Photons and Gravitons in Perturbation Theory: Derivation of Maxwell's and Einstein's Equations

The $S$ matrix for photon and graviton processes is studied in perturbation theory, under the restriction that the only creation and annihilation operators for massless particles of spin $j$ allowed in the interaction are those for the physical states with helicity $\pm j$. The most general covariant fields that can be constructed from such operators cannot represent real photon and graviton interactions, because they give amplitudes for emission or absorption of massless particles which vanish as $p^j$ for momentum $p\to 0$. In order to obtain long-range forces it is necessary to introduce noncovariant "potentials" in the interaction, and the Lorentz invariance of the $S$ matrix requires that these potentials be coupled to conserved tensor currents, and also that there appear in the interaction direct current-current couplings, like the Coulomb interaction. We then find that the potentials for $j=1\mathrm{}$ and $j=2\mathrm{}$ must inevitably satisfy Maxwell's and Einstein's equations in the Heisenberg representation. We also show that although the existence of magnetic monopoles is consistent with parity and time-reversal invariance [provided that $P$ and $T$ are defined to take a monopole into its antiparticle], it is nevertheless impossible to construct a Lorentz-invariant $S$ matrix for magnetic monopoles and charges in perturbation theory.