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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|弦论代写string theory代考|A Covariant Action for Massless and Massive Particles

To include massless particles, we use a trick which works by introducing an auxiliary field $e(\sigma)$ on the word-line. We require that $e d \sigma$ is reparamentrisation invariant:
$$e d \sigma=\tilde{e} d \tilde{\sigma} .$$
This implies that $e$ transforms inversely to a coordinate differential:
$$\tilde{e}(\tilde{\sigma})=e(\sigma) \frac{d \sigma}{d \tilde{\sigma}} .$$
We also require that $e$ does not have any zeros, which makes $e d \sigma$ a one-dimensional volume element. We take $e>0$ for definiteness, and this condition is preserved under reparametrisations which respect the orientation of the world-line.
Using the invariant line element, we write down the following action:
$$S[x, e]=\frac{1}{2} \int e d \sigma\left(\frac{1}{e^2}\left(\frac{d x^\mu}{d \sigma}\right)^2-K\right),$$
where $K$ is a real constant.
The action (1.54) has the following symmetries:

• $S[x, e]$ is invariant under reparametrisations $\sigma \rightarrow \tilde{\sigma}$.
• $S[x, e]$ is invariant under Poincaré transformations $x^\mu \rightarrow \Lambda_v^\mu{ }_v{ }^v+a^\mu$.
The action depends on the fields $x=\left(x^\mu\right)$ and $e$. Performing the variations $x \rightarrow x+\delta x$ and $e \rightarrow e+\delta e$, respectively, we obtain the following equations of motion.
\begin{aligned} \frac{d}{d \sigma}\left(\frac{x^{\prime \mu}}{e}\right) &=0, \ x^{\prime 2}+e^2 K &=0 . \end{aligned}
Exercise 1.9.1 Derive the equations of motion (1.55) and (1.56) by variation of the action (1.54).

## 物理代写|弦论代写string theory代考|Literature

Some remarks in this chapter assume a basic knowledge of general relativity and quantum field theory, which can be found in any standard textbook. Coherent states are frequently used in quantum optics, but do not belong to the standard canon of quantum field theory as used by particle theorists. See, for example, Duncan (2012) who also discusses the subtleties of the particle concept. The world-line formalism is reviewed in Schubert (2001). We also assume some background in Lie groups and Lie algebras. While the material included in particle theory and quantum field theory textbooks should be sufficient to follow the text, we mention some books for further reading: Gilmore (1974), Cahn (1984), Cornwell (1997), Fuchs and Schweigert (1997), Ramond (2010), cover group theory and its applications to particle theory. Sexl and Urbantke (2001) give a detailed discussion of the Lorentz and Poincaré group in the context of special relativity, while Woit (2017) is a comprehensive and pedagogical introduction into group theory as used in quantum theory. Mathematical texts are Humphreys (1972), Fulton and Harris (1991), Bump (2004), and the classic, Weyl (1939).

The study of constrained dynamics is a research subject in its own right, starting with Dirac (1964). Classical textbooks on the subject are Sudarshan and Mukunda (1974) and Sundermeyer (1982). Modern approaches to the quantisation of ‘general gauge theories’, including string theory, use the BRS and BV formalisms which are covered in detail in the monograph Henneaux and Teitelboim (1992), see also the review article Marnelius (1982).

# 弦论代考

## 物理代写|弦论代写string theory代考|A Covariant Action for Massless and Massive Particles

$$e d \sigma=\tilde{e} d \tilde{\sigma} .$$

$$\tilde{e}(\tilde{\sigma})=e(\sigma) \frac{d \sigma}{d \tilde{\sigma}} .$$

$$S[x, e]=\frac{1}{2} \int e d \sigma\left(\frac{1}{e^2}\left(\frac{d x^\mu}{d \sigma}\right)^2-K\right),$$

• $S[x, e]$ 在重新参数化下是不变的 $\sigma \rightarrow \tilde{\sigma}$.
• $S[x, e]$ 在庞加莱变换下是不变的 $x^\mu \rightarrow \Lambda_{v v}^\mu{ }^v+a^\mu$.
操作取决于字段 $x=\left(x^\mu\right)$ 和 $e$. 执行变化 $x \rightarrow x+\delta x$ 和 $e \rightarrow e+\delta e$ ，我们分别得到以下运动方程。
$$\frac{d}{d \sigma}\left(\frac{x^{\prime \mu}}{e}\right)=0, x^{\prime 2}+e^2 K=0 .$$
练习 $1.9 .1$ 通过作用 (1.54) 的变化导出运动方程 (1.55) 和 (1.56)。

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