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物理代写|弦论代写string theory代考|超引力代写supergravity|Actions, symmetries and solutions

如果你也在 怎样代写弦论string theory这个学科遇到相关的难题,请随时右上角联系我们的24/7代写客服。弦论string theory在物理学中是一个理论框架,其中粒子物理学中的点状粒子被称为弦的一维物体取代。弦理论描述了这些弦如何在空间传播并相互作用。在大于弦的距离尺度上,弦看起来就像一个普通的粒子,其质量、电荷和其他属性由弦的振动状态决定。在弦理论中,弦的许多振动状态之一对应于引力子,一种携带引力的量子力学粒子。因此,弦理论是一种量子引力的理论。

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物理代写|弦论代写string theory代考|超引力代写supergravity|Actions, symmetries and solutions

物理代写|弦论代写string theory代考|超引力代写supergravity|Polyakov action

We will consider a point particle and a string propagating in a $D$-dimensional curved spacetime, with Minkowski signature $(-,+, \ldots,+)$. We will employ the natural units $\hbar=c=1$. The point particle sweeps out a 1-dimensional worldline, whereas the string sweeps out a 2-dimensional worldsheet, both in spacetime. The worldline of the point particle will be parametrized by a real number $\tau$, while the worldsheet of the string will be parametrized by two real numbers $\tau$ and $\sigma$. Obviously, $\tau$ is timelike and $\sigma$ is spacelike. The coordinates of the point particle and the string are given by
$$
X^{\mu}=X^{\mu}(\tau), \quad \text { point particle, }
$$
$$
X^{\mu}=X^{\mu}(\tau, \sigma), \quad \text { string. }
$$
We will define
$$
\dot{X}^{\mu}=\frac{\partial X^{\mu}}{\partial \tau}, \quad X^{\mu^{\prime}}=\frac{\partial X^{\mu}}{\partial \sigma} .
$$
The action principle of the point particle is proportional to the invariant length of the worldline, viz
$$
S=-m \int d s=-m \int d \tau \sqrt{-g_{\mu \nu}(X) \dot{X}^{\mu} \dot{X}^{\nu}}=-m \int d \tau \sqrt{-\dot{X}^{2}}
$$

物理代写|弦论代写string theory代考|超引力代写supergravity|Boundary conditions and symmetries

The equations of motion for $X^{\mu}$ in the case of a flat spacetime are:
$$
\delta S=T \int d^{2} \sigma \partial_{\alpha}\left(\sqrt{-h} h^{\alpha \beta} \partial_{\beta} X_{\mu}\right) \delta X^{\mu}-T \int d^{2} \sigma \partial_{\alpha}\left(\sqrt{-h} h^{\alpha \beta} \partial_{\beta} X^{\mu} \delta X_{\mu}\right) .
$$
In the bulk the equations of motion are
$$
\partial_{\alpha}\left(\sqrt{-h} h^{\alpha \beta} \partial_{\beta} X_{\mu}\right)=\sqrt{-h} \nabla^{2} X_{\mu}=0 .
$$
For worldsheets with boundary there is also a surface term:
$$
\delta S=-\left.T \int d \tau \sqrt{-h} \partial^{\sigma} X^{\mu} \delta X_{\mu}\right|{\sigma=0} ^{\sigma=l} . $$ This vanishes if: $$ \partial^{\sigma} X^{\mu}(\tau, 0)=\partial^{\sigma} X^{\mu}(\tau, l)=0 . $$ These are Neumann boundary conditions, i.e. the ends of the open string move freely in spacetime, and the component of the momentum normal to the boundary of the worldsheet vanishes. $$ X^{\mu}(\tau, 0)=X^{\mu}(\tau, l), \quad h{\alpha \beta}(\tau, 0)=h_{\alpha \beta}(\tau, l) .
$$
The fields are periodic which corresponds to a closed string.

物理代写|弦论代写string theory代考|超引力代写supergravity|Closed string solutions

In flat spacetime, i.e. $g_{\mu \nu}(X)=\eta_{\mu \nu}$, the gauge-fixed Polyakov action becomes
$$
S=\frac{1}{2} T \int d^{2} \sigma\left(\dot{X}^{\mu} \dot{X}{\mu}-X^{\prime \mu} X{\mu}^{\prime}\right) \text {. }
$$
The equations of motion are:
$$
\square X^{\mu}=\left(\frac{\partial^{2}}{\partial \sigma^{2}}-\frac{\partial^{2}}{\partial \tau^{2}}\right) X^{\mu}=0
$$
The general solution is
$$
X^{\mu}=X_{R}^{\mu}\left(\sigma^{-}\right)+X_{L}^{\mu}\left(\sigma^{+}\right)
$$
The function $X_{R}^{\mu}$ describes right-moving modes of the string, while the function $X_{L}^{\mu}$ describes left-moving modes.
The worldsheet lightcone coordinates are:
$$
\sigma^{\mp}=\tau \mp \sigma .
$$
The derivatives conjugate to $\sigma^{\pm}$are defined by
$$
\partial_{\pm}=\frac{1}{2}\left(\partial_{\tau} \mp \partial_{\sigma}\right) .
$$
The metric in worldsheet coordinates is
$$
d s^{2}=-h_{\alpha \beta} d \sigma^{\alpha} d \sigma^{\beta}=d \tau^{2}-d \sigma^{2}=d \sigma^{-} d \sigma^{+} .
$$
Thus
$$
\eta_{+-}=\eta_{-+}=-\frac{1}{2}, \quad \eta^{+-}=\eta^{-+}=-2 .
$$

物理代写|弦论代写string theory代考|超引力代写supergravity|Actions, symmetries and solutions

弦论超引力代写

物理代写|弦论代写STRING THEORY代考|超引力代写SUPERGRAVITY|POLYAKOV ACTION

我们将考虑一个点粒子和一个在D维弯曲时空,带有 Minkowski 签名(−,+,…,+). 我们将使用自然单位⁇=C=1. 点粒子扫出一维世界线,而弦扫出二维世界表,两者都在时空中。点粒子的世界线将由实数参数化τ,而字符串的世界表将由两个实数参数化τ和σ. 明显地,τ是及时的并且σ是类太空的。点粒子和字符串的坐标由下式给出
Xμ=Xμ(τ), 点粒子, 
Xμ=Xμ(τ,σ), 细绳。 
我们将定义
X˙μ=∂Xμ∂τ,Xμ′=∂Xμ∂σ.
点粒子的作用原理与世界线的不变长度成正比,即
小号=−米∫ds=−米∫dτ−Gμν(X)X˙μX˙ν=−米∫dτ−X˙2

物理代写|弦论代写STRING THEORY代考|超引力代写SUPERGRAVITY|BOUNDARY CONDITIONS AND SYMMETRIES

运动方程为Xμ在平坦时空的情况下:
$$
\delta S=T \int d^{2} \sigma \partial_{\alpha}\left(\sqrt{-h} h^{\alpha \beta} \partial_{\beta} X_{\mu}\right) \delta X^{\mu}-T \int d^{2} \sigma \partial_{\alpha}\left(\sqrt{-h} h^{\alpha \beta} \partial_{\beta} X^{\mu} \delta X_{\mu}\right) .
$$
In the bulk the equations of motion are
$$
\partial_{\alpha}\left(\sqrt{-h} h^{\alpha \beta} \partial_{\beta} X_{\mu}\right)=\sqrt{-h} \nabla^{2} X_{\mu}=0 .
$$
For worldsheets with boundary there is also a surface term:
$$
\delta S=-\left.T \int d \tau \sqrt{-h} \partial^{\sigma} X^{\mu} \delta X_{\mu}\right|{\sigma=0} ^{\sigma=l} . $$ This vanishes if: $$ \partial^{\sigma} X^{\mu}(\tau, 0)=\partial^{\sigma} X^{\mu}(\tau, l)=0 . $$ These are Neumann boundary conditions, i.e. the ends of the open string move freely in spacetime, and the component of the momentum normal to the boundary of the worldsheet vanishes. $$ X^{\mu}(\tau, 0)=X^{\mu}(\tau, l), \quad h{\alpha \beta}(\tau, 0)=h_{\alpha \beta}(\tau, l) .
$$
字段是周期性的,对应于一个封闭的字符串。

物理代写|弦论代写STRING THEORY代考|超引力代写SUPERGRAVITY|CLOSED STRING SOLUTIONS

在平坦的时空中,即Gμν(X)=这μν, 规范固定的 Polyakov 动作变为
$$
S=\frac{1}{2} T \int d^{2} \sigma\left(\dot{X}^{\mu} \dot{X}{\mu}-X^{\prime \mu} X{\mu}^{\prime}\right) \text {. }
$$
The equations of motion are:
$$
\square X^{\mu}=\left(\frac{\partial^{2}}{\partial \sigma^{2}}-\frac{\partial^{2}}{\partial \tau^{2}}\right) X^{\mu}=0
$$
The general solution is
$$
X^{\mu}=X_{R}^{\mu}\left(\sigma^{-}\right)+X_{L}^{\mu}\left(\sigma^{+}\right)
$$
The function $X_{R}^{\mu}$ describes right-moving modes of the string, while the function $X_{L}^{\mu}$ describes left-moving modes.
The worldsheet lightcone coordinates are:
$$
\sigma^{\mp}=\tau \mp \sigma .
$$
The derivatives conjugate to $\sigma^{\pm}$are defined by
$$
\partial_{\pm}=\frac{1}{2}\left(\partial_{\tau} \mp \partial_{\sigma}\right) .
$$
The metric in worldsheet coordinates is
$$
d s^{2}=-h_{\alpha \beta} d \sigma^{\alpha} d \sigma^{\beta}=d \tau^{2}-d \sigma^{2}=d \sigma^{-} d \sigma^{+} .
$$
Thus
$$
\eta_{+-}=\eta_{-+}=-\frac{1}{2}, \quad \eta^{+-}=\eta^{-+}=-2 .
$$

物理代写|弦论代写string theory代考|超引力代写supergravity

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