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\setcounter{finalframe}{\value{framenumber}} +} + +\newcommand{\backupend}{ + \setcounter{framenumber}{\value{finalframe}} +} + +\author{Física Nuclear y subnuclear } +\title{Interacciones y conservaciones} +%\setbeamercovered{transparent} +%\setbeamertemplate{navigation symbols}{} +%\logo{} +%\institute{} +%\date{} +%\subject{} +\begin{document} + +\begin{frame} +\titlepage +\end{frame} + +%\begin{frame}{Contenido} +% \tableofcontents +%\end{frame} + +\begin{frame}{Conservaciones} + Sistema cuántico descrito por $\hat{H}$ + \begin{equation} + -i\hbar \frac{d\Psi}{dt} = \hat{H}\Psi + \end{equation} + Relaciones de permutación para un operador $\hat{A}$ con observable $A$: + \begin{equation*} + [\mathbf{H},\mathbf{A}] = 0 \rightarrow \frac{d}{dt}\langle A \rangle = 0 + \end{equation*} + La carga + \begin{equation*} + \mathbf{Q}\Psi = q\Psi. + \end{equation*} + Invariancia de norma + \begin{equation*} + \Psi' = e^{i\epsilon Q}\Psi, + \end{equation*} +\end{frame} + +\begin{frame}{Isospín} + \begin{table}[ht!] + \begin{tabular}{|p{0.2\textwidth}|p{0.2\textwidth}|p{0.2\textwidth}|} + \hline + Partícula & $I$ & $I_3$ \\ + \hline + $p$ & $1/2$ & $1/2$ \\ + $n$ & $1/2$ & $-1/2$ \\ + \hline + $\pi^+$ & $1$ & $1$ \\ + $\pi^0$ & $1$ & $0$\\ + $\pi^-$ & $1$ & $-1$ \\ + \hline + $K^+$ & $1/2$ & $1/2$ \\ + $K^0$ & $1/2$ & $-1/2$ \\ + \hline + $\Sigma^+$ & $1$ & $1$ \\ + $\Sigma^0$ & $1$ & $0$ \\ + $\Sigma^-$ & $1$ & $-1$ \\ + \hline + \end{tabular} + \label{tab:lep} + \caption{Valores del número leptónico por familia para los leptones} + \end{table} + +\end{frame} + +\begin{frame}{Relación Gell-Mann-Nishima\footnote{Realmente se amplía a $Y=B-S-C-\hat{B}-T$, pero nos quedaremos con la extrañeza nada más.}} + \begin{align*} + Q &= I_3 + \frac{Y}{2} = I_3 + \frac{B-S}{2},\\ + I_3 &= \frac{1}{2}(N_u-N_d) + \end{align*} +\end{frame} + +\section{Resonancias en hadrones} +\begin{frame}{Resonancia $\Delta(1234)$} + \begin{figure}[ht!] + \begin{center} + \includegraphics[width=0.6\linewidth]{gaussianas.jpg} + \caption{Esquema de la sección eficaz de las colisiones $\pi-N$ a bajas energías. Imagen adaptada de: \href{http://www.flickr.com/photos/77004318@N00/91432761}{"case3b"} por \href{http://www.flickr.com/photos/77004318@N00}{Samuel Foucher} con licencia \href{https://creativecommons.org/licenses/by-sa/2.0/?ref=ccsearch&atype=rich}{CC BY-SA 2.0}} + \label{fig:gauss} + \end{center} + \end{figure} + +\end{frame} + +\begin{frame}{Vida media} + \begin{equation*} + \tau_{\Delta} \approx \frac{\hbar}{\Gamma_{\Delta}c^2}\approx \frac{6.6\times 10^{-22}MeV-sec}{100MeV} \approx 10^{-23} segundos + \end{equation*} +\end{frame} + +\begin{frame}{Resonancia $\rho^0$} + \begin{equation} + \pi^- + p \rightarrow \pi^+ + \pi^- + n + \label{ec:rho} + \end{equation} + + \begin{align*} + \pi^- + p &\rightarrow \rho^0 + n \\ + \text{después } \rho^0 &\rightarrow \pi^+ + \pi^- + \end{align*} + ?`Cómo harían el digrama de Feynmann? Los que estén desde video o en línea les toca dibujarlo en casa o imaginarse a partir de lo que escuchan. +\end{frame} + +\begin{frame}{Tiempo de vida media} + \begin{equation*} + \psi \propto e^{\frac{ic^2}{\hbar} (M_0-i\frac{\Gamma}{2})t}, t>0. + \end{equation*} + + \begin{equation*} + \tau=\frac{\hbar}{\Gamma c^2}. + \end{equation*} +\end{frame} + +\section*{Interacciones} +\begin{frame}{Interacciones electromagnéticas} + \begin{itemize} + \item Una interacción muy estudiada + \item Aproximaciones clásicas + \item Teoría de perturbaciones + \item ?`Si las energías son relativistas y el tratamiento cuántico? + \begin{itemize} + \item Electrodinámica cuántica + \item Radiación multipolar + \end{itemize} + \end{itemize} + +\end{frame} + + +\begin{frame}{Dispersión electromagnética de leptones} + \begin{itemize} + \item Dispersión de M\o{}ller + \begin{equation*} + e^- + e^- \rightarrow e^- + e^- + \end{equation*} + \item Dispersión de Bhabha + \begin{equation*} + e^- + e^+ \rightarrow e^- + e^+ + \end{equation*} + \end{itemize} + +\end{frame} + +\begin{frame}{Dispersión electromagnética de leptones} + \feynmandiagram [large, vertical=b to c] { + a -- [anti fermion, edge label'=\( e^- \)] b -- [ anti fermion, edge label'=\( e^- \)] j, + b -- [photon,edge label'=\(\gamma\)] c, + h -- [fermion, edge label'=\( e^+ \)] c -- [fermion, edge label'=\( e^+ \)] i; + }; + \feynmandiagram [horizontal=a to b] { + i1[particle=\( e^- \)] -- [fermion] a -- [fermion] i2[particle=\( e^+ \)], + a -- [photon, edge label'=\(\gamma\)] b, + f1[particle= \( e^- \)] -- [fermion] b -- [fermion] f2 [particle=\( e^+ \)], + }; + +\end{frame} + +\begin{frame}{Interacción fotón-hadrón y mesones mediadores} + ?`Un fotón puede decaer en un par hadrón anti-hadrón? + \begin{itemize} + \item $\rho^0$, $\omega^0$ y $\phi^0$ + \end{itemize} +\end{frame} + +\begin{frame}{Conservaciones y violaciones} + Conserva + \begin{itemize} + \item Extrañeza + \item Paridad + \item Conjugación + \end{itemize} + +\end{frame} + +\begin{frame}{Interacción débil} + \begin{itemize} + \item Electrodébil + \item Radiación nuclear: $\alpha$, $\beta$ y $\gamma$ + \item $\beta = e^{\pm}$ + \begin{itemize} + \item ?`Elctrones en el núcleo? + \item Espectro de energías continuo + \end{itemize} + \end{itemize} + +\end{frame} + +\begin{frame}{El rincón poético de Vladimir} + \centering + Neutrinos, they are very small.\\ + They have no charge and have no mass\\ + And do not interact at all.\\ + The earth is just a silly ball\\ + To them, through which they simply pass,\\ + Like dustmaids down a drafty hall\\ + Or photons through a sheet of glass.\\ + J. Updike\footnote{De \emph{Telephone Poles and Other Poems}, André Deutch, Londres (1964)} +\end{frame} + +\begin{frame}{Neutrinos} + \begin{itemize} + \item Interacción débil + \item Partículas neutras + \item Recuerden + \begin{align*} + n &\rightarrow p + e^- + \bar{\nu_e},\\ + p &\rightarrow n + e^+ + \nu_e. + \end{align*} + \begin{align*} + \nu_e + n &\rightarrow p + e^-,\\ + \bar{\nu_e} + p &\rightarrow n + e^+, + \end{align*} + \end{itemize} + +\end{frame} + + + +\begin{frame}{Corrientes neutras} + \begin{equation*} + \nu_{\mu} + e^- \rightarrow \nu_{\mu} + e^- + \end{equation*} + + \feynmandiagram [large, vertical=b to c] { + a -- [fermion, edge label'=\( \nu_{\mu} \)] b -- [fermion, edge label'=\( \nu_{\mu} \)] j, + b -- [scalar,edge label'=\(Z^0\)] c, + h -- [fermion, edge label'=\( e^- \)] c -- [fermion, edge label'=\( e^- \)] i; + }; + +\end{frame} + +\begin{frame}{Mezcla de neutrinos} + \begin{equation} + \begin{pmatrix} + \nu_e \\ + \nu_{\mu} \\ + \nu_{\tau} + \end{pmatrix} = + \begin{pmatrix} + V_{e1} & V_{e2} & V_{e3} \\ + V_{\mu 1} & V_{\mu 2} & V_{\mu 3} \\ + V_{\tau 1} & V_{\tau 2} & V_{\tau 3} + \end{pmatrix} + \begin{pmatrix} + \nu_1 \\ + \nu_2 \\ + \nu_3 + \end{pmatrix} + \end{equation} + Matriz Pontecorvo-Maki-Nakagawa-Sakata +\end{frame} + +\begin{frame}{Oscilaciones de neutrinos} + \begin{align*} + \nu_e =& cos\theta_{12} \nu_1 + sen\theta_{12} \nu_2 \\ + \nu_{\mu} =& -sen\theta_{12} \nu_1 + cos\theta_{12} \nu_2 + \end{align*} + + \begin{equation} + \ket{\nu_e(t)} = e^{-iE_1t/\hbar}cos\theta_{12} \nu_1 + e^{-iE_2t/\hbar}sen\theta_{12} \nu_2 + \end{equation} + + \begin{equation} + \mathbb{P}_{\nu_{\mu}}(t) = |\bra{\nu_{\mu}}\ket{\nu_e}(t)|^2 = sen^2 \theta_{12} sen^2\left[ \frac{1}{2} \frac{(E_1 - E_2)t}{\hbar}\right] + \end{equation} +\end{frame} + + +\begin{frame}{Procesos leptónicos} + \begin{equation*} + \mu^+ \rightarrow \bar{\nu_{\mu}} + e^+ + \nu_{e} + \end{equation*} + + \begin{equation*} + \nu_{\tau} + e^- \rightarrow \nu_{\tau} + e^- + \end{equation*} + +\end{frame} + +\begin{frame}{Procesos semileptónicos} + \begin{equation*} + \underset{\bar{u}d}{\pi^-} \rightarrow \underset{u\bar{u}}{\pi^0} + e^- + \bar{\nu_e} + \end{equation*} + + \feynmandiagram [horizontal=a to b] { + i1[particle=\( \bar{u} \)] -- [] a, + a -- [fermion] b, + b -- [] f2 [particle=\( \bar{u} \)], + }; + + + \feynmandiagram [layered layout, horizontal=a to b] { + a [particle=d] -- [fermion] b -- [fermion] f1 [particle=u], + b -- [scalar, edge label'=\(W^{-}\)] c, + c -- [anti fermion] f2 [particle=\(\overline \nu_e\)], + c -- [fermion] f3 [particle=\( e^- \)], + }; + + +\end{frame} + +\begin{frame}{Procesos semileptónicos} + \begin{equation*} + \nu_{\mu} + \underset{udd}{n} \rightarrow \mu^- + \underset{uud}{p} + \end{equation*} + + \begin{equation*} + \nu_{\mu} + p \rightarrow \nu_{\mu} + p + \end{equation*} + +\end{frame} + +\begin{frame}{Procesos hadrónicos} + \begin{multicols}{2} + \begin{align*} + K^+ &\rightarrow \pi^+ + \pi^0 \\ + &\rightarrow \pi^+ + \pi^+ + \pi^- \\ + &\rightarrow \pi^+ + \pi^0 + \pi^0. + \end{align*} + En ninguno cambia la extrañeza. + \begin{figure}[h!] + \feynmandiagram [horizontal=a to b] { + i1[particle=\( u \)] -- [] a, + a -- [fermion] b, + b -- [] f2 [particle=\( u \)], + }; + + + \feynmandiagram [layered layout, horizontal=a to b] { + a [particle=\(\bar{s}\)] -- [anti fermion] b -- [anti fermion] f1 [particle=\(\bar{u}\)], + b -- [scalar, edge label'=\(W^{+}\)] c, + c -- [fermion] f2 [particle=\(u\)], + c -- [anti fermion] f3 [particle=\( \bar{d} \)], + }; + \end{figure} + \end{multicols} +\end{frame} + + + +\begin{frame}{Mamá yo quiero saber de dónde son los muones} + Fuente de muones + \begin{equation} + \pi^+ \rightarrow \mu^+ + \nu_{\mu} + \end{equation} + \begin{itemize} + \item Conservación momento + \item conservación momento angular + \end{itemize} + +\end{frame} + +\begin{frame}{Paridad} + 1956 Lee y Yang: en interacciones débiles no hay evidencia de que se conserve la paridad + + Vectores polares + \begin{align*} + \mathbf{P}(\overrightarrow{r}) &= -\overrightarrow{r} \\ + \mathbf{P}(\overrightarrow{p}) &= -\overrightarrow{p} + \end{align*} + + Vectores axiales + \begin{equation*} + \mathbf{P}(\vec{u}\times \vec{v}) = \vec{u}\times \vec{v}. + \end{equation*} + + \begin{equation*} + \mathbf{P}(\overrightarrow{L}) = \mathbf{P}(\overrightarrow{r}) \times \mathbf{P}(\overrightarrow{p}) = (-\overrightarrow{r}) \times (-\overrightarrow{p})= (\overrightarrow{r}) \times (\overrightarrow{p}) = \overrightarrow{L} + \end{equation*} +\end{frame} + +\begin{frame} + \begin{align*} + \mathbf{P}\square &= +\square \text{ paridad positiva o par} \\ + \mathbf{P}\square &= -\square \text{ paridad negativa o impar} + \end{align*} + + \begin{equation*} + \mathbf{P}\Psi(x) = \Psi(-x) + \end{equation*} + +\end{frame} + +\begin{frame}{Paridad invariante, $[\hat{H},\hat{P}]=0$} + De ser distintas funciones de onda $\Psi(x)$ y $\hat{P}\Psi(x)$ el estado estaría degenreado, la opción: + + \begin{equation*} + \mathbf{\Psi(x)} = \eta_P\Psi(x),\ \eta_p=\pm 1 + \end{equation*} + + \begin{equation*} + \mathbf{P}\ket{\text{estado inicial}} = \mathbf{P}(\ket{a})\mathbf{P}(\ket{b})\mathbf{P}(\ket{\text{movimiento relativo}}) + \label{ec:paridad} + \end{equation*} + + \begin{align*} + \eta_p(\text{estado incial}) &= \eta_p(a) \eta_p(b)\eta_p(\text{movimieno relativo}) \notag \\ + \text{función de onda } \eta_p(\text{estado incial}) &= \eta_p(a) \eta_p(b)(-1)^{\ell} + \label{ec:parorb} + \end{align*} + +\end{frame} + +\begin{frame}{Determinando paridades} + Fijamos $\eta(p)=+1$ + + \begin{equation*} + d+\pi^- \rightarrow n + n + \end{equation*} + Usamos + \begin{equation*} + \eta_p(d) \eta_p(\pi^-)(-1)^{\ell} = \eta_p(n) \eta_p(n)(-1)^{\ell'} + \end{equation*} + Deuterón en el estado base, $\ell=0$, al atrapar al pión, $\ell=0$ + \begin{equation*} + \eta_p(p) \eta_p(n) \eta_p(\pi^-) = -1 + \end{equation*} +\end{frame} + +\begin{frame}{Violaciones de conservación de la paridad} + \begin{itemize} + \item 1924 Laporte propone que hay dos diferentes clases de niveles para los átomos + \item Wigner asocio estas clases son producto de la invariancia respecto a la reflexión espacial + \item Se volvió un dogma, que en 1956 Lee y Yang derribaron + \item Wu descubre la violación de la paridad en decaimientos $\beta$ + \end{itemize} + \begin{align*} + \vec{r} \overset{\hat{P}}{\rightarrow}& -\vec{r} \\ + \vec{J} \overset{\hat{P}}{\rightarrow}& \vec{J} + \end{align*} +\end{frame} + +\begin{frame}{Los neutrinos zurdos} + \begin{figure} + \tikz [ultra thick] + {\draw [red, arrows = {-Stealth[red]}] (0,1) -- node[above=1mm] {$\vec{J}$} (1,1); + \draw [black, arrows = {-Stealth[black]}] (0,1) -- node[above=1mm] {$\vec{p}$} (1.5,1); + \draw [red, arrows = {-Stealth[red]}] (0,0.5) -- node[below=1mm] {$\hat{P}(\vec{J})$} (1,0.5); + \draw [black, arrows = {-Stealth[black]}] (0,0.5) -- node[below=1mm] {$\hat{P}(\vec{p})$} (-1.5,0.5);} + \end{figure} +\end{frame} + +\begin{frame}{Combinación de estados} + \begin{align*} + \ket{\alpha} =& c\ket{par} + d\ket{impar}, \ |c|^2 + |d|^2 =1\\ + \hat{P}\ket{\alpha} =& c\hat{P}\ket{par} + d\hat{P}\ket{impar} \neq \eta_p \ket{\alpha} + \end{align*} +\end{frame} + +\begin{frame}{Conjugación de carga} + \begin{align*} + \mathbf{C} \ket{q_{gen}} &= \ket{-q_{gen}}, \\ + \mathbf{C}^2 &= \mathbf{I} \\ + [Q,C] &\neq 0 + \end{align*} + Pareciera que sólo en partículas neutras, pero tampoco en neutrinos no. +\end{frame} + +\begin{frame}{Inversión del tiempo} + \begin{align*} + t\overset{\mathbf{T}}{\rightarrow}& -t \\ + \vec{x}\overset{\mathbf{T}}{\rightarrow}& \vec{x} \\ + \vec{p}\overset{\mathbf{T}}{\rightarrow}& -\vec{p} \\ + \vec{J}\overset{\mathbf{T}}{\rightarrow}& -\vec{J} +\end{align*} +\end{frame} + +\begin{frame}{Interacción fuerte} + \begin{itemize} + \item Similar a la \emph{QED}, ahora tenemos \emph{QCD} + \item Tres carga: $r$, $g$ y $b$ + \item Gluón carga bicolor $r\bar{g}$, \emph{QCD} es no abeliana + \end{itemize} +\end{frame} + + +\begin{frame}{Bariones pesados} + \begin{itemize} + \item $\Delta^{++}$ + \item $\Omega^-$ + \end{itemize} +\end{frame} + +\begin{frame}{Invariancia de norma} + \begin{itemize} + \item Todas las fuerzas pueden expresarse como teorías de norma + \item Son invariantes ante la transformación de norma + \item La conservación de carga (conservación aditiva) es invariante ante una transformación global + \item AL agregar la dependencia para una carga no estática se mantiene la invariancia incluso en transformación local. + \end{itemize} +\end{frame} + +\begin{frame}{Invariancia de norma - Grados de libertad} + \begin{itemize} + \item Libertad parcial de elegir el potencial electromagnético + \item Parecía una teoría con cabos sueltos, o sólo una peculiaridad matemática + \item La invariancia de norma dicta la forma d ela interacción y los campos vectoriales sin masa. + \end{itemize} +\end{frame} + +\begin{frame}{Potenciales vectoriales y normas covariantes} + $(A_0,\mathbf{A})$ + \begin{align} + D_{\mu}=& (D_0,\mathbf{D})\\ + D_0 =& \frac{1}{c}\frac{\partial}{\partial t} + \frac{iqA_0}{\hbar c}\\ + \mathbf{D} =& \nabla - \frac{iq\mathbf{A}}{\hbar c} + \end{align} +\end{frame} + +\begin{frame}{Movimiento de los campos vectoriales} + \begin{align*} + \frac{1}{c^2}\frac{\partial^2 A_0}{\partial t^2} -\nabla A_0 =& \rho = \psi^* q \psi \\ + \frac{1}{c^2}\frac{\partial^2 A_i}{\partial t^2} -\nabla A_i =& \frac{j_i}{c} = \psi^* \frac{q\vec{v}_i}{c} \psi + \end{align*} + Con la condición para su invariancia de norma: + \begin{equation} + \frac{1}{c^2}\frac{\partial^2\epsilon(\vec{x},t)}{\partial t^2} - \nabla^2 \epsilon(\vec{x},t) = 0 + \end{equation} +\end{frame} + +\begin{frame}{Bosones sin masa} + \begin{itemize} + \item Aparece la necesidad de que los bosones de norma no tengan masa + \item La invariancia local se extiende a la global + \item La fase en una función de onda es arbitraria + \item Pero siempre debe ser la misma fase en todos los puntos del espacio tiempo. + \end{itemize} +\end{frame} + +\begin{frame}{Sentido físico de los potenciales vectoriales} + \begin{itemize} + \item Aparece en la teoría cuántica + \end{itemize} + + En ausencia de campo electromagnético, la ecuación estacionaria de Schrödinger: + \begin{equation} + -\frac{\hbar^2}{2m}\nabla^2 \psi_0 = E\psi_0, + \end{equation} +\end{frame} + +\begin{frame}{Ecuación de Schrödinger con campo vectorial electromagnético estático} + \begin{equation} + -\frac{\hbar^2}{2m}\mathbf{D}^2\psi= -\frac{\hbar^2}{2m} \left( \nabla + \frac{ie\mathbf{A}(\vec{x})}{\hbar c}\right)^2 \psi + \end{equation} + \begin{figure}[ht!] + \begin{center} + \includegraphics[width=0.5\linewidth]{bohm_aharanov.jpg} + \caption{Imagen del experomento de Aharanov-Bohm. Imagen con licencia CC-BY-SA toma de wikipedia} + \label{fig:bohm} + \end{center} + \end{figure} +\end{frame} + +\begin{frame}{Invariancia de norma y teorías no abelianas} + \begin{itemize} + \item La invariancia pide campos vectoriales sin masa + \item Bosones de interacción débil con masa y cargados + \item Bosones de interacción fuerte sin masa, pero cargados + \item Problemas por ejemplo en teoría de perturbaciones + \end{itemize} +\end{frame} + +\begin{frame}{Mecanismo de Higgs} + \begin{itemize} + \item Rompimiento de simetría aproximada + \item Rompimiento de simetría espontáneo + \item Simetrías escondidas por ejemplo en los ferromagnetos + \end{itemize} +\end{frame} + +\begin{frame}{Bosón de Higgs} + \begin{itemize} + \item Campo escalar complejo invariante de norma, $\phi$ y $\phi^*$ + \item Representan mesones escalares $H^+$ y $H^-$ + \begin{equation*} + \phi = \frac{1}{\sqrt{2}}(\phi_1 + i \phi_2) + \end{equation*} + \item Obedecen a la ecuación de Klein-Gordon + \end{itemize} +\end{frame} + +\end{document} diff --git a/pres4.tex~ b/pres4.tex~ new file mode 100644 index 0000000..cfe1d20 --- /dev/null +++ b/pres4.tex~ @@ -0,0 +1,611 @@ +\documentclass[12pt]{beamer} +\usetheme{Berlin} +\usepackage[utf8]{inputenc} +\usepackage[spanish]{babel} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{graphicx} +\usepackage{multimedia} +\usepackage{braket} +\usepackage{multicol} +\usepackage{tikz} +\usepackage{tikz-feynman}[compat=1.1.0] +\usetikzlibrary {arrows.meta} + +\usepackage{appendixnumberbeamer} + +%\setbeamerfont{page number in head}{size=\large} +%\setbeamertemplate{footline}{Diapositiva} +\setbeamertemplate{footline}[frame number] + +\newcommand{\backupbegin}{ + \newcounter{finalframe} + \setcounter{finalframe}{\value{framenumber}} +} + +\newcommand{\backupend}{ + \setcounter{framenumber}{\value{finalframe}} +} + +\author{Física Nuclear y subnuclear } +\title{Interacciones y conservaciones} +%\setbeamercovered{transparent} +%\setbeamertemplate{navigation symbols}{} +%\logo{} +%\institute{} +%\date{} +%\subject{} +\begin{document} + +\begin{frame} +\titlepage +\end{frame} + +%\begin{frame}{Contenido} +% \tableofcontents +%\end{frame} + +\begin{frame}{Conservaciones} + Sistema cuántico descrito por $\hat{H}$ + \begin{equation} + -i\hbar \frac{d\Psi}{dt} = \hat{H}\Psi + \end{equation} + Relaciones de permutación para un operador $\hat{A}$ con observable $A$: + \begin{equation*} + [\mathbf{H},\mathbf{A}] = 0 \rightarrow \frac{d}{dt}\langle A \rangle = 0 + \end{equation*} + La carga + \begin{equation*} + \mathbf{Q}\Psi = q\Psi. + \end{equation*} + Invariancia de norma + \begin{equation*} + \Psi' = e^{i\epsilon Q}\Psi, + \end{equation*} +\end{frame} + +\begin{frame}{Isospín} + \begin{table}[ht!] + \begin{tabular}{|p{0.2\textwidth}|p{0.2\textwidth}|p{0.2\textwidth}|} + \hline + Partícula & $I$ & $I_3$ \\ + \hline + $p$ & $1/2$ & $1/2$ \\ + $n$ & $1/2$ & $-1/2$ \\ + \hline + $\pi^+$ & $1$ & $1$ \\ + $\pi^0$ & $1$ & $0$\\ + $\pi^-$ & $1$ & $-1$ \\ + \hline + $K^+$ & $1/2$ & $1/2$ \\ + $K^0$ & $1/2$ & $-1/2$ \\ + \hline + $\Sigma^+$ & $1$ & $1$ \\ + $\Sigma^0$ & $1$ & $0$ \\ + $\Sigma^-$ & $1$ & $-1$ \\ + \hline + \end{tabular} + \label{tab:lep} + \caption{Valores del número leptónico por familia para los leptones} + \end{table} + +\end{frame} + +\begin{frame}{Relación Gell-Mann-Nishima\footnote{Realmente se amplía a $Y=B-S-C-\hat{B}-T$, pero nos quedaremos con la extrañeza nada más.}} + \begin{align*} + Q &= I_3 + \frac{Y}{2} = I_3 + \frac{B-S}{2},\\ + I_3 &= \frac{1}{2}(N_u-N_d) + \end{align*} +\end{frame} + +\section{Resonancias en hadrones} +\begin{frame}{Resonancia $\Delta(1234)$} + \begin{figure}[ht!] + \begin{center} + \includegraphics[width=0.6\linewidth]{gaussianas.jpg} + \caption{Esquema de la sección eficaz de las colisiones $\pi-N$ a bajas energías. Imagen adaptada de: \href{http://www.flickr.com/photos/77004318@N00/91432761}{"case3b"} por \href{http://www.flickr.com/photos/77004318@N00}{Samuel Foucher} con licencia \href{https://creativecommons.org/licenses/by-sa/2.0/?ref=ccsearch&atype=rich}{CC BY-SA 2.0}} + \label{fig:gauss} + \end{center} + \end{figure} + +\end{frame} + +\begin{frame}{Vida media} + \begin{equation*} + \tau_{\Delta} \approx \frac{\hbar}{\Gamma_{\Delta}c^2}\approx \frac{6.6\times 10^{-22}MeV-sec}{100MeV} \approx 10^{-23} segundos + \end{equation*} +\end{frame} + +\begin{frame}{Resonancia $\rho^0$} + \begin{equation} + \pi^- + p \rightarrow \pi^+ + \pi^- + n + \label{ec:rho} + \end{equation} + + \begin{align*} + \pi^- + p &\rightarrow \rho^0 + n \\ + \text{después } \rho^0 &\rightarrow \pi^+ + \pi^- + \end{align*} + ?`Cómo harían el digrama de Feynmann? Los que estén desde video o en línea les toca dibujarlo en casa o imaginarse a partir de lo que escuchan. +\end{frame} + +\begin{frame}{Tiempo de vida media} + \begin{equation*} + \psi \propto e^{\frac{ic^2}{\hbar} (M_0-i\frac{\Gamma}{2})t}, t>0. + \end{equation*} + + \begin{equation*} + \tau=\frac{\hbar}{\Gamma c^2}. + \end{equation*} +\end{frame} + +\section*{Interacciones} +\begin{frame}{Interacciones electromagnéticas} + \begin{itemize} + \item Una interacción muy estudiada + \item Aproximaciones clásicas + \item Teoría de perturbaciones + \item ?`Si las energías son relativistas y el tratamiento cuántico? + \begin{itemize} + \item Electrodinámica cuántica + \item Radiación multipolar + \end{itemize} + \end{itemize} + +\end{frame} + + +\begin{frame}{Dispersión electromagnética de leptones} + \begin{itemize} + \item Dispersión de M\o{}ller + \begin{equation*} + e^- + e^- \rightarrow e^- + e^- + \end{equation*} + \item Dispersión de Bhabha + \begin{equation*} + e^- + e^+ \rightarrow e^- + e^+ + \end{equation*} + \end{itemize} + +\end{frame} + +\begin{frame}{Dispersión electromagnética de leptones} + \feynmandiagram [large, vertical=b to c] { + a -- [anti fermion, edge label'=\( e^- \)] b -- [ anti fermion, edge label'=\( e^- \)] j, + b -- [photon,edge label'=\(\gamma\)] c, + h -- [fermion, edge label'=\( e^+ \)] c -- [fermion, edge label'=\( e^+ \)] i; + }; + \feynmandiagram [horizontal=a to b] { + i1[particle=\( e^- \)] -- [fermion] a -- [fermion] i2[particle=\( e^+ \)], + a -- [photon, edge label'=\(\gamma\)] b, + f1[particle= \( e^- \)] -- [fermion] b -- [fermion] f2 [particle=\( e^+ \)], + }; + +\end{frame} + +\begin{frame}{Interacción fotón-hadrón y mesones mediadores} + ?`Un fotón puede decaer en un par hadrón anti-hadrón? + \begin{itemize} + \item $\rho^0$, $\omega^0$ y $\phi^0$ + \end{itemize} +\end{frame} + +\begin{frame}{Conservaciones y violaciones} + Conserva + \begin{itemize} + \item Extrañeza + \item Paridad + \item Conjugación + \end{itemize} + +\end{frame} + +\begin{frame}{Interacción débil} + \begin{itemize} + \item Electrodébil + \item Radiación nuclear: $\alpha$, $\beta$ y $\gamma$ + \item $\beta = e^{\pm}$ + \begin{itemize} + \item ?`Elctrones en el núcleo? + \item Espectro de energías continuo + \end{itemize} + \end{itemize} + +\end{frame} + +\begin{frame}{El rincón poético de Vladimir} + \centering + Neutrinos, they are very small.\\ + They have no charge and have no mass\\ + And do not interact at all.\\ + The earth is just a silly ball\\ + To them, through which they simply pass,\\ + Like dustmaids down a drafty hall\\ + Or photons through a sheet of glass.\\ + J. Updike\footnote{De \emph{Telephone Poles and Other Poems}, André Deutch, Londres (1964)} +\end{frame} + +\begin{frame}{Neutrinos} + \begin{itemize} + \item Interacción débil + \item Partículas neutras + \item Recuerden + \begin{align*} + n &\rightarrow p + e^- + \bar{\nu_e}, + p &\rightarrow n + e^+ + \nu_e. + \end{align*} + \begin{align*} + \nu_e + n &\rightarrow p + e^-, + \bar{\nu_e} + p &\rightarrow n + e^+, + \end{align*} + \end{itemize} + +\end{frame} + + + +\begin{frame}{Corrientes neutras} + \begin{equation*} + \nu_{\mu} + e^- \rightarrow \nu_{\mu} + e^- + \end{equation*} + + \feynmandiagram [large, vertical=b to c] { + a -- [fermion, edge label'=\( \nu_{\mu} \)] b -- [fermion, edge label'=\( \nu_{\mu} \)] j, + b -- [scalar,edge label'=\(Z^0\)] c, + h -- [fermion, edge label'=\( e^- \)] c -- [fermion, edge label'=\( e^- \)] i; + }; + +\end{frame} + +\begin{frame}{Mezcla de neutrinos} + \begin{equation} + \begin{pmatrix} + \nu_e \\ + \nu_{\mu} \\ + \nu_{\tau} + \end{pmatrix} = + \begin{pmatrix} + V_{e1} & V_{e2} & V_{e3} \\ + V_{\mu 1} & V_{\mu 2} & V_{\mu 3} \\ + V_{\tau 1} & V_{\tau 2} & V_{\tau 3} + \end{pmatrix} + \begin{pmatrix} + \nu_1 \\ + \nu_2 \\ + \nu_3 + \end{pmatrix} + \end{equation} + Matriz Pontecorvo-Maki-Nakagawa-Sakata +\end{frame} + +\begin{frame}{Oscilaciones de neutrinos} + \begin{align*} + \nu_e =& cos\theta_{12} \nu_1 + sen\theta_{12} \nu_2 \\ + \nu_{\mu} =& -sen\theta_{12} \nu_1 + cos\theta_{12} \nu_2 + \end{align*} + + \begin{equation} + \ket{\nu_e(t)} = e^{-iE_1t/\hbar}cos\theta_{12} \nu_1 + e^{-iE_2t/\hbar}sen\theta_{12} \nu_2 + \end{equation} + + \begin{equation} + \mathbb{P}_{\nu_{\mu}}(t) = |\bra{\nu_{\mu}}\ket{\nu_e}(t)|^2 = sen^2 \theta_{12} sen^2\left[ \frac{1}{2} \frac{(E_1 - E_2)t}{\hbar}\right] + \end{equation} +\end{frame} + + +\begin{frame}{Procesos leptónicos} + \begin{equation*} + \mu^+ \rightarrow \bar{\nu_{\mu}} + e^+ + \nu_{e} + \end{equation*} + + \begin{equation*} + \nu_{\tau} + e^- \rightarrow \nu_{\tau} + e^- + \end{equation*} + +\end{frame} + +\begin{frame}{Procesos semileptónicos} + \begin{equation*} + \underset{\bar{u}d}{\pi^-} \rightarrow \underset{u\bar{u}}{\pi^0} + e^- + \bar{\nu_e} + \end{equation*} + + \feynmandiagram [horizontal=a to b] { + i1[particle=\( \bar{u} \)] -- [] a, + a -- [fermion] b, + b -- [] f2 [particle=\( \bar{u} \)], + }; + + + \feynmandiagram [layered layout, horizontal=a to b] { + a [particle=d] -- [fermion] b -- [fermion] f1 [particle=u], + b -- [scalar, edge label'=\(W^{-}\)] c, + c -- [anti fermion] f2 [particle=\(\overline \nu_e\)], + c -- [fermion] f3 [particle=\( e^- \)], + }; + + +\end{frame} + +\begin{frame}{Procesos semileptónicos} + \begin{equation*} + \nu_{\mu} + \underset{udd}{n} \rightarrow \mu^- + \underset{uud}{p} + \end{equation*} + + \begin{equation*} + \nu_{\mu} + p \rightarrow \nu_{\mu} + p + \end{equation*} + +\end{frame} + +\begin{frame}{Procesos hadrónicos} + \begin{multicols}{2} + \begin{align*} + K^+ &\rightarrow \pi^+ + \pi^0 \\ + &\rightarrow \pi^+ + \pi^+ + \pi^- \\ + &\rightarrow \pi^+ + \pi^0 + \pi^0. + \end{align*} + En ninguno cambia la extrañeza. + \begin{figure}[h!] + \feynmandiagram [horizontal=a to b] { + i1[particle=\( u \)] -- [] a, + a -- [fermion] b, + b -- [] f2 [particle=\( u \)], + }; + + + \feynmandiagram [layered layout, horizontal=a to b] { + a [particle=\(\bar{s}\)] -- [anti fermion] b -- [anti fermion] f1 [particle=\(\bar{u}\)], + b -- [scalar, edge label'=\(W^{+}\)] c, + c -- [fermion] f2 [particle=\(u\)], + c -- [anti fermion] f3 [particle=\( \bar{d} \)], + }; + \end{figure} + \end{multicols} +\end{frame} + + + +\begin{frame}{Mamá yo quiero saber de dónde son los muones} + Fuente de muones + \begin{equation} + \pi^+ \rightarrow \mu^+ + \nu_{\mu} + \end{equation} + \begin{itemize} + \item Conservación momento + \item conservación momento angular + \end{itemize} + +\end{frame} + +\begin{frame}{Paridad} + 1956 Lee y Yang: en interacciones débiles no hay evidencia de que se conserve la paridad + + Vectores polares + \begin{align*} + \mathbf{P}(\overrightarrow{r}) &= -\overrightarrow{r} \\ + \mathbf{P}(\overrightarrow{p}) &= -\overrightarrow{p} + \end{align*} + + Vectores axiales + \begin{equation*} + \mathbf{P}(\vec{u}\times \vec{v}) = \vec{u}\times \vec{v}. + \end{equation*} + + \begin{equation*} + \mathbf{P}(\overrightarrow{L}) = \mathbf{P}(\overrightarrow{r}) \times \mathbf{P}(\overrightarrow{p}) = (-\overrightarrow{r}) \times (-\overrightarrow{p})= (\overrightarrow{r}) \times (\overrightarrow{p}) = \overrightarrow{L} + \end{equation*} +\end{frame} + +\begin{frame} + \begin{align*} + \mathbf{P}\square &= +\square \text{ paridad positiva o par} \\ + \mathbf{P}\square &= -\square \text{ paridad negativa o impar} + \end{align*} + + \begin{equation*} + \mathbf{P}\Psi(x) = \Psi(-x) + \end{equation*} + +\end{frame} + +\begin{frame}{Paridad invariante, $[\hat{H},\hat{P}]=0$} + De ser distintas funciones de onda $\Psi(x)$ y $\hat{P}\Psi(x)$ el estado estaría degenreado, la opción: + + \begin{equation*} + \mathbf{\Psi(x)} = \eta_P\Psi(x),\ \eta_p=\pm 1 + \end{equation*} + + \begin{equation*} + \mathbf{P}\ket{\text{estado inicial}} = \mathbf{P}(\ket{a})\mathbf{P}(\ket{b})\mathbf{P}(\ket{\text{movimiento relativo}}) + \label{ec:paridad} + \end{equation*} + + \begin{align*} + \eta_p(\text{estado incial}) &= \eta_p(a) \eta_p(b)\eta_p(\text{movimieno relativo}) \notag \\ + \text{función de onda } \eta_p(\text{estado incial}) &= \eta_p(a) \eta_p(b)(-1)^{\ell} + \label{ec:parorb} + \end{align*} + +\end{frame} + +\begin{frame}{Determinando paridades} + Fijamos $\eta(p)=+1$ + + \begin{equation*} + d+\pi^- \rightarrow n + n + \end{equation*} + Usamos + \begin{equation*} + \eta_p(d) \eta_p(\pi^-)(-1)^{\ell} = \eta_p(n) \eta_p(n)(-1)^{\ell'} + \end{equation*} + Deuterón en el estado base, $\ell=0$, al atrapar al pión, $\ell=0$ + \begin{equation*} + \eta_p(p) \eta_p(n) \eta_p(\pi^-) = -1 + \end{equation*} +\end{frame} + +\begin{frame}{Violaciones de conservación de la paridad} + \begin{itemize} + \item 1924 Laporte propone que hay dos diferentes clases de niveles para los átomos + \item Wigner asocio estas clases son producto de la invariancia respecto a la reflexión espacial + \item Se volvió un dogma, que en 1956 Lee y Yang derribaron + \item Wu descubre la violación de la paridad en decaimientos $\beta$ + \end{itemize} + \begin{align*} + \vec{r} \overset{\hat{P}}{\rightarrow}& -\vec{r} \\ + \vec{J} \overset{\hat{P}}{\rightarrow}& \vec{J} + \end{align*} +\end{frame} + +\begin{frame}{Los neutrinos zurdos} + \begin{figure} + \tikz [ultra thick] + {\draw [red, arrows = {-Stealth[red]}] (0,1) -- node[above=1mm] {$\vec{J}$} (1,1); + \draw [black, arrows = {-Stealth[black]}] (0,1) -- node[above=1mm] {$\vec{p}$} (1.5,1); + \draw [red, arrows = {-Stealth[red]}] (0,0.5) -- node[below=1mm] {$\hat{P}(\vec{J})$} (1,0.5); + \draw [black, arrows = {-Stealth[black]}] (0,0.5) -- node[below=1mm] {$\hat{P}(\vec{p})$} (-1.5,0.5);} + \end{figure} +\end{frame} + +\begin{frame}{Combinación de estados} + \begin{align*} + \ket{\alpha} =& c\ket{par} + d\ket{impar}, \ |c|^2 + |d|^2 =1\\ + \hat{P}\ket{\alpha} =& c\hat{P}\ket{par} + d\hat{P}\ket{impar} \neq \eta_p \ket{\alpha} + \end{align*} +\end{frame} + +\begin{frame}{Conjugación de carga} + \begin{align*} + \mathbf{C} \ket{q_{gen}} &= \ket{-q_{gen}}, \\ + \mathbf{C}^2 &= \mathbf{I} \\ + [Q,C] &\neq 0 + \end{align*} + Pareciera que sólo en partículas neutras, pero tampoco en neutrinos no. +\end{frame} + +\begin{frame}{Inversión del tiempo} + \begin{align*} + t\overset{\mathbf{T}}{\rightarrow}& -t \\ + \vec{x}\overset{\mathbf{T}}{\rightarrow}& \vec{x} \\ + \vec{p}\overset{\mathbf{T}}{\rightarrow}& -\vec{p} \\ + \vec{J}\overset{\mathbf{T}}{\rightarrow}& -\vec{J} +\end{align*} +\end{frame} + +\begin{frame}{Interacción fuerte} + \begin{itemize} + \item Similar a la \emph{QED}, ahora tenemos \emph{QCD} + \item Tres carga: $r$, $g$ y $b$ + \item Gluón carga bicolor $r\bar{g}$, \emph{QCD} es no abeliana + \end{itemize} +\end{frame} + + +\begin{frame}{Bariones pesados} + \begin{itemize} + \item $\Delta^{++}$ + \item $\Omega^-$ + \end{itemize} +\end{frame} + +\begin{frame}{Invariancia de norma} + \begin{itemize} + \item Todas las fuerzas pueden expresarse como teorías de norma + \item Son invariantes ante la transformación de norma + \item La conservación de carga (conservación aditiva) es invariante ante una transformación global + \item AL agregar la dependencia para una carga no estática se mantiene la invariancia incluso en transformación local. + \end{itemize} +\end{frame} + +\begin{frame}{Invariancia de norma - Grados de libertad} + \begin{itemize} + \item Libertad parcial de elegir el potencial electromagnético + \item Parecía una teoría con cabos sueltos, o sólo una peculiaridad matemática + \item La invariancia de norma dicta la forma d ela interacción y los campos vectoriales sin masa. + \end{itemize} +\end{frame} + +\begin{frame}{Potenciales vectoriales y normas covariantes} + $(A_0,\mathbf{A})$ + \begin{align} + D_{\mu}=& (D_0,\mathbf{D})\\ + D_0 =& \frac{1}{c}\frac{\partial}{\partial t} + \frac{iqA_0}{\hbar c}\\ + \mathbf{D} =& \nabla - \frac{iq\mathbf{A}}{\hbar c} + \end{align} +\end{frame} + +\begin{frame}{Movimiento de los campos vectoriales} + \begin{align*} + \frac{1}{c^2}\frac{\partial^2 A_0}{\partial t^2} -\nabla A_0 =& \rho = \psi^* q \psi \\ + \frac{1}{c^2}\frac{\partial^2 A_i}{\partial t^2} -\nabla A_i =& \frac{j_i}{c} = \psi^* \frac{q\vec{v}_i}{c} \psi + \end{align*} + Con la condición para su invariancia de norma: + \begin{equation} + \frac{1}{c^2}\frac{\partial^2\epsilon(\vec{x},t)}{\partial t^2} - \nabla^2 \epsilon(\vec{x},t) = 0 + \end{equation} +\end{frame} + +\begin{frame}{Bosones sin masa} + \begin{itemize} + \item Aparece la necesidad de que los bosones de norma no tengan masa + \item La invariancia local se extiende a la global + \item La fase en una función de onda es arbitraria + \item Pero siempre debe ser la misma fase en todos los puntos del espacio tiempo. + \end{itemize} +\end{frame} + +\begin{frame}{Sentido físico de los potenciales vectoriales} + \begin{itemize} + \item Aparece en la teoría cuántica + \end{itemize} + + En ausencia de campo electromagnético, la ecuación estacionaria de Schrödinger: + \begin{equation} + -\frac{\hbar^2}{2m}\nabla^2 \psi_0 = E\psi_0, + \end{equation} +\end{frame} + +\begin{frame}{Ecuación de Schrödinger con campo vectorial electromagnético estático} + \begin{equation} + -\frac{\hbar^2}{2m}\mathbf{D}^2\psi= -\frac{\hbar^2}{2m} \left( \nabla + \frac{ie\mathbf{A}(\vec{x})}{\hbar c}\right)^2 \psi + \end{equation} + \begin{figure}[ht!] + \begin{center} + \includegraphics[width=0.5\linewidth]{bohm_aharanov.jpg} + \caption{Imagen del experomento de Aharanov-Bohm. Imagen con licencia CC-BY-SA toma de wikipedia} + \label{fig:bohm} + \end{center} + \end{figure} +\end{frame} + +\begin{frame}{Invariancia de norma y teorías no abelianas} + \begin{itemize} + \item La invariancia pide campos vectoriales sin masa + \item Bosones de interacción débil con masa y cargados + \item Bosones de interacción fuerte sin masa, pero cargados + \item Problemas por ejemplo en teoría de perturbaciones + \end{itemize} +\end{frame} + +\begin{frame}{Mecanismo de Higgs} + \begin{itemize} + \item Rompimiento de simetría aproximada + \item Rompimiento de simetría espontáneo + \item Simetrías escondidas por ejemplo en los ferromagnetos + \end{itemize} +\end{frame} + +\begin{frame}{Bosón de Higgs} + \begin{itemize} + \item Campo escalar complejo invariante de norma, $\phi$ y $\phi^*$ + \item Representan mesones escalares $H^+$ y $H^-$ + \begin{equation*} + \phi = \frac{1}{\sqrt{2}}(\phi_1 + i \phi_2) + \end{equation*} + \item Obedecen a la ecuación de Klein-Gordon + \end{itemize} +\end{frame} + +\end{document}