Off the mass shell and external fields

[ismaili]

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."

Suppose the external fields are treated as a c-number, $$\epsilon_a(x)$$, and they coupled to the current operators $$o_a(t)$$ in this way that the potential looks like,
$$V_\epsilon(t) = V(t) + \sum_a\int d^3x\epsilon_a(\mathbf{x},t)o_a(\mathbf{x},t)$$
where $$V(t)$$ 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 $$S_{\beta\alpha}[\epsilon]$$ with respect to $$\epsilon_a(x), \epsilon_b(y)\cdots$$ at $$\epsilon=0$$ is given by position space diagrams with $$r$$ 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 $$o_a(x)$$? 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.

 

[olgranpappy]

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."

Suppose the external fields are treated as a c-number, $$\epsilon_a(x)$$, and they coupled to the current operators $$o_a(t)$$ in this way that the potential looks like,
$$V_\epsilon(t) = V(t) + \sum_a\int d^3x\epsilon_a(\mathbf{x},t)o_a(\mathbf{x},t)$$
where $$V(t)$$ 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 $$S_{\beta\alpha}[\epsilon]$$ with respect to $$\epsilon_a(x), \epsilon_b(y)\cdots$$ at $$\epsilon=0$$ is given by position space diagrams with $$r$$ 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 $$o_a(x)$$? 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.

hmm... it seems like you are right. Maybe Weinberg is assuming that the fields in the
$$o_a(x)$$
operators are never in the incoming or outgoing states.