- 课题报告：(格式20%，细节80%)
- 平时作业
- 普朗克 普朗克的悲剧人生（上） 普朗克的悲剧人生（下）
- 亥姆霍兹
- Hermann Grassmann Hermann Grassmann_wikipedia
- A. Proca Proca equations of a massive vector boson field
- 类比思想：苏湛, 论物理学中的类比方法 以汤姆森和麦克斯韦为例[J].科学文化评论,2014,11(04):32-50. 原文下载
- 主要内容(Note, Reading Material, Summary详细内容请点击)：
- Maxwell equations与EM统一理论
- 场论的定义与内涵（量子场论; 重要问题; 无穷大来源; 常见场论模型）
- 数学基础（vector calculus; differential form; Stokes定理; Helmholtz decomposition）
- 回顾电磁学：历史电磁学历史名人多为法德两国人
- Biot-Savart定律与knot-theory
- 作业1：讨论Lorentz方程能不能从Maxwell方程中得到
- 作业2：定义linking number \(\mathscr{L}(l_{1},l_{2}):=\frac{1}{4\pi}\oint\frac{d\vec{l_{1}}\times d\vec{l_{2}}}{\left|\vec{l_{1}}-\vec{l_{2}}\right|^{3}}\cdot\left( \vec{l_{1}}-\vec{l_{2}} \right)\)，证明其为整数
- 主要内容(Note, Reading Material, Summary详细内容请点击)：
- 协变、逆变矢量(covariant/contravariant vector)
口诀： "co- is low and that's all you need to know." A. Zee:"Tensor is something transform like a tensor(multilinear vector)." - 电磁张量形式的Maxwell方程
- 微分形式在电磁学其他方面的应用（Lorentz force; Energy-momentum tensor; Lorentz invariant; Dual Field）
- 协变、逆变矢量(covariant/contravariant vector)
- 作业1：验证\(d^2 A = 0\); 由于\(F = dA\),故\(dF = 0\). 并讨论与\(\nabla \cdot \vec{B} = 0, \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}\)的关系
- 作业2：定义Hodge star operator作用形式(Euclidean space) \(\star(dx^{i_1} \wedge dx^{i_2} ... \wedge dx^{i_p} ):=\frac{1}{(n-p)!}\epsilon_{i_1 i_2 ...i_p i_{p+1} ...i_n} dx^{i_{p+1}} \wedge dx^{i_{p+2}} ... \wedge dx^{i_n}\)，验证\(n=1, 2, 3 ,4\)下的所有关系
例：\(n=2\)： \(\star(dx^1)=dx^2, \star(dx^2)=-dx^1\); \(n=3\)： \(\star(dx^1) \rightarrow dx^2 \wedge dx^3, \star(dx^2) \rightarrow dx^1 \wedge dx^3, \star\star(dx^2) \rightarrow \star(dx^1 \wedge dx^3) \rightarrow (dx^2)\) - 作业3：验证Maxwell方程 \(d \star F= \mathbf{J}\)
- 主要内容(Note, Reading Material, Summary详细内容请点击)：
- Dirac 辐射的量子理论
- Second quantization = occupation number representation = canonical quantization
- QFT = 经典场的量子化
"From particles to fields" "From field to particles" - Some models (Harmonic oscillator; Hydrogen model; phonon field)
- 作业1：将proca equations用\(F_{\mu \nu}\)的形式表示
- 作业2：将Monopole修正的Maxwell方程用\(F_{\mu \nu}\)的形式表示(wikipedia)
- 主要内容(Note, Reading Material, Summary详细内容请点击)：
- 利用类比思想和实际运用，讨论\(\left[x,p\right]=i\hbar, \left[\psi(x),\psi^{\dagger}(x')\right]=\delta(x-x'), \left[\psi(x),\pi(x')\right]=i\hbar \delta(x-x'), \left[\psi(x),\dot{\psi(x')}\right]=i\hbar \delta(x-x'), \left[A,E\right] \propto i\hbar \delta(x-x')\)这几个量子化条件
理解\(\left[x,p\right]=i\hbar\)是唯一的量子化条件，可从中推导出其他量子化条件。在处理具体模型时，多从\(\left[\psi(x),\pi(x')\right]=i\hbar \delta(x-x')\)出发 - 实标量场的量子化（phonon field, Klein-Gordon equation）
- 复标量场的量子化（Shrodinger equation）
- 矢量场的量子化（电磁场的Maxwell equations）
- 利用类比思想和实际运用，讨论\(\left[x,p\right]=i\hbar, \left[\psi(x),\psi^{\dagger}(x')\right]=\delta(x-x'), \left[\psi(x),\pi(x')\right]=i\hbar \delta(x-x'), \left[\psi(x),\dot{\psi(x')}\right]=i\hbar \delta(x-x'), \left[A,E\right] \propto i\hbar \delta(x-x')\)这几个量子化条件
- 作业1：在量子化phonon field的过程中\(H=\int dx \left [ \frac{1}{2}\Pi^2 + ...\right]
= \frac{1}{2} \sum_{kk'}(i\omega_k f_k)(i\omega_{k'} f_{k'})\left[a_k e^{i(kx-\omega_k t)} - a_{k}^{\dagger} e^{-i(kx-\omega_k t)}\right] \left[a_{k'} e^{i(k'x-\omega_{k'}t)} - a_{k'}^{\dagger} e^{-i(k'x-\omega_{k'} t)}\right]+...\)
其中有(1)\(a_k e^{i(kx-\omega_k t)}\), (2)\(a_{k}^{\dagger} e^{-i(kx-\omega_k t)}\), (3)\(a_{k'} e^{i(k'x-\omega_{k'}t)}\), (4)\(a_{k'}^{\dagger} e^{-i(k'x-\omega_{k'} t)}\)这四项，请证明(1)*(3)抵消 - 作业2：量子化1d Klein-Gordon场\(L=\int \left(\frac{1}{2}\dot{\phi}^2 - \frac{c^2}{2} (\partial_x)^2 - \frac{m^2}{2} \phi^2 \right) \)：(1) 将其量子化（可参考Michael Stone's book）(2) 计算动量\(P\), 流密度\(J^{\mu}\)
- 作业3：参考该note详细理解电磁场的量子化过程，推导所有结果Quantization of the Free Electromagnetic Field: Photons and Operators
- 主要内容(Note, Reading Material, Summary详细内容请点击)：
- 相互作用情形下的二次量子化：两体相互作用（以Ferimi gas为例），多体相互作用（Slater determinant）
- 路径积分的概念
- Gaussian integral（基本是所有可解析求解的情况）
两个问题：(1)\(dx\) & \(Dx\); (2)无穷维积分. 最终得到\(\int Dx e^{S} = \frac{1}{\sqrt{det M}}\)
- 作业：对于谐振子\(H=\frac{p^2}{2m}+\frac{1}{2}m \omega^2 x^2\)，有\( \left \langle q_N \right | e^{-iHt} \left | q_0 \right \rangle
= \left \langle q_N | n \right \rangle \left \langle n | q_0 \right \rangle e^{-in\hbar \omega t}
= \sum_{n=0}^{\infty} \psi_{n} (q_N) \psi_{n} (q_0) e^{-in\hbar \omega t} \)，讨论其特殊情况\(q_0=0; q_N=0\).
感兴趣的同学可以尝试\(H=\frac{p^2}{2m}+\frac{1}{2}m \omega^2 x^2+i\eta x\)
- 主要内容(Note, Reading Material, Summary详细内容请点击)：
- 补充了计算\( \left \langle q_f \right | e^{-iHt} \left | q_i \right \rangle\)的第二种方法
- 路径积分的应用：研究经典路径和涨落效应的普遍方法
- 自由粒子的经典路径和涨落效应
- double well势中的粒子隧穿问题：soliton solution, Wick rotation, instanton
- 作业1：给出Gal'fand-Yaglom theorem的严格证明.
- 作业2：derive equation:(eq.1-eq.11) in Haldane. Phys. Rev. Lett. 50, 1153(1983). 原文下载
关键点：\(\left\{\varphi_n, S_{n'}^{z}\right\}=\delta_{nn'}\), 即\(\varphi_n, \cos\theta_n\)是一对共轭量
- 主要内容(Note, Reading Material, Summary详细内容请点击)：
- Boson相干态：\( \left | \alpha \right \rangle = e^{-\frac{1}{2}\alpha^{*}\alpha} \sum_n \frac{\alpha^n}{\sqrt{n!}} \left | n \right \rangle\), \(\int d\alpha^{*}d\alpha \frac{1}{\pi} \left | \alpha \right \rangle \left \langle \alpha \right | = 1\) Boson路径积分：\(\int D\overline{\phi}D\phi e^{iS(\overline{\phi}, \phi)}\rightarrow \frac{1}{det(M)}\)
- Fermion相干态：\( \left | \eta \right \rangle = e^{-\eta C^{\dagger}} \left | 0 \right \rangle\), \(\int d\overline{\eta} d\eta e^{-\overline{\eta}\eta} \left | \eta \right \rangle \left \langle \eta \right | = 1\) Fermion路径积分：\(\int D\overline{\eta}D\eta e^{iS(\overline{\eta}, \eta)}\rightarrow det(M)\)
- Spin相干态：\( \left | \vec{n} \right \rangle = (\cos \frac{\theta}{2}, \sin \frac{\theta}{2} e^{i\phi})^T\), \(\int d\Omega \left | \vec{n} \right \rangle \left \langle \vec{n} \right | = 1\) Spin路径积分：\(\int D\vec{n} e^{i\int\left(\sin^2\frac{\theta}{2}\dot{\phi}-H\right)}\)
- 利用Wick rotation(\(\beta=it\))可利用path integral研究统计物理问题
- 作业1：压缩态(squeezed state)路径积分.
- 作业2：推导所有结果 M, S, Wang. Phys. Rev. A. 37, 1036(1988). 原文下载
关键点：(1)path integral使用的都是经典变量（避免了繁复的对易关系）. (2)构造"I". (3)寻找conjugate pair.
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主要内容(Note, Reading Material, Summary详细内容请点击)：
- 路径积分补遗
- Adiabatic invariant: \(I=\oint_{S^{1}} p dq, \ dI=0\)
- Geometry phase: \(\gamma=i\oint\left \langle \varphi(\vec{R}) \right | \frac{\partial}{\partial \vec{R}} \left | \varphi(\vec{R}) \right \rangle \cdot d \vec{R}\)
- Wess-Zumino(WZ) term: \(d\Omega = \vec{n}\cdot (\vec{n_x} \times \vec{n_y}) \)
- Application：(1)Boson, \(H=\omega a^{\dagger} a\).
\(Z = e^{-{\rm ln}(1-e^{-\beta \omega})} = e^{-\sum_n {\rm ln}(i\omega_n+\omega)} \);
(2)Fermion, \(H=\omega c^{\dagger} c\).
\(Z = 1+e^{-\beta \omega} = e^{\sum_n {\rm ln}(i\omega_n+\omega)} \).
详细计算见Simons书上P169 - P173
- 自发对称性破缺(Spontaneous symmetry breaking)
- Background：(1) Magnetism; (2) Superconductivity. 下面从最简单的Scalar field出发
- \(\phi^4\) theory: \(\mathscr{L}=\frac{1}{2}\partial_{\mu}\phi \partial^{\mu}\phi-\frac{m^2}{2} \phi^2 - \frac{m^2}{2}\phi^2 - \frac{\lambda}{4!} \phi^4\), 而\(\left \langle \phi \right \rangle = \sqrt{-\frac{6m^2}{\lambda}} \neq 0\)，\(\phi^4\)会发生SSB，且破缺后\(\left \langle \phi \right \rangle \neq 0\)
- L\(\sigma\)M：\(\mathscr{L}=\frac{1}{2}\partial_{\mu}\phi^{\dagger} \partial^{\mu}\phi - \frac{m^2}{2} \phi^{\dagger} \phi - \frac{\lambda}{4!} (\phi^{\dagger} \phi)^2\) 关键点：处理SSB的两种方式：(1)\(\phi=(v+\sigma,\vec{\pi})\),(2)\(\sqrt{(v+\eta)} \vec{n}\)
- 作业1：Study the SSB in NL\(\sigma\)M
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主要内容(Note, Reading Material, Summary详细内容请点击)：
- Goldstone mode, Goldstone theorem
- Background：Heisenberg model. \(H=J\sum_{\left \langle i,j \right \rangle}\vec{S_i}\cdot \vec{S_j}\)————Magnon(类似phonon), gapless & \(\varepsilon_k=v|k|\)
- Goldstone theorem\(\longleftrightarrow (N-1)\) massless fields，1 massive field
数学表达形式：\(\left\{ \begin{aligned} G\longleftrightarrow \mathscr{L}\ {\rm before\ SSB} \\H\longleftrightarrow \mathscr{L}\ {\rm after\ SSB}\end{aligned} \right. \Longrightarrow {\rm dim}(G/H)={\rm Goldstone\ mode\ number}\) - Application：(1)Magnetism(Goldstone mode=magnon); (2)superfluid; (3)phonon.
- Higgs mechanism(massive gauge field/boson)
- Background：Superconductivity + Maxwell equation.
- Higgs mechanism: eat Goldstone mode, massive gauge boson \(\Leftrightarrow\) proca equation
break gauge invariant(Superfluid(Boson), Superconductivity(Fermion)) - 自由度不变
- London equation & Ginzberg-Landau equation
- London equation: \(\vec{J}=-e^2 v^2 \vec{A}\)，可以唯象解释 Meissner effect & 零电阻效应
- 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: \(\vec{J}=-i\left(\phi^{*} \nabla \phi-\phi \nabla \phi^{*}\right)-2 e|\phi|^{2} \mathbf{A}\)
- 作业1：N=3, \(\phi=\left( \begin{smallmatrix} a \\ b \\ c \end{smallmatrix} \right) + \delta \phi\), show that eigenvalues of \(m_{ij}\rightarrow \left( \begin{smallmatrix} m^2 &\ &\ \\ \ &0&\ \\ \ &\ &0 \end{smallmatrix} \right)\), Thus \((N-1)\) Goldstone modes.
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主要内容(Note, Reading Material, Summary详细内容请点击)：
- 群论基本知识
- 重排定理(rearrangement theorem)
- 子群(subgroup)
- 陪集(coset): left&right coset; Coset decomposition
- 正规/不变子群(normal/invariant subgroup)
- 商群(quotient group)
- 群表示论(group representation theory)
- 将本为operator的群元表示成matrix
- 群的不等价不可约表示
- Application
- 同态基本定理
- 同态基本定理
- Homotopy group & (co)Homology group 概述 同态(Homomorphism)、同构(Isomorphism)、同伦(Homotopy)、同调(Homology)、上同调(Cohomology)、同胚(Homeomorphism=Topological isomorphism)
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主要内容(Note, Reading Material, Summary详细内容请点击)：
- 同伦群(Homotopy group)
- 定义: \(f: M\rightarrow N\)之间的mapping等价类形成群\(\pi(X,Y)\), 更常用地\(\pi(S^n,Y)=\pi_n(Y)\)
- 性质: (1)\(\pi_n(X \times Y) = \pi_n(X) \times \pi_n(Y) = \pi_n(X) \oplus \pi_n(Y)\); (2)\(\pi_n(S^n)=\mathbb{Z}\); (3)Table
- (上)同调群(Cohomology & Homolopy group)
- 同调群
- 上同调群
- de Rham同调 & Stokes定理
- 性质: (1)\(H^r(X \times Y) = \sum_{p+q=r} H^p(X) \times H^q(Y)\)(Kunneth formula); (2)Hurewicz同构定理, 特别地\(H_k(S^n)=\pi_k(S^n)\) if \(k\leq n\); (3)de Rham上同调: \(H_r(M) \simeq H^{n-r}(M)\)
- 具体应用
- Winding number
- Chern number
- 映射度 & pull back: \({\rm deg}(f)\cdot \int_N \omega = \int_M f^{*} \omega\)
- 作业1：Matrix形式的Winding number计算：
1d: \(\frac{1}{2\pi i}\int {\rm Tr}(g^{-1}\partial_x g) dx\); 2d: \(\frac{1}{4\pi}\int {\rm Tr}(g^{-1}\partial_x g) (g^{-1}\partial_y g)\ dxdy\); ... n维: \(\frac{1}{\Omega_n}\int {\rm Tr}(g^{-1}\partial g)^n\ d^n x\)是整数 - 作业2：Chern number：
定义Chern number: \(C=\frac{1}{2 \pi i} \int_{B Z} \operatorname{Tr}\left(P\left[\frac{\partial P}{\partial x}, \frac{\partial P}{\partial y}\right]\right)\) 投影算符\(P=\sum_{\varepsilon_{nk}<0}|\phi_{nk}\rangle\langle\phi_{nk}|\), 这里\(P=\frac{1}{2} (1+\vec{n}\cdot \vec{\sigma})\)
证明：\(C=\frac{1}{4\pi}\oint \vec{n} \cdot\left(\vec{n}_{x} \times \vec{n}_{y}\right)\ dx dy\) - 作业3：Nakahara书上Fig 3.8 & Fig 3.9的例子homework_3
本节课主要参考Nakahara书上： chapter3_Homology_groups, chapter4_Homotogy_groups, chapter6_De_Rham_cohomology_groups.
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主要内容(Note, Reading Material, Summary, 抛砖引玉分享一下我的笔记191213note)：
- 同伦群举例与应用
- \(\pi_1(S^1)=Z\): winding number, SSH model
\(\pi_2(S^2)=Z\): monopole(real space), Topological insulators/superconductors(momentum space)
\(\pi_3(S^2)=Z\): Hopf insulator/Hopf bundle(见作业2)
\(\pi_k(U(n))=\left\{ \begin{aligned} 0,\ {\rm k\ is\ even} \\ Z,\ {\rm k\ is\ odd}\end{aligned} \right. n\geq \frac{1}{2}(1+k)\) - winding number
可以向高维推广, 也可以推广至Matrix(Lie group)
重点：\(S^n \rightarrow M\), 找到\(S^n\)(物理上一般是k-space), \(M\left\{\begin{array}{l}{\vec{B}(\vec{x})} \in S^m\\ {\vec{n}(\vec{x})} \in S^m\\ {H(\vec{x}) \in \text { Lie group }}\end{array}\right.\)
- \(\pi_1(S^1)=Z\): winding number, SSH model
- Topological field theory
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\(S=\int \mathscr{L} dxdt=S_0+S_{topo}\),
\(S_0\)-\({\rm kinetic\ energy}, V(\vec{n}), V(\phi), (\lambda \vec{n})^2, (\lambda \phi)^2\),
\(S_{topo}\)(\(\theta\)-term)\(=i\theta W\)(topological number)
- Particle in a ring(Simons' book problem 3.5)
\(\mathscr{L}=\frac{I}{2} {\dot{\theta}}^2\), Poisson summation, \(\sum_n f(n)=\sum_n \tilde{f}(n)\) \(Z=\sum_n e^{-\beta n^2/2I}\) - Particle in a ring with gauge potential(Simons' book \(\S\)9.1)
\(\mathscr{L}=\frac{M}{2} {\dot{\phi}}^2+A\dot{\phi}\), \(Z=\sum_n e^{-\beta \frac{(n-A)^2}{2M}}\)
注意: (1)Topo term不影响eq of motion, 但却影响能谱;
(2)\(S_{topo}\) is unchanged under Wick rotation, still is imaginary ;
- Particle in a ring(Simons' book problem 3.5)
- 作业1：winding number：
(1) 2d: check \(|\vec{n}|\neq 1\), \(d\Omega=\frac{1}{|\vec{n}|^3} \vec{n}\cdot (\vec{n_x} \times \vec{n_y})\);
(2) 3d: \(n_x=\Delta \sin k_x, n_y=\Delta \sin k_y, n_z=\cos k_x + \cos k_y - \mu\), 计算相图, 扫描\(\Delta, \mu\) - 作业2：Hopf insulatorL.-M. Duan, Hopf insulators and their topologically protected surface states. 原文下载
本节课主要参考资料： Alexander G. Abanov. Topology, geometry and quantum interference in condensed matter physics
Altland, Simons书上对应部分： problem_3.5_Winding_numbers Ch9_Topology
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主要内容(Note, Reading Material, Summary详细内容请点击)：
- Poincare lemma
- closed form:\(d \omega = 0\), exact form: \(\omega = d \eta\)
- Poincare lemma
局域地, \(d \omega = 0\) \(\Rightarrow\) \(\omega = d \eta\), 即局域上所有闭形式都是恰当形式.
\(\mathbb{R}^n\) space is trivial
- Winding number in complex analysis
- 问题引入
\(W\left\{\begin{array}{l} \text{直接积分}, \int_{S^1} \omega=1 \\ \text{Stokes定理}, \int_{S^{1}} d\eta = \int_{\partial S^{1}} \eta = 0 \end{array}\right.\) 矛盾出在\(z=0\)奇点导致Stokes定理失效. - 如何解决
分成上下半区域, 通过Poincare lemma将每个区域上的微分形式都局域地变为恰当形式, 再分别利用Stokes定理
- 问题引入
- Chern-Simons term in IQHE
- 整数Hall效应(IQHE)
\(\sigma_{xy} = ne^2/h = c_1 e^2/h\) - Field theory for the above phys
\(\vec{E}\cdot \vec{B}\)这一项是关键, \(j^{\mu}=\frac{C_{1}}{2 \pi} \epsilon^{\mu \nu \tau} \partial_{\nu} A_{\tau}\)
\(S_{\mathrm{eff}}=\frac{C_{1}}{4 \pi} \int d^{2} x \int d t A_{\mu} \epsilon^{\mu \nu \tau} \partial_{\nu} A_{\tau}\)
参考Xiao-Liang Qi, First Chern number and topological response function in (2+1)d
- Chern character, n-th Chern number, Chern-Simons form, Winding number(详细内容将附上note)
- 整数Hall效应(IQHE)
- 作业1：\(\operatorname{ch}_{n}(\mathcal{F})\)
局域可写成exact form \(\mathrm{d} Q_{2 n-1}(\mathcal{A}, \mathcal{F})\), 请验证\(n=1, 2, 3, 4\)的情况.
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\(\operatorname{ch}_{n}(\mathcal{F})=\mathrm{d} Q_{2 n-1}(\mathcal{A}, \mathcal{F})\)
- 作业2：不同的规范势\(\mathcal{A}\)和\(\mathcal{A}^{\prime}\)定义的Chern-Simons character之间的关系, 请验证\(n=1, 2, 3, 4\)的情况.
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\(Q_{2 n-1}\left(\mathcal{A}^{\prime}, \mathcal{F}^{\prime}\right)-Q_{2 n-1}(\mathcal{A}, \mathcal{F})=Q_{2 n-1}\left(g^{-1} \mathrm{d} g, 0\right)+\mathrm{d} \alpha_{2 n}\)
两道作业题的严格证明见Nakahara_chap11_characteristic_classes
关键是\(\mathscr{L} \rightarrow \mathscr{L}+\frac{df}{dt}\), 多一个全微分, 不改变运动方程, 但却影响物理.
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主要内容(Note, Reading Material, Summary详细内容请点击)：
- 两篇paper
- Enz's paper
U. Enz. Discrete Mass, Elementary Length, and a Topological Invariant as a Consequence of a Relativistic Invariant Variational Principle. 原文下载
\(m_{s}=E_{s} / c^{2}=\frac{4}{c^{2}}(A K)^{1 / 2}\)
即存在time-independent localized with finite energy的解. - Derrick theorem
G. H. Derrick. Comments on Nonlinear Wave Equations as Models for Elementary Particles. 原文下载
\(\text{Conclusion}: \left\{\begin{array}{lll} D \leq 2, & \text{exist} \\ D \geq 3, & \text{no stable time-independent solution of finite energy}(E_g=0) \end{array}\right.\)
Beyond Derrick Theorem: (a)-(d)
- Enz's paper
- 1d model
- \(\phi^4\) theory: \(V(\phi)=A(\phi^2-v^2)^2\)
(时间t, 或是位置x上)tunneling快, 动能大, 势能小; tunneling慢, 动能小, 势能大. 会出现能量极小点\(E_0=\text{min}(E)\), 故有\(E \geq E_0\)的类似形式 - Sine-Gordon model: \(V(\phi)=-V\sin(\phi(x))\)
\(\phi \simeq 0+2\pi n\), 系统有无穷多简并基态. 也是动能、势能平衡, 达到极小.
\(\boxed{E \geq NE_0}\)——\(N\)为整数, winding number
物理图像: 系统可能有多个vacuum, 之间tunneling产生soliton(重点在于搞清楚从哪个vacuum出发, 到哪个vacuum) - \(\phi^4\) theory: \(V(\phi)=A(\phi^2-v^2)^2\)
- 具体计算证明 & 另几例
- Bogomol'ny inequality
\((a\pm b)^2 \geq 0; Tr(A^{\dagger}A)\geq 0\) - Abelian gauge potential
Topo term: \(\vec{E}\cdot \vec{B}\)
\(S=-\frac{1}{2} \int Tr(F_{\mu\nu}^2) d^4 x \geq E_0 / |q|\) - Skyrmion(d=3)
\(E=\int d^{3}x \frac{f_{\pi}^{2}}{4} \operatorname{Tr}\left(\partial_{i}\left(U \partial_{i} U^{\dagger}\right)-\frac{1}{32 e^{2}} \operatorname{Tr}\left(\left[U^{+} \partial_{i} U, U^{\dagger} \partial_{j} U\right]^{2}\right)\right.\geq \frac{f_{\pi}}{4e} q\) - 2d model
无穷远处快速decay, 可视为一点
\(\pi_3(U(N))=Z\)
- Bogomol'ny inequality
本节课主要参考资料：
Ryder. Quantum field theoty 书上对应部分： Ryder. Chapter10 Topological objects in field theory
李政道讲义——场论与粒子物理学 书上对应部分： Tsung-Dao Lee. Chapter7 Soliton
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