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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?

Qmechanic
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IGY
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1 Answers1

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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.

flippiefanus
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