with D=4r/(u2+k2) new degrees of freedom.For example,at values of =+(4r/2)[1-(2/32)+0(2/2)2].(2.32) u2>4m,a virtual photon can dissociate into an ere- pair.This leads to a contribution to D suppressed by However,the resulting spreading of light pulses (an at least a factor of the fine structure constant a1/137 energy nonconserving effect)could be looked for as a and of very short range (10-1 cm)for static fields phenomenon distinct from frequency dispersion of v, (Bjorken and Drell,1965).3 An even weaker cut since it could be observed at a single given frequency. Pulsar signals can be used to give a limit on such An amusing line of speculation is indicated in a series of violations of Postulate(5),but,even if they exist,special a ()Thehhe phott cancellations must occur if the effect on static fields is neutrino-antineutrino pair,producing an effective photon mass to be masked to any given order in u/kuD (where which is different in different Lorentz frames,because there is a filled neutrino sea which is at rest'”only in the“rest frame' D is the dimension of the experimental apparatus). of the Universe. We conclude that,in addition to the basic symmetry

282 REVIEWS OF MODERN PHYSICS.JULY 1971 principles of special relativity and the assumption that wavelength limit (v=c)of fields are linear functions of currents,an eminently reasonable postulate of energy conservation is required -(△p/c)=u2c2/2aw2+o[(uc/a)4门 to deduce that classical electromagnetic theory can be =(u%/8x2)+O(aλ)4门.(3.1) modified in only one way-replacing the Maxwell equation by the proca equation to account for a possible The velocity c is measured to an accuracy of one to ten small photon mass.If the energy postulate is omitted parts in 106 over much of the electromagnetic spec- there is no simple prediction for the effect on statics,but trum.4 The lowest frequency measurement with such there are two remarkable effects on light radiation: precision is that atv=173 MHz (=1.73 m)(Florman, 1955).The one in 105 accuracy of this measurement 1.A narrow-band pulse of light may spread in implies≤2X10-4cm-1=3X10-9eV≡6X10-2g duration or separate into a number of discrete com- Because the effect is quadratic in wavelength,one can ponents in a time (c/Av)r,where r is the original pulse improve considerably on this number by going to length,and Av is the range of light velocities associated lower frequency,even if the measurement is less with the range K of u values in D. accurate.In the 1930's,Mandel'shtam and Papalexi 2.As the spreading occurs,the classical integrated (1944)and their collaborators developed a technique intensity fdx(E2+H2)/8x may increase or decrease for measuring the velocity of long radio waves.5 A dramatically depending on the variation of the sign radio wave of frequency v is sent from a transmitter to of Im D(u2)in K. a receiving station far away.At the receiver,a wave of frequency ()v,for example,is synchronized with the One could also question the postulates of special received wave transmitted back to the original station. relativity and linearity.We don't do so here for two The phase lag of the return wave with respect to the reasons.First,we have no simple way to parameterize original signal has calculable contributions,including deviations from these postulates-too many possi- effects of the apparatus at both ends,plus a term bilities would be opened by discarding them.Second, proportional to the time of travel.Al'pert,Migulin and both postulates have been very successful in quantum Ryazin (1941)-and earlier work cited therein-used electrodynamics,where the accuracy of perturbation this technique to measure the dispersion of long theory validates the use of linearity.Thus,any viola- (102 m)waves travelling over land and sea.Over tions must appear only at very long times or distances. land,the dispersion was quite large (1%),but over This was easy to arrange for our particular version of sea they measured a velocity shift of 7X10-4 between energy nonconservation,but seems nontrivial for the 300-450 m.If this is interpreted as a photon mass other assumptions.Not surprisingly,we feel that the effect,it corresponds to effects (1)and (2)above are also unlikely.In the u≤2X10-8g=7X10-6cm-1=10-10eV.(3.2) remainder of this paper we shall assume there is a single fixed value of 20,except where indicated It is possible that the result of Al'pert,el al.was due to explicitly. instrumental error.However,it would appear difficult It is worth noting that the most common technique to improve enormously on their work because of for deriving massive electrodynamics is the use of a irregularities in the medium through which the wave Lagrangian density (cf.for example,Gintsburg,1963). propagates (the Earth and its atmosphere).We dismiss One simply adds to the u=0 Lagrangian a "photon mass term"proportional to u2AA.This is the most TABLE I.Experimental limits on deviations from Coulomb's law. general modification which vanishes as 0 and in- volves only local coupling (all fields evaluated at the Authors Date ￥(Hz) 4(cm-) same point in spacetime).The Lagrangian approach embodies all of our postulates (1-5),but we hope the Coulomb 1785 10-1 ≈10-1 reader has found it instructive to examine these Robison 1769 0 6×10- 10-2 assumptions separately. Cavendish 1773 0 3X102 ≈103 Maxwell 1873 0 5X10-5 必103 Plimpton el al. 1936 2 2×109 106 III.TERRESTRIAL LIMITS Cochran et al. 1968 102-103 9X10-12 9×10-8 Bartlett et al. 1970 2.5×10310- 10-8 A.Measurement of c Williams et al. 1971 4×10 6×10-16 5X10-0 The most straightforward way to obtain a limit on the photon mass is to look for a variation in c over the 4 The most complete recent summary of precise measurements spectrum.Equation (2.3)shows that long-wavelength of c is Froome and Essen (1969). light will have a velocity differential from the short- sBrief summaries in English are given by Smith-Rose (1942a,b)

284 REVIEWS OF MODERN PHYSICS JULY 1971 in the input to the amplifier is given by check de Broglie's simple numerical calculation.The first published correction appeared in Kobzarev and (△noise2)=4kT△yReZ, (3.12) Okun'(1968).We noticed it in preparing Goldhaber and Nieto (1968). where k is Boltzmann's constant,T is the absolute However,even if one takes advantage of the wider temperature,Av is the bandwidth (or reciprocal of spectrum observations and faster electronics now observation time),and Re 2 is the real part of the available,this method is intrinsically limited.As impedance, Gintsburg (1963)has nicely summarized,this is true Z-=R-1+iC. (3.13) for a number of reasons,but the crucial one is the natural dispersion of light traveling through the inter- Here R is the input resistance,C is the parallel capaci- stellar plasma in a magnetic field.The dispersion tance,and w/2x is the frequency.For large w we have equation for such a system is ReZ≈[R(wC)]1, (3.14) b2=(w2/c2)[1-wp2/(a2±wwB)], (4.4) wp2=4mne2/m;wB=(eB/mc)cos a,(4.5) and the root mean square noise voltage is inversely proportional to frequency. where n is the electron (mass=m)density,and a is the Rather than spoil the reader's pleasure by a second- angle between the magnetic field (B)and the direction hand description of these experiments,we refer him to of propagation.If B is small,Eq.(4.5)yields a disper- the original papers,contenting ourselves with the sion similar to the photon mass effect: (tabular)summary presented in Table I,modified and =dw/dk=c[1-(ωp2/w2)]2≈c[1-(awp/w2)+…]. expanded from that of Bartlett,et al.(1970). The value of Williams et al.may still improve.It (4.6)】 represents the best laboratory limit to date,an improve- Egs.(3.1)and(4.5)imply that we need to compare ment in voltage sensitivity of more than 105 over Plimpton and Lawton and 101s over Cavendish! the "rest frequency"of the photon in cgs units withp: (uc2/i)=8.2X10[u(g)]sec-1, (4.7) IV.EXTRATERRESTRIAL LIMITS (4rne2/m)12=5.6X10[n(cm-3)]2sec-1.(4.8) A.Dispersion in the Speed of Starlight Part of this may be due to the inaccessibility of the original A limit on the photon mass can be obtained by meas- work (de Broglie,1940).It was published in occupied France during World War II,and is difficult to find.In his later works. uring the difference in time of arrival of radiation of e.g.,de Broglie (1957),he quotes the result,but the calculations different frequencies with the same origin.For example, are not repeated.Here it is appropriate to point out that de if blue and red light rays come from the same event,the Broglie has had a life-long interest in the question of a photon mass.He first proposed a set of massive photon equations in difference in time of arrival is de Broglie (1934).After that,besides numerous articles,he has written an extraordinary number of original and revised books that discuss the subject,some of which we have mentioned. 6=∫dlL(1/r)-(1/B)]≈(L/c2)(uB-r),(4.1) For a complete bibliography,the interested reader is referred to the Library of Congress listings under has name.See de Broglie which,from Eq.(3.1),is (1969-1970) During the 1930's de Broglie had a strong influence on many young theorists in Paris including Proca and Petiau.The con- 6=(8r2c)-1u2L(入a2-入x2). (4.2) nection between de Broglie's photon equations and the work of others can be seen by looking at the fundamental (reducible) De Broglie (1940)suggested that this method could 16-dimensional representation of the Duffin-Kemmer-Petiau wave equation for spin-zero and spin-one particles.The 16-dimensional yield a mass limit by using light from a star emerging representation can be defined as a symmetric product space of from behind its dark binary companion.De Broglie two Dirac spaces (that is to say,a composite of two Dirac particle spaces).Contrariwise,the de Broglie equations come from the considered the case (Aa2-Ag2)=0.5X10-8 cm2 (for product of a Dirac particle space with a Dirac antiparticle example,Ar 8000 A,A4000 A),L=10s light years, space.When one reduces the Duffin-Kemmer-Petiau equation and t

286 REVIEWS OF MODERN PHYSICs.JULY 1971 the average of any component of H over the sphere However,the more recent work on the geomagnetic must vanish.Therefore,if true external currents be limit on u does exploit satellite data as well as earth- estimated,detection of any anomalous uniform "ex- based measurements.Furthermore,the Schrodinger ternal"field on the surface of the Earth would demon- method might be applied to other planets in the not too strate a violation of Maxwell's theory of just the form distant future.These facts seem a sufficient defense for produced by a finite photon mass. our classification. We turn to a quantitative derivation of the external Recently the present authors (1968)improved on field effect.Recalling the definition of a magnetic Schrodinger's results by using Cain's fitib(Cain,1966 dipole moment Cain et al.1965,1968;Hendricks and Cain,1965)to De=D=(-)∫drJ×r/c, (4.13) geomagnetic data from earthbound and satellite meas- urements.For epoch 1960.0 Cain obtains values16 of and the static limit of Proca's Eq.(2.18) D=31044yR (-72+)A=J(4π/c), (4.14)） Hxt…=(21士5)Y, we have the dipole vector potential Hext~0=Hxt·(sX2)/八X2|=(14土5)Y A=V x D eHrr (4.15) Hxt0X2=(8±5)Y, (4.19)】 Since H is V x A,the dipole contribution to the earth's where s points towards the south geographic pole field.measured from coordinates centered at the To obtain our mass limit from this number,we had to dipole,12 is take into account the "true external"sources,which H=(Der/r3)[(1+ur+u22)(32-7-2)-u22] were unknown when Schrodinger wrote his paper.As discussed in Goldhaber and Nieto (1968)these include (4.16) ~9y from the quiet-time proton belt,perhaps 15-30 This is to be compared to the ordinary dipole field. due to currents in the geomagnetic tail,and~15y from the hot component of the plasma in the magnetosphere HD=（D/r)(32r-). (4.17) which are all parallel to the dipole moment.In addition, If observations are made near the surface of a sphere there is a true external field antiparallel to the dipole of centered at the dipole,(here it will be the surface of the ~20y at the equator,which is due to the compression earth rR=const),the factors in the first term of of the geomagnetic field by the solar wind.Finally,the Eq.(4.16)would just make D appear to have a slightly interplanetary field of ~5y points in an unknown different value when compared to Eq.(4.17).However. direction at the earth's surface.17 the last term in Eg.(4.16)is new.It will be observed Thus,the total external field parallel to D due to as an apparent external magnetic field antiparallel known sources is

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