## 物理代写|电动力学代写electromagnetism代考|PHYC20014

2023年4月4日

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## 物理代写|电动力学代写electromagnetism代考|The Heisenberg Equations of Motion

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$c_{\mathbf{k}, \lambda}$; after quantisation, (C.0.5) is replaced by $$\dot{\mathrm{c}}{\mathbf{k}, \lambda}(t)=-i \omega \mathbf{c}{\mathbf{k}, \lambda}(t)+\frac{i e \chi_a(k)}{\sqrt{2 \Omega \hbar k c \varepsilon_0}} \dot{\mathbf{q}} \cdot \varepsilon(\mathbf{k})\lambda e^{-i \mathbf{k} \cdot \mathbf{q}}$$ which in integrated form with retarded boundary conditions is $$c{\mathbf{k}, \lambda}(t)=c_{\mathbf{k}, \lambda}\left(t_0\right)+\frac{i e \chi_a(k)}{\sqrt{2 \Omega \hbar k c \varepsilon_0}} \int_{-t_0}^{+\infty} \theta\left(t-t^{\prime}\right) \dot{\mathbf{q}}\left(t^{\prime}\right) \cdot \varepsilon(\mathbf{k})\lambda e^{-i\left(\mathbf{k} \cdot \mathbf{q}\left(t^{\prime}\right)+\omega\left(t-t^{\prime}\right)\right)} \mathrm{d} t^{\prime}$$ 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 $$\Gamma^{\text {out }}=\mathrm{S}^{-1} \Gamma^{\text {in }} \mathrm{S} \text {. }$$ Thus, in (11.27) we replace$c{\mathbf{k}, \lambda}\left(t_0\right)$by$c_{\mathbf{k}, \lambda}^{\text {in }}$and make the lower limit of the integral$-\infty$;$c_{\mathbf{k}, \lambda}(t)$and its adjoint determine the field operators at time$t$. If we attempt to solve the coupled operator equations in the way described in$\$3.8$, we must pay careful attention to the non-commutation of the operators involved. Recalling that $\dot{\mathbf{q}}$ is given by Hamilton’s equation, (3.316), there are two points to mention:

1. The coordinate operators at different times do not commute; hence the product of their exponential factors is not simply equal to an exponential with the exponents added together. The required modification may be computed using the Baker-Campbell-Hausdorff formula.
2. While it is true that $\mathbf{q}\left(t^{\prime}\right)$ and $\mathbf{p}\left(t^{\prime}\right)$ satisfy the fundamental equal-time canonical commutation relation.

## 物理代写|电动力学代写electromagnetism代考|Non-perturbative Ideas

The conventional approach to quantum electrodynamics is based on perturbation theory for the S-matrix as described in Chapters 6,9 and 10 . Such calculations lead to cross sections that can be related to scattering experiments. The difficulties in perturbation theory are of two different sorts. Firstly, the ‘loop’ diagrams like Figure 10.4 allow the involvement of intermediate states with virtual photons of unrestricted momentum, and hence energies far beyond the regime of validity of the non-relativistic theory. These are the ‘ultraviolet’ divergences dealt with by, for example, a maximum momentum cut-off so as to suppress their contributions. The use of a cut-off is a crude realisations of the notion that high-momentum (high-energy) states must be eliminated in order to construct an ‘effective’ theory that is adequate for the low-energy physics of interest. This can be achieved with the systematic use of Feshbach projection (Löwdin’s partitioning technique).

Secondly, charged particles in the field can be associated with an arbitrarily large number of virtual photons with energy close to zero; these require an infrared cutoff. With the full apparatus of covariant QED and an invariant method of calculation (for example, Feynman diagrams) one can extract finite values. When that is done for interacting electrons and photons, the agreement with experiment is remarkable, perhaps the most accurate quantities that can be calculated by quantum mechanics [6]. Nevertheless, the occurrence of infinities is an ugly feature which hints at underlying problems in the formalism of QED. Furthermore, there are important questions in QED which cannot be answered using a perturbation expansion, for example, the demonstration of the existence of a ground state for interacting charges and field required for the stability of bulk matter in the presence of the field, and the nature of the excitations. These require analytical techniques that are not based on perturbation methods. Over the past several decades, a mathematical approach to non-relativistic QED has been developed using the techniques of modern functional analysis; there is now a considerable research literature, and several monographs available too [10]-[12]. This chapter aims to give some introductory remarks about this programme and to indicate some connections with the ideas in the earlier chapters.

The use of the Coulomb gauge condition is the normal choice in the mathematical literature, though as we will see, the PZW transformation makes an appearance. We know from Chapter 9 that the full Hamiltonian for charged particles interacting with the quantised electromagnetic field can be written in the form
$$\mathrm{H}_\lambda=\mathrm{H}_0+\lambda \mathrm{H}_1+\lambda^2 \mathrm{H}_2$$

# 电动力学代考

## 物理代写|电动力学代写electromagnetism代考|The Heisenberg Equations of Motion

1. 不同时间的坐标操作员不上下班; 因此，它们的 指数因子的乘积不仅仅等于指数加在一起的指 数。可以使用 Baker-Campbell-Hausdorff 公式 计算所需的修改。
2. 虽然这是真的 $\mathbf{q}\left(t^{\prime}\right)$ 和 $\mathbf{p}\left(t^{\prime}\right)$ 满足基本的等时正则 对换关系。

## 物理代写|电动力学代写electromagnetism代考|Non-perturbative Ideas

$$\mathrm{H}_\lambda=\mathrm{H}_0+\lambda \mathrm{H}_1+\lambda^2 \mathrm{H}_2$$

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## MATLAB代写

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