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Several path integral representations for the $T$-matrix in nonrelativisticpotential scattering are given which produce the complete Born series whenexpanded to all orders and the eikonal approximation if the quantumfluctuations are suppressed. They are obtained with the help of "phantom"degrees of freedom which take away explicit phases that diverge for asymptotictimes. Energy conservation is enforced by imposing a Faddeev-Popov-likeconstraint in the velocity path integral. An attempt is made to evaluatestochastically the real-time path integral for potential scattering andgeneralizations to relativistic scattering are discussed.

Exact Path-Integral Representations for the $T$-Matrix in Nonrelativistic Potential Scattering