time evolution - Is there any relationship between S-matrix elements and the path integral?

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 ?

Answer

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).