B.20 Chapters 45 and 46

B.20 Chapters 45 and 46

Problem 1:

Show that the Yang-Mills equations (46.9 and 46.11) are Hamilton’s equations for the Yang-Mills Hamiltonian 46.10.

Problem 2:

Show that the matrix $P_{\perp }$ with entries

$$(P_{\perp }{)}_{jk}={\delta }_{jk}-\frac{p_j{p}_k}{|\mathbf{p}{|}^2}$$

acts on momentum vectors in ${\mathbf{R}}^3$ by orthogonal projection on the plane perpendicular to p. Use this to explain why one expects to get the commutation relations of equation 46.18 in Coulomb gauge (the condition $\nabla \nabla \cdot \mathbf{A}=0$ in momentum space says that the vector potential is perpendicular to the momentum).

Problem 3:

Show that the group $SO(2)$ acts on the space of solutions of Maxwell’s equations by

$$\begin{array}{rl}\mathbf{E}(\mathbf{x})& \to \cos \theta \mathbf{E}(\mathbf{x})+\sin \theta \mathbf{B}(\mathbf{x})\\ \mathbf{B}(\mathbf{x})& \to -\sin \theta \mathbf{E}(\mathbf{x})+\cos \theta \mathbf{B}(\mathbf{x})\end{array}$$

For $\theta =\frac{\pi }{2}$ this symmetry interchanges E and B fields and is known as electricmagnetic duality. A much harder problem is to see what the corresponding operator acting on states is (it turns out to be the helicity operator).

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正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

原书 PDF · 印刷页 537、538、539、540、541、542、543、544、545、546、547、548、549、550、551、552、553、554、555、556

来源版本：2025-10-20

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