From 025a3cf303695f8dffacb36a480ac972244dc43e Mon Sep 17 00:00:00 2001 From: jal <jal.ccs@gmail.com> Date: Mon, 1 Mar 2021 21:35:36 -0400 Subject: [PATCH] ajustes en clases --- ejemplo_clase/MT_19.aux | 39 + ejemplo_clase/MT_19.bbl | 17 + ejemplo_clase/MT_19.blg | 46 + ejemplo_clase/MT_19.idx | 0 ejemplo_clase/MT_19.log | 1215 +++++ ejemplo_clase/MT_19.pdf | Bin 0 -> 169151 bytes ejemplo_clase/MT_19.synctex.gz | Bin 0 -> 37928 bytes ejemplo_clase/MT_19.tex | 81 + ejemplo_clase/MT_19.toc | 6 + ejemplo_clase/apendice_glosario.tex | 27 + ejemplo_clase/particulas.bib | 22 + ejemplo_clase/propagador.tex | 65 + ejemplo_clase/propagador_retardado.tex | 61 + ejemplo_clase/prueba.aux | 38 + ejemplo_clase/prueba.bbl | 13 + ejemplo_clase/prueba.blg | 46 + ejemplo_clase/prueba.idx | 0 ejemplo_clase/prueba.log | 1268 +++++ ejemplo_clase/prueba.pdf | Bin 0 -> 172400 bytes ejemplo_clase/prueba.synctex.gz | Bin 0 -> 82181 bytes ejemplo_clase/prueba.tex | 109 + ejemplo_clase/prueba.toc | 6 + ejemplo_clase/prueba_old.tex | 10 + ejemplo_clase/qftdef.tex | 13 + ejemplo_clase/seccion_campo_dirac.log | 5916 +++++++++++++++++++++++ ejemplo_clase/seccion_campo_dirac.tex | 86 + ejemplo_clase/seccion_campo_escalar.tex | 129 + 27 files changed, 9213 insertions(+) create mode 100644 ejemplo_clase/MT_19.aux create mode 100644 ejemplo_clase/MT_19.bbl create mode 100644 ejemplo_clase/MT_19.blg create mode 100644 ejemplo_clase/MT_19.idx create mode 100644 ejemplo_clase/MT_19.log create mode 100644 ejemplo_clase/MT_19.pdf create mode 100644 ejemplo_clase/MT_19.synctex.gz create mode 100644 ejemplo_clase/MT_19.tex create mode 100644 ejemplo_clase/MT_19.toc create mode 100644 ejemplo_clase/apendice_glosario.tex create mode 100644 ejemplo_clase/particulas.bib create mode 100644 ejemplo_clase/propagador.tex create mode 100644 ejemplo_clase/propagador_retardado.tex create mode 100644 ejemplo_clase/prueba.aux create mode 100644 ejemplo_clase/prueba.bbl create mode 100644 ejemplo_clase/prueba.blg create mode 100644 ejemplo_clase/prueba.idx create mode 100644 ejemplo_clase/prueba.log create mode 100644 ejemplo_clase/prueba.pdf create mode 100644 ejemplo_clase/prueba.synctex.gz create mode 100644 ejemplo_clase/prueba.tex create mode 100644 ejemplo_clase/prueba.toc create mode 100644 ejemplo_clase/prueba_old.tex create mode 100644 ejemplo_clase/qftdef.tex create mode 100644 ejemplo_clase/seccion_campo_dirac.log create mode 100644 ejemplo_clase/seccion_campo_dirac.tex create mode 100644 ejemplo_clase/seccion_campo_escalar.tex diff --git a/ejemplo_clase/MT_19.aux b/ejemplo_clase/MT_19.aux new file mode 100644 index 0000000..eafd81c --- /dev/null +++ b/ejemplo_clase/MT_19.aux @@ -0,0 +1,39 @@ +\relax +\catcode `"\active +\catcode `<\active +\catcode `>\active +\@nameuse{es@quoting} +\providecommand \oddpage@label [2]{} +\select@language{spanish} +\@writefile{toc}{\select@language{spanish}} +\@writefile{lof}{\select@language{spanish}} +\@writefile{lot}{\select@language{spanish}} +\citation{mandl2010quantum} +\citation{peskin2018introduction} +\@writefile{toc}{\contentsline {chapter}{\numberline {1}Campo de Dirac}{2}} +\@writefile{lof}{\addvspace {10\p@ }} +\@writefile{lot}{\addvspace {10\p@ }} +\@writefile{toc}{\contentsline {section}{\numberline {1.1}Sobre la representaci\IeC {\'o}n de n\IeC {\'u}mero en fermiones}{3}} +\newlabel{algebra.creacion.destruccion.bosonica}{{1.1}{3}} +\newlabel{operador.numero.r}{{1.2}{3}} +\newlabel{conmutador.operador.numero.r.aDaga}{{1.3}{3}} +\newlabel{conmutador.operador.numero.r.a}{{1.4}{3}} +\citation{asale_2021} +\newlabel{aniquilacion.del.vacio}{{1.5}{4}} +\@writefile{toc}{\contentsline {section}{\numberline {1.2}Sobre la representaci\IeC {\'o}n de n\IeC {\'u}mero para fermiones}{6}} +\@writefile{toc}{\contentsline {section}{\numberline {1.3}El Campo de Dirac}{7}} +\@writefile{toc}{\contentsline {chapter}{\numberline {A}Glosario}{8}} +\@writefile{lof}{\addvspace {10\p@ }} +\@writefile{lot}{\addvspace {10\p@ }} +\newlabel{convencion.metrica}{{A.1}{8}} +\bibstyle{unsrt} +\bibdata{particulas} +\newlabel{J.noether.general}{{A.4}{9}} +\newlabel{momento.lagrangiano.general}{{A.8}{9}} +\newlabel{t.mu.nu.general}{{A.9}{9}} +\bibcite{mandl2010quantum}{1} +\bibcite{peskin2018introduction}{2} +\bibcite{asale_2021}{3} +\newlabel{LastPage}{{}{10}} +\xdef\lastpage@lastpage{10} +\gdef\lastpage@lastpageHy{} diff --git a/ejemplo_clase/MT_19.bbl b/ejemplo_clase/MT_19.bbl new file mode 100644 index 0000000..3f5b61c --- /dev/null +++ b/ejemplo_clase/MT_19.bbl @@ -0,0 +1,17 @@ +\begin{thebibliography}{1} + +\bibitem{mandl2010quantum} +Franz Mandl and Graham Shaw. +\newblock {\em Quantum field theory}. +\newblock John Wiley \& Sons, 2010. + +\bibitem{peskin2018introduction} +Michael Peskin. +\newblock {\em An introduction to quantum field theory}. +\newblock CRC press, 2018. + +\bibitem{asale_2021} +RAE ASALE. +\newblock jurungar | diccionario de la lengua española, 2021. + 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+%\def\changemargin#1#2{\list{}{\rightmargin#2\leftmargin#1}\item[]} +%\let\endchangemargin=\endlist + +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb} + +\usepackage{lipsum} + +\usepackage{blindtext} + +\usepackage{leading} +\leading{18pt} + +%Footer + +\usepackage{lastpage} +\usepackage{fancyhdr} +\setlength{\headheight}{15.2pt} +\pagestyle{fancy} +\renewcommand{\chaptermark}[1]{\markboth{#1}{}} +\fancyhf{} +\fancyhead[LE,LO]{\chaptername~\thechapter:~\leftmark} +%\fancyhead[LO]{\leftmark} +%\fancyhead[LE,RO]{\thepage} + +%\fancyhead[LE,LO]{\thesection} +%\fancyhead[RE,RO]{\chaptername~\thechapter} +\lfoot[\tiny Versión. Fecha: \today]{\tiny Versión. Fecha: \today} +\cfoot[\thepage/\pageref{LastPage}]{\thepage/\pageref{LastPage}} +%\rfoot[<even output>]{\thepage/\pageref{LastPage}} + + +%Tipo de letra +%\renewcommand{\familydefault}{\sfdefault} + + +%Tikz pictures +\usepackage{tikz} +\usetikzlibrary{babel} +\usetikzlibrary{decorations.markings} + +\usepackage{pgfplots} +%\pgfplotsset{compat=1.15} + +%Alinear texto y figuras en las tablas +\usepackage{adjustbox} +\title{Campo de Dirac} +\author{José Antonio López RodrÃguez} +\makeindex + +\input{qftdef.tex} + +\begin{document} +\maketitle +\tableofcontents + +\input{seccion_campo_dirac.tex} + +\appendix +\chapter{Glosario} +\input{apendice_glosario.tex} + +\bibliographystyle{unsrt} +\bibliography{particulas} + +\end{document} + + + + + + diff --git a/ejemplo_clase/MT_19.toc b/ejemplo_clase/MT_19.toc new file mode 100644 index 0000000..5573fec --- /dev/null +++ b/ejemplo_clase/MT_19.toc @@ -0,0 +1,6 @@ +\select@language {spanish} +\contentsline {chapter}{\numberline {1}Campo de Dirac}{2} +\contentsline {section}{\numberline {1.1}Sobre la representaci\IeC {\'o}n de n\IeC {\'u}mero en fermiones}{3} +\contentsline {section}{\numberline {1.2}Sobre la representaci\IeC {\'o}n de n\IeC {\'u}mero para fermiones}{6} +\contentsline {section}{\numberline {1.3}El Campo de Dirac}{7} +\contentsline {chapter}{\numberline {A}Glosario}{8} diff --git a/ejemplo_clase/apendice_glosario.tex b/ejemplo_clase/apendice_glosario.tex new file mode 100644 index 0000000..d655afc --- /dev/null +++ b/ejemplo_clase/apendice_glosario.tex @@ -0,0 +1,27 @@ +\begin{itemize} +\item Definimos la métrica de Minkowski usando la convención +\begin{equation} +\label{convencion.metrica} +\eta_{\mu\nu}=diag(+,-,-,-) +\end{equation} +\item Definimos los operadores de Laplace y de D'Alambert: +\begin{eqnarray} +\nabla^2 & \equiv & \sum_i\partial_i^2 \\ +\square & \equiv & \partial_{0}^{2} - \nabla^2 = \partial_{\mu}\partial^{\mu} +\end{eqnarray} +\item Forma general de las corrientes de Noether para un sistema con campos $\phi_i$, lagrangiano $\mathcal{L}$ y grupo de simetrÃa de la acción con parámetros $\epsilon^{I}$: +\begin{eqnarray}\label{J.noether.general} +{J^{\mu}}&=&\Pi^{i\mu}\delta\phi_i-{T^{\mu}}_{\nu}\delta x^{\nu}, \\ +{J^{\mu}}_{I}&=&\Pi^{i\mu}\frac{\partial\phi_i}{\partial\epsilon^{I}}-{T^{\mu}}_{\nu}\frac{\partial x^{\nu}}{\partial\epsilon^{I}}. +\end{eqnarray} + +Donde +\begin{eqnarray} +\delta x^{\nu} &=& \frac{\partial x^{\nu}}{\partial\epsilon^{I}}\delta\epsilon^{I}, \\ +\delta\phi_i &=& \frac{\partial\phi_i}{\partial\epsilon^{I}}\delta\epsilon^{I}, \\ +\Pi^{i\mu} &=& \frac{\partial\mathcal{L}}{\partial\partial_{\mu}\phi_{i}}, \label{momento.lagrangiano.general}\\ +{T^{\mu}}_{\nu} &=& \Pi^{i\mu}\partial_{\nu}\phi_{i} - \delta^{\mu}_{\nu} \mathcal{L}.\label{t.mu.nu.general} +\end{eqnarray} + + +\end{itemize} \ No newline at end of file diff --git a/ejemplo_clase/particulas.bib b/ejemplo_clase/particulas.bib new file mode 100644 index 0000000..6c9a72e --- /dev/null +++ b/ejemplo_clase/particulas.bib @@ -0,0 +1,22 @@ +@book{mandl2010quantum, + title={Quantum field theory}, + author={Mandl, Franz and Shaw, Graham}, + year={2010}, + publisher={John Wiley \& Sons} +} + +@book{peskin2018introduction, + title={An introduction to quantum field theory}, + author={Peskin, Michael}, + year={2018}, + publisher={CRC press} +} + + +@misc{asale_2021, + title={jurungar | Diccionario de la lengua española}, + url={https://dle.rae.es/jurungar}, + journal={«Diccionario de la lengua española» - Edición del Tricentenario}, + author={ASALE, RAE}, + year={2021} + } diff --git a/ejemplo_clase/propagador.tex b/ejemplo_clase/propagador.tex new file mode 100644 index 0000000..341eae9 --- /dev/null +++ b/ejemplo_clase/propagador.tex @@ -0,0 +1,65 @@ +\documentclass{article} +\usepackage{tikz} +\usetikzlibrary{decorations.markings} +%https://tex.stackexchange.com/questions/361324/adding-arrows-to-a-complex-contour-with-tikz +%https://tex.stackexchange.com/questions/166772/problem-with-babel-and-tikz-using-draw +\begin{document} +\begin{figure} + +\begin{tikzpicture} + [line width=.8pt, + decoration={% + markings, + %mark=at position 0.25 with {\arrow[line width=1pt]{>}}, + mark=at position 0.5 with {\arrow[line width=1pt]{>}}, + %mark=at position 0.75 with {\arrow[line width=1pt]{>}} + } + ] + \draw [help lines,->] (-4,0) -- (4,0) coordinate (xaxis); + \draw [help lines,->] (0,-1)coordinate(-yaxis) -- (0,4) coordinate (yaxis); + \node at (1.5,-0.5) {$\bullet$}; + \node at (-1.5,-0.5) {$\bullet$}; + \draw [postaction=decorate] (3,0) arc (0:180:3) ; + %\draw [postaction=decorate] (-3,0) --(-2,0) arc (180:0:.5) --(1,0) arc (180:0:.5)--(3,0) ; + \draw [postaction=decorate] (-3,0) --(3,0) ; + \node [below] at (xaxis) {$k_0$}; + \node [above] at (yaxis) {$ik_0'$}; + %\node [below left] {$O$}; + %\node at (-1.5,.8) {$\gamma_{1}$}; + %\node at (1.5,.8) {$\gamma_{2}$}; + %\node at (2,3) {$\Gamma_{1}$}; + \node at (3,-.5){$R$}; + \node at (-3,-.5){$-R$}; + \node at (1.5,-0.75){$(\omega_k-i\epsilon)$}; + \node at (-1.5,-0.75){$-(\omega_k+i\epsilon)$}; +\end{tikzpicture} + +\end{figure} + +\begin{tikzpicture} + [yscale=-1,line width=.8pt, + decoration={% + markings, + mark=at position 0.25 with {\arrow[line width=1pt]{>}}, + mark=at position 0.5 with {\arrow[line width=1pt]{>}}, + mark=at position 0.75 with {\arrow[line width=1pt]{>}} + } + ] + \draw [help lines,->] (-4,0) -- (4,0) coordinate (xaxis); + \draw [help lines,->] (0,-1)coordinate(-yaxis) -- (0,4) coordinate (yaxis); + \node at (1.5,0) {$\times$}; + \node at (-1.5,0) {$\times$}; + \draw [postaction=decorate] (3,0) arc (0:180:3) ; + \draw [postaction=decorate] (-3,0) --(-2,0) arc (180:0:.5) --(1,0) arc (180:0:.5)--(3,0) ; + \node [below] at (xaxis) {$Re(z)$}; + \node [above] at (-yaxis) {$Im(z)$}; + \node [below left] {$O$}; + \node at (-1.5,.8) {$\gamma_{1}$}; + \node at (1.5,.8) {$\gamma_{2}$}; + \node at (2,3) {$\Gamma_{2}$}; + \node at (3,-.5){$R$}; + \node at (-3,-.5){$-R$}; + \node at (1.5,-.5){$1$}; + \node at (-1.5,-.5){$-1$}; +\end{tikzpicture} +\end{document} \ No newline at end of file diff --git a/ejemplo_clase/propagador_retardado.tex b/ejemplo_clase/propagador_retardado.tex new file mode 100644 index 0000000..9cd7fde --- /dev/null +++ b/ejemplo_clase/propagador_retardado.tex @@ -0,0 +1,61 @@ +%\begin{center} +\begin{figure} +\begin{tikzpicture} + [scale=0.8,line width=.8pt, + decoration={% + markings, + mark=at position 0.25 with {\arrow[line width=1pt]{>}}, + %mark=at position 0.5 with {\arrow[line width=1pt]{>}}, + mark=at position 0.75 with {\arrow[line width=1pt]{>}} + } + ] + \draw [help lines,->] (-4,0) -- (4,0) coordinate (xaxis); + \draw [help lines,->] (0,-1)coordinate(-yaxis) -- (0,4) coordinate (yaxis); + \node at (1.5,-0.5) {$\bullet$}; + \node at (-1.5,-0.5) {$\bullet$}; + \draw [postaction=decorate] (3,0) arc (0:180:3) ; + %\draw [postaction=decorate] (-3,0) --(-2,0) arc (180:0:.5) --(1,0) arc (180:0:.5)--(3,0) ; + \draw [postaction=decorate] (-3,0) --(3,0) ; + \node [below] at (xaxis) {$k_0$}; + \node [above] at (yaxis) {$ik_0'$}; + %\node [below left] {$O$}; + %\node at (-1.5,.8) {$\gamma_{1}$}; + %\node at (1.5,.8) {$\gamma_{2}$}; + %\node at (2,3) {$\Gamma_{1}$}; + %\node at (3,-.5){$R$}; + %\node at (-3,-.5){$-R$}; + \node at (1.5,-0.95){$\scriptstyle (\omega_k-i\epsilon)$}; + \node at (-1.5,-0.95){$\scriptstyle -(\omega_k+i\epsilon)$}; + \node at (2,3.2){$x_0<0$}; +\end{tikzpicture} +\begin{tikzpicture} + [scale=0.8,yscale=-1,line width=.8pt, + decoration={% + markings, + mark=at position 0.25 with {\arrow[line width=1pt]{>}}, + %mark=at position 0.5 with {\arrow[line width=1pt]{>}}, + mark=at position 0.75 with {\arrow[line width=1pt]{>}} + } + ] + \draw [help lines,->] (-4,0) -- (4,0) coordinate (xaxis); + \draw [help lines,->] (0,-1)coordinate(-yaxis) -- (0,4) coordinate (yaxis); + \node at (1.5,-0.5) {$\bullet$}; + \node at (-1.5,-0.5) {$\bullet$}; + \draw [postaction=decorate] (3,0) arc (0:180:3) ; + %\draw [postaction=decorate] (-3,0) --(-2,0) arc (180:0:.5) --(1,0) arc (180:0:.5)--(3,0) ; + \draw [postaction=decorate] (-3,0) --(3,0) ; + \node [below] at (xaxis) {$k_0$}; + \node [above] at (-yaxis) {$ik_0'$}; + \node [below left] {$O$}; + %\node at (-1.5,.8) {$\gamma_{1}$}; + %\node at (1.5,.8) {$\gamma_{2}$}; + %\node at (2,3) {$\Gamma_{2}$}; + \node at (2,3) {$x_0>0$}; + %\node at (3,-.5){$R$}; + %\node at (-3,-.5){$-R$}; + \node at (1.5,-.95){$\scriptstyle (\omega_k+i\epsilon)$}; + \node at (-1.5,-.95){$\scriptstyle -(\omega_k-i\epsilon)$}; +\end{tikzpicture} +\caption{Contornos de integración para el propagador retardado \label{contornos.propagador.retardado}} +\end{figure} +%\end{center} diff --git a/ejemplo_clase/prueba.aux b/ejemplo_clase/prueba.aux new file mode 100644 index 0000000..48661e8 --- /dev/null +++ b/ejemplo_clase/prueba.aux @@ -0,0 +1,38 @@ +\relax +\catcode `"\active +\catcode `<\active +\catcode `>\active 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+\setlength{\columnsep}{3em} +\usepackage[papersize={10in,6.5in},margin=0.5in]{geometry} +\usepackage[utf8]{inputenc} +\usepackage[spanish]{babel} + + +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb} + +\usepackage{lipsum} + +\usepackage{blindtext} + +\usepackage{leading} +\leading{15pt} + +%Footer + +\usepackage{lastpage} +\usepackage{fancyhdr} +\setlength{\headheight}{15.2pt} +\pagestyle{fancy} +\renewcommand{\chaptermark}[1]{\markboth{#1}{}} +\fancyhf{} +\fancyhead[LE,LO]{\chaptername~\thechapter:~\leftmark} +%\fancyhead[LO]{\leftmark} +%\fancyhead[LE,RO]{\thepage} + +%\fancyhead[LE,LO]{\thesection} +%\fancyhead[RE,RO]{\chaptername~\thechapter} +\lfoot[\tiny Versión. Fecha: \today]{\tiny Versión. Fecha: \today} +\cfoot[\thepage/\pageref{LastPage}]{\thepage/\pageref{LastPage}} +%\rfoot[<even output>]{\thepage/\pageref{LastPage}} + + +%Tipo de letra +%\renewcommand{\familydefault}{\sfdefault} + + +%Tikz pictures +\usepackage{tikz} +\usetikzlibrary{babel} +\usetikzlibrary{decorations.markings} + +\usepackage{pgfplots} +%\pgfplotsset{compat=1.15} + +%Alinear texto y figuras en las tablas +\usepackage{adjustbox} +\title{Campo de Dirac} +\author{José Antonio López RodrÃguez} +\makeindex + +\input{qftdef.tex} + +\begin{document} +\maketitle +\tableofcontents +\chapter{Campo de Dirac} +\input{seccion_campo_dirac.tex} +\chapter{Prueba} +\lipsum[5-20] + +\begin{center} +\begin{figure}[h] +\begin{tikzpicture} +\begin{axis}[ + x tick label style={ + /pgf/number format/1000 sep=}, + xlabel=Year, + enlargelimits=0.05, + legend style={at={(0.5,-0.2)}, + anchor=north,legend columns=-1}, + ybar interval=.7, +] +\addplot + coordinates {(2020,7) (2019,6) + (2018,3.5) (2017,4) (2016,3)}; +\addplot + coordinates {(2020,3) (2019,4) + (2018,4) (2017,6) (2016,7)}; +\legend{A,B} +\end{axis} +\end{tikzpicture} +\caption[jr]{Tomado de: https://tex.stackexchange.com/questions/501238/how-to-center-single-column-pgfplots-bar-graph-in-two-column-document} +\end{figure} +\end{center} + +\lipsum[1-2] + +\section{a} +\lipsum[1-10] +\appendix +\chapter{Glosario} +\input{apendice_glosario.tex} + +\bibliographystyle{unsrt} +\bibliography{particulas} + +\end{document} + + + + + + diff --git a/ejemplo_clase/prueba.toc b/ejemplo_clase/prueba.toc new file mode 100644 index 0000000..6f5fd78 --- /dev/null +++ b/ejemplo_clase/prueba.toc @@ -0,0 +1,6 @@ +\select@language {spanish} +\contentsline {chapter}{\numberline {1}Campo de Dirac}{5} +\contentsline {section}{\numberline {1.1}Sobre la representaci\IeC {\'o}n de n\IeC {\'u}mero en fermiones}{5} +\contentsline {chapter}{\numberline {2}Prueba}{7} +\contentsline {section}{\numberline {2.1}a}{11} +\contentsline {chapter}{\numberline {A}Glosario}{15} diff --git a/ejemplo_clase/prueba_old.tex b/ejemplo_clase/prueba_old.tex new file mode 100644 index 0000000..937211c --- /dev/null +++ b/ejemplo_clase/prueba_old.tex @@ -0,0 +1,10 @@ +\documentclass[12pt,letterpaper]{article} +\usepackage[utf8]{inputenc} +\usepackage[spanish]{babel} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb} +\author{José Antonio López RodrÃguez} +\begin{document} +Hola mundo +\end{document} \ No newline at end of file diff --git a/ejemplo_clase/qftdef.tex b/ejemplo_clase/qftdef.tex new file mode 100644 index 0000000..d975b80 --- /dev/null +++ b/ejemplo_clase/qftdef.tex @@ -0,0 +1,13 @@ +\newcommand{\comm}[2]{ + [#1,#2] +} + +\newcommand{\commEn}[2]{ + \left[#1,#2\right] +} + +\newcommand{\ket}[1]{|#1>} + +\newcommand{\opA}[1]{a_{#1}^{{}}} + +\newcommand{\opAdaga}[1]{a_{#1}^{\dagger}} \ No newline at end of file diff --git a/ejemplo_clase/seccion_campo_dirac.log b/ejemplo_clase/seccion_campo_dirac.log new file mode 100644 index 0000000..4a6c2d8 --- /dev/null +++ b/ejemplo_clase/seccion_campo_dirac.log @@ -0,0 +1,5916 @@ +This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015/Debian) (preloaded format=pdflatex 2021.1.29) 10 FEB 2021 21:19 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**seccion_campo_dirac.tex +(./seccion_campo_dirac.tex +LaTeX2e <2016/02/01> +Babel <3.9q> and hyphenation patterns for 8 language(s) loaded. + +! 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LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.5 U + n detalle a resaltar en la cuantización de una teorÃa de bosones es q... + +You're in trouble here. 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LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.7 T + enemos una serie de operadores de creación y destrucción de modos, et... + +You're in trouble here. 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LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.9 \begin{eqnarray*} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 9--9 +[] + [] + +! Undefined control sequence. +l.10 \comm + {a_r}{a_s^{\dagger}} & = & \delta_{rs}, \\ +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +<recently read> \comm + +l.11 \comm + {a_r}{a_s} & = & 0, \\ +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). 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LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.26 \begin{equation} + \label{ecuacion.KG} +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 26--26 +[] + [] + + +! LaTeX Error: Command \Box not provided in base LaTeX2e. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.27 \left(\Box + + m^2\right)\phi=0. +Load the latexsym or the amsfonts package to define this symbol + + +Overfull \hbox (58.94678pt too wide) detected at line 28 +[][] \OML/cmm/m/it/10 ^^^ \OT1/cmr/m/n/10 = 0\OML/cmm/m/it/10 : + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.30 L + a ecuación (\ref{ecuacion.KG}) implica que el campo $\phi$ tiene solu... + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + +Missing character: There is no L in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no à in font nullfont! +Missing character: There is no ³ in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no ( in font nullfont! +! Undefined control sequence. +<write> ...ference `ecuacion.KG' on page \thepage + \space undefined\on@line . +l.30 La ecuación (\ref{ecuacion.KG} + ) implica que el campo $\phi$ tiene solu... +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). 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LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.32 S + e puede hallar la solución más general de la ecuación (\ref{ecuacio... + +You're in trouble here. 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Undefined control sequence. +<write> ...ference `ecuacion.KG' on page \thepage + \space undefined\on@line . +l.32 ...general de la ecuación (\ref{ecuacion.KG} + ), pasando al espacio de F... +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). 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Undefined control sequence. +<write> ...ference `ecuacion.KG' on page \thepage + \space undefined\on@line . +l.66 ...ogénea de la ecuación (\ref{ecuacion.KG} + ) +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `ecuacion.KG' on page undefined on input line 66. + +Missing character: There is no ) in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 66--67 +[] + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 66--67 +[] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.68 \begin{equation} + \label{ecuacion.KG.con.fuentes} +You're in trouble here. 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LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.74 \begin{eqnarray} + \label{ecuacion.propagador.KG.real} +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 74--74 +[] + [] + + +! LaTeX Error: Command \Box not provided in base LaTeX2e. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.75 && \left(\Box + _x + m^2\right)G(x-x')=\delta^4(x-x'),\\ +Load the latexsym or the amsfonts package to define this symbol + +! Undefined control sequence. +l.76 &\Rightarrow & \phi(x) = \phi_H(x) + \int{\rm + d}^4 x' G(x-x')\rho(x'), +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). 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Undefined control sequence. +<write> ...ference `ecuacion.KG' on page \thepage + \space undefined\on@line . +l.78 ...la ecuación homogénea (\ref{ecuacion.KG} + ). +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `ecuacion.KG' on page undefined on input line 78. + +Missing character: There is no ) in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (27.03615pt too wide) in paragraph at lines 77--79 + $\OML/cmm/m/it/10 ^^^[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 77--79 +[] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.80 C + omo es acostumbrado se escribe la solución de la ecuación (\ref{ecua... + +You're in trouble here. 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Undefined control sequence. +<write> ...n.propagador.KG.real' on page \thepage + \space undefined\on@line . +l.80 ...uación (\ref{ecuacion.propagador.KG.real} + ) en el espacio de Fourier, +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). 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LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.82 \begin{equation} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 82--82 +[] + [] + +! Undefined control sequence. +l.84 G(x) = \frac{1}{(2\pi)^4}\int{\rm + d}^4 k \left[\frac{-1}{k_0^2-\omega_k... +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). 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Undefined control sequence. +<write> ...eference `omega.k.KG' on page \thepage + \space undefined\on@line . +l.86 ...misma definición que en (\ref{omega.k.KG} + ). +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `omega.k.KG' on page undefined on input line 86. + +Missing character: There is no ) in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (11.1447pt too wide) in paragraph at lines 85--87 + $\OML/cmm/m/it/10 ![]$ + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 85--87 +[] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.88 A + partir de (\ref{propagador.KG.fourier}) se definen los propagadores r... + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + +Missing character: There is no A in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ( in font nullfont! +! Undefined control sequence. +<write> ...ropagador.KG.fourier' on page \thepage + \space undefined\on@line . +l.88 A partir de (\ref{propagador.KG.fourier} + ) se definen los propagadores r... +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). 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Undefined control sequence. +<write> ...propagador.retardado' on page \thepage + \space undefined\on@line . +l.88 ... (fig \ref{contornos.propagador.retardado} + ), +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). 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LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.101 X + XXXX +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + +Missing character: There is no X in font nullfont! +Missing character: There is no X in font nullfont! +Missing character: There is no X in font nullfont! +Missing character: There is no X in font nullfont! +Missing character: There is no X in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 101--102 +[] + [] + +(./propagador_retardado.tex + +! LaTeX Error: Environment figure undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.2 \begin{figure} + +Your command was ignored. +Type I <command> <return> to replace it with another command, +or <return> to continue without it. + + +! LaTeX Error: Environment tikzpicture undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.3 \begin{tikzpicture} + +Your command was ignored. +Type I <command> <return> to replace it with another command, +or <return> to continue without it. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.4 [ + scale=0.8,line width=.8pt, +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + +Missing character: There is no [ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +! Undefined control sequence. +l.7 mark=at position 0.25 with {\arrow + [line width=1pt]{>}}, +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no > in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +! Undefined control sequence. +l.9 mark=at position 0.75 with {\arrow + [line width=1pt]{>}} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no > in font nullfont! +Missing character: There is no ] in font nullfont! +! Undefined control sequence. +l.12 \draw + [help lines,->] (-4,0) -- (4,0) coordinate (xaxis); +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no > in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no ; in font nullfont! +! Undefined control sequence. +l.13 \draw + [help lines,->] (0,-1)coordinate(-yaxis) -- (0,4) coordinate (y... +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no > in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no ; in font nullfont! +! Undefined control sequence. +l.14 \node + at (1.5,-0.5) {$\bullet$}; +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no ; in font nullfont! +! Undefined control sequence. +l.15 \node + at (-1.5,-0.5) {$\bullet$}; +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no ; in font nullfont! +! Undefined control sequence. +l.16 \draw + [postaction=decorate] (3,0) arc (0:180:3) ; +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no ; in font nullfont! +! Undefined control sequence. +l.18 \draw + [postaction=decorate] (-3,0) --(3,0) ; +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no ; in font nullfont! +! Undefined control sequence. +l.19 \node + [below] at (xaxis) {$k_0$}; +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no ; in font nullfont! +! Undefined control sequence. +l.20 \node + [above] at (yaxis) {$ik_0'$}; +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no ; in font nullfont! +! Undefined control sequence. +l.27 \node + at (1.5,-0.95){$\scriptstyle (\omega_k-i\epsilon)$}; +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no ; in font nullfont! +! Undefined control sequence. +l.28 \node + at (-1.5,-0.95){$\scriptstyle -(\omega_k+i\epsilon)$}; +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no ; in font nullfont! +! Undefined control sequence. +l.29 \node + at (2,3.2){$x_0<0$}; +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no ; in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{tikzpicture}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.30 \end{tikzpicture} + +Your command was ignored. +Type I <command> <return> to replace it with another command, +or <return> to continue without it. + + +! LaTeX Error: Environment tikzpicture undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.31 \begin{tikzpicture} + +Your command was ignored. +Type I <command> <return> to replace it with another command, +or <return> to continue without it. + +Missing character: There is no [ in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +! Undefined control sequence. +l.35 mark=at position 0.25 with {\arrow + [line width=1pt]{>}}, +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no > in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 7 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +! Undefined control sequence. +l.37 mark=at position 0.75 with {\arrow + [line width=1pt]{>}} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no > in font nullfont! +Missing character: There is no ] in font nullfont! +! Undefined control sequence. +l.40 \draw + [help lines,->] (-4,0) -- (4,0) coordinate (xaxis); +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no > in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no ; in font nullfont! +! Undefined control sequence. +l.41 \draw + [help lines,->] (0,-1)coordinate(-yaxis) -- (0,4) coordinate (y... +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no > in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 4 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no ; in font nullfont! +! Undefined control sequence. +l.42 \node + at (1.5,-0.5) {$\bullet$}; +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no ; in font nullfont! +! Undefined control sequence. +l.43 \node + at (-1.5,-0.5) {$\bullet$}; +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no ; in font nullfont! +! Undefined control sequence. +l.44 \draw + [postaction=decorate] (3,0) arc (0:180:3) ; +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no 8 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no ; in font nullfont! +! Undefined control sequence. +l.46 \draw + [postaction=decorate] (-3,0) --(3,0) ; +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no ; in font nullfont! +! Undefined control sequence. +l.47 \node + [below] at (xaxis) {$k_0$}; +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no ; in font nullfont! +! Undefined control sequence. +l.48 \node + [above] at (-yaxis) {$ik_0'$}; +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no ; in font nullfont! +! Undefined control sequence. +l.49 \node + [below left] {$O$}; +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no [ in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ] in font nullfont! +Missing character: There is no ; in font nullfont! +! Undefined control sequence. +l.53 \node + at (2,3) {$x_0>0$}; +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no ; in font nullfont! +! Undefined control sequence. +l.56 \node + at (1.5,-.95){$\scriptstyle (\omega_k+i\epsilon)$}; +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no ; in font nullfont! +! Undefined control sequence. +l.57 \node + at (-1.5,-.95){$\scriptstyle -(\omega_k-i\epsilon)$}; +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no ( in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no - in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no 9 in font nullfont! +Missing character: There is no 5 in font nullfont! +Missing character: There is no ) in font nullfont! +Missing character: There is no ; in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{tikzpicture}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.58 \end{tikzpicture} + +Your command was ignored. +Type I <command> <return> to replace it with another command, +or <return> to continue without it. + + +! LaTeX Error: \caption outside float. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.59 \caption + {Contornos de integración para el propagador retardado \label{... + +You're in trouble here. 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LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.108 V + eamos las cantidades conservadas debido a la simetrÃa de Poincaré d... + +You're in trouble here. 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Undefined control sequence. +<write> ...n.campo.escalar.real' on page \thepage + \space undefined\on@line . +l.108 ... acción (\ref{accion.campo.escalar.real} + ). La corriente conservada... +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). 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Undefined control sequence. +<write> ...e `J.noether.general' on page \thepage + \space undefined\on@line . +l.108 ...energÃa-impulso (\ref{J.noether.general} + , \ref{t.mu.nu.general}), +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `J.noether.general' on page undefined on input line 1 +08. + +Missing character: There is no , in font nullfont! +! Undefined control sequence. +<write> ...nce `t.mu.nu.general' on page \thepage + \space undefined\on@line . +l.108 ...J.noether.general}, \ref{t.mu.nu.general} + ), +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). 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LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.117 L + as expresiones para las densidades de momentum lineal y angular del c... + +You're in trouble here. 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LaTeX Error: The font size command \normalsize is not defined: + there is probably something wrong with the class file. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.118 + +Your command was ignored. +Type I <command> <return> to replace it with another command, +or <return> to continue without it. + +[1{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map} +! Undefined control sequence. +<write> ...ccion.campo.escalar.real}{{1}{\thepage + }} +l.118 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{accion.campo.escalar.real}{{1}{}} +! Undefined control sequence. +<write> \newlabel{ecuacion.KG}{{2}{\thepage + }} +l.118 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{ecuacion.KG}{{2}{}} +! Undefined control sequence. +<write> ...ion.KG.real.ondas.planas}{{3}{\thepage + }} +l.118 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{solucion.KG.real.ondas.planas}{{3}{}} +! Undefined control sequence. +<write> \newlabel{omega.k.KG}{{5}{\thepage + }} +l.118 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{omega.k.KG}{{5}{}} +! Undefined control sequence. +<write> ...ecuacion.KG.con.fuentes}{{11}{\thepage + }} +l.118 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{ecuacion.KG.con.fuentes}{{11}{}} +! Undefined control sequence. +<write> ...cion.propagador.KG.real}{{12}{\thepage + }} +l.118 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{ecuacion.propagador.KG.real}{{12}{}} +! Undefined control sequence. +<write> ...l{propagador.KG.fourier}{{14}{\thepage + }} +l.118 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{propagador.KG.fourier}{{14}{}} +! Undefined control sequence. +<write> ...propagador.KG.retardado}{{15}{\thepage + }} +l.118 + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +\newlabel{propagador.KG.retardado}{{15}{}} +] + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.119 \begin{eqnarray} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 119--119 +[] + [] + + +Overfull \hbox (55.64914pt too wide) in alignment at lines 119--122 + [][][] [] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.124 C + onviene identificar las propiedades del tensor ${T^{\mu}}_{\nu}$ y la... + +You're in trouble here. 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LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.126 L + a conservación del momentum lineal está garantizada cuando se anula... + +You're in trouble here. 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Emergency stop. +<*> seccion_campo_dirac.tex + +*** (job aborted, no legal \end found) + + +Here is how much of TeX's memory you used: + 34 strings out of 494850 + 685 string characters out of 6178024 + 57945 words of memory out of 5000000 + 3428 multiletter control sequences out of 15000+600000 + 3948 words of font info for 15 fonts, out of 8000000 for 9000 + 62 hyphenation exceptions out of 8191 + 18i,12n,21p,438b,113s stack positions out of 5000i,500n,10000p,200000b,80000s +! ==> Fatal error occurred, no output PDF file produced! diff --git a/ejemplo_clase/seccion_campo_dirac.tex b/ejemplo_clase/seccion_campo_dirac.tex new file mode 100644 index 0000000..b728cf6 --- /dev/null +++ b/ejemplo_clase/seccion_campo_dirac.tex @@ -0,0 +1,86 @@ +\chapter{Campo de Dirac} +El Campo de Dirac es el objeto más importante en la descripción de partÃculas de espÃn $\frac{1}{2}$, fermiones. PartÃculas que obedecen la estadÃstica de Fermi-Dirac. El desarrollo de esta parte sigue la forma de las referencias \cite{mandl2010quantum,peskin2018introduction} + +\section{Sobre la representación de número en fermiones} + +Un detalle a resaltar en la cuantización de una teorÃa de bosones es que el sistema libre es una colección de osciladores armónicos. + +Tenemos una serie de operadores de creación y destrucción de modos, etiquetados por estados, genéricamente $r,s \dots$ + +\begin{eqnarray} + \label{algebra.creacion.destruccion.bosonica} + \comm{a_r}{a_s^{\dagger}} & = & \delta_{rs}, \nonumber\\ + \comm{a_r}{a_s} & = & 0, \\ + \comm{a_r^{\dagger}}{a_s^{\dagger}} & = & 0. \nonumber +\end{eqnarray} + +El operador de número del estado $r$, + +\begin{eqnarray} + N_r & \equiv & a_r^{\dagger}a_r, \label{operador.numero.r} \\ + \comm{N_{r}}{\opAdaga{s}} & = & \delta_{rs}\opAdaga{r} = \delta_{rs}\opAdaga{s}, \label{conmutador.operador.numero.r.aDaga}\\ + \comm{N_{r}}{\opA{s}} &=& -\delta_{rs}\opA{r} = -\delta_{rs}\opA{s}, \label{conmutador.operador.numero.r.a} +\end{eqnarray} + +cuenta el nivel (número de excitaciones o partÃculas) presentes en la configuración $r$. El operador $a_r^{\dagger}$ agrega una partÃcula de configuración $r$. Al contrario, el operador $a_r$ remueve una partÃcula. + +Existe un estado de vacÃo $\ket{0}$, + +\begin{equation}\label{aniquilacion.del.vacio} + a_r \ket{0} = 0;~~~\forall r. +\end{equation} + +\begin{itemize} + \item + El vacÃo es el estado sin excitaciones (partÃculas) (ecuación \ref{aniquilacion.del.vacio}) + \item + El vacÃo tiene la mÃnima energÃa del sistema: cero en la prescripción de orden normal. +\end{itemize} + +Como hemos dicho, los operadores $\opAdaga{r}$ permiten ir agregando grados de libertad de partÃcula libre, con números cuánticos definidos por la etiqueta $r$ (4-momento lineal y espÃn) al estado sobre el cual actúan. En particular, podemos contabilizar que el nuevo estado ha modificado su 4-momento de conforme lo establecido en la etiqueta $r$. + +Por ejemplo, si consideramos el siguiente estado (sin normalizar): + +\begin{equation} + \opAdaga{\vec{p}}\opAdaga{\vec{p}}\opAdaga{\vec{p}~'}\ket{0} \sim \ket{\vec{p},\vec{p},\vec{p}~'}, +\end{equation} +esperamos que represente un estado compuesto por 3 partÃculas libres. 2 de momento lineal $\vec{p}$ y una de momento lineal $\vec{p}~'$. Vamos a jurungar\footnote{tr. coloq. {\it R. Dom. y Ven.} hurgar \cite{asale_2021}} un poco este estado, para convencernos y entender mejor la idea. + +Aplicando el operador $N_{\vec{p}}=\opAdaga{\vec{p}}\opA{\vec{p}}$ nos queda: + +\begin{align*} + \renewcommand{\arraystretch}{1.75} + N_{\vec{p}}\ket{2\vec{p},\vec{p}~'} &\sim \opAdaga{\vec{p}}\opA{\vec{p}}~\opAdaga{\vec{p}}\opAdaga{\vec{p}}\opAdaga{\vec{p}~'}\ket{0} \\ + &= \opAdaga{\vec{p}}~(\opAdaga{\vec{p}}\opA{\vec{p}} + [\opA{\vec{p}}~,~\opAdaga{\vec{p}}])~\opAdaga{\vec{p}}\opAdaga{\vec{p}~'}\ket{0} & ([\opA{\vec{p}}~,~\opAdaga{\vec{p}}]=1)\\ + &= \opAdaga{\vec{p}}~(\opAdaga{\vec{p}}\opA{\vec{p}})~\opAdaga{\vec{p}}\opAdaga{\vec{p}~'}\ket{0} + \opAdaga{\vec{p}}\opAdaga{\vec{p}}\opAdaga{\vec{p}~'}\ket{0} \\ + &= \opAdaga{\vec{p}}\opAdaga{\vec{p}}~(\opAdaga{\vec{p}}\opA{\vec{p}} + [\opA{\vec{p}}~,~\opAdaga{\vec{p}}])~\opAdaga{\vec{p}~'}\ket{0} + \opAdaga{\vec{p}}\opAdaga{\vec{p}}\opAdaga{\vec{p}~'}\ket{0} \\ + &= \opAdaga{\vec{p}}\opAdaga{\vec{p}}~(\opAdaga{\vec{p}}\opA{\vec{p}})~\opAdaga{\vec{p}~'}\ket{0} + 2\opAdaga{\vec{p}}\opAdaga{\vec{p}}\opAdaga{\vec{p}~'}\ket{0} \\ + &= \opAdaga{\vec{p}}\opAdaga{\vec{p}}\opAdaga{\vec{p}}~(\opAdaga{\vec{p}~'}\opA{\vec{p}} + [\opA{\vec{p}}~,~\opAdaga{\vec{p}~'}])~\ket{0} + 2\opAdaga{\vec{p}}\opAdaga{\vec{p}}\opAdaga{\vec{p}~'}\ket{0} & ([\opA{\vec{p}}~,~\opAdaga{\vec{p}~'}]=0) \\ + &= \opAdaga{\vec{p}}\opAdaga{\vec{p}}\opAdaga{\vec{p}}~(\opAdaga{\vec{p}~'}\opA{\vec{p}})~\ket{0} + 2\opAdaga{\vec{p}}\opAdaga{\vec{p}}\opAdaga{\vec{p}~'}\ket{0} & (\opA{\vec{p}}~\ket{0}=0) \\ + &= 2\opAdaga{\vec{p}}\opAdaga{\vec{p}}\opAdaga{\vec{p}~'}\ket{0} \\ + &\sim 2\ket{2\vec{p},\vec{p}~'}. +\end{align*} + +El autovalor del operador número de partÃculas de momento lineal $\vec{p}$ es 2. El estado contiene dos partÃculas de momento lineal $\vec{p}$. + +Se puede obtener el mismo resultado usando las propiedades del operador de número (\ref{conmutador.operador.numero.r.aDaga} y \ref{conmutador.operador.numero.r.a}). Probando para $N_{\vec{p}~'}$, + +\begin{align*} + \renewcommand{\arraystretch}{1.75} + N_{\vec{p}~'}\ket{2\vec{p},\vec{p}~'} &\sim N_{\vec{p}~'}\opAdaga{\vec{p}}\opAdaga{\vec{p}}\opAdaga{\vec{p}~'}\ket{0} & (\comm{N_{\vec{p}~'}}{\opAdaga{\vec{p}}~} = 0) \\ + &= \opAdaga{\vec{p}}\opAdaga{\vec{p}}N_{\vec{p}~'}\opAdaga{\vec{p}~'}\ket{0} \\ + &= \opAdaga{\vec{p}}\opAdaga{\vec{p}}~(\opAdaga{\vec{p}~'}N_{\vec{p}~'} + \comm{N_{\vec{p}~'}}{\opAdaga{\vec{p}~'}})\ket{0} & (\comm{N_{\vec{p}~'}}{\opAdaga{\vec{p}~'}} = \opAdaga{\vec{p}~'}) \\ + &= \opAdaga{\vec{p}}\opAdaga{\vec{p}}\opAdaga{\vec{p}~'}N_{\vec{p}~'} \ket{0} + \opAdaga{\vec{p}}\opAdaga{\vec{p}}\opAdaga{\vec{p}~'}\ket{0} & (N_{\vec{p}~'} \ket{0} = 0) \\ + &= \opAdaga{\vec{p}}\opAdaga{\vec{p}}\opAdaga{\vec{p}~'}\ket{0} \\ + &\sim \ket{2\vec{p},\vec{p}~'}. + \end{align*} + +El autovalor del operador número de partÃculas de momento lineal $\vec{p}~'$ es 1. El estado contiene una partÃcula de momento lineal $\vec{p}~'$. + +\section{Sobre la representación de número para fermiones} +... + +\section{El Campo de Dirac} +En la construcción de los ingredientes de una teorÃa de campos, la simetrÃa de Poincaré juega un papel protagónico. La evidencia observacional y experimental accesible y que acumula nuestro conocimiento indica que esta simetrÃa está presente a un nivel fundamental de la naturaleza y permite definir el conjunto de observadores inerciales. + +Los campos son objetos que dan cuenta de los grados de libertad en cierto sistema fÃsico. En una teorÃa fÃsica exitosa, la forma general de estos objetos es independiente del observador que describe el sistema. Además, las transformaciones de simetrÃa que relacionan a dos observadores relacionan también los campos presentes en la descripción de cada observador, transformándolos de forma que se pueden verificar las observaciones en cada sistema. \ No newline at end of file diff --git a/ejemplo_clase/seccion_campo_escalar.tex b/ejemplo_clase/seccion_campo_escalar.tex new file mode 100644 index 0000000..47d2b76 --- /dev/null +++ b/ejemplo_clase/seccion_campo_escalar.tex @@ -0,0 +1,129 @@ +Consideramos el campo escalar en sus versiones real y complejo. Para comenzar, introducimos la acción del campo escalar real: +\begin{equation}\label{accion.campo.escalar.real} +\mathcal{S}=\int {\rm d}^4 x \mathcal{L}=-\frac{1}{2}\int {\rm d}^4 x \left(\partial_{\mu}\phi\partial^{\mu}\phi - m^2\phi^2\right). +\end{equation} + +El funcional $\mathcal{S}$ se hace extremo si el campo $\phi$ cumple la ecuación de Klein-Gordon, + +\begin{equation}\label{ecuacion.KG} +\left(\Box + m^2\right)\phi=0. +\end{equation} + +La ecuación (\ref{ecuacion.KG}) implica que el campo $\phi$ tiene soluciones de tipo onda plana, con una relación de dispersión de la forma $k^2=k^{\mu}k_{\mu}=m^2$. En la interpretación cuántica del campo de Klein-Gordon podremos decir que la onda plana de vector de onda $k_{\mu}$ corresponde al estado de una partÃcula de momentum lineal definido. + +Se puede hallar la solución más general de la ecuación (\ref{ecuacion.KG}), pasando al espacio de Fourier. Se puede escribir de la siguiente forma, + +\begin{equation}\label{solucion.KG.real.ondas.planas} +\phi(x)=\int {\rm d}^3k(a(\vec{k})f_{\vec{k}}(x)+\bar{a}(\vec{k})\bar{f}_{\vec{k}}(x)), +\end{equation} +donde $a(\vec{k})$ es una función arbitraria y se han hecho las siguientes definiciones +\begin{eqnarray} +f_{\vec{k}}(x) &=& \frac{1}{(2\pi)^{3/2}\sqrt{2\omega_{k}}}\left.e^{-ik_{\mu}x^{\mu}}\right|_{k_0=\omega_k}, \\ +\omega_k &=& +\sqrt{m^2+|\vec{k}|^2}. \label{omega.k.KG} +\end{eqnarray} + +La función $f_{\vec{k}}(x)$, con la definición de $\omega_k$ satisface la ecuación de Klein-Gordon. De igual forma lo hace la expresión (\ref{solucion.KG.real.ondas.planas}). Además, la normalización de las funciones $f_{\vec{k}}(x)$ garantiza que se cumplan las siguientes propiedades, +\begin{eqnarray} +(f_{\vec{k}},f_{\vec{k}'}) &=& \delta^3(\vec{k}-\vec{k}'), \\ +(\bar{f}_{\vec{k}},\bar{f}_{\vec{k}'}) &=& -\delta^3(\vec{k}-\vec{k}'), \\ +(f_{\vec{k}},\bar{f}_{\vec{k}'}) &=& 0, +\end{eqnarray} +donde se ha definido el paréntesis $(f_1,f_2)$ de la siguiente forma +\iffalse +\begin{equation} +(f_1,f_2)=i\int {\rm d}^3x\bar{f}_1{\genfrac{}{}{0pt}{0}{\leftrightarrow}{\partial_{0}}}f_2. +\end{equation} +\fi +\begin{eqnarray} +\renewcommand*{\arraystretch}{0.8} +(f_1,f_2) &=& i\int {\rm d}^3x\bar{f}_1{\begin{array}{c} +\leftrightarrow \\ +\partial_{0}\\ +\\ +\end{array} +}f_2\\ + &=& i\int {\rm d}^3 x (\bar{f}_1\partial_{0}f_2 - (\partial_{0}\bar{f}_1)f_2). +\end{eqnarray} + +Para construir soluciones particulares de la ecuación de Klein-Gordon se construye la función de Green, considerando la forma no-homogénea de la ecuación (\ref{ecuacion.KG}) + +\begin{equation}\label{ecuacion.KG.con.fuentes} +\left(\Box + m^2\right)\phi=\rho. +\end{equation} + +Se define $G(x-x')$ tal que + +\begin{eqnarray}\label{ecuacion.propagador.KG.real} +&& \left(\Box_x + m^2\right)G(x-x')=\delta^4(x-x'),\\ +&\Rightarrow & \phi(x) = \phi_H(x) + \int{\rm d}^4 x' G(x-x')\rho(x'), +\end{eqnarray} +donde $\phi_H(x)$ es solución de la ecuación homogénea (\ref{ecuacion.KG}). + +Como es acostumbrado se escribe la solución de la ecuación (\ref{ecuacion.propagador.KG.real}) en el espacio de Fourier, + +\begin{equation} +\label{propagador.KG.fourier} +G(x) = \frac{1}{(2\pi)^4}\int{\rm d}^4 k \left[\frac{-1}{k_0^2-\omega_k^2}\right]e^{-ikx}, +\end{equation} +donde $\omega_k$ tiene la misma definición que en (\ref{omega.k.KG}). + +A partir de (\ref{propagador.KG.fourier}) se definen los propagadores retardado y avanzado según se defina el contorno de la integral en $k_0$. Por ejemplo, añadiendo una componente infinitesimal imaginaria negativa a las singularidades $k_0=\pm\omega_k$ se definirá el propagador retardado (fig \ref{contornos.propagador.retardado}), + +\begin{eqnarray} +\label{propagador.KG.retardado} +\Delta_{ret}(x) &=& \lim_{\epsilon\rightarrow 0}\frac{1}{(2\pi)^4}\int{\rm d}^4 k \left[\frac{-1}{(k_0+i\epsilon)^2-\omega_k^2}\right]e^{-ikx} \nonumber \\ +&=& \frac{\Theta(x_0)}{(2\pi)^3}\int{\rm d}^3 k~\frac{\sin{\omega_k x_0}}{\omega_k}~e^{i\vec{k}\cdot\vec{x}}. +\end{eqnarray} + +Si en cambio la componente infinitesimal es positiva se define el propagador avanzado +\begin{equation} +\Delta_{av}(x) = \Delta_{ret}(-x). +\end{equation} + +XXXXX + + +%%%% +\input{propagador_retardado.tex} +%%%% + +Veamos las cantidades conservadas debido a la simetrÃa de Poincaré de la acción (\ref{accion.campo.escalar.real}). La corriente conservada dependerá del tensor de energÃa-impulso (\ref{J.noether.general}, \ref{t.mu.nu.general}), + +\begin{eqnarray} +J^{\mu} &=& -{T^{\mu}}_{\nu}\delta x^{\nu}, \\ +{T^{\mu}}_{\nu} &=& - \partial^{\mu}\phi\partial_{\nu}\phi - \delta^{\mu}_{\nu} \mathcal{L}. \label{t.mu.nu} +\end{eqnarray} + +El grupo de Poincaré tiene 10 dimensiones y sendas corrientes conservadas independientes. Estas corrientes quedan empaquetadas en un vector y un tensor antisimétrico de dos Ãndices. + +Las expresiones para las densidades de momentum lineal y angular del campo $\phi$ se obtienen definiendo $\delta x^{\nu}=\epsilon^{\nu}$ y $\delta x^{\nu}={\omega^{\nu}}_{\alpha}x^{\alpha}$ respectivamente. Los 4 parámetros $\epsilon$ corresponden a una traslación infinitesimal arbitraria y los $\omega$ cumplen $\omega_{\mu\nu}=-\omega_{\nu\mu}$, siendo los 6 parámetros independientes del grupo de Lorentz. + +\begin{eqnarray} +p_{\nu} &=& -{T^{0}}_{\nu}, \label{densidad.P.KG.real}\\ +m^{\alpha\nu} &=& x^{[\alpha}p^{\nu]}. \label{densidad.J.KG.real} +\end{eqnarray} + +Conviene identificar las propiedades del tensor ${T^{\mu}}_{\nu}$ y la conservación de la corriente de Noether. + +La conservación del momentum lineal está garantizada cuando se anula la divergencia de la corriente traslacional $J_{t}^{\mu}=-{T^{\mu}}_{\nu}\epsilon^{\nu}$. Por otro lado, la conservación del momento angular va emparejada con la anulación de la corriente rotacional $J_{r}^{\mu}=-{T^{\mu[\nu}}x^{\alpha]}\omega_{\nu\alpha}$. + +Tenemos + +\begin{eqnarray} +\partial_{\mu}J_{t}^{\mu}=0 &\Longrightarrow & \partial_{\mu}{T^{\mu}}_{\nu}=0, \nonumber \\ +\partial_{\mu}T^{\mu\nu} &=& -\left(\Box + m^2\right)\phi~\partial^{\nu}\phi. \\ +\partial_{\mu}J_{r}^{\mu}=0 &\Longrightarrow & \partial_{\mu}(T^{\mu[\nu}x^{\alpha]})=0, \nonumber \\ +\partial_{\mu}(T^{\mu[\nu}x^{\alpha]}) &=& x^{\alpha}\partial_{\mu}T^{\mu\nu} - x^{\nu}\partial_{\mu}T^{\mu\alpha} + 2~T^{[\alpha\nu]}. +\end{eqnarray} + +La conservación del momentum está garantizada por la ecuación de movimiento (\ref{ecuacion.KG}) y la conservación del momento angular requiere además la propiedad de simetrÃa del tensor de energÃa impulso que se deriva de la expresión (\ref{t.mu.nu}). + +Las componentes del vector densidad de momentum lineal (\ref{densidad.P.KG.real}) son las siguientes, + +\begin{eqnarray} +p_{0} &=& \frac{1}{2}\left((\Pi^{0})^2+|\vec{\nabla}\phi|^2+m^2\phi^2\right), \label{densidad.P0.KG.real}\\ +\vec{p} &=& \Pi^{0}~\vec{\nabla}\phi, \label{densidad.P.espacial.KG.real} +\end{eqnarray} +donde como es usual $\Pi^{i\mu}=\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\phi_{i}}$ (\ref{momento.lagrangiano.general}). Se evidencia que la parte temporal del vector densidad de momentum lineal es efectivamente la densidad hamiltoniana. + + -- GitLab