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The Schwinger–Dyson equation on Peskin and Schroder reads (p.308): $$ \!\!\!\!\!\!\!\!\left\langle\left(\frac{\delta}{\delta\phi(x)}\int d^4x'\mathcal L\right)\phi(x_1)...\phi(x_n)\right\rangle = \sum_{i=1}^n \left\langle\phi(x_1)...(i\delta(x-x_i))...\phi(x_n)\right\rangle \tag{9.88} $$

This equation tells us that the classical Euler-Lagrange equations of the field $\phi$ are obeyed for all Green’s functions of $\phi$, up to contact terms arising from the nontrivial commutation relations of field operators.

In my QFT class we're told that those equations describe how fields and particles interact with each other. My question is how do we interpret this equation in a way that this interaction is clear? Can we tell how particles are excited from the associated quantum field from this equation, or can we tell which representation - particles or fields - is more fundamental?

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  • $\begingroup$ The only thing we can measure in the lab are quanta. How you want to describe that in theory is really up to you... as long as you know how to translate the description into an actionable physical hypothesis (usually scattering cross sections or at least poles that give us energy and width of an excitation). $\endgroup$ May 4, 2023 at 2:02
  • $\begingroup$ Keep in mind that fields in QFT are not the same as classical fields.en.wikipedia.org/wiki/Quantum_field_theory . $\endgroup$
    – anna v
    May 4, 2023 at 4:33

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The Schwinger-Dyson equations provide relationship among the summations of all diagrams with the same number of external lines. Diagrammatically, such a summation is represented as a blob with the external lines attached to it. These relationships can be represented in terms of functional derivatives applied to these blobs. When a functional derivative is applied, it basically cuts a line in each of the diagram in the summation, thus producing two extra external lines for the blob.

As an example, consider the sum of all vacuum diagram. It would give a blob without external lines. When I apply the functional derivative, it produces two external lines. The result then represents a blob for the dressed propagator of that field. One can now use the Schwinger Dyson equation to form a self-consistent equation for the propagator.

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  • $\begingroup$ Thanks so much for the answer! I found this reference shows the idea of what you explained: arxiv.org/abs/2006.04908 I'm looking at fig.2 (p.4) on this paper, is it right for me to say the left-hand side of the equation above (9.88) corresponds to $G_{msln}$ in the figure, while the right-hand side is the sum of 2,3,4 point functions and the vertices? $\endgroup$
    – IGY
    May 4, 2023 at 9:32
  • $\begingroup$ It seems to be somewhat more complicated. The have several functional derivatives to open up lines. In your expression there is only one functional derivative. I think the paper is a good place to figure out how it works. $\endgroup$ May 5, 2023 at 3:24

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