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# 物理代写|多体物理作业代写many-body physics代考|Feynman Diagrams and Green Functions

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## 物理代写|多体物理作业代写many-body physics代考|Feynman Diagrams

Feynman introduced a diagrammatic notation which will help us systematically enumerate all of the perturbative contributions which we generate using Wick’s theorem. This diagrammatic notation will have the added benefit of having a simple physical interpretation which will guide our intuition about physical processes.

Suppose we want to construct a matrix element at $n^{\text {th }}$ order in perturbation theory.
We draw a diagram containing $n$ vertices with 4 lines emanating from each vertex. Each such vertex represents a factor of $\left(\partial_{k} u_{k}\right)^{4}$. The lines emanating from the vertices can be connected. Each such connection represents a contraction. We will call such a line an internal line. The remaining (uncontracted) lines – external lines-represent incoming and outgoing phonons. We will adopt the convention that incoming phonon lines enter at the left of the diagram while outgoing phonon lines exit at the right of the diagram.

The first contribution which we considered in chapter 3 (4.47) can be represented as:

Given such a diagram – a Feynman diagram – you can immediately reconstruct the expression which it represents according to the following rules:

• Assign a directed momentum and energy to each line. For external lines, the momentum is directed into or out of the diagram for, respectively, incoming and outgoing phonons.
• For each external line with momentum $\vec{k}$, write $|\vec{k}|$.
• For each internal line with momentum and energy $\vec{p}, \omega$ write:
$$\frac{1}{\rho} \int \frac{d^{3} \vec{p}}{(2 \pi)^{3}} \frac{d \omega}{2 \pi}|\vec{p}|^{2} \frac{i}{\omega^{2}-v_{l}^{2} p^{2}+i \delta}$$
• For each vertex with momenta, energies $\left(\vec{p}{1}, \omega{1}\right), \ldots,\left(\vec{p}{4}, \omega{4}\right)$ directed into the vertex, write:
$$g(2 \pi)^{3} \delta\left(\vec{p}{1}+\vec{p}{2}+\vec{p}{3}+\vec{p}{4}\right) 2 \pi \delta\left(\omega_{1}+\omega_{2}+\omega_{3}+\omega_{4}\right)$$
• Imagine labelling the vertices $1,2, \ldots, n$. Vertex $i$ will be connected to vertices $j_{1}, \ldots, j_{m}(m \leq 4)$ and to external momenta $p_{1}, \ldots, p_{4-m}$. Consider a permutation of these labels. Such a permutation leaves the diagram invariant if, for all vertices $i, i$ is still connected to vertices $j_{1}, \ldots, j_{m}(m \leq 4)$ and to external momenta $p_{1}, \ldots, p_{4-m}$. If $S$ is the number of permutations which leave the diagram invariant, we assign a factor $1 / S$ to the diagram.

## 物理代写|多体物理作业代写many-body physics代考|Loop Integrals

Suppose we have a Feynman diagram with $E$ external lines, $I$ internal lines, and $V$ vertices. Suppose, further, that this diagram has $L$ loops (e.g. the first diagram in figure $5.2$ has one loop, while the third, fourth, and fifth have two loops. The second has no loops.). Then, let’s imagine connecting all of the external lines at a single point so that the Feynman diagram defines a polyhedron with $E+I$ edges; $V+1$ vertices – the extra vertex being the one at which the external lines are connected; and $L+E$ faces – with $E$ faces formed as a result of connecting the external lines. According to Euler’s fomula,
$$(# \text { faces })+(# \text { vertices })-(# \text { edges })=2$$
or,
$$L=I-V+1$$

## 物理代写|多体物理作业代写MANY-BODY PHYSICS代考|Green Functions

In the preceding discussion, we have implicitly assumed that external phonon lines are ‘on shell’, i.e. they satisfy $\omega^{2}=\left(\omega_{p}^{l}\right)^{2}$. It does, however, make sense to relax this requirement and allow even the external phonons to be “off-shell”. One reason is that we may want to define a Feynman diagram – or a set of diagrams – which can be part of a larger diagram. In such a case, the lines which enter this part might not be on-shell.

Consider the diagram of figure 5.4a. The shaded circle represents all possible diagrams with 4 external legs. The first few are shown in figure $5.4 \mathrm{~b}$. We will call Chapter 5: Feynman Diagrams and Green Functions$$G\left(p_{1}, p_{2}, p_{3}, p_{4}\right)$$
(We will use $p$ as a shorthand for $\vec{p}, \omega$.) $G\left(p_{1}, p_{2}, p_{3}, p_{4}\right)$ is defined to include the momentum conserving $\delta$ functions,
$$(2 \pi)^{3} \delta\left(\vec{p}{1}+\vec{p}{2}+\vec{p}{3}+\vec{p}{4}\right) 2 \pi \delta\left(\omega_{1}+\omega_{2}+\omega_{3}+\omega_{4}\right)$$
and a propagator
$$\left|\vec{p}{i}\right|^{2} \frac{i}{\omega{i}^{2}-v_{l}^{2} p_{i}^{2}+i \delta}$$
on each external leg.

## 物理代写|多体物理作业代写MANY-BODY PHYSICS代考|FEYNMAN DIAGRAMS

• 为每条线分配一个定向动量和能量。对于外部线，动量分别被引导进或出图表，用于传入和传出声子。
• 对于每条带动量的外线ķ→， 写|ķ→|.
• 对于每个具有动量和能量的内部线p→,ω写：
1ρ∫d3p→(2圆周率)3dω2圆周率|p→|2一世ω2−在l2p2+一世d
• 对于每个具有动量的顶点，能量 $\left(\vec{p} {1}, \omega {1}\right), \ldots,\left(\vec{p} {4}, \omega {4}\对）d一世r和C吨和d一世n吨这吨H和在和r吨和X,在r一世吨和:$
克2圆周率^{3} \delta\left(\vec{p} {1}+\vec{p} {2}+\vec{p} {3}+\vec{p} {4}\right) 2 \pi \三角洲\左\omega_{1}+\omega_{2}+\omega_{3}+\omega_{4}\right\omega_{1}+\omega_{2}+\omega_{3}+\omega_{4}\right
$$• 想象一下标记顶点1,2,…,n. 顶点一世将连接到顶点j1,…,j米(米≤4)和外部动量p1,…,p4−米. 考虑这些标签的排列。如果，对于所有顶点，这样的排列使图保持不变一世,一世仍然连接到顶点j1,…,j米(米≤4)和外部动量p1,…,p4−米. 如果小号是使图保持不变的排列数，我们分配一个因子1/小号到图。 ## 物理代写|多体物理作业代写MANY-BODY PHYSICS代考|LOOP INTEGRALS 假设我们有一个费曼图和外线，一世内部线路，以及在顶点。进一步假设这个图有大号循环和.G.吨H和F一世rs吨d一世一种Gr一种米一世nF一世G在r和5.2H一种s这n和l这这p,在H一世l和吨H和吨H一世rd,F这在r吨H,一种ndF一世F吨HH一种在和吨在这l这这ps.吨H和s和C这ndH一种sn这l这这ps.. 然后，让我们想象将所有外部线连接到一个点，以便费曼图定义一个多面体和+一世边缘；在+1vertices – 额外的顶点是连接外部线的顶点；和大号+和面对 – 与和由于连接外部线而形成的面。根据欧拉公式， (# \text { faces })+(# \text { vertices })-(# \text { edges })=2(# \text { faces })+(# \text { vertices })-(# \text { edges })=2 或者， 大号=一世−在+1 ## 物理代写|多体物理作业代写MANY-BODY PHYSICS代考|GREEN FUNCTIONS 在前面的讨论中，我们隐含地假设外部声子线是“在壳上”，即它们满足ω2=(ωpl)2. 然而，放宽这一要求并允许外部声子“脱壳”是有意义的。一个原因是我们可能想要定义一个费曼图——或一组图——它可以是更大图的一部分。在这种情况下，进入这部分的行可能不在 shell 上。 考虑图 5.4a 的示意图。阴影圆圈代表所有可能的图表，带有 4 个外部腿。前几个如图5.4 b. 我们将第 5 章：费曼图和格林函数$$
G\left(p_{1}, p_{2}, p_{3}, p_{4}\right)
$$(We will use p as a shorthand for \vec{p}, \omega.) G\left(p_{1}, p_{2}, p_{3}, p_{4}\right) is defined to include the momentum conserving \delta functions,$$
(2 \pi)^{3} \delta\left(\vec{p}{1}+\vec{p}{2}+\vec{p}{3}+\vec{p}{4}\right) 2 \pi \delta\left(\omega_{1}+\omega_{2}+\omega_{3}+\omega_{4}\right)
$$and a propagator$$
\left|\vec{p}{i}\right|^{2} \frac{i}{\omega{i}^{2}-v_{l}^{2} p_{i}^{2}+i \delta}

on each external leg.

## Matlab代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。