12.27主要内容

上节课内容回顾

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. Topo term多来源于此.
1. 1d winding number
   \(W\left\{\begin{array}{l} \text{Complex analysis} \\ \text{Topo term} \end{array}\right.\)
2. \(Tr(\mathcal{F}^n) \Rightarrow \text{Pontryagin class} (d=2n)\)
3. \(Tr(\mathcal{A}\mathcal{F}^n) \Rightarrow \text{Chern class} (d=2n+1)\)

两篇paper

1. 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
   \(F_{A}=A\left[\left(\frac{\partial \theta}{\partial x}\right)^{2}+\left(\frac{\partial \theta}{\partial y}\right)^{2}+\left(\frac{\partial \theta}{\partial z}\right)^{2}-\frac{1}{c^{2}}\left(\frac{\partial \theta}{\partial t}\right)^{2}\right]\)
   \(\delta \int\left(F_{A}+K \sin ^{2} \theta\right) d x d y d z d t=0\)
   \(\frac{\partial^{2} \theta}{\partial x^{2}}-\frac{1}{c^{2}} \frac{\partial^{2} \theta}{\partial t^{2}}=\frac{K}{2 A} \sin 2 \theta\)
   \(m_{s}=E_{s} / c^{2}=\frac{4}{c^{2}}(A K)^{1 / 2}\)
   即存在time-independent localized with finite energy的解.
Enz的工作实际上只考虑了(1+1)d的情况, 那么高维情况下这样的态能否存在？是否有普遍的结论？
2. 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(以下情况也可能出现这种解):
   
   (a)Lagragian中阶数\(>2\);
   (b)first-order spinor equation(Dirac' equation);
   (c)operator;
   (d)periodic in time而不是time-independent

1d model

1. \(\phi^4\) theory: \(V(\phi)=A(\phi^2-v^2)^2\)
   SSB会有global基态(GS), 但我们这里关注的是local能量极小处.
   单个kink: \(\pi_0(\{+v,-v\})=Z_2\); 两个kinks(\(-\infty, +\infty\)连在一起):\(\pi_1(S^1)=Z\)
   (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
2. 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)
这个解不能用SSB解释, 表现在的确存在\(E>0\)的解, 而且是稳定的.

具体计算证明 & 另几例

1. Bogomol'ny inequality
   \((a\pm b)^2 \geq 0; Tr(A^{\dagger}A)\geq 0\)
2. Abelian gauge potential
   Topo term: \(\vec{E}\cdot \vec{B}\)
   \(S=-\frac{1}{2} \int Tr(F_{\mu\nu}^2) d^4 x \geq E_0 / |q|\)
3. Skyrmion(d=3)
   \(E=\int d^{3}x \frac{f_{\pi}^{2}}{4} \operatorname{Tr}\left(\partial_{i}\left(U \partial_{i} U^{\dagger}\right)-\frac{1}{32 e^{2}} \operatorname{Tr}\left(\left[U^{+} \partial_{i} U, U^{\dagger} \partial_{j} U\right]^{2}\right)\right.\)
   \(E \geq \frac{f_{\pi}}{4e} q\)
4. 2d model
   无穷远处快速decay, 可视为一点
   \(\pi_3(U(N))=Z\)

Summary

Summary(191227).jpg

USTC| BBS