Weinberg Volume I Equation (10.5.10) on page 451

Question

I originally asked in physics stackexchange, getting no answers within 2 days. http://physics.stackexchange.com/questions/194355/weinberg-volume-i-equation-10-5-10-on-page-451

I copy it here: $$\Delta'_{\mu\nu}(q)=\Delta_{\mu\nu}(q)+\Delta_{\mu\rho}(q) M^{\rho\sigma}(q) \Delta_{\sigma\nu}(q)$$.

$\Delta_{\mu\nu}(q)$ is the bare photon propagator, and $\Delta'_{\mu\nu}(q)$ is the complete photon propagator. Weinberg says the $M^{\rho\sigma}$ is the matrix element of two currents between vacuum states ($\alpha$ and $\beta$ are taken to be vacuum states in the following), proportional to equation (10.5.1): $$M^{\mu\mu'...}_{\beta\alpha}(q,q',...)=\int d^4x\int d^4x' e^{-iqx}e^{-iq'x'}\times(\Psi^-_b,T\{J^{\mu}(x)J^{\mu'}(x')...\}\Psi^+_a)$$, for arbitrary transition $\alpha\rightarrow\beta$.

It seems related to the form of photon propagator in the Heisenberg picture form. Could you explain this to me? Thank you in advance!

Comments

Isn't it a matter of definition?

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Do you know how to draw the Feynman diagrams (full propagators) of $\langle A^\mu (x_1) A^\nu (x_2)\rangle$ and $\langle J^\rho (y_1) J^\sigma (y_2)\rangle$? The relation is rather transparent in the diagrams.

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@Jia Yiyang I can understand the perturbative form in terms of one-photon-irreducible diagrams, but not aware of the full form in the question. Could you explain it to me? Thanks!

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@ruifeng14, I've written you an answer. By the way to properly @ a user you need to eliminate all the spaces in his/her name. For example to @ me you need to use "@JiaYiyang" instead of "@Jia Yiyang".  

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@JiaYiyang Thanks for your advice.

Answer 1

For a time-ordered product like $\langle T\{O(x_1)O(x_2)\cdots O(x_n)\}\rangle$, the Feynman diagram of it is like this: external lines are the lines that represent $O(x_1)$,$O(x_2)$...$O(x_n)$, and you try to connect these external lines with the internal lines and vertices available, and whenever you have a valid graph, this graph contributes to $\langle T\{O(x_1)O(x_2)\cdots O(x_n)\}\rangle$. Hence $\langle T\{O(x_1)O(x_2)\cdots O(x_n)\}\rangle$ is a sum of all such graphs, and we may graphically represent the sum of all such graphs in a single graph with the summed internal process as a blob. If you apply the above described picture to  $\langle T\{A^\mu (x_1) A^\nu (x_2)\}\rangle$ and $\langle T\{ J^\rho (y_1) J^\sigma (y_2)\}\rangle$, you immediately see the relation between the two:

Namely, the graphs that represent $\langle T\{A^\mu (x_1) A^\nu (x_2)\}\rangle$ can be divided into two classes:

(1)A direct Wick contraction between $A^\mu (x_1)$ and $A^\nu (x_2)$;

(2) $A^\mu (x_1)$ and $A^\nu (x_2)$ are connected to two internal vertices $\int \text{d}y_1 A_\rho(y_1) J^\rho(y_1)$ and $\int \text{d}y_2 A_\sigma(y_2) J^\sigma(y_2)$, and hence all the graphs that come with these two prescribed vertices(as external lines of the graph representing $M^{\rho \sigma}$). Note that $J^\mu(x)=\bar{\psi}(x)\gamma^\mu \psi(x)$, so one external line of $J^\mu$ is really two lines of $\psi(x)$ pinched to the same point $x$. 

Comments

@JiaYiyang Very clear answer! Thank you very much!