So I am starting to read QFT from P&S, and in the very beginning they set out to clarify why a naive approach of writing a single particle relativistic hamiltonian would lead to propagation outside the light cone. I have several questions about their line of argument, some technical and some physical.

  1. Technical question: How do I get from the second to the final step? enter image description here

  2. The phase is stationary when $p = \frac{imx}{\sqrt{x^2 - t^2}}$, but that does not lead me to the same conclusion as P&S in a straightforward way. Am I missing some further approximations? enter image description here

  3. Here P&S talks about causality in terms of measurements outside the light cone affecting the inside. They say that QFT will take care of it by introducing antiparticles, which will cancel out the contribution from the particle. But didn't we just argue that, for real particles at least, the amplitude does not cancel? (since their antiparticle are themselves) So what happens to the causality, or are there no real particles? I am confused. enter image description here

I know its really three questions but they are connected, so it made more sense to pack them into one. Can someone help me out with explicit calculations and explanations?

Thanks in advance.

References: Peskin & Schroeder p.14

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    $\begingroup$ Possible duplicates: physics.stackexchange.com/q/346780/2451 , physics.stackexchange.com/q/194877/2451 and links therein. $\endgroup$ – Qmechanic Apr 7 '18 at 14:26
  • $\begingroup$ The first 2 subquestions seems to be technical exercises. The third subquestion is conceptional. To reopen this post (v2), consider to only ask one subquestion per post. $\endgroup$ – Qmechanic Apr 7 '18 at 14:33
  • $\begingroup$ None of the links you gave answers any of my questions above. Plus, I agree it's too many subquestions but it's related to essentially a single derivation in P&S. Would you please reconsider opening this thread on these grounds? $\endgroup$ – Razor Apr 7 '18 at 15:21