The covariant $S$-matrix formalism of Dyson has been applied to the calculation of the fourth order non-linear polarization of the vacuum, which is related to the lowest order non-linear interaction between electromagnetic fields. The finiteness and the gauge invariance of the interaction are exhibited explicitly by an expression for the fourth-rank vacuum polarization tensor in momentum space.

• Received 12 June 1950

DOI:https://doi.org/10.1103/PhysRev.80.380