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| -rw-r--r-- | sources/know/concept/self-energy/main.tex | 171 |
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diff --git a/sources/know/concept/self-energy/main.tex b/sources/know/concept/self-energy/main.tex new file mode 100644 index 0000000..114b433 --- /dev/null +++ b/sources/know/concept/self-energy/main.tex @@ -0,0 +1,171 @@ +\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} |