Hurwitz numbers from Feynman diagrams

热带化超几何$τ$-函数

Tropicalising hypergeometric $τ$-functions

本文为加权Hurwitz数发展了热带几何框架，通过对应定理用热带覆叠表达该数，将已知结构结果推广至所有加权情形，研究多项式性并推导壁交叉公式，同时引入椭圆加权Hurwitz数并证明其热带镜像对称。

一个推广的Ramanujan主定理在Feynman图计算中的应用

A generalized Ramanujan Master Theorem applied to the evaluation of Feynman diagrams

拉马努金大师定理是印度数学家S。Ramanujan求一类定积分的方法。这里使用这种方法来计算与特定费曼图相关的积分值。

Equivalence between the Arquès-Walsh sequence formula and the number of connected Feynman diagrams for every perturbation order in the fermionic many-body problem

From the perturbative expansion of the exact Green function, an exact counting formula is derived to determine the number of different types of connected Feynman diagrams. This formula coincides with the Arquès-Walsh sequence formula in the rooted map theory, supporting the topological connection between Feynman diagrams and rooted maps. A classificatory summing-terms approach is used, in connection to discrete mathematical theory.

电子-光子相互作用：费曼图与接触点

Electron-photon interaction: Feynman diagrams and contact points

本文为费曼图定义了正问题、反问题及接触点的概念，针对电子-光子相互作用给出了正反问题的解，并讨论了费曼图与经典着色图之间的关系。

Graph Minors and the Linear Reducibility of Feynman Diagrams

We look at a graph property called reducibility which is closely related to a condition developed by Brown to evaluate Feynman integrals. We show for graphs with a fixed number of external momenta, that reducibility with respect to both Symanzik polynomials is graph minor closed. We also survey the known forbidden minors and the known structural results. This gives some structural information on those Feynman diagrams which are reducible.

Feynman diagrams and their algebraic lattices

We present the lattice structure of Feynman diagram renormalization in physical QFTs from the viewpoint of Dyson-Schwinger-Equations and the core Hopf algebra of Feynman diagrams. The lattice structure encapsules the nestedness of diagrams. This structure can be used to give explicit expressions for the counterterms in zero-dimensional QFTs using the lattice-Moebius function. Different applications for the tadpole-free quotient, in which all appearing elements correspond to semimodular lattices, are discussed.

Field Calculus: Quantum and Statistical Field Theory without the Feynman Diagrams

For a given base space $M$ (spacetime), we consider the Guichardet space over the Guichardet space over $M$. Here we develop a ''field calculus'' based on the Guichardet integral. This is the natural setting in which to describe Green function relations for Boson systems. Here we can follow the suggestion of Schwinger and develop a differential (local field) approach rather than the integral one pioneered by Feynman. This is helped by a DEFG (Dyson-Einstein-Feynman-Guichardet) shorthand which greatly simplifies expressions. This gives a convenient framework for the formal approach of Schwinger…

费曼图的梅林-巴恩斯表示，微分方程的线性系统和多项式解

Mellin-Barnes representations of Feynman diagrams, linear systems of differential equations, and polynomial solutions

我们认为，费曼图的梅林-巴恩斯表示可以用于获得线性系统的齐次微分方程的原始费曼图与传播的任意权力，而不求助于集成的部分技术。这些微分方程组可以用于（i）基本函数集的微分约化和（ii）计算主积分的个数。

标量场理论中单圈Feynman图的约化

Reduction of One Loop Feynman Diagrams in Scalar Field Theory

这是一个历史笔记。1979年，我们在一份俄罗斯期刊上发表了一篇论文，名为Vestnik Leingradskogo Gosudarstvennogo Universiteta。我们考虑了大质量标量量子场论。评价了单回路费曼图。定理证明了一个具有多条内部线[大于时空维数]的回路图可以表示为一个内部线数等于时空维数[乘以树形图]的回路图。这是英文翻译。

费曼图与拉克斯对方程

Feynman Diagrams and Lax Pair Equations

我们找到了一个对应于Hopf代数特征群的Connes-Kreimer Birkhoff分解的拉克斯对方程.这个流保持了反项的局部性。特别地，我们得到了Feynman规则给出的特征的流，并将此流与重整化群流联系起来。

费曼图和KdV层次

Feynman diagrams and the KdV hierarchy

曲线的模空间上的稳定上同调类的相交数的生成级数满足弦方程和KdV族。Kontsevich对这一结果的原始证明使用了矩阵模型和矩阵艾里方程。威滕然后改写孔采维奇的结果在Virasoro代数，通过巧妙的混合费曼图技术和积分的部分与一些“相当可怕的选择”的被积函数。在这份说明中，我们将展示如何威滕的强大的选择可以交换标准的费曼图操作，只要一个适当扩大类的费曼图发生在证明。

Geometrical splitting and reduction of Feynman diagrams

A geometrical approach to the calculation of N-point Feynman diagrams is reviewed. It is shown that the geometrical splitting yields useful connections between Feynman integrals with different momenta and masses. It is demonstrated how these results can be used to reduce the number of variables in the occurring functions.

An exact solution method for the enumeration of connected Feynman diagrams

We completely generalize previous results related to the counting of connected Feynman diagrams. We use a generating function approach, which encodes the Wick contraction combinatorics of the respective connected diagrams. Exact solutions are found for an arbitrary number of external legs, and a general algorithm is implemented for this calculus. From these solutions, we calculate many asymptotics expansion terms for a simple analytical tool (Taylor expansion theorem). Our approach offers new perspectives in the realm of Feynman diagrams enumeration (or zero-dimensional quantum field theory).

Feynman Diagrams and Rooted Maps

The Rooted Maps Theory, a branch of the Theory of Homology, is shown to be a powerful tool for investigating the topological properties of Feynman diagrams, related to the single particle propagator in the quantum many-body systems. The numerical correspondence between the number of this class of Feynman diagrams as a function of perturbative order and the number of rooted maps as a function of the number of edges is studied. A graphical procedure to associate Feynman diagrams and rooted maps is then stated. Finally, starting from rooted maps principles, an original definition of the genus of …

具有有限个连通Feynman图的拓扑场论

A Topological Field Theory With a Finite Number of Connected Feynman Diagrams

构造了一种新的拓扑场论，它的特征是类似于非交换的Chern-Simons场论的立方相互作用，但仍保留了交换情形的简单性.这个理论的微扰展开实际上只包含两个连通的费曼图，即传播子和一个三顶点。除了高斯链接数，威尔逊圈振幅产生进一步的拓扑不变量，其物理和数学意义进行了研究。

Feynman图应用的微分几何和物理基础

The Differential Geometry and Physical Basis for the Applications of Feynman Diagrams

本文回顾了规范理论的发展，最终发展到杨-米尔斯理论，以及微分几何在场论中的应用，包括纤维丛上的联系。最后，我们看到前面是如何用来解释费曼图出现在费曼邮票在2005年5月发行。第2版包括费曼图，第3版纠正了错别字，第4版包括了一个附录，用于推导杨-米尔斯变换和场强。第5版显示，该文章已发表在AMS的通知中，并于2009年7月以中文翻译出现在中国科学院的期刊上。它还纠正了一些错别字，并在附录中增加了杨-米尔斯场强的启发式推导。

Group invariants for Feynman diagrams

It is well-known that the symmetry group of a Feynman diagram can give important information on possible strategies for its evaluation, and the mathematical objects that will be involved. Motivated by ongoing work on multi-loop multi-photon amplitudes in quantum electrodynamics, here I will discuss the usefulness of introducing a polynomial basis of invariants of the symmetry group of a diagram in Feynman-Schwinger parameter space.

与Feynman图有关的几个无穷级数

Some infinite series related to Feynman diagrams

给出了费曼图计算中出现的一些无穷级数的结果，其中许多级数与欧拉级数相似。这些包括一维、二维和三维系列。这些级数的和可以用超几何函数的各种积分表示来计算，并表示为$ζ(2)$，$ζ(3)$，加泰罗尼亚常数$G$和${\rm Cl}_2(π/3)$，其中${\rm Cl}_2(θ)$是克劳森函数。

Bold Feynman diagrams and the Luttinger-Ward formalism via Gibbs measures. Part I: Perturbative approach

Many-body perturbation theory (MBPT) is widely used in quantum physics, chemistry, and materials science. At the heart of MBPT is the Feynman diagrammatic expansion, which is, simply speaking, an elegant way of organizing the combinatorially growing number of terms of a certain Taylor expansion. In particular, the construction of the `bold Feynman diagrammatic expansion' involves the partial resummation to infinite order of possibly divergent series of diagrams. This procedure demands investigation from both the combinatorial (perturbative) and the analytical (non-perturbative) viewpoints. In …

Periods of Feynman Diagrams and GKZ D-Modules

We study differential equations for Feynman amplitudes and we show that the corresponding D-module is isomorphic to a GKZ D-modules. We show that the sheaf of solutions to the D-module is isomorphic to a certain relative homology and the amplitudes are periods of a relative motive. Using these ideas, we develop a method of regularization which specializes to dimensional regularization and analytic regularization.

费曼图、微分约化与超几何函数

Feynman Diagrams, Differential Reduction, and Hypergeometric Functions

我们将提出一些（正式的）论点，任何费曼图可以理解为一个特定的情况下，霍恩型多变量超几何函数。这种类型的方法来评价费曼图的优点和缺点进行了讨论。

A recursive enumeration of connected Feynman diagrams with an arbitrary number of external legs in the fermionic non-relativistic interacting gas

In this work, we generalize a recursive enumerative formula for connected Feynman diagrams with two external legs. The Feynman diagrams are defined from a fermionic gas with a two-body interaction. The generalized recurrence is valid for connected Feynman diagrams with an arbitrary number of external legs and an arbitrary order. The recurrence formula terms are expressed in function of weak compositions of non-negative integers and partitions of positive integers in such a way that to each term of the recurrence correspond a partition and a weak composition. The foundation of this enumeration …

现代求和方法与二圈和三圈费曼图的计算

Modern Summation Methods and the Computation of 2- and 3-loop Feynman Diagrams

通过基于差域的符号求和方法，我们提出了一个通用的策略，将确定的多和，例如，从超几何项和调和和到不定套和与积。我们成功地完成了这项任务，我们所有的具体计算2-loop和3-loop大量的单尺度费曼图与当地运营商插入。

行人和数学家的费曼图

Feynman diagrams for pedestrians and mathematicians

这是一个简单的数学介绍到费曼图技术，这是一个标准的物理工具，写微扰展开的路径积分附近的临界点的行动。我从一个有限维的情况下（实际上属于多变量微积分比物理）严格的治疗，然后使用一个简单的“字典”将这些结果翻译到一个无限维的情况。标准方法，如规格固定和Faddeev-Popov鬼也包括在内。由此产生的费曼图系列通常可以严格地使用，而不需要任何初始物理理论的参考（人们可以“扫到地毯下”）。这个想法是说明了一个例子的陈-西蒙斯理论，导致通用的有限型不变量的节点和3-流形。

Counting the number of master integrals for sunrise-type Feynman diagrams via Mellin-Barnes representation

A number of irreducible master integrals for L-loop sunrise-type and bubble Feynman diagrams with generic values of masses and external momenta are explicitly evaluated via Mellin-Barnes representation.

超几何函数微分约简算法在费曼图计算中的应用

Differential Reduction Algorithms for Hypergeometric Functions Applied to Feynman Diagram Calculation

我们描述了微分归约算法在费曼图计算中的应用。我们说明的过程中的广义超几何函数的上下文中，并给出一个例子的一种类型的q-环泡图。

New results for a two-loop massless propagator-type Feynman diagram

We consider the two-loop massless propagator-type Feynman diagram with an arbitrary (non-integer) index on the central line. We analytically prove the equality of the two well-known results existing in the literature which express this diagram in terms of ${}_3F_2$-hypergeometric functions of argument $-1$ and $1$, respectively. We also derive new representations for this diagram which may be of importance in practical calculations.

Feynman图的微分约化方法

Differential Reduction Techniques for the Evaluation of Feynman Diagrams

稳定化约化方法在评价QCD中出现的高阶微扰图和LHC中的混合QCD-电弱辐射修正中将是重要的。微分归约技术对于将超几何函数与参数的移位值相关联是有用的。我们提出了一个命题有关的主积分的数量在扩展的费曼图的微分减少的导数的数量。

Feynman图的超几何展开和微分约化

The Epsilon Expansion of Feynman Diagrams via Hypergeometric Functions and Differential Reduction

在大型强子对撞机和预期的未来对撞机上，对混合电弱和QCD过程进行辐射校正所需的高阶图将需要相关费曼图的数值稳定表示。超几何表示提供了一个分析框架，这是有用的推导这种稳定的表示。我们讨论了费曼图的减少，掌握积分，并比较积分的部分方法，微分减少超几何函数。我们描述的问题，建设高阶项的展开，并在这样的扩展所产生的功能的特点。

消失圈与费曼图的奇点分析

Vanishing cycles and analysis of singularities of Feynman diagrams

本研究利用Mayer-Vietoris谱序列分析费曼圈积分的消失圈并完成分类，推导了奇点附近渐近展开的显式系数，为轻子-强子散射实验辐射修正中的高阶费曼图精确计算提供了框架基础。

关于量纲调节费曼图的ε展开的几点注记

Some remarks on the epsilon-expansion of dimensionally regulated Feynman diagrams

本文讨论了维数调节的费曼积分的ε展开式的构造问题。对于某些类型的图，ε展开式的任意项可以用与多项式相关的对数正弦积分来表示。它示出了如何解析这些功能的延续，可以构造在广义尼尔森多项式。

规范理论振幅来自三次标量费曼图

Gauge Theory Amplitudes from Cubic Scalar Feynman diagrams

本文通过Witten开弦场论在场论极限中揭示了规范理论振幅与标量量子场论三价费曼图重整化积分元的联系。

Cut-and-join structure and integrability for spin Hurwitz numbers

Spin Hurwitz numbers are related to characters of the Sergeev group, which are the expansion coefficients of the Q Schur functions, depending on odd times and on a subset of all Young diagrams. These characters involve two dual subsets: the odd partitions (OP) and the strict partitions (SP). The Q Schur functions Q_R with R\in SP are common eigenfunctions of cut-and-join operators W_Δwith Δ\in OP. The eigenvalues of these operators are the generalized Sergeev characters, their algebra is isomorphic to the algebra of Q Schur functions. Similarly to the case of the ordinary Hurwitz numbers, the …

多圈Feynman积分与共形量子力学

Multi-loop Feynman integrals and conformal quantum mechanics

提出了多圈费曼图D维积分解析计算的新代数方法。我们表明，已知的分析方法的评价多圈费曼积分，如积分的部分和星三角关系的方法，可以大大简化使用这种代数方法。为了证明多圈费曼图的分析评价的代数方法的优点，我们计算梯形图的无质量$ϕ^3$理论。使用我们的代数方法，我们表明，特殊类的费曼图的评价问题减少到特定的量子力学问题的格林函数的计算。特别地，$ϕ^3$标量场理论中阶梯无质量图的积分由共形量子力学的格林函数给出。

费曼图参数表示中的分子

Numerators in Parametric Representations of Feynman Diagrams

本文利用 Dodgson 恒等式简化费曼图参数表示中分子的处理，并在无质量情形下通过约化为真空图补全，显著降低分母中 Symanzik 多项式的最大幂次。

Exact maximum-entropy estimation with Feynman diagrams

A classical longstanding open problem in statistics is finding an explicit expression for the probability measure which maximizes entropy with respect to given constraints. In this paper a solution to this problem is found, using perturbative Feynman calculus. The explicit expression is given as a sum over weighted trees.

用多重指标重整化费曼图

Renormalising Feynman diagrams with multi-indices

本文提出用多重指标构造新的 Hopf 代数，直接对原始测度展开式进行 BPHZ 重整化以获得量子场论重整化测度，并以 Φ⁴ 测度为例验证方法。

“统一的第六根”与费曼图：超几何函数逼近的观点

"Sixth root of unity" and Feynman diagrams: hypergeometric function approach point of view

本文简要讨论了广义超几何函数的ε展开式所产生的超越常数及其与“六次方单位根”的关系。"

超几何函数、超几何展开式和费曼图

Hypergeometric functions, their epsilon expansions and Feynman diagrams

我们回顾了费曼图的超几何函数方法。特别考虑到建设的洛朗扩展。作为一个例子，我们描述了一个收集的物理上重要的单圈顶点图，这种方法是有用的。