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

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。

弦论string theory是一个广泛而多样的学科,它试图解决基础物理学的一些深层次问题。弦理论为数学物理学贡献了许多进展,这些进展被应用于黑洞物理学、早期宇宙宇宙学、核物理学和凝聚态物理学中的各种问题,它也刺激了纯数学的一些重大发展。由于弦理论有可能提供对引力和粒子物理学的统一描述,它是万物理论的候选者,是描述所有基本力量和物质形式的独立数学模型。尽管在这些问题上做了很多工作,但目前还不知道弦理论在多大程度上描述了现实世界,也不知道该理论在选择其细节方面允许多大的自由度。

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

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

We consider a relativistic particle moving in a $D$-dimensional curved spacetime manifold with metric $g_{\mu \nu}$ of Minkowski signature, viz
$$
g_{\mu \nu}=(-1,+1, \ldots,+1)
$$
The coordinates of this particle in spacetime are denoted by $X^{\mu}$. The infinitesimal line element is then defined by
$$
d s^{2}=-g_{\mu \nu}(X) d X^{\mu} d X^{\nu} .
$$
The worldlines, i.e. trajectories in spacetime, of this particle are timelike geodesics of the spacetime manifold. Let $\tau$ be an affine parameter on the worldline which can be identified with the proper time of the particle. Since the worldline is a timelike geodesic and since timelike geodesics by construction are the curves with maximum proper time, i.e. minimum spacetime length, we can immediately conclude that the action of the particle must be proportional to the line element $s$. The constant of proportionality can be easily shown to be equal to the mass of the particle. We have then
$$
S=-m \int d s
$$

We can rewrite this action as
$$
S_{0}=-m \int d \tau \sqrt{-g_{\mu \nu}(X) \dot{X}^{\mu} \dot{X}^{\nu}}, \quad \dot{X}^{\mu}=\frac{d X^{\mu}}{d \tau} .
$$
We are immediately faced with two problems: (1) quantization of this action is very difficult because of the square root, and (2) in this form this action cannot be used to describe massless particles. These two problems are solved at once by introducing the metric $e$ on the worldline as an independent variable as follows
$$
\tilde{S}{0}=\frac{1}{2} \int d \tau\left(e^{-1} g{\mu \nu}(X) \dot{X}^{\mu} \dot{X}^{\nu}-m^{2} e\right) .
$$
By substituting with the equation of motion of $e$ in $\tilde{S}{0}$ we obtain the action $S{0}$ which shows explicitly that these two actions are classically equivalent.

The action $\tilde{S}{0}$ or $S{0}$ enjoys, almost obviously, by construction, the following symmetries:

  • Poincaré invariance in $D$ dimensions.
  • Diffeomorphism invariance which is given by the reparametrizations
    $$
    \tau \longrightarrow \tau^{\prime}=f(\tau) \text {. }
    $$

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

The relativistic particle which appears as a non-perturbative topological configuration in type IIA string theory is a superparticle and as such it is invariant under supersymmetric transformations in addition to its invariance under the abovementioned symmetries (Poincaré and diffeomorphisms). This supersymmetric particle or superparticle is also called a $D_{0}$-brane. It enjoys $\mathcal{N}=2$ supersymmetry in $D=10$ dimensions which is expressed in terms of two Majorana-Weyl spinors with opposite chiralities. The Majorana condition means that the spinor is real (neutral) while the Weyl condition means that the spinor has a definite chirality (either left-handed or right-handed). These two conditions can be simultaneously imposed only in 10 dimensions.

Let us then consider in what follows a superparticle with $\mathcal{N}=2$ supersymmetry in $D=10$ dimensions corresponding to two Majorana-Weyl spinors $\Theta^{a A}$ where the index $A$ denotes the number of supersymmetries, i.e. $A=1, \ldots, \mathcal{N}$, while the index $a$ denotes the number of components of the spinor in spacetime, i.e. $a=1, \ldots, 2^{D / 2}$.
The bosonic particle is described by the bosonic coordinates $X^{\mu}$, whereas the superparticle is described in addition to these by the fermionic coordinates $\Theta^{a A}$. The bosonic coordinates are real numbers, whereas the fermionic coordinates are Grassmannian numbers. The supersymmetric transformations, which are a generalization of Poincaré (translation and Lorentz) transformations, view the bosonic and fermionic coordinates $X^{\mu}$ and $\Theta^{a A}$ collectively as the coordinates of a superspace. These transformations mix between the bosonic and fermionic coordinates as follows. The supersymmetric transformation appears as a translation in the fermionic direction:
$$
\Theta^{a A} \longrightarrow \Theta^{a A}+\delta \Theta^{a A}, \quad \delta \Theta^{a A}=\epsilon^{a A} .
$$

The constant Majorana-Weyl spinor $\epsilon^{a A}$ is the supersymmetry parameter. The effect of the supersymmetric transformation on $X^{\mu}$ must be linear in $\epsilon^{a A}$, it must be a vector, and it must be linear. There is one solution which meets these three requirements given by
$$
X^{\mu} \longrightarrow X^{\mu}+\delta X^{\mu}, \quad \delta X^{\mu}=\bar{\epsilon}^{A} \Gamma^{\mu} \Theta^{A} .
$$
The $\Gamma^{\mu}$ are the Dirac matrices in $D=10$, i.e. $\mu=0,1, \ldots, 9$. They are 32 dimensional Hermitian matrices satisfying
$$
\Gamma^{\mu} \Gamma^{\nu}+\Gamma^{\nu} \Gamma^{\mu}=2 g^{\mu \nu} .
$$
The chirality matrix satisfies
$$
\Gamma_{11}=\Gamma_{0} \Gamma_{1} \ldots \Gamma_{9}
$$
It satisfies
$$
\Gamma_{11}^{2}=1, \quad\left{\Gamma_{\mu}, \Gamma_{11}\right}=0 .
$$
The set of Poincaré transformations form a group called the Poincaré group. The set of Poincaré plus supersymmetric transformations form the super-Poincaré group. All these symmetries are global symmetries of the action since they do not depend on the affine parameter $\tau$ of the worldline.

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

The next question is to determine the action of the superparticle. First, we note that the action $S_{0}$ depends on the velocities $\dot{X}^{\mu}$. The variation of these velocities under supersymmetry is
$$
\begin{aligned}
\delta \dot{X}^{\mu} &=\bar{\epsilon}^{A} \Gamma^{\mu} \dot{\Theta}^{A} \
&=\delta \bar{\Theta}^{A} \cdot \Gamma^{\mu} \dot{\Theta}^{A} \
&=\delta\left(\bar{\Theta}^{A} \Gamma^{\mu} \dot{\Theta}^{A}\right)
\end{aligned}
$$
We conclude immediately that the expression
$$
\Pi_{0}^{\mu}=\dot{X}^{\mu}-\bar{\Theta}^{A} \Gamma^{\mu} \dot{\Theta}^{A}
$$
is invariant under supersymmetric transformations. We get then the supersymmetric action by making in $S_{0}$ the replacement $\dot{X}^{\mu} \longrightarrow \Pi_{0}^{\mu}$. We get the action
$$
S_{1}=-m \int d \tau \sqrt{-g_{\mu \nu}(X) \Pi_{0}^{\mu} \Pi_{0}^{\nu}} .
$$
In terms of the metric $e$ on the worldline of the superparticle we obtain the action
$$
\tilde{S}{1}=\frac{1}{2} \int d \tau\left(e^{-1} g{\mu \nu}(X) \Pi_{0}^{\mu} \Pi_{0}^{\nu}-m^{2} e\right)
$$

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

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|BOSONIC PARTICLE

我们考虑一个相对论粒子在D具有度量的维弯曲时空流形Gμν闵可夫斯基签名,即
Gμν=(−1,+1,…,+1)
该粒子在时空中的坐标表示为Xμ. 然后将无穷小的线元素定义为
ds2=−Gμν(X)dXμdXν.
这个粒子的世界线,即时空中的轨迹,是时空流形的类时测地线。让τ是世界线上的一个仿射参数,可以用粒子的适当时间来识别。由于世界线是类时测地线,并且由于构造时类测地线是具有最大固有时间(即最小时空长度)的曲线,我们可以立即得出结论,粒子的作用必须与线元素成比例s. 比例常数可以很容易地证明等于粒子的质量。那时我们有
小号=−米∫ds

我们可以将这个动作重写为
小号0=−米∫dτ−Gμν(X)X˙μX˙ν,X˙μ=dXμdτ.
我们立即面临两个问题:1由于平方根,这个动作的量化非常困难,并且2在这种形式中,这个动作不能用来描述无质量的粒子。这两个问题通过引入度量一下子解决了和在世界线上作为自变量如下
$$
\tilde{S} {0}=\frac{1}{2} \int d \tau\left(e^{-1} g {\mu \nu}X\dot{X}^{\mu} \dot{X}^{\nu}-m^{2} e\right) 。
$$
代入运动方程和在 $\tilde{S} {0}在和这b吨一种一世n吨H和一种C吨一世这nS {0}$ 明确表明这两个动作在经典上是等价的。

动作 $\tilde{S} {0}这rS {0}$ 几乎很明显地具有以下对称性:

  • 庞加莱不变性D方面。
  • 由重新参数化给出的微分同胚不变性
    τ⟶τ′=F(τ). 

物理代写|弦论代写STRING THEORY代考|超引力代写SUPERGRAVITY|THE SUPERPARTICLE

在 IIA 型弦理论中以非微扰拓扑结构出现的相对论粒子是超粒子,因此除了在上述对称性下的不变性外,它在超对称变换下也是不变的é磷这一世nC一种r它是一种ndd一世FF和这米这rpH一世s米s. 这种超对称粒子或超粒子也称为D0-膜。它喜欢ñ=2超对称性D=10以两个具有相反手性的 Majorana-Weyl 旋量表示的尺寸。马约拉纳条件意味着旋量是真实的n和你吨r一种一世而外尔条件意味着旋量具有确定的手性和一世吨H和r一世和F吨−H一种nd和d这rr一世GH吨−H一种nd和d. 这两个条件只能在 10 个维度上同时施加。

然后让我们考虑一下超粒子之后的情况ñ=2超对称性D=10对应于两个 Majorana-Weyl 旋量的尺寸θ一种一种索引在哪里一种表示超对称的数量,即一种=1,…,ñ, 而指数一种表示旋量在时空中的分量数,即一种=1,…,2D/2.
玻色子粒子由玻色子坐标描述Xμ,而超粒子除了这些之外还由费米子坐标描述θ一种一种. 玻色子坐标是实数,而费米子坐标是格拉斯曼数。超对称变换,是 Poincaré 的推广吨r一种ns一世一种吨一世这n一种nd大号这r和n吨和变换,查看玻色子和费米子坐标Xμ和θ一种一种统称为超空间的坐标。这些变换在玻色子和费米子坐标之间混合如下。超对称变换表现为费米子方向的平移:
θ一种一种⟶θ一种一种+dθ一种一种,dθ一种一种=ε一种一种.

常数 Majorana-Weyl 旋量ε一种一种是超对称参数。超对称变换的影响Xμ必须是线性的ε一种一种,它必须是一个向量,而且它必须是线性的。有一种解决方案可以满足以下三个要求
Xμ⟶Xμ+dXμ,dXμ=ε¯一种Γμθ一种.
这Γμ是狄拉克矩阵D=10, IEμ=0,1,…,9. 它们是 32 维 Hermitian 矩阵,满足
ΓμΓν+ΓνΓμ=2Gμν.
手性矩阵满足
Γ11=Γ0Γ1…Γ9
它满足
\Gamma_{11}^{2}=1, \quad\left{\Gamma_{\mu}, \Gamma_{11}\right}=0 。\Gamma_{11}^{2}=1, \quad\left{\Gamma_{\mu}, \Gamma_{11}\right}=0 。
一组庞加莱变换形成一个称为庞加莱群的群。庞加莱加超对称变换的集合形成超庞加莱群。所有这些对称性都是动作的全局对称性,因为它们不依赖于仿射参数τ的世界线。

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

下一个问题是确定超粒子的作用。首先,我们注意到动作小号0取决于速度X˙μ. 在超对称下这些速度的变化是
dX˙μ=ε¯一种Γμθ˙一种 =dθ¯一种⋅Γμθ˙一种 =d(θ¯一种Γμθ˙一种)
我们立即得出结论,表达式
圆周率0μ=X˙μ−θ¯一种Γμθ˙一种
在超对称变换下是不变的。然后我们通过 make in 得到超对称作用小号0替代品X˙μ⟶圆周率0μ. 我们得到了行动
小号1=−米∫dτ−Gμν(X)圆周率0μ圆周率0ν.
在指标方面和在超粒子的世界线上,我们得到动作
$$
\tilde{S} {1}=\frac{1}{2} \int d \tau\left(e^{-1} g {\mu \nu}X\Pi_{0}^{\mu}\Pi_{0}^{\nu}-m^{2}e\right)
$$

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

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