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在理论物理学中，量子场论（QFT）是一个理论框架，它结合了经典场论、狭义相对论和量子力学。QFT在粒子物理学中被用来构建亚原子粒子的物理模型，在凝聚态物理学中被用来构建类粒子的模型。
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我们提供的量子场论Quantum field theory及其相关学科的代写，服务范围广, 其中包括但不限于:
- Statistical Inference 统计推断
- Statistical Computing 统计计算
- Advanced Probability Theory 高等概率论
- Advanced Mathematical Statistics 高等数理统计学
- (Generalized) Linear Models 广义线性模型
- Statistical Machine Learning 统计机器学习
- Longitudinal Data Analysis 纵向数据分析
- Foundations of Data Science 数据科学基础
物理代写|量子场论代写Quantum field theory代考|Simulating the Anharmonic Oscillator
Since a CPU only knows numbers, we rescale the dimensionful constants and coordinates in the action (4.39) with powers of the dimensionful lattice constant $\varepsilon$ to arrive at dimensionless lattice quantities $\left(m_{L}, \mu_{L}, \lambda_{L}, q_{L}\right)$ :
$$
q_{L}=q / \varepsilon, \quad m_{L}=\varepsilon m, \quad \mu_{L}=\varepsilon^{3} \mu \quad \text { and } \quad \lambda_{L}=\varepsilon^{5} \lambda
$$
In terms of these dimensionless quantities, the lattice action takes the simple form
$$
S(\boldsymbol{q})=\sum_{j=0}^{n-1}\left{\frac{m_{L}}{2}\left(q_{j+1}-q_{j}\right){L}^{2}+\mu{L} q_{j, L}^{2}+\lambda_{L} q_{j, L}^{4}\right}
$$
In a local Metropolis algorithm, we test a new configuration $\boldsymbol{q}^{\prime}$ which differs from the old configuration $q$ only on one lattice point, say lattice point $j$. Then the difference of the two actions is
$$
\begin{aligned}
S\left(\boldsymbol{q}^{\prime}\right)-S(\boldsymbol{q}) \approx &\left(q_{j}^{\prime}-q_{j}\right){L}\left{-m{L}\left(q_{j+1}+q_{j-1}\right){L}\right.\ &\left.+\left(q{j}^{\prime}+q_{j}\right){L}\left{m{L}+\mu_{L}+\lambda_{L}\left(q_{j}^{2}+q_{j}^{2}\right)_{L}\right}\right}
\end{aligned}
$$
Here the problem arises how to determine energies or lengths in physical units. They are only given in terms of the unknown lattice constant $\varepsilon$ which does not even enter the lattice action $S(\boldsymbol{q})$. Thus one first calculates some observable (e.g., example an energy) which is then compared to the experimentally known value to set the scale.
Alternatively one may express all dimensionful quantities in units of a known and fixed unit of length $\ell$,
$$
\varepsilon=a \ell, \quad q=\tilde{q} \ell, \quad m=\tilde{m} / \ell, \quad \mu=\tilde{\mu} / \ell^{3} \quad \text { and } \quad \lambda=\tilde{\lambda} / \ell^{5}
$$
In this system of units
$$
\begin{aligned}
S\left(\boldsymbol{q}^{\prime}\right)-S(\boldsymbol{q})=&\left(\tilde{q}{j}^{\prime}-\tilde{q}{j}\right)\left{-\frac{\tilde{m}}{a}\left(\tilde{q}{j+1}+\tilde{q}{j-1}\right)\right.\
&\left.+\left(\tilde{q}{j}^{\prime}+\tilde{q}{j}\right)\left{\frac{\tilde{m}}{a}+a \tilde{\mu}+a \tilde{\lambda}\left(\tilde{q}{j}^{n 2}+\tilde{q}{j}^{2}\right)\right}\right}
\end{aligned}
$$
This formula is used in the header file stdanho. h on page 80 which is called by the program anharmonic1.c listed on p. 77. The parameters $m, \mu, \lambda$, the number of lattice points (in the C-program $n$ is renamed $N$ ), and the lattice constant $a$ are all defined in another header file constants. hon p. 79. In order to have uncorrelated configurations, only one out of $M A$ configuration is measured. In addition, since a Markov chain needs some time to reach equilibrium, we start measuring configurations only after $M A \times M E$ sweeps through the lattice.
To measure the square of $\psi_{0}(q)$ on the interval $[-I N T E R V, I N T E R V]$ with the help of (4.49), we divide the interval into BIN bins. With the parameter DELTA, we adjust the amplitude of a tentative coordinate change during a local update according to $q^{\prime}=q+D E L T A \times(1-2 r)$, where $r$ is a uniform random number on $[0,1]$. The program anharmonic $1 .$. generates the histogram of the probability distribution. In Fig. $4.4$ we plotted the so obtained density $\left|\psi_{0}(q)\right|^{2}$, both for the harmonic and the anharmonic oscillator.
物理代写|量子场论代写Quantum field theory代考|Hybrid Monte Carlo Algorithm
The powerful hybrid Monte Carlo algorithm has been developed by S. DUANE ET AL. [10]. A recommendable introduction can be found in [11], and in our presentation, we follow in part the reviews $[12,13]$. The algorithm represents a combination of molecular dynamics (see [14]) and Metropolis algorithm. It aims at global updates of whole configurations with reasonable large acceptance rates, in order to minimize the time required to generate independent configurations.
Molecular dynamics (MD) simulations are frequently and successfully used to study (classical) many-body systems and are applied to problems in material science, astrophysics, and biophysics. In molecular dynamics simulations, one solves the Hamiltonian cquations of motion
$$
\dot{q}{i}=\frac{\partial H}{\partial p{i}} \quad, \quad \dot{p}{i}=-\frac{\partial H}{\partial q{i}}
$$
numerically and makes use of the ergodic hypothesis, according to which the statistical ensemble averages are equal to time averages. From an initial configuration, represented by a point $\left(\boldsymbol{q}{0}, \boldsymbol{p}{0}\right)$ in phase space, one obtains a unique solution of Hamilton’s equations of motion. Without any numerical errors, the energy is a constant of motion, and this simple observation will be relevant in what follows.
In Euclidean quantum mechanics discretized on an imaginary-time lattice $\Lambda$ with $n$ points ${x}$, we introduce an extended phase space with dimension $2 n$. Each point in extended phase space consists of a broken-line path characterized by variables $\left{q_{x}\right}$ and their canonically conjugated momenta $\left{p_{x}\right}$. To construct a Markov process with global updates and high acceptance rate, we introduce the following auxiliary Hamiltonian in extended phase space:
$$
H(\boldsymbol{q}, \boldsymbol{p})=\frac{\boldsymbol{p}^{2}}{2}+S(\boldsymbol{q}), \quad \boldsymbol{p}^{2}=\sum_{x \in \Lambda} p_{x}^{2}
$$
物理代写|量子场论代写Quantum field theory代考|Implementing the HMC Algorithm
To fulfil the condition of detailed balance, we need a time-reversible and symplectic integrator to numerically solve the fictitious Hamiltonian dynamics. Using the naive forward difference operator $\dot{f}(\tau) h \approx f(\tau+h)-f(\tau)$ in the discretized equations of motion leads to the time-irreversible prescription
$$
\begin{aligned}
&\boldsymbol{q}(\tau+h)=\boldsymbol{q}(\tau)+h \boldsymbol{p}(\tau) \
&\boldsymbol{p}(\tau+h)=\boldsymbol{p}(\tau)+h \boldsymbol{F}(\boldsymbol{q}(\tau))
\end{aligned}
$$
with force $\boldsymbol{F}=-\nabla_{q} S$. This prescription must not be used in any HMC algorithm. In most simulation the time-reversible leapfrog integration is used instead. Here one first moves forward with the momenta a half step in fictitious time. Then one moves forward several time-steps alternately with coordinates $\boldsymbol{q}$ and momenta $\boldsymbol{p}$. The last move of the momenta is again a half step in fictitious time.
量子场论代考
物理代写|量子场论代写Quantum field theory代考|Simulating the Anharmonic Oscillator
由于 CPU 只知道数字，我们用维数晶格常数的幂重新调整动作 (4.39) 中的维数常数和坐标e达到无量纲晶格数量(米大号,μ大号,λ大号,q大号) :
q大号=q/e,米大号=e米,μ大号=e3μ 和 λ大号=e5λ
就这些无量纲量而言，晶格作用采用简单形式
\left 的分隔符缺失或无法识别\left 的分隔符缺失或无法识别
在本地 Metropolis 算法中，我们测试一个新的配置q′与旧配置不同q只在一个格点上，说格点j. 那么这两个动作的区别是
\left 的分隔符缺失或无法识别\left 的分隔符缺失或无法识别
这里的问题是如何确定物理单位的能量或长度。它们仅根据未知的晶格常数给出e它甚至没有进入晶格动作小号(q). 因此，首先计算一些可观察的（例如，例如能量），然后将其与实验已知的值进行比较以设置比例。
或者，可以用已知的固定长度单位表示所有量纲量ℓ,
e=一个ℓ,q=q~ℓ,米=米~/ℓ,μ=μ~/ℓ3 和 λ=λ~/ℓ5
在这个单位制中
\left 的分隔符缺失或无法识别\left 的分隔符缺失或无法识别
此公式用于头文件 stdanho. h 在第 80 页上，它由 p 中列出的程序 anharmonic1.c 调用。77.参数米,μ,λ, 格点数（在 C 程序中n已重命名ñ)，以及晶格常数一个都是在另一个头文件常量中定义的。汉山口 79. 为了有不相关的配置，只有一个米一个配置进行测量。此外，由于马尔可夫链需要一些时间才能达到平衡，我们只在之后才开始测量配置米一个×米和扫过格子。
测量平方ψ0(q)在区间[−我ñ吨和R在,我ñ吨和R在]在 (4.49) 的帮助下，我们将区间划分为 BIN 箱。使用参数 DELTA，我们根据下式调整局部更新期间暂定坐标变化的幅度q′=q+D和大号吨一个×(1−2r)， 在哪里r是一个统一的随机数[0,1]. 程序非谐波1.. 生成概率分布的直方图。在图。4.4我们绘制了如此获得的密度|ψ0(q)|2，对于谐波和非谐波振荡器。
物理代写|量子场论代写Quantum field theory代考|Hybrid Monte Carlo Algorithm
S. DUANE 等人开发了强大的混合蒙特卡罗算法。[10]。推荐的介绍可以在 [11] 中找到，在我们的演示文稿中，我们部分遵循了评论[12,13]. 该算法代表了分子动力学（参见 [14]）和 Metropolis 算法的组合。它旨在以合理的大接受率对整个配置进行全局更新，以最大限度地减少生成独立配置所需的时间。
分子动力学 (MD) 模拟经常成功地用于研究（经典）多体系统，并应用于材料科学、天体物理学和生物物理学中的问题。在分子动力学模拟中，可以求解哈密顿运动方程
q˙一世=∂H∂p一世,p˙一世=−∂H∂q一世
在数值上并利用遍历假设，根据该假设，统计集合平均值等于时间平均值。从初始配置开始，由一个点表示(q0,p0)在相空间中，人们获得了哈密尔顿运动方程的唯一解。没有任何数值误差，能量是一个运动常数，这个简单的观察将与以下内容相关。
在欧几里得量子力学中，在虚时格上离散化Λ和n积分X，我们引入了一个扩展的相空间，维数2n. 扩展相空间中的每个点都由一条以变量为特征的折线路径组成\left 的分隔符缺失或无法识别\left 的分隔符缺失或无法识别和他们典型的共轭动量\left 的分隔符缺失或无法识别\left 的分隔符缺失或无法识别. 为了构建具有全局更新和高接受率的马尔可夫过程，我们在扩展相空间中引入了以下辅助哈密顿量：
H(q,p)=p22+小号(q),p2=∑X∈ΛpX2
物理代写|量子场论代写Quantum field theory代考|Implementing the HMC Algorithm
为了满足详细平衡的条件，我们需要一个时间可逆的辛积分器来数值求解虚构的哈密顿动力学。使用朴素的前向差分算子F˙(τ)H≈F(τ+H)−F(τ)在离散的运动方程中导致时间不可逆的处方
q(τ+H)=q(τ)+Hp(τ) p(τ+H)=p(τ)+HF(q(τ))
用力F=−∇q小号. 此规定不得用于任何 HMC 算法。在大多数模拟中，使用时间可逆的越级积分代替。这里第一个在虚拟时间中以半步的势头向前移动。然后与坐标交替向前移动几个时间步q和动量p. 动量的最后一步又是虚拟时间的半步。
