### 物理代写|量子场论代写Quantum field theory代考|PHYS4125

statistics-lab™ 为您的留学生涯保驾护航 在代写量子场论Quantum field theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写量子场论Quantum field theory代写方面经验极为丰富，各种代写量子场论Quantum field theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|量子场论代写Quantum field theory代考|The Delta Function

Besides reviewing the delta function, this section introduces the idea of a smooth cutoff, how to get rid of the troublesome part of an integral.
Mathematically, the delta “function” $\delta$ is simply the distribution given by
$$\delta(\zeta)=\zeta(0)$$
for any test function $\zeta \in \mathcal{S}$. Pretending that the delta “function” is actually a true function we will shamelessly write $(1.12)$ as
$$\zeta(0)=\int \mathrm{d} x \zeta(x) \delta(x)$$
The name “delta function” is historical. Physicists have been using this object long before distributions were invented.

Exercise 1.4.1 (a) Convince yourself that it makes perfect sense to say that $\delta(x)=0$ if $x \neq 0$.
(b) Make sure that you understand that despite the terminology, the delta function $\delta$ is not a function in the mathematical sense and that the quantity $\delta(0)$ makes no sense.

(c) Convince yourself from (1.13) that, in the words of physicists, “the delta function $\delta$ is the function of $x$ which is equal to zero for $x \neq 0$ and to infinity for $x=0$, but in such a way that its integral is 1”.
(d) Convince yourself that the derivative of the delta function $\delta$, i.e. the distribution $\delta^{\prime}$ given by $\delta^{\prime}(\zeta)=-\zeta^{\prime}(0)$ “does not look at all like a function”.
For $a \neq 0$ let us define $\delta(a x)$ by
$$\int \mathrm{d} x \zeta(x) \delta(a x):=\frac{1}{|a|} \int \mathrm{d} x \zeta(x / a) \delta(x)=\frac{1}{|a|} \zeta(0)$$
so that
$$\delta(a x)=\frac{1}{|a|} \delta(x)$$
and in particular $\delta(-x)=\delta(x) .^{13}$

## 物理代写|量子场论代写Quantum field theory代考|The Fourier Transform

Besides reviewing some basic facts about Fourier transforms, this section provides the first example of certain calculations common in physics.

The Fourier transform will play a fundamental role. ${ }^{17}$ Let us temporarily denote by $\mathcal{F}{m}$ the Fourier transform, that is $$\mathcal{F}{m}(f)(x)=\frac{1}{\sqrt{2 \pi}} \int \mathrm{d} y \exp (-\mathrm{i} x y) f(y),$$
where the subscript $m$ reminds you that this is the way mathematicians like to define it (whereas our choice of normalization will be different). The right-hand side is defined for $f$ integrable, and in particular for a Schwartz function $f \in \mathcal{S}$. Using integration by parts in the first equality, and differentiation under the integral sign in the second one, we obtain the fundamental facts that for any test function $f$,
$$\mathcal{F}{m}\left(f^{\prime}\right)(x)=\mathrm{i} x \mathcal{F}{m}(f)(x) ; \mathcal{F}{m}(x f)=\mathrm{i} \mathcal{F}{m}(f)^{\prime},$$
where we abuse notation by denoting by $x f$ the function $x \mapsto x f(x)$. An essential fact is that the Fourier transform of a test function is a test function. The details of the proof are a bit tedious, and are given in Section L.1. 18
The Plancherel formula is the equality
$$\left(\mathcal{F}{m}(f), \mathcal{F}{m}(g)\right)=(f, g),$$
for $f, g \in \mathcal{S}$, where $(f, g)=\int \mathrm{d} x f(x)^{} g(x)$. It is very instructive to “prove” this formula the way a physicist would, since this is a very simplified occurrence of the type of computations that are ubiquitous in Quantum Field Theory: \begin{aligned} \left(\mathcal{F}{m}(f), \mathcal{F}{m}(g)\right) &=\frac{1}{2 \pi} \int \mathrm{d} x\left(\int \mathrm{d} y_{1} \exp \left(-\mathrm{i} x y_{1}\right) f\left(y_{1}\right)\right)^{} \int \mathrm{d} y_{2} \exp \left(-\mathrm{i} x y_{2}\right) g\left(y_{2}\right) \ &=\frac{1}{2 \pi} \iint \mathrm{d} y_{1} \mathrm{~d} y_{2} f\left(y_{1}\right)^{*} g\left(y_{2}\right) \int \mathrm{d} x \exp \left(\mathrm{i} x\left(y_{1}-y_{2}\right)\right) \end{aligned} $=\iint \mathrm{d} y_{1} \mathrm{~d} y_{2} f\left(y_{1}\right)^{} g\left(y_{2}\right) \delta\left(y_{1}-y_{2}\right)$ $=\int \mathrm{d} y_{2} f\left(y_{2}\right)^{} g\left(y_{2}\right)$
$=(f, g)$,

## 物理代写|量子场论代写Quantum field theory代考|Basic Setting

We must first stress that one does not “prove” the basic concepts of Quantum Mechanics, or of any physical theory. One builds models, and the ultimate test of the validity of such models is whether they predict correctly the results of experiments. Still, one strives for mathematical consistency and elegance. ${ }^{2}$

The purpose of Mechanics is to describe the state of mechanical systems and to determine their time-evolution. Consider, for example, one of the simplest mechanical systems: a massive dimensionless point. Its state at a given time is described by its position and velocity. That this, indeed, is the correct way to specify even such a simple system is by itself a deep fact: it is the position and the velocity (and not, say, the acceleration) at a given time that determine the future motion of the point. This fact is delicate enough that apparently it is not understood by the general public, which seems to still believe that an astronaut stepping out of the International Space Station (ISS) will start falling toward Earth. ${ }^{3}$

As we pointed out, it is not easy to relate the principles of Quantum Mechanics to actual physical experiments. On the positive side, this means that there is no real loss in starting to learn them even with little knowledge of Classical Mechanics. A very brief introduction to Classical Mechanics will be given in Sections $6.4$ and $6.5$.

Principle $1^{4}$ The state of a physical system is described by a unit vector in a complex ${ }^{5}$ Hilbert space $\mathcal{H}$

This Hilbert space is called the state space, and the unit vector is called the state vector. The state space will always be either finite-dimensional or separable (i.e. admitting a countable orthonormal basis). As in the case of the lowly classical massive point, the correct description of the state of a system encompasses a huge amount of wisdom. The fact that it is done by a vector in Hilbert space allows some of the most surprising features of Quantum Mechanics. It makes physical sense to consider linear combinations of different states. ${ }^{6}$ The principles of Quantum Mechanics, and this one in particular, really do not appeal to our everyday intuition. There seems to be no remedy to this situation.

## 物理代写|量子场论代写Quantum field theory代考|The Delta Function

d(G)=G(0)

G(0)=∫dXG(X)d(X)
“delta函数”这个名字是历史性的。早在分布发明之前，物理学家就一直在使用这个对象。

(b) 确保您了解尽管有术语，但 delta 函数d不是数学意义上的函数，并且数量d(0)没有意义。

(c) 从 (1.13) 中说服自己，用物理学家的话来说，“δ 函数d是函数X这等于零X≠0和无穷大X=0, 但其积分为 1”。
(d) 说服自己 delta 函数的导数d，即分布d′由d′(G)=−G′(0)“看起来一点也不像函数”。

∫dXG(X)d(一个X):=1|一个|∫dXG(X/一个)d(X)=1|一个|G(0)

d(一个X)=1|一个|d(X)

## 物理代写|量子场论代写Quantum field theory代考|The Fourier Transform

F米(F)(X)=12圆周率∫d是经验⁡(−一世X是)F(是),

F米(F′)(X)=一世XF米(F)(X);F米(XF)=一世F米(F)′,

Plancherel 公式是等式

(F米(F),F米(G))=(F,G),

(F米(F),F米(G))=12圆周率∫dX(∫d是1经验⁡(−一世X是1)F(是1))∫d是2经验⁡(−一世X是2)G(是2) =12圆周率∬d是1 d是2F(是1)∗G(是2)∫dX经验⁡(一世X(是1−是2))=∬d是1 d是2F(是1)G(是2)d(是1−是2) =∫d是2F(是2)G(是2)
=(F,G),

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。