Posted on Categories:Particle Physics, 物理代写, 粒子物理

# 物理代写|粒子物理代写Particle Physics代考|PHY408 Tensor calculus

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 物理代写|粒子物理代写Particle Physics代考|Tensor calculus

Tensor calculus. The only noticeable difference between the tensor calculus in the Lorentz group and the corresponding one in any real orthogonal group is the non-Euclidean metric in $\mathbb{M}^4$ which distinguishes between upper and lower indices. So, a contravariant four-vector $V^\mu$ with $\mu=0,1,2,3$, is defined as a set of four quantities transforming as:
$$V^{\prime \mu}=\Lambda^\mu{ }\nu V^\nu$$ Similarly, a set of four quantities $U\mu$ define a covariant 4-vector if they transform under a Lorentz transformation as
$$U_\mu^{\prime}=U_\nu\left(\Lambda^{-1}\right)^\nu{ }\mu \equiv \Lambda\mu^\nu U_\nu, \Lambda_\mu^\nu=\eta_{\mu \lambda} \Lambda_\rho^\lambda \eta^{\rho \nu}$$
The metric $\eta_{\mu \nu}$ and its inverse $\eta^{\mu \nu}$ can be used to lower and raise indices, respectively. These definitions extend to general contravariant or covariant or mixed tensors with arbitrary numbers of upper (contravariant) and/or lower (covariant) indices. The $4^{n+m}$ quantities $T_{\nu_1 \nu_2 \ldots \nu_m}^{\mu_1 \mu_2 \ldots \mu_n}$ are the components of a mixed tensor with $n$ contravariant and $m$ covariant indices, iff under Lorentz transformations they transform according to
$$T_{\nu_1 \nu_2 \ldots \nu_m}^{\mu_1 \mu_2 \ldots \mu_n}=\Lambda_{\alpha_1}^{\mu_1} \Lambda_{\alpha_2}^{\mu_2} \ldots \Lambda_{\alpha_n}^{\mu_n}\left(\Lambda^{-1}\right){\nu_1}^{\beta_1}\left(\Lambda^{-1}\right){\nu_2}^{\beta_2} \ldots\left(\Lambda^{-1}\right){\nu_m}^{\beta_m} T{\beta_1 \beta_2 \ldots \beta_m}^{\alpha_1 \alpha_2 \ldots \alpha_n}$$
A quantity which is invariant under Lorentz transformations is called a Lorentz scalar. A tensor without any index or with all its indices contracted is a Lorentz scalar.

In analogy to tensors of the rotation group $O(3)$, Lorentz tensors decompose into irreducible ones transforming independently with the help of the operations of symmetrisation, anti symmetrisation and trace.

## 物理代写|粒子物理代写Particle Physics代考|The Lie algebra of the Lorentz group

The Lie algebra of the Lorentz group. The proper Lorentz group contains the spatial rotations as a subgroup. Indeed, the matrices
$$\Lambda=\left(\begin{array}{ll} 1 & \ & R \end{array}\right)$$
with the three-by-three matrices $R$ being rotation matrices $R^T R=R R^T=1$, satisfy (5.131). The rotation matrix in the ” $x^1-x^2$ plane” by angle $\omega_{12}=\theta$ is
$$R\left(\omega_{12}=\theta\right)=\left(\begin{array}{ccc} \cos \theta & \sin \theta & 0 \ -\sin \theta & \cos \theta & 0 \ 0 & 0 & 1 \end{array}\right)$$
and the corresponding generator defined in general in (5.21) is
$$\mathcal{J}^{12}=\left(\begin{array}{cccc} 0 & 0 & 0 & 0 \ 0 & 0 & \mathrm{i} & 0 \ 0 & -\mathrm{i} & 0 & 0 \ 0 & 0 & 0 & 0 \end{array}\right)$$
Similarly, the generators of rotations in the 2-3 and 3-1 planes are

$$\mathcal{J}^{23}=\left(\begin{array}{cccc} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & \mathrm{i} \ 0 & 0 & -\mathrm{i} & 0 \end{array}\right) \quad \text { and } \mathcal{J}^{13}=\left(\begin{array}{cccc} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & \mathrm{i} \ 0 & 0 & 0 & 0 \ 0 & -\mathrm{i} & 0 & 0 \end{array}\right)$$
The Lorentz boost with velocity $V$ in the $x^1$-direction is given by
$$t^{\prime}=\gamma(V)(t-V x), \quad x^{\prime}=\gamma(V)(x-V t), \quad y^{\prime}=y, \quad z^{\prime}=z$$
with
$$\gamma(V)=\frac{1}{\sqrt{1-V^2}}$$
and similarly for boosts in the other two directions. From (5.143) we obtain for the generators of boosts
$$\mathcal{J}^{01}=-\mathrm{i}\left(\begin{array}{llll} 0 & 1 & 0 & 0 \ 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \end{array}\right), \mathcal{J}^{02}=-\mathrm{i}\left(\begin{array}{llll} 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \end{array}\right), \mathcal{J}^{03}=-\mathrm{i}\left(\begin{array}{llll} 0 & 0 & 0 & 1 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 \end{array}\right)$$

## 物理代写|粒子物理代写粒子物理代考|张量微积分

$$V^{\prime \mu}=\Lambda^\mu{ }\nu V^\nu$$类似地，一组四个量$U\mu$定义一个协变四向量，如果它们在洛伦兹变换下变换为
$$U_\mu^{\prime}=U_\nu\left(\Lambda^{-1}\right)^\nu{ }\mu \equiv \Lambda\mu^\nu U_\nu, \Lambda_\mu^\nu=\eta_{\mu \lambda} \Lambda_\rho^\lambda \eta^{\rho \nu}$$

$$T_{\nu_1 \nu_2 \ldots \nu_m}^{\mu_1 \mu_2 \ldots \mu_n}=\Lambda_{\alpha_1}^{\mu_1} \Lambda_{\alpha_2}^{\mu_2} \ldots \Lambda_{\alpha_n}^{\mu_n}\left(\Lambda^{-1}\right){\nu_1}^{\beta_1}\left(\Lambda^{-1}\right){\nu_2}^{\beta_2} \ldots\left(\Lambda^{-1}\right){\nu_m}^{\beta_m} T{\beta_1 \beta_2 \ldots \beta_m}^{\alpha_1 \alpha_2 \ldots \alpha_n}$$

## 物理代写|粒子物理代写粒子物理学代考|洛伦兹群的李代数

$$\Lambda=\left(\begin{array}{ll} 1 & \ & R \end{array}\right)$$
， 3 × 3矩阵$R$是旋转矩阵$R^T R=R R^T=1$，满足(5.131)。“$x^1-x^2$平面”中通过角度$\omega_{12}=\theta$的旋转矩阵为
$$R\left(\omega_{12}=\theta\right)=\left(\begin{array}{ccc} \cos \theta & \sin \theta & 0 \ -\sin \theta & \cos \theta & 0 \ 0 & 0 & 1 \end{array}\right)$$
，对应的生成器在(5.21)中一般定义为
$$\mathcal{J}^{12}=\left(\begin{array}{cccc} 0 & 0 & 0 & 0 \ 0 & 0 & \mathrm{i} & 0 \ 0 & -\mathrm{i} & 0 & 0 \ 0 & 0 & 0 & 0 \end{array}\right)$$

$$\mathcal{J}^{23}=\left(\begin{array}{cccc} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & \mathrm{i} \ 0 & 0 & -\mathrm{i} & 0 \end{array}\right) \quad \text { and } \mathcal{J}^{13}=\left(\begin{array}{cccc} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & \mathrm{i} \ 0 & 0 & 0 & 0 \ 0 & -\mathrm{i} & 0 & 0 \end{array}\right)$$
$x^1$方向上速度$V$的洛伦兹升力由
$$t^{\prime}=\gamma(V)(t-V x), \quad x^{\prime}=\gamma(V)(x-V t), \quad y^{\prime}=y, \quad z^{\prime}=z$$
with
$$\gamma(V)=\frac{1}{\sqrt{1-V^2}}$$

$$\mathcal{J}^{01}=-\mathrm{i}\left(\begin{array}{llll} 0 & 1 & 0 & 0 \ 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \end{array}\right), \mathcal{J}^{02}=-\mathrm{i}\left(\begin{array}{llll} 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \end{array}\right), \mathcal{J}^{03}=-\mathrm{i}\left(\begin{array}{llll} 0 & 0 & 0 & 1 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 \end{array}\right)$$

## 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。