# Phantom_Ghost

1. 1天前
2019-01-14 22:16:06
Phantom_Ghost 更新于 非欧几何是向量空间不

这也要封？我认为只要是没有逾越理性讨论这点原则都不可以实施封禁处罚，别人能不能理解那是他自己的事，你解释得好不好也是你自己的事情，自由理性讨论是每个用户在学术论坛上应享有的基本权利。如果觉得对方不能理解而懒得进一步交流你完全可以不回复不再参与讨论。

2. 6天前
2019-01-10 03:50:36

我想看看天文专业的人是怎么理解卡西尼裂缝之类的东西

另外也不用给自己增添负担强行活跃，顺其自然来兴致就写点内容好了

3. 2周前
2018-12-30 03:31:35
Phantom_Ghost 更新于 Conformal Field Theory

#### Exercise sheet 4

[attachment:5c27cb8e95f45]

4. 2018-12-30 03:11:58
Phantom_Ghost 更新于 Conformal Field Theory

#### 5. Introduction to CFT

Noether current with charges
Canonical approach: Poincar$\acute{\text{e}}$ transformation+dilatation

$\tilde{x}^\mu=x^\mu+\xi^\mu(x)$

$J^\mu=\int_{\Sigma_t}(\xi^\nu T^\mu_{\;\nu}-\frac{1}{4}\xi^{\alpha}_{\;,\alpha}\underset{\text{virial current}}{\underbrace{V^\mu}})$ , $T^0_{\;\mu}=P_\mu\sim-i\hbar\partial_\mu$

$\phi(x)=\lambda^\delta\tilde{\phi}(\tilde{x})$ , $\delta\phi(x)=[J_0,\phi(x)]=-i(\xi^\mu\partial_\mu\phi-\frac{1}{n}\xi^\alpha_{\;\alpha}\Delta\phi)$ ($\frac{1}{n}\xi^\alpha_{\;\alpha}=\lambda$ for $\tilde{x}^\mu=x^\mu+\lambda x^\mu$ dilatation)

Conceptually, conformal transformation is the composition of conformal diffeomorphism and Weyl transformation.

Conformal diffeomorphism: $\tilde{x}^\mu=x^\mu+\xi^\mu(x)$ , $(1+\theta)g^{\mu\nu}=\tilde{g}^{\mu\nu}+\xi_{\mu,\nu}+\xi_{\nu,\mu}$ , $\tilde{\phi}(x)=\phi(x)-\xi^\mu\partial_\mu\phi$

Weyl transformation: $\tilde{x}=x$ , $\tilde{g}^{\mu\nu}=(1-\theta)g^{\mu\nu}$ , $\tilde{\phi}(x)=(1+\Delta\sigma)\phi(x)$

$\text{Weyl trans.}\circ\text{conformal Diff.}$: $\tilde{x}^\mu=x^\mu+\xi^\mu$ , $\tilde{g}^{\mu\nu}=g^{\mu\nu}$ , $\tilde{\phi}(x)=\phi(x)-\xi^\mu\partial_\mu\phi(x)+\Delta\sigma\phi(x)$ ($\sigma=\xi^\alpha_{\;,\alpha}$)

Explicit example:
$T_{\mu\nu}=\partial_\mu\phi\partial_\nu\phi$ , $T^\mu_{\;\mu}=\partial^\mu\partial_\mu(\phi^2)$

$\Rightarrow$ $V_\mu=\partial_\mu(\varphi^2)$ , $V_0=2(\partial_0\phi)\phi=2\pi_\phi\phi$ , $[\pi_\phi\phi(x),\phi(y)]=-\delta(x-y)\phi(x)$.

Include special conformal transformation: Noether current from $T_{\mu\nu}$ and $V_\mu$
$\partial^\mu(x^\nu T_{\mu\nu}+\xi^\alpha_{\;,\alpha}V_\mu)$ , if $V_\mu=\partial_\nu L^{\mu\nu}$ ($L^{\mu\nu}$ local field)

$\delta x^\mu=\xi^\mu=2(c\cdot x)x^\mu-x^2 c^4$

$J^\mu=\xi^\nu T^\mu_{\;\nu}-2(c\cdot x)V^\mu+2c_\nu L^{\nu\mu}$

Example: free scalar field in $\mathbb{R}^n$
$T^\alpha_{\;\alpha}=\partial_\mu\partial^\mu(\phi^2)$ , $L^{\mu\nu}=g^\mu\nu\phi^2$



Question: Can scale invariance imply conformal invariance ?

$n=2$ + unitarity (+ Lorentz invariance + conservation) show that $T^\mu_{\;\mu}=0$

$\tilde{T}_{\mu\nu}(p)=\int d^2x e^{ip\cdot x}T_{\mu\nu}(x)$

$\langle 0|\tilde{T}_{\mu\nu}(p)\tilde{T}_{\alpha\beta}(-p)|0\rangle=(p_\mu p_\nu-\eta_{\mu\nu p^2})(p_\alpha p_\beta-\eta_{\alpha\beta}p^2)\frac{f(p^2)}{p^2}+(\nu\leftrightarrow\alpha)\frac{g(p^2)}{p^2}$

$\langle 0|T^{\mu}_{\;\mu}(p)T^\alpha_{\;\alpha}(-p)|0\rangle=(f+2g)p^2$

$\langle 0|T^{\mu}_{\;\mu}(x)T^\alpha_{\;\alpha}(y)|0\rangle=(f+2g)\square\delta^{(2)}(x-y)$

This can be removed by certain improvement term such as $S[\phi]=\int[\partial_\alpha\phi\partial^\alpha\phi+V(\phi)]+\underset{\text{geo. local counter term}}{\underbrace{\int\sqrt{g}R}}$

$\Rightarrow$ $\langle 0|(T_\text{imp})^{\mu}_{\;\mu}(x)(T_\text{imp})^\alpha_{\;\alpha}(y)|0\rangle=0$ , $\sum_n\langle 0|(T_\text{imp})^{\mu}_{\;\mu}(x)|n\rangle\langle n|(T_\text{imp})^\alpha_{\;\alpha}(y)|0\rangle\geq 0$ (insert complete set of states)

Counter example: (violation of unitarity)
$S[\phi]=\int(\phi^2\partial_\alpha\partial^\alpha\phi+\phi\square\phi)$

$T_{\mu\nu}=\phi^2(\partial_\mu\phi\partial_\nu\phi-\frac{1}{2}\eta_{\mu\nu}\partial_\alpha\phi\partial^\alpha\phi)$ , $T^\mu_{\;\mu}=\phi^2\partial_\alpha\phi\partial^\alpha\phi=\partial_\mu V^\mu$ , $V_\mu=\phi^2\partial_\mu(\phi^2)$

$\Rightarrow$ $\partial^\mu V_\mu=\partial^\mu\phi^2\partial_\mu\phi^2+\phi^2\partial^\mu\partial_\mu\phi^2$

e.o.m. $\frac{\delta}{\delta\phi}\partial_\mu(\phi^2\partial_\alpha\phi)=0$ , $\frac{\delta}{\delta\phi}\phi\partial_\alpha\phi\partial^\alpha=0$

Maxwell theory ($n=3$)
$S[A_\mu]=\int\,d^3x\,F_{\mu\nu}F^{\mu\nu}$ , $T^{\mu}_{\;\mu}=\partial_\mu V^\mu\rightarrow A_\nu F^{\mu\nu}$ , $\int F_{\mu\nu}F^{\mu\nu}=\int \partial_\alpha\phi\partial^\alpha\phi$

$\varepsilon^{\mu\nu\lambda}F_{\mu\nu}=B^\lambda$ , $\partial_{[\mu}F_{\lambda\rho]}=0$ $\Rightarrow$ $B_\lambda=\partial_\lambda\phi$ , $\partial^\mu F_{\mu\nu}=0$ $\Rightarrow$ $\square\phi=0$

5. 5周前
2018-12-06 07:44:47

-besetzt-

6. 2018-12-06 07:42:37

[attachment:5c086200654b5]

[attachment:5c0861ffd4fd6]

[attachment:5c08620202fee]

[attachment:5c0862045ed76]

[attachment:5c08620644023]

[attachment:5c0862061b654]

[attachment:5c086207cc5bc]

[attachment:5c08620db556f]

[attachment:5c08620ee7b1a]

[attachment:5c086211c6d61]

[attachment:5c08672e4a5db]

[attachment:5c086212761c0]

7. 2018-12-06 07:39:28

This is a note written for a seminar of Lie algebra. Please feel free to read and leave comments and suggestions. If you find any typos and errors please inform me.

So far it has not yet been finished and will be updated in the succeeding period of time.

[attachment:5c08616ce022c]

[attachment:5c08616c90cd9]

8. 8周前
2018-11-17 10:00:27
Phantom_Ghost 更新于 Symplectic Geometry

#### Exercise Sheet 5

[attachment:5bef7634e4b31]

9. 2月前
2018-11-13 06:19:02
Phantom_Ghost 更新于 Symplectic Geometry

....



Theorem(Poincaré-Birkhoff): If $h: A\to A$ is an area preserving homeomorphism satisfying the twist (boundary) condition, then $h$ has at least two fixed points.

Examples:
(i) $(r,\varphi)$ polar coordinate on $A$ and $h(r,\varphi)=(r,\varphi+\alpha)$ is area preserving and its lifts to universal cover are translations $\Rightarrow$ $h$ does not satisfy the twist condition and if $\alpha\neq 2\pi\mathbb{Z}$ then there are no fixed points.

(ii) $h(r,\varphi)=(\rho(r),\varphi+r-\frac{a+b}{2})$ where $\rho(r)$ is a function.

[attachment:5be9fc3525c3b]

This satisfies the twist condition and has no fixed points (for $a,b\notin 2\pi\mathbb{Z}$), $h$ is not area preserving.

(iii) Let $H$ be a Hamilton function on $A$, whose level sets (the trajectories of Hamiltonian flow) are like in the figure bellow:

assume: along $\partial A$, $\nabla H\neq 0$. Hamiltonian flow satisfies the twist condition, it is area preserving and for short time the Hamiltonian flow has two fixed points (i.e. the critical points of $H$).