Terrestrial and Extraterrestrial Limits on The Photon Mass

Abstract

We give a review of methods used to set a limit on the mass $\ensuremath{\mu}$ of the photon. Direct tests for frequency dependence of the speed of light are discussed, along with more sensitive techniques which test Coulomb's Law and its analog in magnetostatics. The link between dynamic and static implications of finite $\ensuremath{\mu}$ is deduced from a set of postulates that make Proca's equations the unique generalization of Maxwell's. We note one hallowed postulate, that of energy conservation, which may be tested severely using pulsar signals. We present the merits of the old methods and of possible new experiments, and discuss other physical implications of finite $\ensuremath{\mu}$. A simple theorem is proved: For an experiment confined in dimensions D, effects of finite $\ensuremath{\mu}$ are of order ${(\ensuremath{\mu}\mathrm{D})}^{2}$---there is no "resonance" as the oscillation frequency $\ensuremath{\omega}$ approaches $\ensuremath{\mu}$ ($\ensuremath{\hbar}=c=1$). The best results from past experiments are (a) terrestrial measurements of $c$ at different frequencies $\ensuremath{\mu}\ensuremath{\le}2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}43}\mathrm{g}\ensuremath{\equiv}7\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}{\mathrm{cm}}^{\ensuremath{-}1}\ensuremath{\equiv}{10}^{\ensuremath{-}10} \mathrm{eV}$ (b) measurements of radio dispersion in pulsar signals (whistler effect) $\ensuremath{\mu}\ensuremath{\le}{10}^{\ensuremath{-}44}\mathrm{g}\ensuremath{\equiv}3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}{\mathrm{cm}}^{\ensuremath{-}1}\ensuremath{\equiv}6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}12} \mathrm{eV}$ (c) laboratory tests of Coulomb's law $\ensuremath{\mu}\ensuremath{\le}2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}47}\mathrm{g}\ensuremath{\equiv}6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}{\mathrm{cm}}^{\ensuremath{-}1}\ensuremath{\equiv}{10}^{\ensuremath{-}14} \mathrm{eV}$ (d) limits on a constant "external" magnetic field at the earth's surface $\ensuremath{\mu}\ensuremath{\le}4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}48}\mathrm{g}\ensuremath{\equiv}{10}^{\ensuremath{-}10}{\mathrm{cm}}^{\ensuremath{-}1}\ensuremath{\equiv}3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}15} \mathrm{eV}.$ Observations of the Galactic magnetic field could improve the limit dramatically.