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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|Metropolis–Hastings Algorithm
In the original Metropolis algorithm, the test probability (jump probability) is symmetric in $\omega$ and $\omega^{\prime}$. Most common is a normal distribution centered at $\omega$ the variance of which must be tuned to get a reasonable acceptance rate for new configurations. A generalization due to Hastings allows for asymmetric test distributions. Typically one takes a distribution $T\left(\omega, \omega^{\prime}\right)$ that is the same for all configurations $\omega^{\prime}$ that can be reached from $\omega[6,7]$. The test probability of the
remaining configurations is set to zero. Thus, if $N$ is the number of accessible configurations, then
$$
T\left(\omega, \omega^{\prime}\right)= \begin{cases}1 / N & \text { if } \omega \rightarrow \omega^{\prime} \text { is possible } \ 0 & \text { otherwise }\end{cases}
$$
We choose the acceptance rate
$$
A\left(\omega, \omega^{\prime}\right)=\min \left(\frac{P_{\omega^{\prime}} T\left(\omega^{\prime}, \omega\right)}{P_{\omega} T\left(\omega, \omega^{\prime}\right)}, 1\right)
$$
for which $W$ in (4.23) fulfills the condition for detailed balance. In fact, the condition
$$
P_{\omega} T\left(\omega, \omega^{\prime}\right) \times \min \left(\frac{P_{\omega^{\prime}} T\left(\omega^{\prime}, \omega\right)}{P_{\omega} T\left(\omega, \omega^{\prime}\right)}, 1\right)=P_{\omega^{\prime}} T\left(\omega^{\prime}, \omega\right) \times \min \left(\frac{P_{\omega} T\left(\omega, \omega^{\prime}\right)}{P_{\omega^{\prime}} T\left(\omega^{\prime}, \omega\right)}, 1\right)
$$
is fulfilled both for $P_{\omega^{\prime}} T\left(\omega^{\prime}, \omega\right)$ larger and smaller $P_{\omega} T\left(\omega, \omega^{\prime}\right)$.
A good choice of the initial configuration $\omega$ may save computing time. For example, at high temperatures the degrees of freedom are uncorrelated, and we choose the variables at random in contrast to low temperatures where they are strongly correlated.
We now discuss a particular implementation of the algorithm for a onedimensional quantum mechanical system discretized on time-lattice with $n$ points. We choose an initial configuration $q=\left(q_{1}, \ldots, q_{n}\right)$. The first lattice-variable $q_{1}$ is altered or remains unchanged according to the following rules:
- Suggest a provisional change of $q_{1}$ to a randomly chosen $q_{1}^{\prime}$.
- If the action decreases, that is, $\Delta S<0$, then permanently replace $q_{1}$ by $q_{1}^{\prime}$.
- If the action increases, choose an uniformly distributed random number $r \in$ $[0,1]$. The suggestion $q_{1}^{\prime}$ is accepted if $\exp (-\Delta S)>r$. Otherwise the latticevariable $q_{1}$ remains unaltered.
- Proceed with the variables $q_{2}, q_{3}, \ldots$ in the same way till all variables have been tested.
- If the last lattice point is reached, a “sweep through the lattice” or a Monte Carlo iteration is finished, and one starts again with the first lattice point.
物理代写|量子场论代写Quantum field theory代考|Heat Bath Algorithm
For the heat bath algorithm, the transition probability $W\left(\omega, \omega^{\prime}\right)$ depends only on the final state $\omega^{\prime}$ such that the condition of detailed balance (4.21) implies $W\left(\omega, \omega^{\prime}\right) \propto$ $P_{\omega^{\prime}}$. The normalization conditions for $P$ and $W$ lead to
$$
W\left(\omega, \omega^{\prime}\right)=P_{\omega^{\prime}}
$$
The algorithm is particularly useful when the equilibrium distribution $P$ can be integrated or summed up easily. Let us first apply the heat bath algorithm to estimate one-dimensional integrals of the form $\langle O\rangle=\int O(x) P(x) \mathrm{d} x$ with fixed $P$ and varying $O$. Thus we need random numbers distributed according to $P(x)$. To this end we first generate uniformly distributed random numbers $y_{i}$ on the unit interval and consider the preimages $\left{F^{-1}\left(y_{i}\right)\right}$ of these numbers. Here $F$ denotes the monotonically increasing anti-derivative of the probahility density,
$$
F(x)=\int_{-\infty}^{x} P(u) \mathrm{d} u \in[0,1]
$$
Because of the identity
$$
y_{2}-y_{1}=\int_{F^{-1}\left(y_{1}\right)}^{F^{-1}\left(y_{2}\right)} P(u) \mathrm{d} u,
$$
these preimages are distributed according to $P$. This is made clear in Fig. 4.2.
Box-Muller Method
This method has been introduced by George Box and Mervin Muller [8] and nicely illustrates how to extend the previous algorithm to higher dimensions. Uniformly distributed random points $\boldsymbol{y}$ in the unit square are mapped into normally distributed random numbers $\boldsymbol{x} \in \mathbb{R}^{2}$ with mean $\overline{\boldsymbol{x}}$ and variance $\sigma$. Thus we demand
$$
\mathrm{d}^{2} y=\operatorname{det}\left(\frac{\partial y_{i}}{\partial x_{j}}\right) \mathrm{d}^{2} x=P(\boldsymbol{x}) \mathrm{d}^{2} x, \quad P(\boldsymbol{x})=\frac{1}{2 \pi \sigma^{2}} \mathrm{e}^{-(\boldsymbol{x}-\bar{x})^{2} / 2 \sigma^{2}}
$$
We introduce polar coordinates $x_{1}-\bar{x}{1}=r \cos \varphi$ and $x{2}-\bar{x}{2}=r \sin \varphi$ and set $\varphi=2 \pi y{2}$. Assuming that $r$ only depends on $y_{1}$, we arrive at
$$
\mathrm{d}^{2} y=\frac{\mathrm{d} y_{1}}{\mathrm{~d} r} \mathrm{~d} r \frac{\mathrm{d} \varphi}{2 \pi}=\frac{1}{2 \pi \sigma^{2}} \mathrm{e}^{-r^{2} / 2 \sigma^{2}} r \mathrm{~d} r \mathrm{~d} \varphi \quad \text { or } \quad \frac{\mathrm{d} y_{1}}{\mathrm{~d} r}=\frac{r}{\sigma^{2}} \mathrm{e}^{-r^{2} / 2 \sigma^{2}}
$$
物理代写|量子场论代写Quantum field theory代考|The Anharmonic Oscillator
We return to one-dimensional quantum mechanical systems at imaginary time and discretized on a time-lattice. They are characterized by their Euclidean lattice action
$$
S(q)=\varepsilon \sum_{j=0}^{n-1}\left{\frac{m}{2} \frac{\left(q_{j+1}-q_{j}\right)^{2}}{\varepsilon^{2}}+V\left(q_{j}\right)\right}
$$
In particular we shall consider the anharmonic oscillator with quartic potential
$$
V(q)=\mu q^{2}+\lambda q^{4}
$$
in more detail. The choice of the number $n$ of lattice points and of the lattice constant $\varepsilon$ is limited mainly by two aspects:
- $\varepsilon$ should be sufficiently small to be near the continuum limit $\varepsilon \rightarrow 0$.
- The quantities of interest should fit into the interval $n \varepsilon$. For instance, the width of the ground state should be less than $n \varepsilon$.
If $\lambda_{0}$ is a typical length scale of the system at hand, then the quantities $n$ and $\varepsilon$ should satisfy constraints of the type
$$
\varepsilon \lesssim \frac{\lambda_{0}}{10} \quad \text { and } \quad n \varepsilon \gtrsim 10 \lambda_{0}
$$
Another problem concerns the size of statistical fluctuations in any Monte Carlo simulation. The relative standard deviation of a random variable $O$ is
$$
\Delta_{O}=\sqrt{\frac{\left\langle O^{2}\right\rangle-\langle O\rangle^{2}}{\langle O\rangle^{2}}} \propto \text { (number of lattice points) }{ }^{-1 / 2}
$$
As an estimate for the expectation value $\langle O\rangle$, we take
$$
\bar{O}=\frac{1}{M} \sum_{\mu=1}^{M} O\left(q_{\mu}\right)
$$
with Boltzmann-distributed configurations $\boldsymbol{q}{\mu}$. Depending on the initial configuration, the Markov chain may need some “time” to equilibrate. In the simulations of the anharmonic oscillator presented below, equilibrium is reached after approximately $10-100$ sweeps through the lattice. In addition, since configurations of successive sweeps are correlated, only every MA’th sweep is used to estimate expectation values. The number $M A$ should be larger than the relevant autocorrelation time – the time over which the values $O\left(q{\mu}\right)$ are correlated. Different random variables may have vastly different auto-correlation times. As a general rule, they are large for spatially averaged quantities.
量子场论代考
物理代写|量子场论代写Quantum field theory代考|Metropolis–Hastings Algorithm
在原始的 Metropolis 算法中，测试概率（跳跃概率）在ω和ω′. 最常见的是正态分布，集中在ω必须调整其方差以获得新配置的合理接受率。由于黑斯廷斯的概括允许不对称的测试分布。通常需要一个分布吨(ω,ω′)这对于所有配置都是相同的ω′可以从ω[6,7]. 的测试概率
其余配置设置为零。因此，如果ñ是可访问配置的数量，那么
吨(ω,ω′)={1/ñ 如果 ω→ω′ 是可能的 0 否则
我们选择接受率
一个(ω,ω′)=分钟(磷ω′吨(ω′,ω)磷ω吨(ω,ω′),1)
为此在（4.23）式满足详细平衡的条件。事实上，条件
磷ω吨(ω,ω′)×分钟(磷ω′吨(ω′,ω)磷ω吨(ω,ω′),1)=磷ω′吨(ω′,ω)×分钟(磷ω吨(ω,ω′)磷ω′吨(ω′,ω),1)
都满足磷ω′吨(ω′,ω)越来越大磷ω吨(ω,ω′).
初始配置的不错选择ω可以节省计算时间。例如，在高温下，自由度是不相关的，我们随机选择变量，而不是在它们强相关的低温下。
我们现在讨论在时间格上离散化的一维量子力学系统的算法的特定实现n点。我们选择一个初始配置q=(q1,…,qn). 第一个格变量q1按下列规则变更或保持不变：
- 建议临时更改q1到一个随机选择的q1′.
- 如果作用减小，即Δ小号<0，然后永久替换q1经过q1′.
- 如果动作增加，选择一个均匀分布的随机数r∈ [0,1]. 建议q1′被接受，如果经验(−Δ小号)>r. 否则格变量q1保持不变。
- 继续变量q2,q3,…以同样的方式测试所有变量。
- 如果到达最后一个格点，则完成“扫过格”或蒙特卡罗迭代，然后从第一个格点重新开始。
物理代写|量子场论代写Quantum field theory代考|Heat Bath Algorithm
对于热浴算法，转移概率在(ω,ω′)仅取决于最终状态ω′使得详细平衡条件（4.21）意味着在(ω,ω′)∝ 磷ω′. 归一化条件磷和在导致
在(ω,ω′)=磷ω′
该算法在平衡分布时特别有用磷可以很容易地整合或总结。让我们首先应用热浴算法来估计形式的一维积分⟨○⟩=∫○(X)磷(X)dX与固定磷和变化○. 因此，我们需要根据分布的随机数磷(X). 为此我们首先生成均匀分布的随机数是一世在单位间隔上并考虑原像\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些数字中。这里F表示概率密度的单调递增反导数，
F(X)=∫−∞X磷(在)d在∈[0,1]
因为身份
是2−是1=∫F−1(是1)F−1(是2)磷(在)d在,
这些原像是根据磷. 这在图 4.2 中很清楚。
Box-Muller 方法
George Box 和 Mervin Muller [8] 介绍了该方法，并很好地说明了如何将先前的算法扩展到更高的维度。均匀分布的随机点是在单位平方被映射成正态分布的随机数X∈R2平均X―和方差σ. 因此我们要求
d2是=这(∂是一世∂Xj)d2X=磷(X)d2X,磷(X)=12圆周率σ2和−(X−X¯)2/2σ2
我们引入极坐标 $x_{1}-\bar{x} {1}=r \cos \varphi一个ndx {2} – \ bar {x} {2} = r \ sin \ varphi一个nds和吨\varphi=2 \pi y {2}.一个ss在米一世nG吨H一个吨r○nl是d和p和nds○n你_{1},在和一个rr一世在和一个吨d2是=d是1 dr drd披2圆周率=12圆周率σ2和−r2/2σ2r dr d披 或者 d是1 dr=rσ2和−r2/2σ2$
物理代写|量子场论代写Quantum field theory代考|The Anharmonic Oscillator
我们回到虚时间的一维量子力学系统，并在时格上离散化。它们的特点是欧几里得晶格作用
\left 的分隔符缺失或无法识别\left 的分隔符缺失或无法识别
特别是我们将考虑具有四次势的非谐振荡器
在(q)=μq2+λq4
更详细。号码的选择n晶格点的和晶格常数的e主要受限于两个方面：
- e应该足够小以接近连续极限e→0.
- 感兴趣的数量应适合区间ne. 例如，基态的宽度应该小于ne.
如果λ0是手头系统的典型长度尺度，那么数量n和e应该满足类型的约束
e≲λ010 和 ne≳10λ0
另一个问题涉及任何蒙特卡洛模拟中统计波动的大小。随机变量的相对标准差○是
Δ○=⟨○2⟩−⟨○⟩2⟨○⟩2∝ （格点数） −1/2
作为期望值的估计⟨○⟩， 我们采取
○¯=1米∑μ=1米○(qμ)
具有玻尔兹曼分布的配置qμ. 根据初始配置，马尔可夫链可能需要一些“时间”来平衡。在下面介绍的非谐波振荡器的模拟中，大约在10−100扫过格子。此外，由于连续扫描的配置是相关的，因此只有每个 MA’th 扫描用于估计期望值。号码米一个应该大于相关的自相关时间——值的时间○(qμ)是相关的。不同的随机变量可能具有非常不同的自相关时间。作为一般规则，它们对于空间平均量来说很大。
