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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|Fixed Points of Markov Chains
Every stochastic matrix has the eigenvalue 1 . The corresponding right eigenvector is given by $\sim(1,1, \ldots, 1)^{T}$. But since $W$ acts from the right, we are interested in the left eigenvector with eigenvalue 1 . We follow [1] and consider the series
$$
\boldsymbol{p}{n}=\frac{1}{n} \sum{j=0}^{n-1} \boldsymbol{p} W^{j} .
$$
Since stochastic vectors form a compact set, the series has a convergent subsequence
$$
\frac{1}{n_{k}} \sum_{0}^{n_{k}-1} p W^{j} \longrightarrow \boldsymbol{P}
$$
We multiply this subsequence by $W$ from the right,
$$
\frac{1}{n_{k}} \sum_{1}^{n_{k}} \boldsymbol{p} W^{j} \longrightarrow \boldsymbol{P W} .
$$
In the difference of the last two formulas, only two terms in the left-hand sides remain, and we obtain in the limit $n_{k} \rightarrow \infty$
$$
\frac{1}{n_{k}}\left(\boldsymbol{p}-\boldsymbol{p} W^{n_{k}}\right) \longrightarrow \boldsymbol{P}-\boldsymbol{P} W
$$
Since the left-hand side converges to zero, this implies the eigenvalue equation
$$
\boldsymbol{P} W=\boldsymbol{P}
$$
which means that every stochastic matrix $W$ has at least one fixed point $\boldsymbol{P}$, i.e., a left eigenvector with eigenvalue 1 .
We now assume that $W$ has at least one column with minimal element greater than a positive number $\delta$. This means that all configurations can jump with nonvanishing probability to at least one configuration. Stochastic matrices with this property are called attractive, and for an attractive matrix, the series $W^{n} p$ converges for any stochastic vector $\boldsymbol{p}$. Note that $W$ in (4.6) is not attractive and this explains why the associated Markov chain does not converge. For the proof of convergence, we first note that for two real numbers $p$ and $p^{\prime}$ we have
$$
\left|p-p^{\prime}\right|=p+p^{\prime}-2 \min \left(p, p^{\prime}\right),
$$
and this relation implies
$$
\left|\boldsymbol{p}-\boldsymbol{p}^{\prime}\right|=2-2 \sum_{\omega} \min \left(p_{\omega}, p_{\omega}^{\prime}\right)
$$
for two stochastic vectors. We used the $\ell_{1}$-norm $|\boldsymbol{p}|=\sum_{\omega}\left|p_{\omega}\right|$.
Next we wish to prove that an attractive $W$ acts on vectors $\Delta=\left(\Delta_{1}, \Delta_{2}, \ldots\right)$ with
$$
|\Delta| \equiv \sum\left|\Delta_{\omega}\right|=2 \quad \text { and } \quad \sum \Delta_{\omega}=0
$$ in a contractive way. Let us assign the stochastic vector $\boldsymbol{e}{\omega}$ to each $\omega$ which represents the probability distribution for finding $\omega$ with probability one. Thus all entries of $e{\omega}$ vanish with the exception of entry $\omega$, which is 1 . The stochastic vectors $\left{\boldsymbol{e}{\omega}\right}$ form an orthonormal basis. In a first step, we prove that $W$ is contractive for difference vectors $\boldsymbol{e}{\omega}-\boldsymbol{e}{\omega}{ }^{\prime}$. Thus we apply the identity (4.9) to the stochastic vectors $\boldsymbol{e}{\omega} W$ and $\boldsymbol{e}{\omega} W$, i.e., to rows of $W$ belonging to $\omega$ and $\omega^{\prime}$. In case of an attractive $W$, we find for $\omega \neq \omega^{\prime}$ $$ \begin{aligned} \left|\boldsymbol{e}{\omega} W-\boldsymbol{e}{\omega^{\prime}} W\right| &=2-2 \sum{\omega^{\prime \prime}} \min \left{W\left(\omega, \omega^{\prime \prime}\right), W\left(\omega^{\prime}, \omega^{\prime \prime}\right)\right} \
& \leq 2-2 \delta=(1-\delta) \underbrace{\left|\boldsymbol{e}{\omega}-\boldsymbol{e}{\omega^{\prime}}\right|}_{=2} \text { with } 0<\delta<1
\end{aligned}
$$
物理代写|量子场论代写Quantum field theory代考|Detailed Balance
The condition of detailed balance is a simple and physically well-founded constraint on a Markov chain which implies the fixed point equation (4.8). Detailed balance means a balance between any two configurations: the equilibrium probability for $\omega$, multiplied by the jump probability from $\omega$ to $\omega^{\prime}$, is equal to the equilibrium probability of $\omega^{\prime}$ multiplied by the jump probability from $\omega^{\prime}$ to $\omega$,
$$
P_{\omega} W\left(\omega, \omega^{\prime}\right)=P_{\omega^{\prime}} W\left(\omega^{\prime}, \omega\right)
$$
If in equilibrium the configuration $\omega$ is more likely occupied than the configuration $\omega^{\prime}$, then the transition amplitude from $\omega$ to $\omega^{\prime}$ is less than the amplitude for the reverse transition.
The condition of detailed balance guarantees that $\boldsymbol{P}$ is a fixed point of the chain,
$$
\sum_{\omega} P_{\omega} W\left(\omega, \omega^{\prime}\right)=\sum_{\omega} P_{\omega^{\prime}} W\left(\omega^{\prime}, \omega\right)=P_{\omega^{\prime}}
$$ but it does not fix $W$ uniquely. We may use the residual freedom to choose simple and efficient algorithms. In particular the fast Metropolis and heat bath algorithms are universally applicable and are often used. For statistical systems which can be dualized, there exist the more efficient cluster algorithms, which do not suffer from the problem of the so-called critical slowing down. These and other Monte Carlo algorithms will be introduced and applied in this book. More material can be found in the textbooks [2-5].
物理代写|量子场论代写Quantum field theory代考|Acceptance Rate
The probabilities $W(\omega, \omega)$ of not jumping are not constrained by the condition of detailed balance. Thus we may change the probabilities $W\left(\omega, \omega^{\prime}\right)$ for the transition between different configurations without violating the sum rule (4.1) if we only readjust the unconstrained $W(\omega, \omega)$.
We factorize the transition probability into the product of a test probability and an acceptance rate
$$
W\left(\omega, \omega^{\prime}\right)=T\left(\omega, \omega^{\prime}\right) A\left(\omega, \omega^{\prime}\right)
$$
where $T\left(\omega, \omega^{\prime}\right)$ is the probability of testing the new configuration $\omega^{\prime}$ with given initial configuration $\omega$. If $\omega^{\prime}$ is tested, then the quantity $0 \leq A\left(\omega, \omega^{\prime}\right) \leq 1$ corresponds to the probability that the transition to $\omega^{\prime}$ is accepted. Note that the conditions
$$
\frac{T\left(\omega, \omega^{\prime}\right) A\left(\omega, \omega^{\prime}\right)}{T\left(\omega^{\prime}, \omega\right) A\left(\omega^{\prime}, \omega\right)}=\frac{P_{\omega^{\prime}}}{P_{\omega}}
$$
do not fix the ratios of acceptance rates. A good Monte Carlo algorithm requires the best possible choice for these rates. For too small rates, only a tiny fraction of jumps is accepted, and the system is stuck in its initial configuration. We waste valuable computing time without passing through the configuration space. Thus, in many cases, one sets the greater of two acceptance rates $A\left(\omega, \omega^{\prime}\right)$ and $A\left(\omega^{\prime}, \omega\right)$ equal to 1. The smaller rate is chosen such that the condition $(4.24)$ is satisfied.
量子场论代考
物理代写|量子场论代写Quantum field theory代考|Fixed Points of Markov Chains
每个随机矩阵都有特征值 1 。对应的右特征向量由下式给出∼(1,1,…,1)吨. 但是由于在从右侧开始，我们对特征值为 1 的左侧特征向量感兴趣。我们遵循 [1] 并考虑系列
pn=1n∑j=0n−1p在j.
由于随机向量形成一个紧集，该系列有一个收敛的子序列
1nķ∑0nķ−1p在j⟶磷
我们将这个子序列乘以在从右边，
1nķ∑1nķp在j⟶磷在.
最后两个公式的差，只剩下左边两项，我们在极限中得到nķ→∞
1nķ(p−p在nķ)⟶磷−磷在
由于左侧收敛到零，这意味着特征值方程
磷在=磷
这意味着每个随机矩阵在至少有一个固定点磷，即特征值为 1 的左特征向量。
我们现在假设在至少有一列的最小元素大于一个正数d. 这意味着所有配置都可以以非消失概率跳转到至少一种配置。具有这种性质的随机矩阵称为有吸引力的，对于有吸引力的矩阵，级数在np收敛于任何随机向量p. 注意在在（4.6）中没有吸引力，这解释了为什么相关的马尔可夫链不收敛。为了证明收敛性，我们首先注意到对于两个实数p和p′我们有
|p−p′|=p+p′−2分钟(p,p′),
这种关系意味着
|p−p′|=2−2∑ω分钟(pω,pω′)
对于两个随机向量。我们使用了ℓ1-规范|p|=∑ω|pω|.
接下来我们要证明一个有吸引力的在作用于向量Δ=(Δ1,Δ2,…)和
|Δ|≡∑|Δω|=2 和 ∑Δω=0以一种收缩的方式。让我们分配随机向量和ω每个ω表示找到的概率分布ω概率为一。因此所有条目和ω除进入外消失ω，即 1 。随机向量\left 的分隔符缺失或无法识别\left 的分隔符缺失或无法识别形成一个标准正交基。第一步，我们证明在对于差分向量是收缩的和ω−和ω′. 因此我们将恒等式（4.9）应用于随机向量和ω在和和ω在，即，到行在属于ω和ω′. 在有吸引力的情况下在, 我们发现ω≠ω′\left 的分隔符缺失或无法识别\left 的分隔符缺失或无法识别
物理代写|量子场论代写Quantum field theory代考|Detailed Balance
详细平衡的条件是对马尔可夫链的简单且物理上有根据的约束，这意味着不动点方程（4.8）。详细平衡是指任意两种配置之间的平衡：ω，乘以跳跃概率ω至ω′, 等于平衡概率ω′乘以跳跃概率ω′至ω,
磷ω在(ω,ω′)=磷ω′在(ω′,ω)
如果处于平衡状态，配置ω比配置更可能被占用ω′，然后从跃迁幅度ω至ω′小于反向转换的幅度。
详细余额条件保证磷是链的一个不动点，
∑ω磷ω在(ω,ω′)=∑ω磷ω′在(ω′,ω)=磷ω′但它不能解决在独一无二。我们可以利用剩余自由度来选择简单有效的算法。特别是快速 Metropolis 和热浴算法普遍适用并且经常使用。对于可以对偶的统计系统，存在更有效的聚类算法，这些算法不存在所谓的临界减速问题。本书将介绍和应用这些和其他蒙特卡罗算法。更多的材料可以在教科书中找到[2-5]。
物理代写|量子场论代写Quantum field theory代考|Acceptance Rate
概率在(ω,ω)不跳跃不受详细平衡条件的限制。因此我们可以改变概率在(ω,ω′)对于不同配置之间的转换，如果我们只重新调整不受约束的在(ω,ω).
我们将转换概率分解为测试概率和接受率的乘积
在(ω,ω′)=吨(ω,ω′)一个(ω,ω′)
在哪里吨(ω,ω′)是测试新配置的概率ω′具有给定的初始配置ω. 如果ω′被测试，然后数量0≤一个(ω,ω′)≤1对应于过渡到的概率ω′被接受。注意条件
吨(ω,ω′)一个(ω,ω′)吨(ω′,ω)一个(ω′,ω)=磷ω′磷ω
不要固定接受率的比率。一个好的蒙特卡洛算法需要这些速率的最佳选择。对于太小的速率，只接受一小部分跳转，并且系统卡在其初始配置中。我们在没有经过配置空间的情况下浪费了宝贵的计算时间。因此，在许多情况下，一个设置两个接受率中的较大者一个(ω,ω′)和一个(ω′,ω)等于 1。选择较小的比率使得条件(4.24)很满意。
