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<2x10-4g. different for photons polarized L or to k (Jackson, In the next section extraterrestrial methods are 1962;unreferenced assertions on electromagnetism in reviewed.The first method is a variation on the terres- this paper may be found in Jackson's book): trial velocity of light experiments.Dispersion in the speed of starlight is inferred from the difference in Eol=(w/uc)Eo⊥rest, arrival times of different colors of light from the same H=(k/u)XEo rest, astronomical event.We then discuss the limits that can be obtained by studying the effects that a massive Eou=Eoll rest. (2.4) photon would have on the earth's magnetic field.This If u is much smaller thank,the field of a longitudi- yields the lowest limit to date,u<4X10-48 g.Another nal (photon will be smaller than that of a transverse technique considered is the study of long period (L)photon by the factor uc/w.Since power absorbed hydromagnetic waves in plasma.If the photon has a by electric charges is proportional to E,we infer that finite mass,then such waves are damped below a critical scattering cross sections of longitudinal photons will be frequency depending on u and the plasma characteristics suppressed compared to those of transverse photons by In the next section the physical effects of longitudinal photons are derived.We close in Sec.VI with a dis- a factor (uc/)2;this weak coupling explains how the longitudinal polarization,if it exists,could have escaped cussion of possible future experiments,their efficacy detection up to the present.The phantom longitudinal in improving present limits,and the physical implica- photon is the second consequence of nonzero u. tions of the results. Finally,we consider the limit of static fields.For these fields,we have=(k2+u2)1=0,implying k=iu, II.ELECTRODYNAMICS WITH FINITE hence,exponential decay of static fields with a range A.Heuristic Discussion This behavior is familiar from Yukawa's model for interaction of nucleons through pion exchange.The The assertion of a definite nonzero photon mass is exponential deviation from Coulomb's law,and its equivalent to the specification of a free-electromagnetic magnetic analog,provide the most sensitive current test for a photon mass.In the next section we find the 1A remarkably similar discussion was given by Cap (1953) postulates required to link this third effect rigorously (See also Marochnik,1968). with the previous consequences of finite u
GOLDHABER AND NIETO Limits on the Photon Mass 279 B.Deductive Approach If we ignore parity-violating terms as required by We adopt the following postulates: Postulate 3 above,we may write Eq.(2.11)more simply as (1)The electromagnetic field is defined through its Fa8(k)=-iD(k)(ko]8-kgJa), (2.12) action on a test charge g by the Lorentz force law, F=g[E+(v/c)XH]. (2.5) where D is an invariant function of k,and the right- hand side is the most general antisymmetric tensor built This law determines the behavior of E and H under out of J and its derivatives,i.e.,linear in Ja and an Lorentz transformations:they may be identified as arbitrary function of ka.Thus,the requirements of independent components of the antisymmetric 4-tensor Poincare invariance (including parity)are sufficient Fas by to deduce the homogeneous Maxwell equations,which Fo=Ei may be written Fij=esmH (2.6) keaB(k)=0, (2.13) The force law in standard notation becomes and are obviously satisfied by the above form Eq. dps/dr=quFas. (2.7) (2.12).To state this another way,we have now shown from invariance requirements alone that the fields (2)The electromagnetic field at point x in space- may be derived from a 4-vector potential: time is linear in the charge and current densities,and in the derivatives of these densities,all evaluated at FaB(k)=-iTkaAg(k)-kgAa(k)], earlier points x'.Further,this linear relationship is A.(k)=D(k)J.(k) (2.14) Poincare covariant (translation invariant and Lorentz covariant): Next,we study the properties of D(k).Since D is Lorentz invariant,we shall assume that it is a function Fas(x)=f d4xDaB8(x-x)018(x') only of the invariant quantity kak,even for +terms with higher derivatives.(2.8) complex ka,giving D(k)=D(k2).This can be proven from our postulates.?Let us consider k=0.The condi- This latter requirement is applied to assure invariance tion D(/<0)=0 implied by Postulate 2 in turn of the theory under the transformations of special implies that if the inverse Fourier transform D(t)= relativity.The quantity Dasts must be an invariant (2r)-∫k exp(-k·x)D(w,k=0)exists,then tensor.There are only two possibilities: D(@,k=0)is analytic in the upper-half complex plane.Further,the requirement that D is real implies Dasis()=D(x)(gavg8-ga8g8)+D(x)eo86,(2.9) D()=D*(-w*).Translated into the variable2= 02/c2-k2=02c2,these results imply that D(k2)is ana- where e is the completely antisymmetric 4-tensor.The presence of D implies parity violation or magnetic lytic in the entire complex k2 plane except for the posi- tive real k2 axis,and any discontinuity across this axis is sources,depending on the point of view.The reason is that D produces a pseudovector E field,and a vector imaginary.Unless there is a purely local current-current interaction,D(k2)must go to zero as k2 goes to infinity. H field. (3)We shall assume there are no magnetic sources We exclude the local interaction since it is not present in the Maxwell theory. or parity-violating terms in the theory.This eliminates terms like D. We then may use Cauchy's theorem to write a dis- persion relation for D by integrating over its imaginary (4)Finally,we insist that the dependence of the discontinuity theory on a small photon mass,be such that as 40 there is a smooth transition to the Maxwell theory. D(k2)=T- du Im D(u2) (2.15) It is easiest to find the consequences of these postulates 2-2 in "momentum space".Define (k a 4-vector) If Im D has a delta function,then D has a pole. Fas(k)=fdix exp (ikx)FaB(x), Before considering the most general case,let us specialize by assuming Im D consists of a single delta Das(k)=f dx exp (ikx)Dain(x), function at a particular value u2,giving 了a(k)=∫dx exp(i·x)J.(x). (2.10) (-k2+u2)Fa8=(4n/c)(-i)(kaJ8-kgJa) Then,the convolution integral Eq.(2.8)becomes (☐十2)Fa8=(4x/c)(0aJs-0gJa) (2.16) FaB=DoBlv(-ik)J8 This can be shown as a trivial example of the discussion in +terms with more factors of the 4-vector k.(2.11) Streater and Wightman (1964)
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)
GOLDHABER AND NIETO Limits on the Photon Mass 283 in Sec.VI the possibility of further improvement using aa limit (Maxwell,1873) more controlled environment of smaller dimensions. 9<1/21600. (3.4) B.Deviations from Coulomb's Law Plimpton and Lawton (1936)performed an improved The inverse square force law was first announced in version of the Cavendish-Maxwell experiment.They 1785 by Coulomb (1788),who used a torsion balance to took two concentric conducting spheres of radii a=2.5 ft,and b=2.0 ft,grounded them,and then charged the measure directly the repulsive force between two like charges.A significantly better test of Coulomb's law outer sphere to V=3000 V.Actually,for technical was devised by Cavendish in 1773 (sic!).6 Cavendish reasons,the voltage was quasistatic,having a frequency of 130 cycles/min.A galvanometer which connected set up two concentric conducting spheres connected by a wire.He charged the outer sphere,and then dis- the two spheres and which could be observed through a connected the wire.If there were a deviation from the conducting window indicated△V=Φ(a)-Φ(b)<10t inverse square law,then upon removing the outer V.Using the theory of Maxwell(1873),this meant sphere,one would find a calculable charge on the inner 9<(△V/)F(a,b)≈2X10-9 (3.5 sphere.Cavendish's experimental limits on such a where charge allowed him to say that,if the correct law is F(a,b)=nln[(+1)/(-1)]-ln[4n2/(n2-1)], F=a2/r2+g, (3.3) u=a/b.(3.6 theng is less than 1/50.Surprisingly,Cavendish To convert this result to a limit on the photon mass, never published this result.A public description had to we note that from Eg.(2.17)the potential between wait until Maxwell (1873)included it in his great the two spheres is given in the static limit by treatise." Maxwell improved the result in a new experiment (72-2)Φ=0， (3.7) or The only modification was that the outer shell was grounded instead of removed,and the inner globe was Φ(r)ax(er一er）/2ur (3.8) tested for charge through a small hole.Maxwell also Equation (3.8)is normalized by taking derived the theory of such an experiment for an arbi- trary central force law.From his null result he obtained V=Φ() △V=Φ(a)-Φ(b) (3.9) Then an expansion in powers of (ua)and (ub)yields 6 Actually,the discovery of the inverse-square law for elec- tricity precedes Coulomb by quite a number of years.In 1755 △V/V=2(a2-b)+o[(ua)门. (3.10 Benjamin Franklin noted that a cork lowered inside a charged silver can was not attracted to the side of it,as he thought it Substitution of Plimpton and Lawton's experimental would be.He wrote to John Lining that,"You require the reason; I do not know it."After Franklin later wrote to Joseph Priestley, results into Eq.(3.10)gives Priestley repeated the experiments and reported them at the end of his great work of 1767.There Priestley made the brilliant u≤106cm-1 deduction that the experiment implied that the electric force law was the same as the law of gravitational attraction,i.e. =2X10-1ⅡeV it was an inverse square law.Two years later in 1769 the Scotsman John Robison made the first.experimental determination of =4X10-44g (3.11) the law.Robison had been inspired by the speculation of AEpinus that there was an inverse square law,and AEpinus in turn had which was the best laboratory limit until recently. been inspired to this speculation by the two charge theory of Franklin.By balancing the electrical and gravitational forces Within the past four years,Cochran and Franken acting on a sphere,Robinson obtained a result of g=2.06.Thus, (1967,1968);Bartlett,Goldhagen,and Phillips (1969 Robison preceded the experiments of Cavendish that we mention 1970);and Williams,Faller and Hill (1970a,b;1971) below.But,except for an unremembered lecture,he did not make public his results until 1803.Consult:Franklin (1774), have surpassed the Plimpton-Lawton result by one,two, Priestley (1767),AEpinus (1759),and Robison (1803) and better than three orders of magnitude,respectively. Finally,we mention that the inverse square law for magnetic forces was discovered by Johann Tobias Mayer in 1760,by The three experiments are similar to the older one in Johann Heinrich Lambert in 1766-1776, and in its fullness by principle.Aside from advances in quality of available Coulomb in 1785.Consult Mottelay (1922). equipment,the first essential improvement is the use of 7A description of Cavendish's experiments,taken from his a "lock-in"detector to observe oscillations in AV in manuscripts is containe in Cavendish (1879).A.D.Dolgov and V.I.Zakharov (1971)have pointed out that the Cavendish synchronism with oscillations in the applied potential technique was more sensitive to u0 than the raw data indicate. V.The second improvement is to increase the oscillation The reason is that,as we now know,conductors on the Earth's frequency,reducing thermal,or Tohnson,noise in the surface are at an absolute potential of about 105 V because of charge separation between earth and ionosphere.Therefore,the relevant frequency band:the Johnson (1928)8 noise V inEq.(3.10)is 10 V for the static experiments of Cavendish and Maxwell.The corresponding limits are improved by a con- siderable factor,though remaining inferior to the nonstatic s The theory of Johnson's experimental discovery was presented result of Plimpton and Lawton (1936). by Nyquist (1928)
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<10-3 sec.Then one gets into irreducible representations,and makes the added assumption to set the parity that the wave function transforms as the product of two Dirac wave functions,one obtains an identically zero, 4≤0.78X10-9g (4.3) one-dimensional scalar equation,a five-dimensional pseudoscalar which for plane-wave and diagonal matrix elements is equivalent to the Klein-Gordon equation),and a 10-dimensional spin-one Interestingly,in the last step of his numerical equation (which is equivalent to the Proca equation) As one calculation,de Broglie made a mistake of order 105 could surmise from the parity change in going to the Dirac quoting a limit of 10-44g.What is more amusing, antiparticle space,with the added assumption of transforming as the product of a Dirac particle and an antiparticle wave function however,is that this number was quoted (de Broglie, the de Broglie equation decomposes into a one-dimensiona 1957,p.59)and requoted(Bass and Schrodinger,1955), pseudoscalar,a five-dimensional scalar,and a 10-dimensional but for 28 years no one publicly took the trouble to axial vector equation.See also Duffin (1938),Kemmer (1939), and Petiau (1936,1949)
GOLDHABER AND NIETO Limits on the Photon Mass 285 Equations (4.7)and (4.8)show that the dispersion An amusing particular case of Eq.(4.11)arises for due to a plasma of one electron per cm2 would equal the =-2.This case is intriguing because,as discussed dispersion of a photon with the Plimpton-Lawton in Sec.II.B,such a form of D would produce exact mass.Despite this limitation,as Feinberg (1969)has cancellation of the lowest-order effects (proportional to pointed out,the observed dispersion in arrival time of 2)in electrostatics or magnetostatics.For this special radio signals from pulsars provides the most stringent form,the limit Eq.(4.12)on Au2 becomes a limit on "dynamic"'test of the photon mass to date.These data 2.If combined with Eq.(4.10),this result implies (assuming u=0)may be used to deduce an average h1≤10-2cm-/42,41<<5X109cm-1. interstellar plasma density of <0.028 electron/cm3 for We conclude that pulsar data do more than give the radiation from the Crab pulsar NP0532.If the disper- best dynamic (velocity-dispersion)limit on u.They sion is partly a photon mass effect,then we have also provide stringent limits on violation of energy conservation associated with a "multicomponent" 4≤10-4g=3X10-7cm-1=6X10-12eV.(4.9) photon having two or more different small masses. Feinberg also makes the interesting point that pulse B.Magnetostatic Effects arrival times show no sign of any dispersion,except that implied by the simple quadratic formula Eq.(4.6), 1.Schrodinger's External Field Method over the whole range of frequency from radio to optical. For the Crab pulsar the departure from Eq.(4.6)is Schrodinger (1943b),following an observation of △/c<10-l4(△arrival<10-8sec). McConnell,proposed a method using the earth's static We may ask for limits on the kind of low mass magnetic field that has yielded the best photon mass "structure"of the photon associated with violation of limit to date.Let us begin with a discussion of the principles on which the method is based.As mentioned energy conservation,for example,two poles in D(k2) separated by Au2.As discussed at the end of Sec.II.B in Sec.II.A,the qualitative effect of a photon of mass such structure could show itself in two ways.The first u on static fields is to cause an extra "Yukawa"de- crease in field strength as e-",where r is distance from is by a spreading in duration of low frequency pulses the source.However,we also have seen that even in At/T=Au2[2(/c)2]1.Here T is the flight time from massive electrodynamics,the divergence of the mag- source to receiver of the radiation.The most accurate netic field H must vanish.This is simply a consequence test on this phenomenon is again supplied by the Crab pulsar,with its very narrow (<1 msec)pulse peaks. of reflection symmetry in electrodynamics,and the absence of magnetic sources:Consider the magnetic In fact,there is an observed pulse broadening of about 10 msec at 74 MHz frequency (Rankin et al.,1970; field at a point r produced by electric currents J at points r'.Any divergence of H would be a pseudoscalar Rankin,private communication).This broadening is function of I and r-r',but there is no such function believed due to"scintillations''or fluctuating irregulari- ties in the interstellar medium.If one assumes that part The dependence only on r-r'is a consequence of assuming that the equations of physics are independent of it is due to a photon mass spread Au2,then we learn that of the choice of origin of coordinates.10 Applying these thoughts to the magnetic dipole field of the Earth,we (Au2)1/255X10-9 cm-1(pulse broadening).(4.10) note V.H=0 means that the flux in each field line is conserved.Now a field line is farther out at the mag- The second effect of structure in D is variation of netic equator than it is near the pole.Hence,the intensity of the signal at a given frequency,as a function Yukawa exponential decrease affects the field line most of time after emission.If the power in the signal is at the equator.To keep constant flux,the field pattern distributed smoothly over a broad range of frequencies, must change shape,allowing flux lines to move in some- then this effect can also be detected by an oscillation of what at the equator.This compression of the equatorial intensity as a function of frequency at a fixed receiver. field lines has the effect,on a sphere of fixed radius,of For the case of two poles in D,one at u2 with residue increasing the field at the equator relative to the field (1+e)-1,and one at ua2 with residue (1+e)-le,one may at the pole.The effect is the same as that of a constant derive the modulation of intensity as a function of external field parallel to Hequatorial.Of course,the field frequency: of the Earth is not pure magnetic dipole.However,for the massless Maxwell theory it is a theorem that only a I(@)=1-{1-[(1-e)/(1+e)]}sin2(△kr/2), true external current can produce a uniform field over △k=(412-2)[2(w/c)]于1+0(4). (4.11) the surface of a sphere.11 In the absence of such currents, For the Crab pulsar,the lack of conspicuous oscillations 10 These comments simply express in familiar three-dimensional terms the arguments of Sec.II.B,which were given in relativistic, down to v=w/2=74 MHz implies eAkr<4m at v=100 four-dimensional notation. MHz or,if e is of order unity, This is most easily proven by noting that if there are no external currents,H outside the sphere is the gradient of a solution of the Laplace equation,and then expanding that (Au2)12<10-cm-(intensity oscillations).(4.12) solution in spherical harmonics
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 <40y.Subtracting this from 2.Hext to the direction of the dipole.The ratio of the"external in Eq.(4.18)gives an upper limit on the antiparallel field"(Hxt)to the dipole field at the equator (HpE)is external field which could be due to a finite photon mass,Hoxt(antiparallel)<20y. Hex/HDE=(uR)2/[1+4R+3(uR)2](4.18) The significance of this limit depends crucially on the Using the 1922 magnetic surveys discussed by Schmidt reliability of the fit of Cain to the geomagnetic field. (1924),13 Schrodinger obtained a ratio of (Hex/HDE)= In Goldhaber and Nieto (1968)we made a number of (539)/(31089y),where 1y=10-5 G.14 Putting this into observations concerning this reliability:(1)The Eq.(4.18)yields u=2.76X10-10cm-1.Later,Bass and existence in Eq.(4.19)of external components perpen- Schrodinger (1955)argued that multiplying this dicular to D is hard to explain in any known physical estimate by a factor of 2 would give a reliable upper model;(2)There appears to be an irreducible "noise" limit4≤5.5×10-10cm-1≡1.9X10-7g. in data from earthbound observatories of about 100 y, The critical reader will have noticed that Schrodinger's in large part due to magnetic anomalies in the earth's results did not use extraterrestrial measurements at all. crust;(3)Earthbound data are very sparse in Asia and In fact the ingenuity of the external field method is much of the southern hemisphere.Despite this,the fit precisely that it requires only ground observations is made by an expansion in spherical harmonics of a potential whose gradient gives H,even though these 1s The orientation of the earth's dipole moment is given by Finch and Leaton (1957).A more recent value is to be found 1sOther reports on this study are contained in Hendricks and in Cain and Hendricks (1968).In the usual physics convention, Cain (1966),and Cain et al.(1965). this dipole points to the southern hemisphere. 16 In consulting the literature,care should be taken to dif. See also Chapman and Bartels (1940). ferentiate between the sign conventions used for the spherical 1 The equations we are using are in cgs electrostatic units, harmonics,and between north-seeking and north poles.See, so that magnetic fields are not measured in the Gauss of cgs for example,Table 3 in Fougere (1965),as well as Cain et al. electromagnetic units.In Sec.IV.B.1,this does not matter sin we are calculating ratios of fields.However,in Sec.IV.B.2, (1967),Appendix.This is a later fit than that of Cain (1966), but it does not calculate possible external terms. one must be sure to insert the correct unit conversion factor 17A more detailed discussion on these points and pertinent (which is essentially c). references can be found in Goldhaber and Nieto (1968)