11.22主要内容

Goldstone mode, Goldstone theorem

1. Background：Heisenberg model. \(H=J\sum_{\left \langle i,j \right \rangle}\vec{S_i}\cdot \vec{S_j}\)————Magnon(类似phonon), gapless & \(\varepsilon_k=v|k|\)
2. Before SSB
   \(\mathscr{L}= \frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi - V(\phi)\), 其中\(V(\phi) = \frac{m^{2}}{2} \phi^2 - \frac{\lambda}{4!} \phi^{4}\)
   After SSB
   \(\mathscr{L}= \mathscr{L}_0 + {\rm cubic\ terms} + {\rm quartic\ terms}\), 其中\(\mathscr{L}_0 = \frac{1}{2} \partial_{\mu}\pi \partial^{\mu}\pi - \frac{1}{2} \partial_{\mu}\vec{\sigma} \cdot \partial^{\mu}\vec{\sigma} - m^2\pi^2\)
   \((N-1)\) massless fields，1 massive field
3. Application：
   (1)Magnetism(Goldstone mode=magnon); (2)superfluid; (3)phonon.
4. Goldstone Theorem(数学表达)：
   \(\left\{ \begin{aligned} G\longleftrightarrow \mathscr{L}\ {\rm before\ SSB} \\H\longleftrightarrow \mathscr{L}\ {\rm after\ SSB}\end{aligned} \right. \Longrightarrow {\rm dim}(G/H)={\rm Goldstone\ mode\ number}\)

Higgs mechanism(massive gauge field/boson)

1. Background
   Superconductivity + Maxwell equation.(1) London equation; (2) Ginzberg-Landau equation; (3) BCS theory
2. N=2, \(\phi=\left( \begin{smallmatrix} \phi_1 \\ \phi_2 \end{smallmatrix} \right) = \phi_1 + i\phi_2\)
   Before SSB,
   \(\mathscr{L}= -\frac{1}{4} F_{\mu \nu} F^{\mu \nu} + \left(D_{\mu} \phi \right)^{\ast}\left(D^{\mu} \phi\right) - V(\phi)\), 其中\(V(\phi) = \frac{m^{2}}{2} \phi^{\dagger} \phi - \frac{\lambda}{4!} \left(\phi^{\dagger} \phi\right)^{2}\)
   eat Goldstone mode,
   \(\mathscr{L}=\frac{1}{2}\partial_{\mu} \phi_{1}\partial^{\mu} \phi_{1}-m^2\phi_{1}^{2}-\frac{1}{4}F_{\mu \nu} F^{\mu \nu}-\sqrt{2}ve(A_{\mu}-g_{\mu})\partial^{\mu}\phi_2 + e^2 v^2 A_{\mu} A^{\mu}\), 其中\(g_{\mu}=\frac{1}{2\sqrt{2}ev}\partial_{\mu} \phi_2\)
   产生massive boson
3. massive gauge boson \(\Leftrightarrow\) proca equation(只能写成\(A\)的形式，而不能写成\(\vec{E}, \vec{B}\)的形式)
   break gauge invariant(Superfluid(Boson), Superconductivity(Fermion))
4. 自由度不变
   before SSB: \(\psi=(\phi_1, \phi_2, {\rm Boson}(2))\)共4个自由度
   after SSB: \(\psi=(\phi_1, 0, {\rm Boson}(2+1))\)共4个自由度

London equation & Ginzberg-Landau equation

1. London equation: \(\vec{J}=-e^2 v^2 \vec{A}\)，可以唯象解释 Meissner effect & 零电阻效应
   Meissner effect: \(\left\{ \begin{aligned} \nabla \times \vec{B}&=\vec{J}=-e^2 v^2 \vec{A} \\ \nabla \cdot \vec{B}&=0\end{aligned} \right. \Longrightarrow \nabla^{2} \vec{B}=k^{2} \vec{B} \Longrightarrow B_{x}=B_{0} \mathrm{e}^{-k x}\)
   零电阻效应: \(\left\{ \begin{aligned}\vec{E}&=-\partial \vec{A}/ \partial t=0 \\\vec{E}&= R\vec{J}({\rm Ohm's\ law})\end{aligned} \right. \Longrightarrow R=0\)
   London equation经典文献：
   1935. F, London. The electromagnetic equations of the supraconductor. Proc. R. Soc. Lond. A, 149(866), 71-88(1935). 原文下载
   1936. H, London. An Experimental Examination of the Electrostatic Behaviour of Supraconductors. Proc. R. Soc. Lond. A, 155, 102-110(1936). 原文下载
   1937. F, London. A New Conception of Supraconductivity. Nature, 140, 793-796(1937). 原文下载
   
   
2. Ginzberg-Landau equation \(\mathscr{L}=-\left[(\mathbf{\nabla}-\mathrm{i} e \vec{A}) \phi\right] \cdot \left(\mathbf{\nabla}+\mathrm{i} e \vec{A}) \phi^{*}\right]-m^{2}|\phi|^{2}-\lambda|\phi|^{4}-\frac{1}{2}(-\vec{E}^2+\vec{B}^2)\)
   current: \(\vec{J}=-i\left(\phi^{*} \nabla \phi-\phi \nabla \phi^{*}\right)-2 e|\phi|^{2} \mathbf{A}\)

其他补充

1. 课前Basel problem \(\sum \frac{1}{n^2}=\frac{\pi^2}{6}\) Basel problem_Wikepedia
2. 本节内容主要参考Ryder_chapter8 Spontaneous symmetry breaking and the Weinberg-Salam model

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