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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Particle Interactions

So far we have considered free particles. Interactions can be introduced by adding terms which couple the particle to external fields. The most important examples are the following:

• If the force $f^\mu$ has a potential, $f_\mu=-\partial_\mu V(x)$, then the equation of motion (1.18) follows from the action
$$S=-m \int \sqrt{-\dot{x}^2} d \tau-\int V(x(\tau)) d \tau$$
• If $f^\mu$ is the Lorentz force acting on a particle with charge $q$, that is $f^\mu=q F^{\mu v} \dot{x}v$, then the action is $$S=-m \int \sqrt{-\dot{x}^2} d \tau+q \int A\mu d x^\mu$$
In the second term, the vector potential $A_\mu$ is integrated along the world-line of the particle
$$\int A_\mu d x^\mu=\int A_\mu(x(\tau)) \frac{d x^\mu}{d \tau} d \tau$$
The resulting equation of motion is
$$m \ddot{x}\mu=q F{\mu v} \dot{x}^v,$$
where $F_{\mu v}=\partial_\mu A_v-\partial_v A_\mu$ is the field strength tensor. Equation (1.44) is the manifestly covariant version of the Lorentz force
$$\frac{d \vec{p}}{d t}=q(\vec{E}+\vec{v} \times \vec{B})$$ The coupling to gravity can be obtained by replacing the Minkowski metric $\eta_{\mu v}$ by a general pseudo-Riemannian metric $g_{\mu \nu}(x)$ :
• $$• S=-m \int d \tau \sqrt{-g_{\mu \nu}(x) \dot{x}^\mu \dot{x}^v} . •$$
• The resulting equation of motion is the geodesic equation
• $$• \ddot{x}^\mu+\Gamma_{v \rho}^\mu \dot{x}^v \dot{x}^\rho=0, •$$
• with affine curve parameter $\tau$.

## 物理代写|弦论代写string theory代考|Canonical Momenta and Hamiltonian

From the action (1.37)
$$S=\int L d \sigma=-m \int d \sigma \sqrt{-x^{\prime 2}},$$
we obtain the following canonical momentum vector:
$$\pi^\mu=\frac{\partial L}{\partial x^{\prime} \mu}=m \frac{x^{\prime \mu}}{\sqrt{-x^{\prime 2}}}=m \dot{x}^\mu .$$
A new feature compared to the action (1.20) is that the components of the canonical momentum are not independent, but subject to the constraint
$$\pi^\mu \pi_\mu=-m^2 \text {. }$$
Since canonical and kinetic momenta agree, $\pi^\mu=p^\mu$, we can interpret the constraint as the mass shell condition $p^2=-m^2$. The Hamiltonian associated to (1.37) is
$$H=\pi^\mu \dot{x}_\mu-L=0 .$$
Thus the Hamiltonian is not equal to the total energy, but rather vanishes. Since $H \propto p^2+m^2$, the Hamiltonian does not vanish identically, but only for the subset of configurations which satisfy the mass shell condition. Thus, $H=0$ is a constraint which needs to be imposed on top of the dynamical field equations. This is sometimes denoted $H \simeq 0$, and one says that the Hamiltonian is weakly zero. This type of constraint arises when mechanical or field theoretical systems are formulated in a manifestly Lorentz covariant or manifestly reparametrisation invariant way. We will not need to go deeply into constrained dynamcis, because the constraints we will encounter can be imposed as initial conditions. We will demonstrate this explicitly for strings later, see Section 2.2.5.

Remark: Hamiltonian and time evolution. For those readers who are familiar with the formulation of mechanics using Poisson brackets, we add that while the Hamiltonian is weakly zero, it still generates the infinitesimal time evolution of physical quantities. Similarly, in the quantum version of the theory, the infinitesimal time evolution of an operator in the Heisenberg picture is still given by its commutator with the Hamiltonian. For the point particle, the only constraint is the vanishing of the Hamiltonian.

By accepting that constraints are the prize to pay for a covariant formalism, we can describe relativistic massive particles in a Lorentz covariant and reparametrisation invariant way. But we still need to find a way to include massless particles.

# 弦论代考

## 物理代写|弦论代写string theory代考|Particle Interactions

• 如果力 $f^\mu$ 有潜力， $f_\mu=-\partial_\mu V(x)$, 那么运动方程 (1.18) 来自于动作
$$S=-m \int \sqrt{-\dot{x}^2} d \tau-\int V(x(\tau)) d \tau$$
• 如果 $f^\mu$ 是作用在带电粒子上的洛伦兹力 $q$ ，那是 $f^\mu=q F^{\mu v} \dot{x} v$ ，那么动作是
$$S=-m \int \sqrt{-\dot{x}^2} d \tau+q \int A \mu d x^\mu$$
在第二项中，矢量势 $A_\mu$ 沿粒子的世界线积分
$$\int A_\mu d x^\mu=\int A_\mu(x(\tau)) \frac{d x^\mu}{d \tau} d \tau$$
得到的运动方程是
$$m \ddot{x} \mu=q F \mu v \dot{x}^v,$$
在哪里 $F_{\mu v}=\partial_\mu A_v-\partial_v A_\mu$ 是场强张量。方程 (1.44) 是洛伦兹力的明显协变版本
$$\frac{d \vec{p}}{d t}=q(\vec{E}+\vec{v} \times \vec{B})$$
可以通过替换 Minkowski 度量来获得与重力的耦合 $\eta_{\mu v}$ 由一般伪黎㵋度量 $g_{\mu \nu}(x)$ :
• $\$ \$$• S=-m \backslash int d \backslash tau \backslash sqrt \left{-g_{-} \wedge{m u \backslash n u}(x) \backslash \operatorname{dot}{x}^{\wedge} \backslash m u \backslash \operatorname{dot}{x}^{\wedge} v\right} 。 • \ \$$
• 由此产生的运动方程是测地线方程
• \$\$
• $\$ \$$• 具有仿射曲线参数 \tau. ## 物理代写|弦论代写string theory代考|Canonical Momenta and Hamiltonian 从行动 (1.37)$$
S=\int L d \sigma=-m \int d \sigma \sqrt{-x^{\prime 2}},
$$我们得到以下典型动量向量:$$
\pi^\mu=\frac{\partial L}{\partial x^{\prime} \mu}=m \frac{x^{\prime \mu}}{\sqrt{-x^{\prime 2}}}=m \dot{x}^\mu .
$$与动作 (1.20) 相比的一个新特征是规范动量的分量不是独立的，而是受约束的$$
\pi^\mu \pi_\mu=-m^2 .
$$由于规范动量和动量一致， \pi^\mu=p^\mu ，我们可以将约束解释为质量壳条件 p^2=-m^2. 与 (1.37) 相关的哈密顿量是$$
H=\pi^\mu \dot{x}_\mu-L=0 .


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