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1

In vacuum (or everywhere else, really), Coulomb's law takes the form $\boldsymbol\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon}$, whereas in a polarizable material it is convenient to use $\boldsymbol\nabla \cdot \mathbf{D} = \rho_\mathrm{free}$. The $4\pi$ vs $\epsilon$ has more to do with units. As for the sign, can you give a reference?

2

The entirety of the modern quantum mechanics literature uses inner products that are linear in the second argument, and antilinear in the first one. Mathematicians often use the other convention, but I've never seen it used in physics. This is of course pure convention, but you will find grief, at least when you try to publish, if you go against the flock ...

0

an interesting question, no doubts. Indeed most of the time we rely on conservation of quantum numbers but there is an underlying structure to this. In principle changing a particle with an antiparticle amounts to a CP transformation. This is a symmetry for electromagnetic and (supposedly) strong interactions. Because of this, it doesn't really matter what ...

2

Yes, you'll gain extra hours, but you'll lose them on the way back, unless you keep going round. Let's assume you're in a plane flying along the equator, moving at 800 km/h (in the direction of the earth's travel) - a normal jetplane speed. The earth is rotating so that a point stationary on the equator moves at 1600 km/h. That means that, for every km you ...

3

It started with conservation of quantum numbers, from baryon number when we did not know about quarks, to lepton number, when we discovered the positron.For the neutrino momentum and energy conservation played a role too, since it is only seen as a missing mass. In time the symmetries in the assignments of the quantum numbers became more and more evident ...

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