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\documentclass[11pt]{article}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{physics}
\usepackage{feynmp}
\DeclareGraphicsRule{*}{mps}{*}{}
\begin{document}
\begin{center}
{\Huge \sc Feynman diagrams}
\end{center}
\noindent
Full Green's function
\begin{align}
G(b,a)
&= \quad
\begin{fmffile}{0order1}
\begin{fmfgraph*}(30,20)
\fmfipair{i,o}
\fmfiequ{i}{(0w,0.3h)}
\fmfiequ{o}{(1w,0.3h)}
\fmfi{fermion}{i -- o}
\fmfiv{decor.shape=circle,decor.size=4,label=$a$,label.angle=-90}{i}
\fmfiv{decor.shape=circle,decor.size=4,label=$b$,label.angle=-90}{o}
\end{fmfgraph*}
\end{fmffile}
\quad + \quad
\begin{fmffile}{1order1}
\begin{fmfgraph*}(50,20)
\fmfipair{i,m,t,t',o}
\fmfiequ{i}{(0w,0.3h)}
\fmfiequ{m}{(0.5w,0.3h)}
\fmfiequ{t}{(0.5w,0.9h)}
\fmfiequ{t'}{(0.5w,1.3h)}
\fmfiequ{o}{(1w,0.3h)}
\fmfi{fermion}{i -- m}
\fmfi{boson}{m -- t}
\fmfi{plain}{t{left} .. {right}t'}
\fmfi{plain}{t'{right} .. {left}t}
\fmfi{fermion}{m -- o}
\fmfiv{decor.shape=circle,decor.size=4,label=$a$,label.angle=-90}{i}
\fmfiv{decor.shape=circle,decor.size=4}{m}
\fmfiv{decor.shape=circle,decor.size=4}{t}
\fmfiv{decor.shape=circle,decor.size=4,label=$b$,label.angle=-90}{o}
\end{fmfgraph*}
\end{fmffile}
\quad + \quad
\begin{fmffile}{1order2}
\begin{fmfgraph*}(70,20)
\fmfipair{i,a,b,o}
\fmfiequ{i}{(0w,0.3h)}
\fmfiequ{a}{(0.33w,0.3h)}
\fmfiequ{b}{(0.67w,0.3h)}
\fmfiequ{o}{(1w,0.3h)}
\fmfi{fermion}{i -- a}
\fmfi{boson}{a -- b}
\fmfi{fermion}{a{up} .. {down}b}
\fmfi{fermion}{b -- o}
\fmfiv{decor.shape=circle,decor.size=4,label=$a$,label.angle=-90}{i}
\fmfiv{decor.shape=circle,decor.size=4}{a}
\fmfiv{decor.shape=circle,decor.size=4}{b}
\fmfiv{decor.shape=circle,decor.size=4,label=$b$,label.angle=-90}{o}
\end{fmfgraph*}
\end{fmffile}
\quad + \:\: \cdots
\end{align}
Self-energy definition
\begin{align}
\Sigma(y,x)
=\!\!\!\!
\begin{fmffile}{selfenergy}
\begin{fmfgraph*}(20,20)
\fmfright{v}
\fmfv{label=$x\!\!\to\!\!y$,label.angle=-90,label.dist=10}{v}
\fmfblob{15}{v}
\end{fmfgraph*}
\end{fmffile}
&\quad\: \equiv \:\:\:\:
\begin{fmffile}{hartree}
\begin{fmfgraph*}(25,20)
\fmfleft{i}
\fmfright{o}
\fmf{boson}{i,o}
\fmf{fermion}{o,o}
\fmfv{label=$\:\:x\!\!=\!\!y$,label.angle=-90}{i}
\fmfdot{i}
\fmfdot{o}
\end{fmfgraph*}
\end{fmffile}
\qquad + \:\:\:
\begin{fmffile}{fock}
\begin{fmfgraph*}(40,20)
\fmfipair{i,o}
\fmfiequ{i}{(0,0.3h)}
\fmfiequ{o}{(w,0.3h)}
\fmfi{boson}{i -- o}
\fmfi{fermion}{i{up} .. tension 1.5 .. {down}o}
\fmfiv{decor.shape=circle,decor.size=4,label=$x$,label.angle=-90,label.dist=5}{i}
\fmfiv{decor.shape=circle,decor.size=4,label=$y$,label.angle=-90,label.dist=5}{o}
\end{fmfgraph*}
\end{fmffile}
\:\:\: + \:\:\:
\begin{fmffile}{pairbubble}
\begin{fmfgraph*}(40,20)
\fmfipair{i,a,b,o}
\fmfiequ{i}{(0,0h)}
\fmfiequ{a}{(0,1h)}
\fmfiequ{b}{(1w,1h)}
\fmfiequ{o}{(1w,0h)}
\fmfi{fermion}{i -- o}
\fmfi{boson}{i -- a}
\fmfi{boson}{b -- o}
\fmfi{fermion}{a{up} .. tension 2.5 .. {down}b}
\fmfi{fermion}{b{down} .. tension 2.5 .. {up}a}
\fmfiv{decor.shape=circle,decor.size=4,label=$x$,label.angle=-90,label.dist=5}{i}
\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{a}
\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{b}
\fmfiv{decor.shape=circle,decor.size=4,label=$y$,label.angle=-90,label.dist=5}{o}
\end{fmfgraph*}
\end{fmffile}
\:\:\: + \: \cdots
\end{align}
Dyson equation
\begin{align}
G(b,a)
= \quad
\begin{fmffile}{fullgf}
\begin{fmfgraph*}(50,20)
\fmfleft{i}
\fmfright{o}
\fmf{heavy}{i,o}
\fmfv{label=$a$,label.angle=-90}{i}
\fmfv{label=$b$,label.angle=-90}{o}
\fmfdot{i}
\fmfdot{o}
\end{fmfgraph*}
\end{fmffile}
\quad = \quad
\begin{fmffile}{freegf}
\begin{fmfgraph*}(50,20)
\fmfleft{i}
\fmfright{o}
\fmf{fermion}{i,o}
\fmfv{label=$a$,label.angle=-90}{i}
\fmfv{label=$b$,label.angle=-90}{o}
\fmfdot{i}
\fmfdot{o}
\end{fmfgraph*}
\end{fmffile}
\quad + \quad
\begin{fmffile}{dyson}
\begin{fmfgraph*}(110,20)
\fmfleft{i}
\fmfright{o}
\fmf{fermion}{i,v}
\fmf{heavy}{v,o}
\fmfv{label=$a$,label.angle=-90}{i}
\fmfv{label=$b$,label.angle=-90}{o}
\fmfdot{i}
\fmfdot{o}
\fmfv{label=$x\!\to\!y$,label.angle=-90,label.dist=10}{v}
\fmfblob{15}{v}
\end{fmfgraph*}
\end{fmffile}
\end{align}
\end{document}
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