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)