## 物理代写|弦论代写string theory代考|Explicit Solutions – Periodic Boundary Conditions

So far we have extracted information without solving the equations of motion explicitly. We now turn to this problem, which requires to select solutions of the two-dimensional wave equation (2.52) which satisfy the boundary conditions.

We start with periodic boundary conditions. The most general solution of the twodimensional wave equation which is periodic in $\sigma^1$ can be parametrised in the form
$$X^\mu(\sigma)=a^\mu+b^\mu \sigma^0+\sum_{n \neq 0} c_n^\mu e^{-2 i n \sigma^{-}}+\sum_{n \neq 0} d_n^\mu e^{-2 i n \sigma^{+}},$$
where $a^\mu, b^\mu \in \mathbb{R}$ and $\left(c_n^\mu\right)^=c_{-n}^\mu$ and $\left(d_n^\mu\right)^=d_{-n}^\mu$, since $X^\mu$ is real. The term linear in $\sigma^0$ is allowed by the boundary conditions and solves the wave equation.

The conventional parametrisation of the solution used in string theory looks somewhat different from (2.63):
$$X^\mu(\sigma)=x^\mu+L_S^2 p^\mu \sigma^0+\frac{i}{2} L_S \sum_{n \neq 0} \frac{1}{n} \alpha_n^\mu e^{-2 i n \sigma^{-}}+\frac{i}{2} L_S \sum_{n \neq 0} \frac{1}{n} \tilde{\alpha}_n^\mu e^{-2 i n \sigma^{+}},$$

where $x^\mu, p^\mu \in \mathbb{R}$ and $\left(\alpha_n^\mu\right)^=\alpha_{-n}^\mu$ and $\left(\tilde{\alpha}n^\mu\right)^=\tilde{\alpha}{-n}^\mu$. The string length $L_S$ is defined as ${ }^{12}$
$$L_S=\frac{1}{\sqrt{\pi T}} .$$
Note that the coordinates $\sigma^\alpha$ and the coefficients $\alpha_n^\mu, \tilde{\alpha}_n^\mu$ are dimensionless, while $p^\mu$ has the dimension of an inverse length, given that $c=1$ and $\hbar=1$. To see explicitly that $X^\mu$ splits into left- and right-moving parts, as in (2.52), note that $p^\mu \sigma^0=\frac{1}{2} p^\mu\left(\sigma^{+}+\sigma^{-}\right)$

While looking more complicated than (2.63), equation (2.64) is better adapted to the physical interpretation of the coefficients. To see this, we compute the total momentum:
$$P^\mu=T \int_0^\pi d \sigma^1 \dot{X}^\mu=p^\mu .$$
Thus, the coefficient $p^\mu$ of the term proportional to $\sigma^1$ is equal to the total momentum. Next, we compute the motion of the centre of mass:
$$x_{C M}^\mu=\frac{1}{\pi} \int_0^\pi d \sigma^1 X^\mu(\sigma)=x^\mu+p^\mu \sigma^0 .$$
For time-like $p^\mu$ we can match this with the world-line of a massive relativistic particle,
$$x^\mu(\tau)=x^\mu(0)+\frac{d x^\mu}{d \tau}(0) \tau,$$
and conclude that the centre of mass of a string behaves like a relativistic particle and moves, in the absence of forces, on a straight line in Minkowski space.

## 物理代写|弦论代写string theory代考|Explicit Solutions – Neumann Boundary Conditions

We now turn to open strings. The solution of the two-dimensional wave equation with Neumann boundary conditions is

$$X^\mu(\sigma)=x^\mu+L_S^2 p^\mu \sigma^0+i L_S \sum_{n \neq 0} \frac{1}{n} \alpha_n^\mu e^{-i n \sigma^0} \cos \left(n \sigma^1\right),$$
where $x^\mu, p^\mu \in \mathbb{R}$ and $\left(\alpha_m^\mu\right)^*=\alpha_{-m}^\mu$. There is only one set of Fourier coefficients, since left- and right-moving waves couple through the boundary conditions and combine into standing waves. We can, of course, re-write (2.73) in the form $X=$ $X_L\left(\sigma^{+}\right)+X_R\left(\sigma^{-}\right):$
$$X_{L / R}^\mu\left(\sigma^{\pm}\right)=\frac{1}{2} x^\mu+L_S^2 p_{L / R}^\mu \sigma^{\pm}+L_S \sum_{n \neq 0} \frac{1}{n} \alpha_{n(\mathrm{~L} / R)}^\mu e^{-i n \sigma^{\pm}},$$
subject to
$$p_L^\mu=p_R^\mu=\frac{1}{2} p^\mu, \quad \alpha_{n(\mathrm{~L})}^\mu=\alpha_{n(\mathrm{R})}^\mu .$$
Exercise 2.2.10 Show that the ends of an open string must move with the speed of light.

We mentioned before that with Neumann boundary conditions there is only one set of conserved charges $L_m$. Here is their explicit form in terms of Fourier coefficients:
\begin{aligned} L_m &=\frac{T}{2} \int_0^\pi d \sigma^1\left(e^{i m \sigma^1} T_{++}+e^{-t m \sigma^1} T_{–}\right)=\frac{T}{16} \int_{-\pi}^\pi e^{i m \sigma^1}\left(\dot{X}+X^{\prime}\right)^{\prime} \ &=\frac{1}{2} \pi T \sum_n \alpha_{m-n} \alpha_n, \end{aligned}
where we defined $\alpha_0=p$. The canonical Hamiltonian is $H=L_0$. As for closed strings, the Hamiltonian constraint $H=L_0=0$ is the mass shell condition:
$$M^2=-p^2=2 \pi T N .$$

## 物理代写|弦论代写string theory代考|Explicit Solutions – Periodic Boundary Conditions

$$X^\mu(\sigma)=a^\mu+b^\mu \sigma^0+\sum_{n \neq 0} c_n^\mu e^{-2 i n \sigma^{-}}+\sum_{n \neq 0} d_n^\mu e^{-2 i n \sigma^{+}},$$

$$X^\mu(\sigma)=x^\mu+L_S^2 p^\mu \sigma^0+\frac{i}{2} L_S \sum_{n \neq 0} \frac{1}{n} \alpha_n^\mu e^{-2 i n \sigma^{-}}+\frac{i}{2} L_S \sum_{n \neq 0} \frac{1}{n} \tilde{\alpha}n^\mu e^{-2 i n \sigma^{+}},$$ 在哪里 $x^\mu, p^\mu \in \mathbb{R}$ 和 $\left(\alpha_n^\mu\right)^{=} \alpha{-n}^\mu$ 和 $\left(\tilde{\alpha} n^\mu\right)^{=} \tilde{\alpha}-n^\mu$. 字符串长度 $L_S$ 定义为 ${ }^{12}$
$$L_S=\frac{1}{\sqrt{\pi T}} .$$

$$x^\mu(\tau)=x^\mu(0)+\frac{d x^\mu}{d \tau}(0) \tau,$$

## 物理代写|弦论代写string theory代考|Explicit Solutions – Neumann Boundary Conditions

$$X^\mu(\sigma)=x^\mu+L_S^2 p^\mu \sigma^0+i L_S \sum_{n \neq 0} \frac{1}{n} \alpha_n^\mu e^{-i n \sigma^0} \cos \left(n \sigma^1\right),$$

$$X_{L / R}^\mu\left(\sigma^{\pm}\right)=\frac{1}{2} x^\mu+L_S^2 p_{L / R}^\mu \sigma^{\pm}+L_S \sum_{n \neq 0} \frac{1}{n} \alpha_{n(\mathrm{~L} / R)}^\mu e^{-i n \sigma^{\pm}},$$

$$p_L^\mu=p_R^\mu=\frac{1}{2} p^\mu, \quad \alpha_{n(\mathrm{~L})}^\mu=\alpha_{n(\mathrm{R})}^\mu .$$

$$L_m=\frac{T}{2} \int_0^\pi d \sigma^1\left(e^{i m \sigma^1} T_{++}+e^{-t m \sigma^1} T_{-}\right)=\frac{T}{16} \int_{-\pi}^\pi e^{i m \sigma^1}\left(\dot{X}+X^{\prime}\right)^{\prime} \quad=\frac{1}{2} \pi T \sum_n \alpha_{m-n} \alpha_n,$$

$$M^2=-p^2=2 \pi T N .$$

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