In Weinberg's book, he said "Feynman graphs with lines off the mass shell are just a special case of a wider generalization of the Feynman rules that takes into account the effects of various possible external fields."(adsbygoogle = window.adsbygoogle || []).push({});

Suppose the external fields are treated as a c-number, [tex]\epsilon_a(x)[/tex], and they coupled to the current operators [tex]o_a(t)[/tex] in this way that the potential looks like,

[tex]V_\epsilon(t) = V(t) + \sum_a\int d^3x\epsilon_a(\mathbf{x},t)o_a(\mathbf{x},t)[/tex]

where [tex]V(t)[/tex] is the interaction used in the Dyson series.

Now he argued something that I don't understand:

"It follows then that the rth variational derivative of the scattering matrix [tex]S_{\beta\alpha}[\epsilon][/tex] with respect to [tex]\epsilon_a(x), \epsilon_b(y)\cdots[/tex] at [tex]\epsilon=0[/tex] is given by position space diagrams with [tex]r[/tex] additional vertices, to which are attached some internal lines, and NO external lines."

I don't understand why there are no external lines attached to the vertices formed by [tex]o_a(x)[/tex]? I think there is no reason why the external lines are ruled out?

Any help would be appreciated.

These materials are in the page 287 of Weinberg's book.

Thanks.

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# Off the mass shell and external fields

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