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4

Yes, OP is right: The curl $\vec{\nabla}\times \vec{B}$ of the $\vec{B}$-field (4.65) is non-zero and violates Ampere's Law due to the presence of the Dirac string. However, the Dirac string is a gauge artifact from using a globally defined magnetic vector potential (MVP) $$\vec{A}~=~\lim_{\epsilon\searrow 0^+}\begin{pmatrix}y\cr -x\cr 0\end{pmatrix}\frac{q/... 3 This is indeed true and is what is called the gauge principle. It tells us that if we make a global symmetry local, we need to add a corresponding gauge field such that the total Lagrangian still remains invariant under this local gauge transformation. This is a new dynamical field which has its own equations of motion and can couple to the fermion leading ... 0 I will try to say something about the string theory part of the answer. Here there are some examples where an infinite number vector bundles/sheaves are important: 1) Holography: All known examples of quantum field theories with gravity duals have large N limits. For large N dualities to work, a gigantic number N of color charges are needed. In the ... 3 The following is just an example where you can use gauge groups other than SU(N). While the discovery of the Higgs was and still remains a huge step towards our better understanding of particle physics there still exists the question of whether or not the Higgs is an elementary particle or a bound state of a strongly coupled sector in higher energies. ... 1 This notation is known as the Feynman or Dirac slash notation. The symbol under the slash must be a Lorentz four-vector, and the slash implies that this four-vector should be contracted with the four-vector of Dirac gamma matrices:$$ {A\!\!\!/} = \gamma ^{\mu }A_{\mu }. $$(This identity uses the Einstein summation convention, so repeated indices are ... 2 The BRST symmetry encodes the gauge symmetry. Yes. The x-dependent/local gauge-parameter \alpha^a(x) in the gauge formulation (which doesn't contain ghosts) is replaced by an x-dependent ghost field c^a(x) and an x-independent/global Grassmann-odd parameter \epsilon in the BRST formulation. So the BRST symmetry is an x-independent/global ... 0 Following Frederic Thomas's answer pointing out the error in my example of not accounting for rotation of the coordinates, I am now able to answer the question of the constraint on the gauge field: Under the symmetry transformation x \to x', A(x) \to A'(x') may be expressed A(x') + \nabla \lambda(x'). Using Frederic Thomas's expression for A'(x') we ... 1 The relation of magnetic field and the vector potential is independent of the coordinate system. My first attempt I got the same result, however, soon I realized that the coordinates also have to undergo the same rotation. So if one has the following vector potential$$ \mathbf{A}(x,y,z) = \left( \begin{array}{c} 0, & Bx, & 0 \end{array}\right)^T  ...

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The principle of stationary action always implies the EL equations (1) with partial derivatives, so (1) is a safe bet. By imposing further conditions on the theory, the EL equations (2) with covariant derivatives may hold as well, cf. this related Phys.SE post.

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I won't give a full derivation -- for that I will point to the literature (or really I'll point to my comment where I gave a ref to the literature), where this calculation has been done many times. Instead I'll give an outline. We start with the Einstein-Hilbert action$^\star$ $$S = \frac{1}{16\pi G} \int {\rm d}^4 x \sqrt{-g} R$$ ...

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