## 物理代写|电动力学代写electromagnetism代考|A Coherent State Formulation of the PZW Representation

First, we consider the relationship of the Coulomb gauge and the Power-ZienauWoolley representation of non-relativistic QED in terms of generalised coherent states [2]. We noted in $\S 7.3$ that unitary operators of the form of (9.109) can be presented in the form of a product of the familiar coherent state displacement operators or boson translators. Since the generator $\mathrm{F}$ is a product of charged particle and field variables and is proportional to the charge $e$, the resulting coherent state ‘parameters’ involve mixtures of particle and field variables and $e$, and only make sense for the interacting system. ${ }^3$ Using the mode expansion of the Coulomb gauge vector potential, (7.122), and proceeding to the continuum limit, the transformation operator $(9.109)$ may be cast as
$$\Lambda_{\mathcal{C}}=\exp \left[\sum_{\lambda=1,2} \int\left(\alpha(\mathbf{k}: \mathcal{C})\lambda \mathrm{c}(\mathbf{k})\lambda^{+}-\alpha(\mathbf{k}: \mathcal{C})\lambda^* \mathrm{c}(\mathbf{k})\lambda\right) \mathrm{d}^3 \mathbf{k}\right],$$
where for each mode $\mathbf{k}, \lambda$ and path $\mathcal{C}$, the coherent state ‘parameter’ is
$$\alpha(\mathbf{k}: \mathcal{C})\lambda=-i \sqrt{\frac{1}{2 \varepsilon_0 \hbar c k(2 \pi)^3}} \mathbf{P}(\mathbf{k}: \mathcal{C}) \cdot \hat{\boldsymbol{\varepsilon}}(\mathbf{k})\lambda$$
Here $\mathbf{P}(\mathbf{k}: \mathcal{C})$ is the Fourier transform of the polarisation field for the specified path evaluated at the wave vector $\mathbf{k}$.
The transformed annihilation and creation operators for the mode $\mathbf{k}, \lambda$ are
\begin{aligned} & \mathrm{C}(\mathbf{k}: \mathcal{C})\lambda=\mathrm{c}(\mathbf{k})\lambda+\alpha(\mathbf{k}: \mathcal{C})\lambda \ & \mathrm{C}(\mathbf{k}: \mathcal{C})\lambda^{+}=\mathrm{c}(\mathbf{k})\lambda^{+}+\alpha(\mathbf{k}: \mathcal{C})\lambda^* \end{aligned}
and one still has
$$\left[\mathrm{C}(\mathbf{k}: \mathcal{C})\lambda, \mathrm{C}\left(\mathbf{k}^{\prime}: \mathcal{C}\right){\lambda^{\prime}}^{+}\right]=\delta_{\lambda, \lambda^{\prime}} \delta^3\left(\mathbf{k}-\mathbf{k}^{\prime}\right)$$
We can define a new vacuum state, $\Psi_0(\mathcal{C})$, by setting
$$\mathrm{C}(\mathbf{k}: \mathcal{C})\lambda\left|\Psi_0(\mathcal{C})\right\rangle=0, \forall \mathbf{k}$$ to give a new representation of the Fock space for the field. A straightforward generalisation from the single-mode case (Chapter 7) to the continuum limit shows that $\Psi_0(\mathcal{C})$ is related to the free-field vacuum $\Psi_0$ by \begin{aligned} &\left|\Psi_0(\mathcal{C})\right\rangle=\exp \left(-\frac{1}{2} \int \sum{\lambda=1,2}\left|\alpha(\mathbf{k}: \mathcal{C})\lambda\right|^2 \mathrm{~d}^3 \mathbf{k}\right) \ & \times \exp \left(\int \sum{\lambda=1,2} \alpha(\mathbf{k}: \mathcal{C})\lambda \mathrm{C}(\mathbf{k}: \mathcal{C})\lambda^{+} \mathrm{d}^3 \mathbf{k}\right)\left|\Psi_0\right\rangle \end{aligned}

In the Heisenberg representation the state vector is constant in time, and the problems arise in the relationship between operators at different times, and so we need to look at the integration of the equations of motion. Let us revisit $\$ 3.8$and Appendix$\mathrm{C}$from a quantum mechanical perspective in which the classical variables${\mathbf{q}, \mathbf{p}, \mathbf{A}, \boldsymbol{\pi}}$are reinterpreted as operators in the usual way. Equations (3.315), (3.316), with the Poisson brackets replaced by commutators, are the Heisenberg equations of motion for the charged particle; likewise the Fourier variables for the field are to be reinterpreted as the photon annihilation and creation operators. Consider first the equation of motion for the annihilation operator$c_{k, \lambda}$; after quantisation,$($C. 0.5$)$is replaced by $$\dot{\mathrm{c}}{\mathbf{k}, \lambda}(t)=-i \omega \mathrm{c}{\mathbf{k}, \lambda}(t)+\frac{i e \chi_a(k)}{\sqrt{2 \Omega \hbar k c \varepsilon_0}} \dot{\mathbf{q}} \cdot \varepsilon(\mathbf{k})\lambda e^{-i \mathbf{k} \cdot \mathbf{q}}$$ which in integrated form with retarded boundary conditions is $$\mathrm{c}{\mathbf{k}, \lambda}(t)=\mathrm{c}{\mathbf{k}, \lambda}\left(t_0\right)+\frac{i e \chi_a(k)}{\sqrt{2 \Omega \hbar k c \varepsilon_0}} \int{-t_0}^{+\infty} \theta\left(t-t^{\prime}\right) \dot{\mathbf{q}}\left(t^{\prime}\right) \cdot \varepsilon(\mathbf{k})\lambda e^{-i\left(\mathbf{k} \cdot \mathbf{q}\left(t^{\prime}\right)+\omega\left(t-t^{\prime}\right)\right)} \mathrm{d} t^{\prime}$$ Here$\mathbf{q}$and$\dot{\mathbf{q}}$are now to be interpreted as non-commuting operators, and$\omega=k c$. In the Heisenberg picture a scattering process is described in terms of in- and outoperators corresponding to the physical situations at$t=-\infty$and$t=+\infty$, respectively. The S-matrix provides the relationship between the in- and out-operators according to $$\Gamma^{\text {out }}=\mathrm{S}^{-1} \Gamma^{\text {in }} \mathrm{S}$$ Thus, in (11.27) we replace$c{k, \lambda}\left(t_0\right)$by$c_{k, \lambda}^{\text {in }}$and make the lower limit of the integral$-\infty$;$c_{k, \lambda}(t)$and its adjoint determine the field operators at time$t$. # 电动力学代考 ## 物理代写|电动力学代写electromagnetism代考|A Coherent State Formulation of the PZW Representation 首先，我们根据广义相干态考虑库仑规范与非相对论 QED 的 Power-ZienauWoolley 表 示之间的关系 [2]。我们注意到 § 7.3(9.109) 形式的酉算子可以以熟悉的相干态位移算子 或玻色子翻译器的乘积形式呈现。由于发电机$\mathrm{F}$是带电粒子和场变量的乘积，与电荷成 正比e，由此产生的相干状态”参数“涉及粒子和场变量的混合和$e$, 并且只对交互系统有 意义。${ }^3$使用库仑规范矢量势的模式展开 (7.122)，并继续到连续极限，变换算子$(9.109)$可以投为 $$\Lambda_{\mathcal{C}}=\exp \left[\sum_{\lambda=1,2} \int\left(\alpha(\mathbf{k}: \mathcal{C}) \lambda c(\mathbf{k}) \lambda^{+}-\alpha(\mathbf{k}: \mathcal{C}) \lambda^* \mathrm{c}(\mathbf{k}) \lambda\right) \mathrm{d}^3 \mathbf{k}\right],$$ 每种模式在哪里$\mathbf{k}, \lambda$和路径$\mathcal{C}$，相干态”参数”是 $$\alpha(\mathbf{k}: \mathcal{C}) \lambda=-i \sqrt{\frac{1}{2 \varepsilon_0 \hbar c k(2 \pi)^3}} \mathbf{P}(\mathbf{k}: \mathcal{C}) \cdot \hat{\varepsilon}(\mathbf{k}) \lambda$$ 这里$\mathbf{P}(\mathbf{k}: \mathcal{C})$是在波矢量处评估的指定路径的极化场的傅立叶变换$\mathbf{k}$. 模式的转化的湮和创造算子$\mathbf{k}, \lambda$是 $$\mathrm{C}(\mathbf{k}: \mathcal{C}) \lambda=\mathrm{c}(\mathbf{k}) \lambda+\alpha(\mathbf{k}: \mathcal{C}) \lambda \quad \mathrm{C}(\mathbf{k}: \mathcal{C}) \lambda^{+}=\mathrm{c}(\mathbf{k}) \lambda^{+}+\alpha(\mathbf{k}: \mathcal{C}) \lambda^*$$ $$\left[\mathrm{C}(\mathbf{k}: \mathcal{C}) \lambda, \mathrm{C}\left(\mathbf{k}^{\prime}: \mathcal{C}\right) \lambda^{\prime+}\right]=\delta_{\lambda, \lambda^{\prime}} \delta^3\left(\mathbf{k}-\mathbf{k}^{\prime}\right)$$ 我们可以定义一个新的真空状态，$\Psi_0(\mathcal{C})$，通过设置 $$\mathrm{C}(\mathbf{k}: \mathcal{C}) \lambda\left|\Psi_0(\mathcal{C})\right\rangle=0, \forall \mathbf{k}$$ 为该字段提供 Fock 空间的新表示。从单模情况 (第 7 章) 到连续极限的直接概括表明$\Psi_0(\mathcal{C})$与自由场真空有关$\Psi_0$经过 $$\left|\Psi_0(\mathcal{C})\right\rangle=\exp \left(-\frac{1}{2} \int \sum \lambda=1,2|\alpha(\mathbf{k}: \mathcal{C}) \lambda|^2 \mathrm{~d}^3 \mathbf{k}\right) \quad \times \exp \left(\int \sum \lambda=1,2 \alpha(\mathbf{k}\right.$$ ## 物理代写|电动力学代写electromagnetism代考|The Heisenberg Equations of Motion 在海森堡表示中，状态向量在时间上是恒定的，不同时间的算子之间的关系会出现问 题, 所以我们需要看运动方程的积分。让我们重温$\$3.8$ 和附录C从量子力学的角度来 看，经典变量 $\mathbf{q}, \mathbf{p}, \mathbf{A}, \boldsymbol{\pi}$ 以通常的方式被重新解释为运算符。方程 (3.315)、(3.316) 中 的泊松括号由换向器代替，是带电粒子的海森堡运动方程; 同样，该场的傅立叶变量将 被重新解释为光子湮灭和产生算子。

$$\dot{c} \mathbf{k}, \lambda(t)=-i \omega c \mathbf{k}, \lambda(t)+\frac{i e \chi_a(k)}{\sqrt{2 \Omega \hbar k c \varepsilon_0}} \dot{\mathbf{q}} \cdot \varepsilon(\mathbf{k}) \lambda e^{-i \mathbf{k} \cdot \mathbf{q}}$$

$$\mathbf{c k}, \lambda(t)=\mathrm{ck}, \lambda\left(t_0\right)+\frac{i e \chi_a(k)}{\sqrt{2 \Omega \hbar k c \varepsilon_0}} \int-t_0{ }^{+\infty} \theta\left(t-t^{\prime}\right) \dot{\mathbf{q}}\left(t^{\prime}\right) \cdot \varepsilon(\mathbf{k}) \lambda e^{-i\left(\mathbf{k} \cdot \mathbf{q}\left(t^{\prime}\right)+\omega\left(t-t^{\prime}\right)\right)} \mathrm{d} t^{\prime}$$

$$\Gamma^{\text {out }}=\mathrm{S}^{-1} \Gamma^{\text {in }} \mathrm{S}$$

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