GOLDHABER AND NIETO Limils on the Photon Mass 289 mechanisms.For example,if one takes the order-of- parable with the radius of the sun!For tempera- magnitude lower mass limit that is obtained from the tures of interest in astronomical processes,the factor Schrodinger method,then there was no reason,in (w/uc)2 is sufficient to guarantee the complete trans- Patel's analysis,for these waves to be damped. parency of stellar material to longitudinal photons, In order to establish these waves as being near the even assuming a mean free path of 10-13 cm for trans- critical frequency,one would have to observe higher verse photons. frequency waves propagating for at least a few wave- Longitudinal photons would not supply an important lengths,and lower frequency waves not propagating. mechanism for energy loss by stars because the produc- Also,the damping distance is greater than u for tion rate is far too small.A nucleus which can emit a Alfven waves propagating along Ho,so that meaningful 1 kV x ray,for example,would emit a longitudinal x ray values for a mass limit would require propagation with probability (uc/)210-36 and absolute rate of distances sufficient for attenuation to occur.This would perhaps 10-2/sec,again taking longer than the age of make 2X10-10 cm-1 about the best limit obtainable the Universe.Thus,even with the highest densities and with this type of data,assuming no uncertainties about longest time scales known,longitudinal photons would the medium of propagation. have negligible effect on thermodynamic systems.23 To sum up,although this idea is very interesting,a The possibility of detecting longitudinal photons is an convincing mass limit would require the solution of intriguing one.For example,a capacitor might be many difficulties. placed inside a conducting sphere:Any incident longitudinal wave would pass through the sphere and V.LONGITUDINAL PHOTONS deposit a small fraction of its energy in the capacitor. As mentioned briefly in Sec.II.A.,u0 implies the Capacitors outside the sphere would receive comparable existence of a third degree of freedom for the photon: energy,but for transverse waves they would receive longitudinal polarization.This immediately suggests much more than the shielded capacitor.The difficulty potentially significant effects,of which the most with this suggestion lies in conceiving of a sufficiently dramatic is a fifty per cent increase in the stored energy powerful source for L waves.The sun is the most of a system of photons in thermodynamic equilibrium obvious candidate.From the earlier discussion,the sun with a reservoir at temperature T.Since Planck's radia- might be opaque to longitudinal photons with w=10s tion law is known to be quite precise,this effect is not Hz.Bass(1956,1963)has pointed out that if the usual present.Can we not conclude that u must vanish?The thermodynamic considerations are applied to black answer was given most vividly by Bass and Schrodinger body radiation from a slab of thickness X,and the (1955)[See also Stueckelberg (1957)]who pointed mean free path of longitudinal photons of a given out that the approach to equilibrium of longitudinal frequency is A,then the L-wave radiation from the slab photons in a cavity is very slow,having a time scale will have an intensity comparable with the age of the Universe.They con- IL=[1-exp (-X/A)JIr, (5.2) sidered a closed box with perfectly conducting walls, and asked how long it would take for energy origi- where I is the intensity of either transverse polar- nally stored in transverse waves to be partitioned ization.Let us assume for the application at hand that equally among all three polarizations.While they the factor in parentheses is unity for w=10s Hz;then came up with a very long time for any reasonably It will be just half of the ordinary black body radiation. big container,the basis of their calculation is open A rough estimate indicates that the L-wave radiation to question,since even the best known conductors arriving at the earth would produce a root mean square are not good enough to reflect longitudinal waves. voltage across a capacitor with 1-m2 area about 105 The key concept here is the skin depth of the con- times smaller than the Tohnson noise across the capacitor ductor.It is straightforward to show that the skin (with a parallel resistance of 108 )Thus the chances depth for longitudinal waves of angular frequency w of observing thermal longitudinal photons are remote is given by (Gertsenshtein and Solovei,1969;Kroll, indeed.Instead,one must hypothesize an implausible 1971) coherent charge oscillation in the sun at low frequency (@s10s Hz)in order to predict an observable effect. 6z≥(a/uc)26r, (5.1) It might be worthwhile to scan the sky in search for such a coherent source of L waves. where or is the skin depth for transverse waves.The One may ask whether the statement that reaction or strict equality holds only for low conductivity and,in emission rates for longitudinal photons are suppressed fact,Sr becomes infinite in the limit of infinite con- ductivity (Kroll,1971)!Let us consider photons of by a factor (uc/w)2 compared to those for transverse photons remains true for all quantum transitions in typical frequency for a system at a temperature of atomic or smaller systems.In fact it is not strictly true. 10-*K,near the lowest attainable at present.This would mean >10s Hz,a skin depth or certainly a An early criticism along these lines was voiced by Wigner bigger than 10-5 cm,and therefore >1010 cm,com- (1956)