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一个推广 Weinberg III讨论的refinement CFT中的类似物，待补充

现在mathjax启用braket等包还是用不了； multi line等环境未被正确识别，目前需要手动用放到math mode中

%Abbreviations of Greek letters def \a {\alpha} def \b {\beta} def \g {\gamma} def \G {\Gamma} def \d {\delta} def \D {\Delta} def \e {\epsilon} def \ve {\varepsilon} def \m {\mu} def \n {\nu} def \k {\kappa} def \l {\lambda} def \L {\Lambda} def \s {\sigma} def \S {\Sigma} def \r {\rho} def \o {\omega} def \O {\Omega} def \th {\theta} def \Th {\Theta} def \t {\tau} def \z {\zeta} %Abbreviations of mathbb fonts newcommand{\R}{\mathbb{R}} newcommand{\N}{\mathbb{N}} newcommand{\Z}{\mathbb{Z}} newcommand{\C}{\mathbb{C}} newcommand{\V}{\mathbb{V}} %%vector spaces newcommand{\W}{\mathbb{W}} %%vector spaces %Abbreviations of Lie algebras newcommand{\glie}{\mathfrak{g}} %%generic Lie algebra newcommand{\gllie}{\mathfrak{gl}} %%general linear newcommand{\sllie}{\mathfrak{sl}} %%special linear newcommand{\solie}{\mathfrak{so}} %%special orthogonal newcommand{\splie}{\mathfrak{sp}} %%symplectic newcommand{\nlie}{\mathfrak{n}} %%nilpotent or solvable algebra newcommand{\plie}{\mathfrak{p}} %%nilpotent or solvable algebra newcommand{\hlie}{\mathfrak{h}} %%abelian factor newcommand{\klie}{\mathfrak{k}} %%maximal compact subalgebra newcommand{\blie}{\mathfrak{b}} %%Borel subalgebra %Others newcommand{\greenfunction}[1]{\langle #1\rangle} %%green functions renewcommand{\vev}[1]{\langle #1\rangle} %%green functions renewcommand{\H}{\mathcal{H}} %%Hilbert spaces newcommand{\Ccat}{\mathcal{C}} %%categories newcommand{\Acat}{\mathcal{A}} %%categories newcommand{\Bcat}{\mathcal{B}} %%categories newcommand{\p}{\partial} %%partial derivatives newcommand{\nn}{\nonumber} %due to old version of MathJax newcommand{\set}[1]{{#1}} newcommand{\ket}[1]{\vert #1 \rangle}

%Abbreviations of Greek letters def \a {\alpha} def \b {\beta} def \g {\gamma} def \G {\Gamma} def \d {\delta} def \D {\Delta} def \e {\epsilon} def \ve {\varepsilon} def \m {\mu} def \n {\nu} def \k {\kappa} def \l {\lambda} def \L {\Lambda} def \s {\sigma} def \S {\Sigma} def \r {\rho} def \o {\omega} def \O {\Omega} def \th {\theta} def \Th {\Theta} def \t {\tau} def \z {\zeta} %Abbreviations of mathbb fonts newcommand{\R}{\mathbb{R}} newcommand{\N}{\mathbb{N}} newcommand{\Z}{\mathbb{Z}} newcommand{\C}{\mathbb{C}} newcommand{\V}{\mathbb{V}} %%vector spaces newcommand{\W}{\mathbb{W}} %%vector spaces %Abbreviations of Lie algebras newcommand{\glie}{\mathfrak{g}} %%generic Lie algebra newcommand{\gllie}{\mathfrak{gl}} %%general linear newcommand{\sllie}{\mathfrak{sl}} %%special linear newcommand{\solie}{\mathfrak{so}} %%special orthogonal newcommand{\splie}{\mathfrak{sp}} %%symplectic newcommand{\nlie}{\mathfrak{n}} %%nilpotent or solvable algebra newcommand{\plie}{\mathfrak{p}} %%nilpotent or solvable algebra newcommand{\hlie}{\mathfrak{h}} %%abelian factor newcommand{\klie}{\mathfrak{k}} %%maximal compact subalgebra newcommand{\blie}{\mathfrak{b}} %%Borel subalgebra %Others newcommand{\greenfunction}[1]{\langle #1\rangle} %%green functions renewcommand{\vev}[1]{\langle #1\rangle} %%green functions renewcommand{\H}{\mathcal{H}} %%Hilbert spaces newcommand{\Ccat}{\mathcal{C}} %%categories newcommand{\Acat}{\mathcal{A}} %%categories newcommand{\Bcat}{\mathcal{B}} %%categories newcommand{\p}{\partial} %%partial derivatives newcommand{\nn}{\nonumber} %due to old version of MathJax newcommand{\set}[1]{{#1}}

%Abbreviations of Greek letters def \a {\alpha} def \b {\beta} def \g {\gamma} def \G {\Gamma} def \d {\delta} def \D {\Delta} def \e {\epsilon} def \ve {\varepsilon} def \m {\mu} def \n {\nu} def \k {\kappa} def \l {\lambda} def \L {\Lambda} def \s {\sigma} def \S {\Sigma} def \r {\rho} def \o {\omega} def \O {\Omega} def \th {\theta} def \Th {\Theta} def \t {\tau} def \z {\zeta} %Abbreviations of mathbb fonts newcommand{\R}{\mathbb{R}} newcommand{\N}{\mathbb{N}} newcommand{\Z}{\mathbb{Z}} newcommand{\C}{\mathbb{C}} newcommand{\V}{\mathbb{V}} %%vector spaces newcommand{\W}{\mathbb{W}} %%vector spaces %Abbreviations of Lie algebras newcommand{\glie}{\mathfrak{g}} %%generic Lie algebra newcommand{\gllie}{\mathfrak{gl}} %%general linear newcommand{\sllie}{\mathfrak{sl}} %%special linear newcommand{\solie}{\mathfrak{so}} %%special orthogonal newcommand{\splie}{\mathfrak{sp}} %%symplectic newcommand{\nlie}{\mathfrak{n}} %%nilpotent or solvable algebra newcommand{\hlie}{\mathfrak{h}} %%abelian factor newcommand{\klie}{\mathfrak{k}} %%maximal compact subalgebra newcommand{\blie}{\mathfrak{b}} %%Borel subalgebra %Others %\newcommand{\bra}[1]{\langle #1 \vert} %\newcommand{\ket}[1]{\vert #1 \rangle} %\newcommand{\braket}[2]{\langle #1\vert #2\rangle} %\newcommand{\greenfunction}[1]{\langle #1\rangle} %%green functions newcommand{\vev}[1]{\langle #1\rangle} %%green functions renewcommand{\H}{\mathcal{H}} %%Hilbert spaces newcommand{\Ccat}{\mathcal{C}} %%categories newcommand{\Acat}{\mathcal{A}} %%categories newcommand{\Bcat}{\mathcal{B}} %%categories newcommand{\p}{\partial} %%partial derivatives newcommand{\nn}{\nonumber} %Abbreviations of Lie algebras newcommand{\glie}{\mathfrak{g}} %%generic Lie algebra newcommand{\gllie}{\mathfrak{gl}} %%general linear newcommand{\sllie}{\mathfrak{sl}} %%special linear newcommand{\solie}{\mathfrak{so}} %%special orthogonal newcommand{\splie}{\mathfrak{sp}} %%symplectic newcommand{\nlie}{\mathfrak{n}} %%nilpotent or solvable algebra newcommand{\hlie}{\mathfrak{h}} %%abelian factor newcommand{\klie}{\mathfrak{k}} %%maximal compact subalgebra newcommand{\blie}{\mathfrak{b}} %%Borel subalgebra Shadow transform as an intertwining operator to be added Shadow transform from momentum space Gillioz, Momentum-space conformal blocks on the ight cone 这篇文章讨论如何通过动量空间的完备关系来得到shadow transform。下面我会通过1d CFT的计算来展示这个过程。 一般性的讨论 1d OPE代数， begin{equation} \phi_{i}\left(x_{1}\right) \phi_{j}\left(x_{2}\right)=\sum_{k} c_{i j}^{k} D_{i j}^{k}\left(x_{1}, x_{2},\p_2\right) \phi_{k}\left(x_{2}\right) end{equation} 平移的作用下不是平凡的， D_{i j}^{k}\left(x_{1}, x_{2},\p_2\right)=\frac{1}{|x_{12}|^{\D_{i}+\D_{j}-\D_{k}}}F_{11}\left(\D_{i}-\D_{j}+\D_{k}, 2 \D_{k} ; x_{12} \partial_{2}\right)]

newcommand{\set}[1]{{#1}} newcommand{\dd}[1]{\mathrm{d} #1 } newcommand{\pdv}[2]{\frac{\partial #1}{\partial #2}} Same as in Markdown. Abbreviations of Greek letters renewcommand{\a}{\alpha} renewcommand{\b}{\beta} newcommand{\g}{\gamma} newcommand{\G}{\Gamma} renewcommand{\d}{\delta} newcommand{\D}{\Delta} newcommand{\e}{\epsilon} newcommand{\ve}{\varepsilon} newcommand{\m}{\mu} newcommand{\n}{\nu} renewcommand{\k}{\kappa} renewcommand{\l}{\lambda} renewcommand{\L}{\Lambda} newcommand{\s}{\sigma} renewcommand{\S}{\Sigma} renewcommand{\r}{\rho} renewcommand{\o}{\omega} renewcommand{\O}{\Omega} renewcommand{\th}{\theta} newcommand{\Th}{\Theta} renewcommand{\t}{\tau} newcommand{\z}{\zeta} Abbreviations of mathbb fonts newcommand{\mbf}{\mathbf} newcommand{\mb}{\mathbb} newcommand{\mc}{\mathcal} newcommand{\mf}{\mathfrak} newcommand{\R}{\mathbb{R}} newcommand{\N}{\mathbb{N}} newcommand{\Z}{\mathbb{Z}} newcommand{\C}{\mathbb{C}} newcommand{\V}{\mathbb{V}} \newcommand{\W}{\mathbb{W}} % Abbreviations of mathcal fonts newcommand{\cP}{\mathcal P} newcommand{\cO}{\mathcal O} newcommand{\cA}{\mathcal A} newcommand{\cB}{\mathcal B} newcommand{\cC}{\mathcal C} newcommand{\cD}{\mathcal D} newcommand{\cH}{\mathcal H} newcommand{\cL}{\mathcal L} Abbreviations of Lie algebras newcommand{\glie}{\mathfrak{g}} %%generic Lie algebra newcommand{\gllie}{\mathfrak{gl}} %%general linear newcommand{\sllie}{\mathfrak{sl}} %%special linear newcommand{\solie}{\mathfrak{so}} %%special orthogonal newcommand{\splie}{\mathfrak{sp}} %%symplectic newcommand{\nlie}{\mathfrak{n}} %%nilpotent or solvable algebra newcommand{\plie}{\mathfrak{p}} %%nilpotent or solvable algebra newcommand{\hlie}{\mathfrak{h}} %%abelian factor newcommand{\klie}{\mathfrak{k}} %%maximal compact subalgebra newcommand{\blie}{\mathfrak{b}} %%Borel subalgebra Others newcommand{\greenfunction}[1]{\langle #1\rangle} % green functions renewcommand{\vev}[1]{\langle #1\rangle} \newcommand{\p}{\partial} % partial derivatives renewcommand{\inf}{\infty} % Infinity newcommand{\nn}{\nonumber} newcommand{\Diff}{\operatorname{Diff}} newcommand{\Aut}{\operatorname{Aut}} newcommand{\Ind}{\operatorname{Ind}} DeclareMathOperator{\sign}{sign} renewcommand{\hbar}{\bar{h}} newcommand{\zbar}{\bar{z}} newcommand{\integral}[3][]{\int_{#3}\dd[#1]{#2}} rep: representation, module irrep: irreducible representation spec: spectrum

newcommand{\set}[1]{{#1}} newcommand{\pdv}[2]{\frac{\partial #1}{\partial #2}} newcommand{\myalign}[1]{\begin{aligned}#1\end{aligned}} newcommand{\myarray}[2]{\begin{array}{#1}#2\end{array}} renewcommand{\vev}[1]{\langle #1\rangle} newcommand{\p}{\partial} newcommand{\n}{\nabla} renewcommand{\inf}{\infty} newcommand{\Diff}{\operatorname{Diff}} newcommand{\Aut}{\operatorname{Aut}} newcommand{\End}{\operatorname{End}} newcommand{\Ind}{\operatorname{Ind}} newcommand{\sign}{\operatorname{sign}} renewcommand{\hbar}{\bar{h}} newcommand{\wbar}{\bar{w}} newcommand{\zbar}{\bar{z}} renewcommand{\a}{\alpha} renewcommand{\b}{\beta} newcommand{\g}{\gamma} newcommand{\G}{\Gamma} renewcommand{\d}{\delta} newcommand{\D}{\Delta} newcommand{\e}{\epsilon} newcommand{\ve}{\varepsilon} renewcommand{\k}{\kappa} renewcommand{\l}{\lambda} renewcommand{\L}{\Lambda} newcommand{\s}{\sigma} renewcommand{\S}{\Sigma} renewcommand{\r}{\rho} renewcommand{\o}{\omega} renewcommand{\O}{\Omega} renewcommand{\th}{\theta} newcommand{\Th}{\Theta} renewcommand{\t}{\tau} newcommand{\z}{\zeta} newcommand{\mbf}[1]{\mathbf{#1}} newcommand{\mb}[1]{\mathbb{#1}} newcommand{\mc}[1]{\mathcal{#1}} newcommand{\mf}[1]{\mathfrak{#1}} newcommand{\R}{\mathbb{R}} newcommand{\N}{\mathbb{N}} newcommand{\Z}{\mathbb{Z}} newcommand{\C}{\mathbb{C}} newcommand{\V}{\mathbb{V}} newcommand{\W}{\mathbb{W}} newcommand{\cP}{\mathcal{P}} newcommand{\cO}{\mathcal{O}} newcommand{\cA}{\mathcal{A}} newcommand{\cB}{\mathcal{B}} newcommand{\cC}{\mathcal{C}} newcommand{\cD}{\mathcal{D}} newcommand{\cH}{\mathcal{H}} newcommand{\cL}{\mathcal{L}} newcommand{\glie}{\mathfrak{g}} newcommand{\gllie}{\mathfrak{gl}} newcommand{\sllie}{\mathfrak{sl}} newcommand{\olie}{\mathfrak{o}} newcommand{\solie}{\mathfrak{so}} newcommand{\isolie}{\mathfrak{iso}} newcommand{\splie}{\mathfrak{sp}} newcommand{\nlie}{\mathfrak{n}} newcommand{\plie}{\mathfrak{p}} newcommand{\blie}{\mathfrak{b}} newcommand{\hlie}{\mathfrak{h}} newcommand{\klie}{\mathfrak{k}}

这个帖子存放一些技巧。 Markdown 和 LaTeX 的相互转换问题目前没有很好的解决方案。 在 MathJax 中启用一些包的方式为 \require{}，其支持的包可见 MathJax 文档 。 关于 MathJax 的一些介绍可见 SE 的帖子。 Markdown 基础语法可见 Markdown 中文文档。 交换图的绘制， LaTeX 中我用 tikz-cd 和 tikz 在线编辑器画简单的图，MathJax 的待补充，我好像看到 Colliot 有介绍。 physics package 文档 基于 AxoDraw 的绘图界面 JaxoDraw，优点是绘图中兼容 LaTeX，缺点是 UI 太老，功能单一。 尤亦庄写的一个常见包的推荐。 放弃 Lyx 和 Typora。

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