### 物理代写|量子场论代写Quantum field theory代考|PHYS8302

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|Thermal Correlation Functions

The energies of excited states are encoded in the thermal correlation functions. These functions are expectation values of products of the position operator
$$\hat{q}{\mathrm{E}}(\tau)=\mathrm{e}^{\tau \hat{H} / \hbar} \hat{q} \mathrm{e}^{-\tau \hat{H} / \hbar}, \quad \hat{q}{\mathrm{E}}(0)=\hat{q}(0),$$
at different imaginary times in the canonical ensemble,
$$\left\langle\hat{q}{\mathrm{E}}\left(\tau{1}\right) \cdots \hat{q}{\mathrm{E}}\left(\tau{n}\right)\right\rangle_{\beta} \equiv \frac{1}{Z(\beta)} \operatorname{tr}\left(\mathrm{e}^{-\beta \hat{H}} \hat{q}{\mathrm{E}}\left(\tau{1}\right) \cdots \hat{q}{\mathrm{E}}\left(\tau{n}\right)\right)$$
The normalizing function $Z(\beta)$ is the partition function (2.56). From the thermal two-point function
\begin{aligned} \left\langle\hat{q}{\mathrm{E}}\left(\tau{1}\right) \hat{q}{\mathrm{E}}\left(\tau{2}\right)\right\rangle_{\beta} &=\frac{1}{Z(\beta)} \operatorname{tr}\left(\mathrm{e}^{-\beta \hat{H}} \hat{q}{\mathrm{E}}\left(\tau{1}\right) \hat{q}{\mathrm{E}}\left(\tau{2}\right)\right) \ &=\frac{1}{Z(\beta)} \operatorname{tr}\left(\mathrm{e}^{-\left(\beta-\tau_{1}\right) \hat{H}} \hat{q} \mathrm{e}^{-\left(\tau_{1}-\tau_{2}\right) \hat{H}} \hat{q} \mathrm{e}^{-\tau_{2} \hat{H}}\right) \end{aligned}
we can extract the energy gap between the ground state and the first excited state. For this purpose we use orthonormal energy eigenstates $|n\rangle$ to calculate the trace and in addition insert the resolution of the identity operator $\mathbb{1}=\sum|m\rangle\langle m|$. This yields
$$\langle\ldots\rangle_{\beta}=\frac{1}{Z(\beta)} \sum_{n, m} \mathrm{e}^{-\left(\beta-\tau_{1}+\tau_{2}\right) E_{n}} \mathrm{e}^{-\left(\tau_{1}-\tau_{2}\right) E_{\mathrm{m}}}\langle n|\hat{q}| m\rangle\langle m|\hat{q}| n\rangle$$
Note that in the sum over $n$ the contributions from the excited states are exponentially suppressed at low temperatures $\beta \rightarrow \infty$, implying that the thermal two-point function converges to the Schwinger function in this limit:
$$\left\langle\hat{q}{\mathrm{E}}\left(\tau{1}\right) \hat{q}{\mathrm{E}}\left(\tau{2}\right)\right\rangle_{\beta} \stackrel{\beta \rightarrow \infty}{\longrightarrow} \sum_{m>0} \mathrm{e}^{-\left(\tau_{1}-\tau_{2}\right)\left(E_{m}-E_{0}\right)}|\langle 0|\hat{q}| m\rangle|^{2}=\left\langle 0\left|\hat{q}{\mathrm{E}}\left(\tau{1}\right) \hat{q}{\mathrm{E}}\left(\tau{2}\right)\right| 0\right\rangle$$

## 物理代写|量子场论代写Quantum field theory代考|The Harmonic Oscillator

We wish to study the path integral for the Euclidean oscillator with discretized time. The oscillator is one of the few systems for which the path integral can be calculated explicitly. For more such system, the reader may consult the text [19]. But the results for the oscillator are particularly instructive with regard to lattice field theories considered later in this book. So let us discretize the Euclidean time interval $[0, \tau]$ with $n$ sampling points separated by a lattice constant $\varepsilon=\tau / n$. For the Lagrangian
$$L=\frac{m}{2} \dot{q}^{2}+\mu q^{2}$$
the discretized path integral over periodic paths reads
\begin{aligned} Z &=\int \mathrm{d} q_{1} \cdots \mathrm{d} q_{n}\left(\frac{m}{2 \pi \varepsilon}\right)^{n / 2} \exp \left{-\varepsilon \sum_{j=0}^{n-1}\left(\frac{m}{2}\left(\frac{q_{j+1}-q_{j}}{\varepsilon}\right)^{2}+\mu q_{j}^{2}\right)\right} \ &=\left(\frac{m}{2 \pi \varepsilon}\right)^{n / 2} \int \mathrm{d} q_{1} \cdots \mathrm{d} q_{n} \exp \left(-\frac{1}{2}(\boldsymbol{q}, \mathrm{A} q)\right) \end{aligned}
where we assumed $q_{0}=q_{n}$ and introduced the symmetric matrix
$$\mathrm{A}=\frac{m}{\varepsilon}\left(\begin{array}{cccccc} \alpha & -1 & 0 & \cdots & 0 & -1 \ -1 & \alpha & -1 & \cdots & 0 & 0 \ & & \ddots & & & \ & & & \ddots & & \ 0 & 0 & \cdots & -1 & \alpha & -1 \ -1 & 0 & \cdots & 0 & -1 & \alpha \end{array}\right), \quad \alpha=2\left(1+\frac{\mu}{m} \varepsilon^{2}\right)$$
This is a Toeplitz matrix in which each descending diagonal from left to right is constant. This property results from the invariance of the action under lattice translations. For the explicit calculation of $Z$, we consider the generating function
\begin{aligned} Z[j] &=\left(\frac{m}{2 \pi \varepsilon}\right)^{n / 2} \int \mathrm{d}^{n} q \exp \left{-\frac{1}{2}(\boldsymbol{q}, \mathrm{A} q)+(\boldsymbol{j}, \boldsymbol{q})\right} \ &=\frac{(m / \varepsilon)^{n / 2}}{\sqrt{\operatorname{det} \mathrm{A}}} \exp \left{\frac{1}{2}\left(j, \mathrm{~A}^{-1} \boldsymbol{j}\right)\right} \end{aligned}

## 物理代写|量子场论代写Quantum field theory代考|Problems

2.1 (Gaussian Integral) Show that
$$\int \mathrm{d} z_{1} \mathrm{~d} \bar{z}{1} \ldots \mathrm{d} z{n} \mathrm{~d} \bar{z}{n} \exp \left(-\sum{i j} \bar{z}{i} A{i j} z_{j}\right)=\pi^{n}(\operatorname{det} \mathrm{A})^{-1}$$
with A being a positive Hermitian $n \times n$ matrix and $z_{i}$ complex integration variables.
2.2 (Harmonic Oscillator) In (2.43) we quoted the result for the kernel $K_{\omega}\left(\tau, q^{\prime}, q\right)$ of the $d$-dimensional harmonic oscillator with Hamiltonian
$$\hat{H}=\frac{1}{2 m} \hat{p}^{2}+\frac{m \omega^{2}}{2} \hat{q}^{2}$$
at imaginary time $\tau$. Derive this formula.
Hint: Express the kernel in terms of the eigenfunctions of $\hat{H}$, which for $\hbar=m=$ $\omega=1$ are given by
$$\exp \left(-\xi^{2}-\eta^{2}\right) \sum_{n=0}^{\infty} \frac{\alpha^{n}}{2^{n} n !} H_{n}(\xi) H_{n}(\eta)=\frac{1}{\sqrt{1-\alpha^{2}}} \exp \left(\frac{-\left(\xi^{2}+\eta^{2}-2 \xi \eta \alpha\right)}{1-\alpha^{2}}\right)$$
The functions $H_{n}$ denote the Hermite polynomials.
Comment: This result also follows from the direct evaluation of the path integral.
2.3 (Free Particle on a Circle) A free particle moves on an interval and obeys periodic boundary conditions. Compute the time evolution kernel $K\left(t_{b}-t_{a}, q_{b}, q_{a}\right)=$ $\left\langle q_{b}, t_{b} \mid q_{a}, t_{a}\right\rangle$. Use the familiar formula for the kernel of the free particle (2.26) and enforce the periodic boundary conditions by a suitable sum over the evolution kernel for the particle on $\mathbb{R}$.

## 物理代写|量子场论代写Quantum field theory代考|Thermal Correlation Functions

q^和(τ)=和τH^/ℏq^和−τH^/ℏ,q^和(0)=q^(0),

⟨q^和(τ1)⋯q^和(τn)⟩b≡1从(b)tr⁡(和−bH^q^和(τ1)⋯q^和(τn))

⟨q^和(τ1)q^和(τ2)⟩b=1从(b)tr⁡(和−bH^q^和(τ1)q^和(τ2)) =1从(b)tr⁡(和−(b−τ1)H^q^和−(τ1−τ2)H^q^和−τ2H^)

⟨…⟩b=1从(b)∑n,米和−(b−τ1+τ2)和n和−(τ1−τ2)和米⟨n|q^|米⟩⟨米|q^|n⟩

⟨q^和(τ1)q^和(τ2)⟩b⟶b→∞∑米>0和−(τ1−τ2)(和米−和0)|⟨0|q^|米⟩|2=⟨0|q^和(τ1)q^和(τ2)|0⟩

## 物理代写|量子场论代写Quantum field theory代考|The Harmonic Oscillator

\begin{aligned} Z &=\int \mathrm{d} q_{1} \cdots \mathrm{d} q_{n}\left(\frac{m}{2 \pi \varepsilon}\right)^{ n / 2} \exp \left{-\varepsilon \sum_{j=0}^{n-1}\left(\frac{m}{2}\left(\frac{q_{j+1}-q_ {j}}{\varepsilon}\right)^{2}+\mu q_{j}^{2}\right)\right} \ &=\left(\frac{m}{2 \pi \varepsilon} \right)^{n / 2} \int \mathrm{d} q_{1} \cdots \mathrm{d} q_{n} \exp \left(-\frac{1}{2}(\boldsymbol{q }, \mathrm{A} q)\right) \end{对齐}\begin{aligned} Z &=\int \mathrm{d} q_{1} \cdots \mathrm{d} q_{n}\left(\frac{m}{2 \pi \varepsilon}\right)^{ n / 2} \exp \left{-\varepsilon \sum_{j=0}^{n-1}\left(\frac{m}{2}\left(\frac{q_{j+1}-q_ {j}}{\varepsilon}\right)^{2}+\mu q_{j}^{2}\right)\right} \ &=\left(\frac{m}{2 \pi \varepsilon} \right)^{n / 2} \int \mathrm{d} q_{1} \cdots \mathrm{d} q_{n} \exp \left(-\frac{1}{2}(\boldsymbol{q }, \mathrm{A} q)\right) \end{对齐}

\begin{对齐} Z[j] &=\left(\frac{m}{2 \pi \varepsilon}\right)^{n / 2} \int \mathrm{d}^{n} q \exp \左{-\frac{1}{2}(\boldsymbol{q}, \mathrm{A} q)+(\boldsymbol{j}, \boldsymbol{q})\right} \ &=\frac{(m / \varepsilon)^{n / 2}}{\sqrt{\operatorname{det} \mathrm{A}}} \exp \left{\frac{1}{2}\left(j, \mathrm{~A }^{-1} \boldsymbol{j}\right)\right} \end{aligned}\begin{对齐} Z[j] &=\left(\frac{m}{2 \pi \varepsilon}\right)^{n / 2} \int \mathrm{d}^{n} q \exp \左{-\frac{1}{2}(\boldsymbol{q}, \mathrm{A} q)+(\boldsymbol{j}, \boldsymbol{q})\right} \ &=\frac{(m / \varepsilon)^{n / 2}}{\sqrt{\operatorname{det} \mathrm{A}}} \exp \left{\frac{1}{2}\left(j, \mathrm{~A }^{-1} \boldsymbol{j}\right)\right} \end{aligned}

## 物理代写|量子场论代写Quantum field theory代考|Problems

2.1（高斯积分）证明

∫d和1 d和¯1…d和n d和¯n经验⁡(−∑一世j和¯一世一个一世j和j)=圆周率n(这⁡一个)−1
A 是正厄米特n×n矩阵和和一世复杂的积分变量。
2.2（谐波振荡器）在（2.43）中，我们引用了内核的结果ķω(τ,q′,q)的d具有哈密顿量的维谐振子

H^=12米p^2+米ω22q^2

2.3 （圆周上的自由粒子） 自由粒子在一个区间上移动并服从周期性边界条件。计算时间演化核ķ(吨b−吨一个,qb,q一个)= ⟨qb,吨b∣q一个,吨一个⟩. 对自由粒子的核使用熟悉的公式 (2.26)，并通过粒子在演化核上的适当总和来强制周期性边界条件R.

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