path integral formulation scattering amplitude probability amplitude Hamiltonian generator of time evolution scattering action
The path integral formulation of QFT is concerned with the direct computation of the scattering amplitude of a certain interaction process, rather than the establishment of operators and state spaces. To calculate the probability amplitude for a system to evolve from some initial state 




|


ϕ

I


⟩


{\displaystyle |\phi _{I}\rangle }

 at time t = 0 to some final state 




|


ϕ

F


⟩


{\displaystyle |\phi _{F}\rangle }

 at t = T, the total time T is divided into N small intervals. The overall amplitude is the product of the amplitude of evolution within each interval, integrated over all intermediate states. Let H be the Hamiltonian (i.e. generator of time evolution), then
