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I've seen the path integral formulation now in a couple contexts (propagator in quantum mechanics, and coherent state functional integral in many body physics). I'm now struggling with how to actually compute path integrals. It seems that for Gaussian actions it is a kind of continuous generalization of Gaussian integration. I was wondering if anyone had any references that explain computing path integrals in this context and perhaps offer practice in evaluating them.

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    $\begingroup$ Well, except for the Gaussian case (where it works exactly as for ordinary Gaussian integrals), one rarely explicitly computes the path integral. (And in the cases I know, the trick to evaluate it is usually to get around actually performing the integration) $\endgroup$
    – ACuriousMind
    Commented May 15, 2015 at 23:50
  • $\begingroup$ Take a look at the "related" links on the right hand margin of the page: ----->. Some good stuff there. $\endgroup$
    – Floris
    Commented May 16, 2015 at 0:28
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    $\begingroup$ I don't like to give book recommendations, but Hagen Kleinert has made a cottage industry out of the explicit solution of path integrals and has published a number of editions of his textbook "Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets" with an ever increasing number of applications. I would agree with the others here that a good look at Kleinert's book you will probably remove your interest in spending any more time on path integrals than necessary to move on to better methods. $\endgroup$
    – CuriousOne
    Commented May 16, 2015 at 4:19
  • $\begingroup$ Kleinert (as well as Grosche's big handbook of path integral solutions) is indeed quite a nice book to have, but from what I can remember none of them or close to none are QFT path integrals, it's mostly QM path integrals. $\endgroup$
    – Slereah
    Commented Sep 15, 2015 at 23:04

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You can refer to chap 9 of "An introduction to quantum field theory" of Peskin & Scroeder, which includes a detailed calculation of path integral using the original physical definition of path integral. After the brutal treatment, they will show you more modern treatment using generating functional.

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Another good source on path integrals and in particular their computation is A.Zee's book "Quantum field theory in a nutshell". Path integrals are introduced in the first chapter, so the prerequisite for reading the first chapter (and actually most of the rest of the book) are basics in theoretical mechanics (Lagrange and Hamilton's formalism) and basic knowledge about quantum mechanics. Furthermore Zee claims that one only needs to know to compute Gaussian integrals in all variations to do a path integral. He also shows that by developing Z(J) of an interacting QFT as double Taylor series in $J$ and the coupling constant of the chosen theory one is led to the introduction of Feynman diagrams which is really nice. Zee's book is rather intuitive, but less rigorous. For a more rigorous formalism Srednicki's book on QFT would be better. But the latter has less computation examples and requires a higher level of understanding (i.e. is more abstract).

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