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Reading Peskin&Schroeder I've made the following curious observation: Comparing S-matrix elements to the definition of the path-integral they look remarkably similar:

$$_{out}\langle \mathbf{p}_1 \mathbf{p}_1| \mathbf{k}_A \mathbf{k}_B\rangle_{in}= \lim_{T\rightarrow \infty} \langle \mathbf{p}_1 \mathbf{p}_1| e^{-iH(2T)}|\mathbf{k}_A \mathbf{k}_B\rangle=\langle \mathbf{p}_1 \mathbf{p}_1| S|\mathbf{k}_A \mathbf{k}_B\rangle \tag{4.70 +4.71}$$

compared to:

$$\lim_{T\rightarrow \infty} \langle \phi_b(x)| e^{-iH(2T)}|\phi_a(x)\rangle = \int {\cal D}\phi \exp\left[i\int_{-T}^{T} d^4x \cal{L}\right] \tag{9.14}\equiv Z$$

They are just the time evolution operator sandwiched between appropriate quantum states. I even guess that multi particle states like $|\mathbf{k}_A\mathbf{k}_B\rangle$ can be developed in field states $|\phi_a(x)\rangle$ with some a-priori unknown coefficients. So the transformation from one to the other does not look easy at all or is even impossible. But nevertheless, as the definitions look so similar, I ask the question: Are S-matrix elements related to the path integral or is it just a silly question, i.e. my observation is an accidental coincidence ?

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Yes, you can write down S-matrix elements directly in terms of path integrals. This was figured out by L. Fadeev and is explained in his 1975 Les Houches lecture notes. A review of his work is also in Bailin and Love's gauge field theory textbook.

References: Ludwig Faddeev, Introduction to Functional Methods, p. 1-39 in Methods in Field Theory, North Holland, 1976 (and references therein).

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    $\begingroup$ Thank you for the reference. I got finally access to it. $\endgroup$ – Frederic Thomas Nov 17 '19 at 20:01
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I believe your question is answered by the LSZ reduction formula. I shall give a brief overview of what I understand of the topic bellow focusing on a scalar field for simplicity. Corrections are welcome.

The path integral allows you to compute time-ordered correlation functions:

$$\langle \operatorname{T}[\phi(x_1)\cdots \phi(x_n)]\rangle=\int \varphi(x_1)\cdots \varphi(x_n)\dfrac{e^{iS[\varphi]}\mathfrak{D}\varphi}{Z}$$

Now, these correlation functions are exactly what enter the evaluation of the $\mathcal{S}$-matrix through the LSZ reduction formula. The reference that discusses this in the most basic level I believe is "Quantum Field Theory and the Standard Model" by Matthew Schwartz. The discussion is in Chapter 6.

The LSZ reduction formula states that $\mathcal{S}$-matrix elements between in and out states can be written as $$\langle \text{out}|\mathcal{S}|\text{in}\rangle=\left[-i \int d^4 x_1 e^{-ip_1x_1}(\Box+m^2)\right]\cdots \left[-i \int d^4 x_m e^{-ip_mx_m}(\Box+m^2)\right] \left[i \int d^4 y_1 e^{iq_1y_1}(\Box+m^2)\right]\cdots\left[i\int d^4y_n e^{iq_n y_n}(\Box+m^2)\right]\langle \operatorname{T}[\phi(x_1)\cdots \phi(x_m)\phi(y_1)\cdots \phi(y_n)]\rangle$$

Here the initial state is $|\text{in}\rangle = |p_1,\cdots, p_m\rangle$ and the final state is $|\text{out}\rangle=|q_1,\dots, q_n\rangle$.

In that sense the $\mathcal{S}$-matrix elements can be obtained out of that big operator acting on time-ordered correlation functions.

Now one usually defines the functional measure in perturbation theory in terms of the free theory measure that one knows how to make sense.

This is a very basic overview that I believe is able to guide you on what you ask. I believe it is highly recommended to study also some more advanced references concerning the LSZ reduction formula, in particular Weinberg's "The Quantum Theory of Fields, Vol. 1" is a very good choice. If I'm not mistkaen, Peskin & Schroeder also has a more in depth discussion than Schwartz.

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