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 +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&' for momentum p ~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 6nd that the potentials for j = 1 and j = 2 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 tprovided that I' and T are dered to take a monopole into its anti-particle), it is nevertheless impossible to construct a Lorentz-invariant S matrix for magnetic monopoles and charges in perturbation theory. with H (t) the interaction Hamiltonian in the interaction representation H'(t) = exp(iEPt) H' exp(— iHrt), (1.2) where H~ is the free-particle Hamiltonian and H' the interaction. The operator H'(t) is some function of the creation and annihilation operators of free particles, and we know that these operators transform according to the various familiar representations of the inhomogeneous Lorentz group. ' Among these representations are those characterized by mass m= 0 and spin j = 1 or 2, and we