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• (Generalized) Linear Models 广义线性模型
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• 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|Action, Equations of Motion, and Bounday Conditions

If we replace point particles by strings, that is by one-dimensional extended objects, then the world-line is replaced by a world-sheet $\Sigma$, which is a surface embedded into Minkowksi space: 1
$$X: \Sigma \ni P \longrightarrow X(P) \in \mathbb{M} .$$
On space-time we choose linear coordinates associated to a Lorentz frame, denoted $X=\left(X^\mu\right)$, where $\mu=0,1, \ldots, D-1$. On the world-sheet we choose local coordinates $\sigma=\left(\sigma^0, \sigma^1\right)=\left(\sigma^\alpha\right)$. Depending on the global structure of the world-sheet, it may or may not be possible to cover $\Sigma$ with a single coordinate system. We will have to verify that physical quantites, such as the action and the equations of motion are covariant with respect to reparametrisations. For the time being we will only consider free strings which do not split or join. In this case, the world-sheet $\Sigma$ has the topology of a strip (open strings; see Figure 2.1) or of a cylinder (closed strings).

At each point of $\Sigma$ we can choose a time-like tangent vector (‘the direction towards the future’) and a space-like tangent vector (‘the direction along the string’). We adopt the convention that the coordinate $\sigma^0$ is time-like (that is, the corresponding tangent vector $\partial_0=\frac{\partial}{\partial \sigma^0}$ is a time-like vector), while the coordinate $\sigma^1$ is space-like: ${ }^2$
$$\dot{X}^2 \leq 0, \quad\left(X^{\prime}\right)^2>0 .$$
Here we use the following notation for tangent vectors:
$$\dot{X}=\left(\partial_0 X^\mu\right)=\left(\frac{\partial X^\mu}{\partial \sigma^0}\right), X^{\prime}=\left(\partial_1 X^\mu\right)=\left(\frac{\partial X^\mu}{\partial \sigma^1}\right) .$$
We also make conventional choices for the range of the world-sheet coordinates. The space-like coordinate takes values $\sigma^1 \in[0, \pi]$, whereas the time-like coordinate takes values $\sigma^0 \in\left[\sigma_{(1)}^0, \sigma_{(2)}^0\right] \subset \mathbb{R}$. The limiting case $\sigma^0 \in \mathbb{R}$ is allowed and describes the asymptotic time evolution of a string from the infinite past to the infinite future. ${ }^3$
The Nambu-Goto action is the direct generalisation of (1.37), and thus proportional to the area of the world-sheet $\Sigma$, measured with the metric induced on $\Sigma$ by the Minkowski metric:
$$S_{\mathrm{NG}}[X]=-T A(\Sigma)=-T \int_{\Sigma} d^2 A$$

## 物理代写|弦论代写string theory代考|D-branes

Let us have a closer look into Dirichlet boundary conditions and D-branes. For concreteness we consider open strings ending on a $p$-dimensional $\mathrm{D}$-brane, or D-p-brane for short, which is located at $x^{p+1}=x_0^{p+1}, x^{p+2}=x_0^{p+2}, \ldots, x^{D-1}=x_0^{D-1}$. This means that we impose Neumann boundary conditions for the $p$ space-like coordinates $x^1, \ldots, x^p$, and for time $x^0$. These are the direction parallel to the $(p+1)$ dimensional world-volume of the D-p-brane. Dirichlet boundary conditions are imposed for the transverse coordinates $x^{p+1}, \ldots, x^{D-1}$. Thus, the ends of open strings with such boundary conditions are located on the same D-p-brane (see Figure 2.2).

For $p=D-1$, we have Neumann boundary conditions in all directions and the $\mathrm{D}$ – $(D-1)$-brane is space-filling. The other extreme is the D-0-brane or D-particle where Dirichlet boundary conditions are imposed in all spatial directions. D-1-branes are strings, which are called D-strings to distinguish them from the fundamental strings defined by the Nambu-Goto action. D-2-branes are membranes, and D-pbranes with $p>2$ are higher-dimensional versions of membranes.

As a generalisation we can consider configurations with more than one D-brane. The simplest configuration is two parallel D-p-branes located at different positions in the $(p+1)$-direction, as in Figure 2.2:
$$x_{(1)}^{p+1} \neq x_{(2)}^{p+1}, \quad x_{(1)}^{p+k}=x_{(2)}^{p+k}, k=2, \ldots k=D-(p+1) .$$
As a further generalisation, we can consider configurations involving any number of D-branes, and with different values for $\mu$. This is certainly possible as far as imposing boundary conditions is concerned, but we need to treat the D-branes as dynamical objects if we want to preserve momentum conservation. This raises difficult dynamical question, since now D-branes can move and collide. Moreover, ‘string theory’ now seems to be a theory of strings and D-branes. One way of thinking about D-branes is as solitons, understood in a suitably relaxed sense. In field theory the term soliton refers to solutions of the field equations that behave like particles, which means that they are and remain localised, have finite total energy, and are regular. Solitons break translations invariance, which a D-branes also does transverse to its world-volume directions. Since D-branes with $p>0$ are infinitely extended, we need to replace the condition of finite energy by finite energy per world-volume, but this is natural since D-branes are translation invariant along their world-volume. Therefore we could compactify these directions (impose a periodic identification) and obtain a finite mass point-like object in the remaining non-compact directions. ${ }^6$ In the supersymmetric Type-II string theories one can show explicitly that there are,for specific values of $p$, static solutions of the low-energy effective field theory, ${ }^7$ called supergravity $p$-branes, which correspond to D-p-branes.

# 弦论代考

## 物理代写|弦论代写string theory代考|Action, Equations of Motion, and Bounday Conditions

$$X: \Sigma \ni P \longrightarrow X(P) \in \mathbb{M} .$$

$$\dot{X}^2 \leq 0, \quad\left(X^{\prime}\right)^2>0$$

$$\dot{X}=\left(\partial_0 X^\mu\right)=\left(\frac{\partial X^\mu}{\partial \sigma^0}\right), X^{\prime}=\left(\partial_1 X^\mu\right)=\left(\frac{\partial X^\mu}{\partial \sigma^1}\right)$$

Nambu-Goto 动作是 (1.37) 的直接推广，因此与世界表的面积成正比 $\Sigma$ ，用㶦导的度量标准测量 $\Sigma$ 通过 Minkowski 度量:
$$S_{\mathrm{NG}}[X]=-T A(\Sigma)=-T \int_{\Sigma} d^2 A$$

## 物理代写|弦论代写string theory代考|D-branes

$$x_{(1)}^{p+1} \neq x_{(2)}^{p+1}, \quad x_{(1)}^{p+k}=x_{(2)}^{p+k}, k=2, \ldots k=D-(p+1) .$$

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assignmentutor™您的专属作业导师
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