### 物理代写|量子场论代写Quantum field theory代考|PHYS5125

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• 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|Dimension

The numerical value of many physical quantities depends on the unit one chooses to measure them. My height is $1.8 \mathrm{~m}$, or $180 \mathrm{~cm}$, or $1.90 \times 10^{-16}$ light-years. The use of light-years here as a unit is weird, but not so much more than the use of centimeters to measure distances at the scale of a nucleon as many textbooks do. (A nucleon has a size of about $10^{-15} \mathrm{~m}=$ $10^{-13} \mathrm{~cm}$.) Tradition unfortunately has more weight than rationality in these matters.

The concept of “physical dimension” (which definitely differs from dimension in the mathematical sense) expresses how the numerical value of a physical quantity depends on the units you choose to measure it. A distance has dimension $[l]$ where $l$ stands of course for length. If you increase the unit of length by a factor 100 , the corresponding measure decreases by a factor $100: 100 \mathrm{~cm}=1 \mathrm{~m}$. Then a surface has dimension $\left[l^{2}\right]:(100)^{2} \mathrm{~cm}^{2}=$ $1 \mathrm{~m}^{2}$. A volume has dimension $\left[l^{3}\right]: 1 \mathrm{~km}^{3}=\left(10^{3}\right)^{3} \mathrm{~m}^{3}=10^{9} \mathrm{~m}^{3}$. The unit of time can be chosen independently from the unit of length. Time has dimension [t], so speed, which is a distance divided by a time, has dimension $\left[l t^{-1}\right]$. Thus $1 \mathrm{~m} / \mathrm{s}=3,600 \mathrm{~m} / \mathrm{h}=3.6 \mathrm{~km} / \mathrm{h}$. Acceleration, which is a change of speed divided by a time, has dimension $\left[l t^{-2}\right]$. It is of course a convention to choose time and length as fundamental quantities. One could make other choices, such as choosing time and speed as fundamental quantities. This is indeed basically what is actually done. Since 1983 , in the international system the speed of light is defined to be exactly
$$c=299,792,458 \mathrm{~m} / \mathrm{s}$$
and this serves as a definition of the meter given the unit of time. ${ }^{1}$
A formula in physics must give a correct result independently of the system of units used. This is a strong constraint. This is why it often makes sense to multiply or divide quantities of different dimensions, but it never makes sense to add them. As we learn in kindergarten,you do not add pears with bananas. Furthermore, when a quantity occurs in a formula as the argument of, say, an exponential, it must be dimensionless, i.e. its value must be independent of the unit system. To understand a formula in physics it always helps to check that it makes sense with respect to dimension, a task we will perform many times.

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

Since to enjoy this topic one has to read the work of physicists, it is best to adopt their notation from the beginning. Complex numbers play a central role, and the conjugate of a complex number $a$ is denoted by $a^{*}$. Even some of the best authors let the reader decide

whether $i$ denotes a complex number with $i^{2}=-1$ or an integer index. Since this requires no extra work, the complex number will be denoted by $i$, so that $i^{*}=-i$.

When working with complex Hilbert spaces we adopt the convention that the inner product $(\cdot,$, is anti-linear in the first variable (while often mathematicians use the convention that it is anti-linear in the second variable). One says that the inner product is sesqui-linear. That is, as another example of our notation for complex conjugation, we write
$$(a x, y)=a^{}(x, y)$$ for any vectors $x, y$ and any complex number $a$. Moreover $$(y, x)=(x, y)^{} .$$
The norm $|x|$ of a vector $x$ is given by $|x|^{2}=(x, x)$, and we recall the Cauchy-Schwarz inequality
$$|(x, y)|^{2} \leq|x|^{2}|y|^{2}$$
where $|a|$ denotes the modulus of the complex number $a$. A basic example of a complex Hilbert space ${ }^{5}$ is the space $\mathbb{C}^{n}$, where the inner product is defined by $(x, y)=\sum_{i \leq n} x_{i}^{} y_{i}$, with the obvious notation $x=\left(x_{i}\right){i \leq n}$. Another very important example is the space $L^{2}(\mathbb{R})$ of complex-valued functions $f$ on the real line for which $\int{\mathbb{R}}|f|^{2} \mathrm{~d} x=\int_{\mathbb{R}}|f(x)|^{2} \mathrm{~d} x<\infty$, where $|f(x)|$ denotes the modulus of $f(x)$. The inner product is then given by $(f, g)=$ $\int_{\mathbb{R}} f^{} g \mathrm{~d} x$. A physicist would actually write
$$(f, g)=\int_{-\infty}^{\infty} \mathrm{d}^{1} x f(x)^{*} g(x),$$
where the superscript 1 refers to the fact that one integrates for a one-dimensional measure. The reason for which the $\mathrm{d}^{1} x$ is put before the function to integrate is that this makes the formula easier to parse when there are multiple integrals. We will use this convention systematically. We will not however mention the dimension in which we integrate when this dimension is equal to one.

An operator $A$ on a finite-dimensional Hilbert space $\mathcal{H}$ is simply a linear map $\mathcal{H} \rightarrow \mathcal{H}$. Its adjoint $A^{\dagger}$ is defined by
$$\left(A^{\dagger}(x), y\right)=(x, A(y))$$
for all vectors $x, y$. (Mathematicians would use the notation $A^{*}$ rather than $A^{\dagger}$.)

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

Laurent Schwartz invented the theory of distributions to give a rigorous meaning to many formal calculations of physicists. The theory of distributions is a fully rigorous part of mathematical analysis. In the main text however we will use only the very basics of this theory at a purely informal level. In Appendix $L$ the reader may find an introduction to rigorous methods.

We will consider distributions on $\mathbb{R}^{n}$ but here we assume $n=1$. The central object is the space $\mathcal{S}=\mathcal{S}(\mathbb{R})$ of rapidly decreasing functions, called also test functions or Schwartz functions. A complex-valued ${ }^{7}$ function $\zeta$ on $\mathbb{R}$ is a test function if it has derivatives of all orders and if for any integers $k, n \geq 0$ one has 8
$$\sup _{x}\left|x^{n} \zeta^{(k)}(x)\right|<\infty .$$
A distribution is simply a linear functional (which also satisfies certain regularity conditions which will not concern us before we reach Appendix L, as they will be satisfied in all the examples we will consider). That is, a distribution $\Phi$ is a complex linear map from $\mathcal{S}$ to $\mathbb{C}$, and for each test function $\zeta$ the number $\Phi(\zeta)$ makes sense. Such a distribution should actually be called a tempered distribution, but we will simply say “distribution” since we will

hardly consider any other type of distribution. Tempered distributions are also known under the name of generalized functions. This name has the advantage of explaining the point of the theory of distributions: it generalizes the theory of functions. Indeed, a sufficiently well-behaved function ${ }^{9} f$ defines a distribution (= generalized function) $\Phi_{f}$ by the formula
$$\Phi_{f}(\zeta)=\int \mathrm{d} x \zeta(x) f(x)$$

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

“物理维度”的概念（与数学意义上的维度绝对不同）表达了物理量的数值如何取决于您选择测量它的单位。距离有维度[l]在哪里l当然代表长度。如果将长度单位增加 100 倍，则相应的度量会减少一个倍数100:100 C米=1 米. 然后一个表面有尺寸[l2]:(100)2 C米2= 1 米2. 卷有维度[l3]:1 ķ米3=(103)3 米3=109 米3. 时间单位可以独立于长度单位来选择。时间有维度[t]，所以速度，即距离除以时间，有维度[l吨−1]. 因此1 米/s=3,600 米/H=3.6 ķ米/H. 加速度是速度的变化除以时间，它有维度[l吨−2]. 选择时间和长度作为基本量当然是一种惯例。人们可以做出其他选择，例如选择时间和速度作为基本量。这确实基本上是实际所做的。自1983年以来，在国际体系中，光速被精确定义为

C=299,792,458 米/s

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

(一个X,是)=一个(X,是)对于任何向量X,是和任何复数一个. 而且

(是,X)=(X,是).

|(X,是)|2≤|X|2|是|2

(F,G)=∫−∞∞d1XF(X)∗G(X),

(一个†(X),是)=(X,一个(是))

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

Laurent Schwartz 发明了分布理论，为物理学家的许多形式计算赋予了严格的含义。分布理论是数学分析的一个完全严谨的部分。然而，在正文中，我们将仅在纯粹的非正式层面上使用该理论的基础知识。在附录中大号读者可能会找到有关严格方法的介绍。

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## MATLAB代写

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