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Right at the bottom of page 52 of David Tong's QFT notes we have just defined the time ordered Dyson formula, David Tong then shows the expansion of $(3.20)$ however an extra second-order term has appeared. The second-order term in the expansion of the exponential has two terms and the integration limits have been changed from what we see in $(3.18)$.

I can't figure out why this would be the case and this change is introduced with little/no explanation. I imagine it comes from the introduction of the time ordering symbol in some way but exactly how is unclear to me, could someone please briefly explain this?

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    $\begingroup$ Questions should be as self-contained as possible. So think about including the most important equations. $\endgroup$
    – Tobias Fünke
    Jun 7, 2022 at 10:18

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In Tong's lecture notes, (3.18) is not a correct answer, it does not solve (3.16) as one could naively think. He is trying to motivate the final result and raise the issue of ordering problems. Dyson's formula (3.20) rigorously solves (3.16) by modifying (3.18) using time ordering.

The proof of Dyson's formula is presented on the next page.

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  • $\begingroup$ The way he introduces this formula as "Expanding out the expression (3.20) we now have..." makes it seem like this is just what you get when you Taylor expand (3.20). Are you saying it's just introduced for the sake of showing (3.22) (which is pretty standard in most books)? If so I think the wording is exceptionally misleading, thank you for your answer though. $\endgroup$
    – Charlie
    Jun 7, 2022 at 10:33
  • $\begingroup$ When he expands out (3.20) he does expand out the exponential using its Taylor expansion. (3.18) is just there to show that a first naive idea does not work, and is to be contrasted to the real solution expansion given in (3.23). He's just trying to make the differences explicit. $\endgroup$
    – LPZ
    Jun 7, 2022 at 10:46

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