%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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二十多年来AdS/CFT和十年多年来Conformal Bootstrap的进展极大的加深了我们对于非微扰物理的理解，而这或多或少起源于有效理论的范式在各个物理领域中的实践。有效理论的应用依然在拓宽，另一方面在中文区相关的资源似乎并不够充足。因此译者想翻译这方面的始祖文章之一，Wilson的《重整化群和\epsilon展开》，另一篇被翻译的相关的简短介绍是《重整化群和临界现象》。 译者第一次尝试翻译，人名不翻译，一些在文献中第一次出现的名词会用括号标注，初稿可能跳过一些不想翻译的段落，以后逐步完善。 重整化群和epsilon展开 Kenneth G. Wilson, J. Kogut 目录 介绍 重整化群和物理中的连贯性问题 目前的一些参考文献 Ising模型的一些基本结论 a 系统在临界温度附近的基本性质 对解析性的探寻：1. 平均场理论；2. Kadanoff理论 重整化群的平凡例子：Gauss型模型 s^4模型 .简化版的重整化群变换 \epsilon展开和一个非平凡的不动点 线性化方程和对\nu的计算 s^4模型（续） 无关变量和\epsilon展开 完整的重整化群变换 近似的递推公式 Polyakov的推导 一些数值结果 a 关于近似递推公式进一步的结果 自旋n点函数的标度定理 slow transients和对\epsilon的消除 临界指数的Feynman图计算（\epsilon展开） 在4-\epsilon维时空中张量算子的量纲 统计力学和场论的联系 精确重整化群的微分形式 a 重整化群变换的拓扑（不动点，轨迹和子空间） 不动点，子空间和重整化 多不动点，区域和普适性 不动点和反常量纲 目前为止对4维\phi^4理论非平凡不动点的徒劳探寻 结语 致谢附录：精确重整化群方程的简单解参考文献

这个「编辑」按钮终于不再是摆设！

Haag Kastler axioms (Lorentzian, flat) sterwalder Schrader axioms(Euclidean, flat) Bruneffi-Fredehhagen-Verch axioms(Lorentzian, curved) or any spacelikehypersurface, ass a Hilbert space, lobally hyperbolic Atiyah-Segal axioms(Euclidean, curved) linear algebra uppose some vector space V over R, (V): C -valued quad forms on V st existing real basis, form is diagonal, eigenvalue in C-0 sum arg(\lambda_i)<\pi \lambda\in C-R_<0 Q_C(V) \subset GL(d,C)/O(d,C), aka ,non-deg sym matrices. Facts: Q is contractible closure in is homotopic to S^1 domain of holomorphy W\subset V, restriction Q(V)\to Q(W) Definition of unitary QFT ordisms_C-met \to nulear Frechet spaces and nuclear maps oth sides are non-unital mor: M,g_C) compact real-analytic manifold with oriented bdry, _C is sections of the Q-bundle. obj: d-1 manifold + collar + metric on collar, neaclear space: proj limits of Hilbert spaces axioms: holomorphicity semi-continuity positivity or unitarity Holomorphicity _\sigma: wave functionals is holomorphic on infinite dim manifold Met_C(collar of \sigma) phi_M: holomorphic on Met_C(M) ovariance Diff(Collar_sigma) acts on H_\sigma eg partial M=0 _M=C, \phi_M is just partition function. g_C \to Z(M,g_C) is holo on the infinite-dim manifold nvariant under Diff(M) M,g_C \to M_C, g^hol_C consider all maps {\tilde{i}: M\to M_C: \tilde{i} close to i } conclusion: _C nstead of allowable metrices \simto allowable totally real submanifold

Borokhov, Topological disorder operators in three-dimensional conformal field theory 发点是3d QED（嗯，数值bootstrap经常讨论的一个场论，可以看Rychkov的综述），场内容包括U(1) gauge field A和N_f个fermion (complex dim=2，我分不清是哪种表示)，这个东西的fermion path integral可能有反常，称为parity anomaly，相消条件是， -\frac{N_f}{2}\in\mathbb{Z} 是Chern-Simons level，用新引入的Chern-Simons term， frac{ik}{4\pi}\int d^3 x \, \epsilon_{123}A_1\partial_2 A_3 消除odd N_f时的反常。具体计算技术我没掌握，以后要做相关的再说。 d QED特殊在于耦合常数的量纲[g]=[m]^{1/2}，微扰场论计算是被\frac{g^2}{\mu}组织的，\mu是某个能量尺度，因此可以期待是super-renormalizable，并且在紫外流到自由场论，在红外强耦合。 篇文章(2002)之前有一些large N的讨论，N充分大的时候红外变得弱耦合，可以按照1/N展开。large N的一个结论是在红外[F]=2，在紫外[F]=1，也能说明红外的强耦合性质。