Recently I have been searching for good introductory books to Quantum Field Theory in the hope of getting a good enough basis for string theory in about a semester or two. There exist two different approaches to QFT, canonical quantization and the path integral approach. The book I just purchased takes the canonical quantization approach. Is one approach preferred for string theory over the other? Do I need to find another book that takes the path integral approach or is the canonical quantization fine?
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1$\begingroup$ I think the correct answer is that you really ought to be very familiar with both. Path integrals often sweep important technical issues under the rug; I think it's difficult to achieve a deep understanding of path integrals without first carefully working through canonical quantization. But ultimately path integrals are often simpler to work with and can give more physical insight. $\endgroup$– ZackCommented Aug 30, 2022 at 18:38
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$\begingroup$ You need to be familiar with both of them. In particular, since you have string theory in mind, path integral quantization of string theory is, in my opinion, the nicest approach, because you have gauge invariance in the worldsheet theory. When you have gauge invariance functional integration is the best path to follow, employing the so-called Faddeev-Popov procedure. For example, non-abelian gauge theories, like QCD, are studied using this method, since in that case canonical quantization becomes quite cumbersome. $\endgroup$– GoldCommented Aug 30, 2022 at 20:00
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You need to be very familiar with both. Knowledge of only one formalism will not suffice for string theory.
On the one hand, the string theory path integral is heavily used to compute partition functions and amplitudes, as well as anomalies and their cancellations.
On the other hand, constructing the spectrum of states of a string theory is most easily done within canonical formalism. Furthermore, on the string worldsheet you have a CFT and CFT techniques are oftentimes phrased in canonical formalism.
Moreover, sometimes in string theory you encounter peculiar discrepancies if you try to compare the two formalisms naively, that are saved by quite elegant solutions. You need to know both to appreciate the beauty and depth of quantum field theory that saves string theory those days.