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Quantization refers to the procedure or methodology for replacing a classical system by a quantum system. If the question is about the quantized or discrete behavior of a phenomenon use the [discrete] tag instead.
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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 …