Path Integral Representation of S-matrix

The S-matrix of gas atom and crystal at finite times reads 

The S-matrix may be expressed through the conventional path integral in coordinate representation by the Fourier transform in Pf j (Bogdanov et al. 1989)  

Gf [r(t)] stands for the amplitude of the i - f transition in phonon mode m. Analogous expressions for the scattering amplitude in coordinate representation were written by different authors, to mention (Pechukas and Davis 1972 Newns 1985 Nourtier 1985 Billing 1990) for example. 

After integration by parts in the exponent, this expression becomes regular in the limit tj -4 —oo, tf - +oo. Including the integration over rj, rf into the measure yields 

The trajectories in this path integral meet boundary conditions p(—oo) = pj, 

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