290 REVIEWS OF MODERN PHYSICS.JULY 1971 For example,it is well known that a transition between are even bigger,and so do not alter the point made two states,each with angular momentum zero,cannot now.The result obtained for J:/implies for the total proceed by emission of a single transverse photon. cross sections of photons However,it can go by L-wave emission,albeit so slowly that double-photon emission is much more likely.The gL/o=(/m。)2. (5.9) rate for such a transition is suppressed by a factor (ur)2 Thus,the ratio of mean free paths AL/Ar goes as compared with an allowed electric dipole transition of (w/uc)2 for low frequencies,but as (m/u)2 for high the same frequency.Here r is a characteristic (length) (>1023 Hz)in this model.24 It would require an dimension of the quantum system in question.For a extreme modification to alter the conclusion that photon-induced transition to an arbitrary state f,the longitudinal cross sections are infinitesimal at all quantum mechanical matrix element is frequencies except w of order uc. Tri=-e(f Jali), (5.3) We conclude that,if longitudinal photons exist,their observation will be difficult if not impossible,and their where i is the initial target state.The polarization effects are negligible,even on an astronomical scale.36 vector e may be normalized by the condition ee=-1. For transverse polarizations this means VI.CONCLUSIONS r2=(0,1,0,0) A.Future Experiments r2四=(0,0,1,0) (5.4 Let us re-examine each of the categories discussed so far and see if the results could be improved. and for the longitudinal polarization 1.Terrestrial Measurements of c eL=(|k/u,0,0,w/uc), (5.5) Comparison of past results here with those of other where we take the photon momentum in the g direction, methods is not encouraging.To give a feeling for the and an arbitrary coordinate 4-vector would be written difficulty in making improvements,consider the classic as (cl,x,y,2).Current conservation implies the technique of measuring c by observing the normal mode vanishing of T if the photon 4-momentum k is frequency of a cavity with conducting walls.If the substituted for e.This immediately gives cavity is a cube of side a then the lowest mode fre- Tr=(-kc/uw十a/uc)(f|J.|i》 quency (which would be transverse in the u=0 case) is completely independent of u!The solution for the =(uc/w)(fJ:i) (5.6) electric field E in this u-0 mode is but,for transverse photons, E=ZEo cosh u(z-a/2)sin (wx/a)sin (y/a)cos wt, Tx)=(f Jli). (5.7) (6.1a) Thus,if transverse and longitudinal matrix elements with (6.1b) of J are comparable,then amplitudes for absorbing or w=V2Tc/a. emitting longitudinal photons are suppressed by The only effect of u is an almost imperceptible bulge in (uc/),and rates or cross sections are suppressed by the magnitude of E in the middle of the box.If we think the square of this factor.In the classical (Thomson) of this field as a superposition of travelling waves,then limit,matrix elements of/are independent of direction, these waves still have phase velocity c,independent of and the suppression factor is exactly (uc/a)2.The only But they are no longer truly transverse since E obeys way to get large longitudinal cross sections is to have the divergence relation much bigger matrix elements for longitudinal currents, or else for more longitudinal-than transverse-photon V.E=-u2V=uEo sinh uLs-(a/2)] induced transitions. X sin (x/a)sin (ry/2)cos wt0.(6.2) A specific model for high-energy cross sections of longitudinal photons on hadrons is supplied by the The reason for this remarkable result,which tends to "vector meson-current identity,"(see,for example, go unnoticed in 4=0 electrodynamics,is that the Sakurai,1969,1969a),which states that the electro- phase velocity of a wave enclosed by conducting walls magnetic current operator is a linear combination of is determined by the parameter c that appears when the fields which create the p,w,and mesons.Since 24 Data from inelastic electron scattering do not decisively p-hadron cross sections appear to approach a constant determine the asymptotic behavior of/r=(0.2+0.25).See Taylor (1969) spin-independent value at high energies,we require for In particular,we do not agree with Bass (1956,1963)who consistent normalization on the scattering amplitudes proposed longitudinal photon emission as a mechanism for cooling of the earth's interior.A crucial element in his calculation is the (A|J:|B)=(/mc)(A|J|B, (5.8) mean free path of transverse photons in the earth.To obtain this he uses a formula of Eddington designed to apply to stellar where mp is the p-meson mass in inverse length units. plasma.The calculated path is about 1010 times too short, The w and make small contributions,but their masses since the most elementary considerations imply a path of about one micron