294 REVIEWS OF MODERN PHYSICS JULY 1971 now that pulsars have appeared as possible sources for the Yamaguchi limit 10-21 cm-1.The last number cosmic rays (Lingenfelter,1969).There is convincing represents the ultimate possibility for an improved evidence that part of the Galactic field is slowly vary- limit among suggestions we have seen. ing across distances of the order of the thickness of a spiral arm.This comes from Faraday rotation measure- B.Discussion ments for radio waves from sources outside the Galaxy. I.Special Relativity with Finite Photon Mass The Faraday rotation shows a systematic dependence on Galactic latitude and longitude,albeit with some Before summarizing the status of the search for a violent fluctuations (Berge and Seielstad,1967;Parker, photon mass,let us digress briefly to consider a peda- 1969,70).However,the actual magnitude of H,and gogical point.We have made critical use of special the relative size of any more rapidly varying field,are relativity in arguing that the Proca Eqs.(2.19)repre- still subject to large uncertainty (Verschuur,1970). sent practically unique modifications of Maxwellian Nevertheless,it is reasonable to suppose that these electrodynamics.However,special relativity itself was great uncertainties will be partly or wholly resolved, first presented as a consequence of the constancy of the and to speculate on the consequences for a photon speed of light.27 What new postulate must we adopt in mass limit.An admirable effort in this direction,and order to restore the deductions of special relativity the only one of which we are aware,is a recent paper theory,in a world with u>o! by Williams and Park (1971).Adopting plausible pa- One answer is quite straightforward,if less appealing rameters for the plasma and H field in a spiral arm of in its simplicity than c=constant:Postulate:Given any the galaxy,they make the very reasonable assumption two inertial frames,the first traveling at velocity v that a"straightened"spiral arm-a cylinder of similar with respect to the second,there exists a frequency vo, dimensions-would behave similarly.The critical point depending onv and the desired accuracy e,such that in the calculation is that the plasma has a very high any light wave of frequency greater than ro in either resistivity for currents traveling perpendicular to H. frame will have a speed between c and c-e in both They suppose that H is parallel to the axis of the arm frames. and decreases slowly away from the axis.These assump- We leave it to the reader to verify this postulate by tions imply a large azimuthal vector potential A explicit construction of the function vo(,e).The usual rH/2,where r is the distance from the axis.To keep H derivation of special relativity can now be applied, slowly varying there must be a large current perpen- using the Lorentz invariant limit yo. dicular to H,cancelling the Proca pseudocurrent Jp= The above postulate is equivalent,in the special case -(c/4)A.Because this current is perpendicular,it of light,to the Einstein assumption that there is an suffers high resistivity103 sec,leading to a inique limiting velocity (c)for all phenomena.In a dissipation of the stored energy at a rate (c22). world with u0 but the neutrino mass equal to zero, Insisting on a field lifetime of 10y,Williams and Park then the velccity of the neutrino would be equal to c. obtain a limit 10-18 cm-1.It is amusing to note that This view makes the structure of space-time more in their cylinder model a special modification could fundamental than interactions.For example,it rules drastically alter the result.Adding a smoothly varying out the possibility that strongly interacting particles azimuthal field H comparable to H.in magnitude,one could have one limiting velocity while electromagneti- can obtain a field pattern obeying the remarkable con- cally interacting particles had another.(Otherwise dition AH=V xA!In this case the current required acausal effects could occur.) to cancel the Proca pseudocurrent travels parallel to H and faces much less resistance.Using the approach of 2.Monopoles andu=0 Williams and Park,one obtains a limit us10-u cm-1. Another amusing side issue is the combination of For this case,a better limit can be got in another way. twc fascinating speculations;a finite u and the existence There is a maximum current the plasma can support, of magnetic monopoles.If both these dreams come true Jmax=Vne,where V is the velocity of the charges,and one could not apply the ideas of gauge invariance ne the density of electron or ion charge in the plasma. (Dirac,1931,1948)or rotational invariance(Goldhaber, ForH≈10-6G,V≤105cm/sec,andn≤1/cm3(Allen, 1965)to derive the famous Dirac quantization condition 1963;Verschuur,1970),we would get us10-15 cm-1, that eg/hc is a half-integer (with e and g the elementary assuming R≈1021cm. electric and magnetic charges,respectively).In fact, It is probable that a conservative treatment of data the approach of Dirac would not allow any gauge- now available would produce a convincing limit of two invariant theory including a particle which is a source or three orders of magnitude better than the geo- for a 1/r2 magnetic field with additional "Yukawa" magnetic value at far less cost than previously men-falloff.It is another interesting exercise for the reader tioned approaches.In any case,future "experiments' to verify this difficulty.Along these lines,it is worth in Galactic magnetohydrodynamics would open the noting that finite u does not disturb another quan- door to a hierarchy of improved limits,depending on tization condition,the quantization of magnetic flux the detailed character of the results,ranging from 10-1 cm-1 through the Williams-Park 10-18 cm-1 to (1957),p.61. Our discussion of this matter is similar to that of de Broglie