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In Tom Bank's book Modern Quantum Field Theory he says,

Consider a classical machine (an emission source) that has probability amplitude $J_E(x)$ of producing a particle at position $x$ in space-time, and an absorption source, which has amplitude $J_A(x)$ to absorb the particle. Assume that the particle propagates freely between emission and absorption, and has mass $m$. The standard rules of quantum mechanics tell us that the amplitude (to leading order in perturbation theory in the sources) for the entire process is (remember our natural units!) $$A_{AE}=\int d^4x d^4y\langle x|e^{ −iH(x^0 −y^0)}|y\rangle J_A(x)J_E(y),\tag{1}$$ where $|x\rangle$ is the state of the particle at spatial position $x$.

What does he have in mind when he says "by standard rules of quantum mechanics"? In other words, by which standard rules of QM did he obtain Eqn.(1)?

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    $\begingroup$ $\langle x | e^{-i H(x^0-y^0)} | y \rangle$ is the amplitude for the particle to propagate from position $y$ to position $x$. Multiplying by $J_A(x) J_E(y)$ gives the probability that a particle is emitted at $y$, propagates for time $x^0-y^0$, and is absorbed at $x$. Summing over $x$ and $y$ gives the total emission-absorbtion probability. $\endgroup$ – Elliot Schneider Apr 6 '17 at 15:40
  • $\begingroup$ @user81003 That should be an answer, not a comment. $\endgroup$ – pfnuesel Apr 6 '17 at 16:56

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