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