Bern-Kosower Rule for Scalar QED

We derive a full Bern-Kosower-type rule for scalar QED starting from quantum field theory: we derive a set of rules for calculating $S$-matrix elements for any processes at any order of the coupling constant. Gauge-invariant set of diagrams in general is first written in the worldline path-integral expression. Then we integrate over $x(\tau)$, and the resulting expression is given in terms of correlation function on the worldline $\left< x(\tau) x(\tau') \right>$. Simple rules to decompose the correlation function into basic elements are obtained. Gauge transformation known as integration by parts technique can be used to reduce the number of independent terms before integration over proper-time variables. The surface terms can be omitted provided the external scalars are on-shell. Also, we clarify correspondence to the conventional Feynman rule, which enabled us to avoid any ambiguity coming from the infinite dimensionality of the path-integral approach.