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)