1. Notes
1.1. Special relativity
The Minkowski spacetime is a \(4\)-dimensional real affine space \(\mathcal{M}\). An affine coordinate chart on \(\mathcal{M}\) is a choice of basis \(e_0, \dots, e_3\) of \(\mathbb{R}^4\) and an origin point \(O\in \mathcal{M}\), uniquely determining \(x^\lambda(P)\) for any \(P\in \mathcal{M}\); we further set \(x^0 = ct, x^1 = x, x^2 = y, x^3 = z\) where \(c>0\), and equip \(\mathcal{M}\) with the Minkowski inner product \(2\)-form \(ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2 = \eta_{\lambda\mu}dx^\lambda dx^\mu\).
The transformations of \(\mathbb{R}^4\) which preserve the form of \(\eta\) compose the Poincare group.
2. TODO To read
2.1. TODO Principal G-bundles
2.1.1. TODO Mangiarotti [1]
The entire chapter 6 contains all the relevant information.
I need to read it carefully because it is new material and confusing. For example:
- ☑ why is there both a left Lie algebra \(\mathfrak{g}_l\) and a right Lie algebra \(\mathfrak{g}_r\)? Answer: the bracket \([u, v]\) is defined for two vector fields of \(G\); if left or right invariant the answer differs by a minus. Thus the notation merely involves whether one speaks of the left or the right bracket.
- ☑
What is a reduction of a principal G-bundle? See Remark 6.5.2 in p. 183. In particular Theorem 6.5.1 says that the structure group \(G\) reduces to \(H\) if and only if \(P\) has an atlas with \(H\)-valued transition functions.
A reduction simplifies a principal G-bundle.
- ☑
Why is the Maurer-Cartan formula \(d\phi(\epsilon_1, \epsilon_2) = -\frac{1}{2}\phi([\epsilon_1, \epsilon_2])\) holding true? In this vein, I might be aided by Chern [2] chapter 3. Ah; Mangiarotti has written the formula wrong by a factor of \(\frac{1}{2}\) (actually, it's a convention).
There's a general formula for \(d\omega(\epsilon_1, \dots, \epsilon_n)\). Applied to a \(2\)-form it yields:
\begin{align} d\phi(\epsilon_1, \epsilon_2) = \frac{1}{2}\epsilon_1\phi(\epsilon_2) - \frac{1}{2}\epsilon_2\phi(\epsilon_1) - \frac{1}{2}\phi([\epsilon_1, \epsilon_2]). \end{align}For left-invariant forms and left-invariant vector fields \(\epsilon_1, \epsilon_2 \in \mathfrak{g}_l\), the first two terms are zero.
2.2. TODO Special relativity
2.2.1. TODO Wald [3]
Chapter 8 for special relativity and electromagnetism and chapter 9 for electromagnetism as a connection in special and general relativity as well as electromagnetism as a \(U(1)\) theory.
2.2.2. TODO Carroll [4]
Chapter 1 details special relativity. Chapter 4 for general relativity.
2.2.3. TODO Hughston, Tod [5]
Chapter 3 on special relativity contains a lot of pictures.
- p. 9 motivates the stress tensor from the ideal fluid example; derives Euler's equations of motion and conservation equations of energy. Furthermore, explains that the tensor requires faith/imagination when generalized in general relativity. The three-body problem in GR becomes a two-body problem: the third body is gravitational radiation!
- Continue from 3.2.
2.2.4. TODO Frè [6]
Chapter 1 introduces special relativity rigorously. Spinors in Appendix A.
2.3. TODO Naber [7]
Chapter 3 for spinors. Rewrites Maxwell's equations into
\begin{align} \nabla^{A\dot{X}}\phi_{AB} & = 0, & A = 0,1, \quad \dot{X} = \dot{1}, \dot{0}. \end{align}Generalizations of the above equations such as the Weyl neutrino equation:
\begin{align} \nabla^{A\dot{X}}\phi_A & = 0, & \dot{X} = \dot{1}, \dot{0}. \end{align}Requires understanding the spin map \(\Spin : \SL(2, \mathbb{C}) \to \mathcal{L}\), where \(\mathcal{L}\) is the Lorentz group.
2.4. TODO Naber [8]
Chapter 6 talks about connections on principal G-bundles. Interesting in p. 344 the equivalence between connections is discussed, called the moduli space of connections on a bundle. Yang-Mills functional in 6.3. Note that there is a lot of discussion around the quarternionic Hopf bundle and \(\SL(2, \mathbb{H})\). The Atiyah-Hitchin-Singer theorem in §6.5 has the moduli space of connections for that bundle computed as \(\SL(2, \mathbb{H})/{\Sp(2)}\).
2.5. TODO Terek [9]
This article is interesting; it introduces magnetic geometry, and it might be a nice addition to my knowledge of classical electromagnetism.
2.6. TODO Woit [10] for groups related to physics
This book contains a lot of analysis of groups that are related to physics.
2.7. TODO Nair [11] for an intro to QFT, QED, etc
2.8. TODO Guillemin, Sternberg [12] for some geometry on Kepler's problem
There is a lot of geometry on this problem. It is quite impressive yet appears elementary, so perhaps a worthwhile read.
2.8.1. TODO Weinstein [13] geometry of Poisson manifolds
Guillemin mentions Poisson manifolds, there's interesting questions about them explored in this paper, in particular how they're foliated by symplectic manifolds of various dimensions, and how the structure constants of a Lie algebra can define a Poisson manifold.