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## 物理代写|粒子物理代写Particle Physics代考|The Klein-Gordon Equation

We start with the simplest case, the equation for a real, scalar field. In our notation of Chapter 5 , it belongs to the trivial one-dimensional $(0,0)$ representation of the Lorentz algebra. In this case the elements which are at our disposal are the field itself $\phi$ and the four-vector operator of derivation $\partial_\mu$. It is clear that the lowest order, non-trivial, relativistically covariant equation, which can be built with these quantities is
$$\left(\partial_\mu \partial^\mu+m^2\right) \phi(x)=0$$
This is the Klein-Gordon equation. In our usual system of units $\hbar=c=1$ the parameter $m^2$ has the dimensions of $[\mathrm{M}]^2$. We shall often call this equation the massive Klein-Gordon equation, although at this stage we have no real justification for this name. The $m^2$ is just a parameter which can take any real value. ${ }^1$ This equation can be derived by the variational principle applied to the action
$$S[\phi]=\int \mathrm{d}^4 x \mathcal{L}(x)=\frac{1}{2} \int \mathrm{d}^4 x\left(\partial_\mu \phi(x) \partial^\mu \phi(x)-m^2 \phi^2(x)\right)$$
where
$$\mathcal{L}=\frac{1}{2}\left(\partial_\mu \phi(x) \partial^\mu \phi(x)-m^2 \phi^2(x)\right)$$
is the Lagrangian density. The canonical momentum associated to $\phi(x)$ is given by
$$\pi(x)=\frac{\partial \mathcal{L}}{\partial\left(\partial_0 \phi(x)\right)}=\partial_0 \phi(x)$$

and the Hamiltonian density by
$$\mathcal{H}=\frac{1}{2}\left[\pi^2(x)+\left(\partial_i \phi(x)\right)^2+m^2 \phi^2(x)\right]$$
Equation (7.1) admits plane wave solutions

## 物理代写|粒子物理代写Particle Physics代考|The Green’s functions

The solution of a linear homogeneous wave equation is always rather trivial. However, in practice we are often interested in the dynamics of the field $\phi(x)$ coupled to a given external source described by a function $j(x)$. The corresponding equation of motion is
$$\left(\square+m^2\right) \phi(x)=j(x)$$
It is an equation of hyperbolic type. As a second order differential equation, its solutions are determined by the Cauchy data, the value of the function and its first derivatives on a surface, called the Cauchy surface. In practice, every time we use local coordinates, we will take as a Cauchy surface the hyperplane $\mathbb{R}^3=\left{(t, \boldsymbol{x}) \in M^4 \mid t=0\right}$ or its time translations. By relativistic invariance, the properties of the solutions are independent of this particular choice and we can choose any space-like hypersurface.
Since the equation (7.10) is linear, the solution for a general $j(x)$ will be given by the superposition principle starting from the solution of the equation corresponding to a point source
$$\left(\square+m^2\right) G(x, y)=\delta^4(x-y)$$
In physics, these solutions are the so-called elementary solutions or Green functions. $G(x, y)$ is the field produced by a point source, which appears at the point $\boldsymbol{y}$ instantaneously at time $y_0$. We are particularly interested in those solutions that are translationally invariant. The general solution of $(7.10)$ will then be of the form
$$\phi(x)=\phi_0(x)+\int \mathrm{d}^4 y G(x-y) j(y)$$
where $\phi_0(x)$ is a solution of the homogeneous equation $\left(\square+m^2\right) \phi_0(x)=0$, which is fixed by the Cauchy data.

## 物理代写粒子物理代写Particle Physics代考|The Klein-Gordon Equation

$$\left(\partial_\mu \partial^\mu+m^2\right) \phi(x)=0$$

$$S[\phi]=\int \mathrm{d}^4 x \mathcal{L}(x)=\frac{1}{2} \int \mathrm{d}^4 x\left(\partial_\mu \phi(x) \partial^\mu \phi(x)-m^2 \phi^2(x)\right)$$

$$\mathcal{L}=\frac{1}{2}\left(\partial_\mu \phi(x) \partial^\mu \phi(x)-m^2 \phi^2(x)\right)$$

$$\pi(x)=\frac{\partial \mathcal{L}}{\partial\left(\partial_0 \phi(x)\right)}=\partial_0 \phi(x)$$

$$\mathcal{H}=\frac{1}{2}\left[\pi^2(x)+\left(\partial_i \phi(x)\right)^2+m^2 \phi^2(x)\right]$$

## 物理代写|粒子物理代写Particle Physics代晏|The Green’s functions

$$\left(\square+m^2\right) \phi(x)=j(x)$$

$$\left(\square+m^2\right) G(x, y)=\delta^4(x-y)$$

$$\phi(x)=\phi_0(x)+\int \mathrm{d}^4 y G(x-y) j(y)$$

## 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。