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Path integral formulation (Due to Feynman) is a major formulation of Quantum Mechanics along with Matrix mechanics (Due to Heisenberg and Pauli), Wave Mechanics (Due to Schrodinger), and Variational Mechanics (Due to Dirac). DO NOT USE THIS TAG for line/contour integrals.
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S-Matrix Elements in Path Integral Formalism
For your question 1.: note that the integral in $Z[J]$ is performed over only paths connecting the initial state $q_i$ to the final state $q_f$, i.e., $Z[J]$ actually depends on $q_i$ and $q_f$. So, y …