Posted on Categories:Quantum mechanics, 物理代写, 量子力学

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## 物理代写|量子力学代写Quantum mechanics代考|The thermal propagator

The thermal propagator is the time-ordered propagator in imaginary time. It is equivalent to the Euclidean correlation function in cylindrical geometry. Denote the thermal propagator by:
$$G_{T}^{(0)}(x, \tau)=\langle\phi(x, \tau) \phi(\mathbf{0}, 0)\rangle_{T}$$
It has the Fourier expansion
$$\langle\phi(\boldsymbol{x}, \tau) \phi(\mathbf{0}, 0)\rangle_{T}=\frac{1}{\beta} \sum_{n=-\infty}^{\infty} \int \frac{d^{d} p}{(2 \pi)^{d}} \frac{e^{i \omega_{n} \tau+i p \cdot x}}{\omega_{n}^{2}+\boldsymbol{p}^{2}+m^{2}}$$
where, once again, $\omega_{n}=2 \pi T n$ are the Matsubara frequencies.
We will now obtain two useful expressions for the thermal propagator. The expressions follow from doing the momentum integrals first. The Matsubara frequencies act as mass terms of a field in one dimension lower. This observation allows us to identify the integrals in eq. (5.210) with the Euclidean propagators of an infinite number of fields, each labeled by an integer $n$, in $d$ Euclidean dimensions with mass squared equal to
$$m_{n}^{2}=m^{2}+\omega_{n}^{2}$$

## 物理代写|量子力学代写Quantum mechanics代考|Second quantization and the many-body problem

Let us consider now the problem of a system of $N$ identical nonrelativistic particles. For the sake of simplicity, assume that the physical state of each particle $j$ is described by its position $x_{j}$ relative to some reference frame. This case is easy to generalize.

The wave function for this system is $\Psi\left(x_{1}, \ldots, x_{N}\right)$. If the particles are identical, then the probability density, $\left|\Psi\left(x_{1}, \ldots, x_{N}\right)\right|^{2}$, must be invariant under arbitrary exchanges of the labels that we use to identify (or designate) the particles. In quantum mechanics, particles do not have well-defined trajectories. Only the states of a physical system are well defined. Thus, even though at some initial time $t_{0}$ the $N$ particles may be localized to a set of well-defined positions $x_{1}, \ldots, x_{N}$, they will become delocalized as the system evolves. Furthermore, the Hamiltonian itself is invariant under a permutation of the particle labels. Hence, permutations constitute a symmetry of a many-particle quantum mechanical system. In other words, in quantum mechanics, identical particles are indistinguishable. In particular, the probability density of any eigenstate must remain invariant if the labels of any pair of particles are exchanged.

If we denote by $P_{j k}$ the operator that exchanges the labels of particles $j$ and $k$, the wave functions must satisfy
$$P_{j k} \Psi\left(x_{1}, \ldots, x_{j}, \ldots, x_{k}, \ldots, x_{N}\right)=e^{i \phi} \Psi\left(x_{1}, \ldots, x_{j}, \ldots, x_{k}, \ldots, x_{N}\right)$$
Under a further exchange operation, the particles return to their initial labels, and we recover the original state. This sample argument then requires that $\phi=0, \pi$, since $2 \phi$ must not be an observable phase. We then conclude that there are two possibilities: either $\Psi$ is even under permutation and $P \Psi=\Psi$, or $\Psi$ is odd under permutation and $P \Psi=-\Psi$.

## 物理代写|量子力学代写Quantum mechanics代考|The thermal propagator

$$G_{T}^{(0)}(x, \tau)=\langle\phi(x, \tau) \phi(\mathbf{0}, 0)\rangle_{T}$$

$$\langle\phi(\boldsymbol{x}, \tau) \phi(\mathbf{0}, 0)\rangle_{T}=\frac{1}{\beta} \sum_{n=-\infty}^{\infty} \int \frac{d^{d} p}{(2 \pi)^{d}} \frac{e^{i \omega_{n} \tau+i p-x}}{\omega_{n}^{2}+\boldsymbol{p}^{2}+m^{2}}$$

$$m_{n}^{2}=m^{2}+\omega_{n}^{2}$$

## 物理代写|量子力学代写Quantum mechanics代考|Second quantization and the many-body problem

$$P_{j k} \Psi\left(x_{1}, \ldots, x_{j}, \ldots, x_{k}, \ldots, x_{N}\right)=e^{i \phi} \Psi\left(x_{1}, \ldots, x_{j}, \ldots, x_{k}, \ldots, x_{N}\right)$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:Quantum field theory, 物理代写, 量子场论

## 物理代考|量子场论代考QUANTUM FIELD THEORY代考|PHYSICS332 Canonical quantization in field theory

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## 物理代考|量子场论代考QUANTUM FIELD THEORY代考|Canonical quantization in field theory

We will now apply the axioms of quantum mechanics to a classical field theory. The result will be a QFT. For the sake of simplicity, we first consider the case of a scalar field $\phi(x)$. We have seen before that, given a Lagrangian density $\mathcal{L}\left(\phi, \partial_{\mu} \phi\right)$, the Hamiltonian can be found once the canonical momentum $\Pi(x)$ is defined:
$$\Pi(x)=\frac{\delta \mathcal{L}}{\delta \partial_{0} \phi(x)}$$
On a given time surface $x_{0}$, the classical Hamiltonian is
$$H=\int d^{3} x\left[\Pi\left(\boldsymbol{x}, x_{0}\right) \partial_{0} \phi\left(\boldsymbol{x}, x_{0}\right)-\mathcal{L}\left(\phi, \partial_{\mu} \phi\right)\right]$$
We quantize this theory by assigning to each dynamical variable of the classical theory a hermitian operator that acts on the Hilbert space of the quantum states of the system. Thus, the field $\hat{\phi}(\boldsymbol{x})$ and the canonical momentum $\widehat{\Pi}(\boldsymbol{x})$ are operators acting on a Hilbert space. These operators obey equal-time canonical commutation relations:
$$[\hat{\phi}(x), \widehat{\Pi}(y)]=i \hbar \delta(x-y)$$

## 物理代考|量子场论代考QUANTUM FIELD THEORY代考|Quantization of the free scalar field theory

We will now quantize the theory of a relativistic scalar field $\phi(x)$. In particular, we consider a free real scalar field $\phi$ whose Lagrangian density is
$$\mathcal{L}=\frac{1}{2}\left(\partial_{\mu} \phi\right)\left(\partial^{\mu} \phi\right)-\frac{1}{2} m^{2} \phi^{2}$$
The quantum mechanical Hamiltonian $\widehat{H}$ for a free real scalar field is
$$\widehat{H}=\int d^{3} x\left[\frac{1}{2} \widehat{\Pi}^{2}(x)+\frac{1}{2}(\nabla \hat{\phi}(x))^{2}+\frac{1}{2} m^{2} \hat{\phi}^{2}(x)\right]$$
where $\hat{\phi}$ and $\widehat{\Pi}$ satisfy the equal-time commutation relations (in units with $\hbar=c=1$ ):
$$\left[\hat{\phi}\left(x, x_{0}\right), \widehat{\Pi}\left(y, x_{0}\right)\right]=i \delta(x-y)$$
In the Heisenberg representation, $\hat{\phi}$ and $\widehat{\Pi}$ are time-dependent operators, while the states are time independent. The field operators obey the equations of motion:
$$i \partial_{0} \hat{\phi}\left(\boldsymbol{x}, x_{0}\right)=\left[\hat{\phi}\left(\boldsymbol{x}, x_{0}\right), \widehat{H}\right], \quad i \partial_{0} \widehat{\Pi}\left(\boldsymbol{x}, x_{0}\right)=\left[\widehat{\Pi}\left(\boldsymbol{x}, x_{0}\right), \widehat{H}\right]$$

## 物理代考|量子场论代考QUANTUM FIELD THEORY代考|Canonical quantization in field theory

X0，经典哈密顿量是
H=∫d3X[圆周率(X,X0)∂0φ(X,X0)−大号(φ,∂μφ)]

[φ^(X),圆周率^(是)]=一世ℏd(X−是)

## 物理代考|量子场论代考QUANTUM FIELD THEORY代考|Quantization of the free scalar field theory

H^对于一个自由的实标量场是
H^=∫d3X[12圆周率^2(X)+12(∇φ^(X))2+12米2φ^2(X)]
φ^和圆周率^满足等时交换关系（单位为ℏ=C=1 ):
[φ^(X,X0),圆周率^(是,X0)]=一世d(X−是)

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。