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