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I am answering the question formulated after the "edit" in a newer version of the text because that one seems well-defined. Indeed, a situation with a uniform field $\vec E$ may be said to be "uniform" or translationally invariant in space. Noether's theorem says that this "uniformity" (spatial translational invariance) implies the existence of a conserved ...

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No, we know enough of the "bulk properties" of antimatter to rule this out. Antimatter interacts with the electromagnetic field in exactly the same way as regular matter, just with the opposite charge. Therefore, antimatter should be detectable using most of the techniques we use to detect regular matter in astronomy. This works even if the antimatter is a ...

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Your reasoning is essentially correct, apart from the last paragraph. To conclude, note that Newton's equation: $$\ddot {\mathbf r}(t) = \mathbf f(\mathbf r (t)),$$ with initial condition $\mathbf x (0)=(x_0,0,0)$, $\dot {\mathbf x} (0)=(0,0,0)$ can be solved by puttin $y(t)=z(t)\equiv 0$, thus reducing to a one dimensional problem: $$\ddot x (t)=f(x(t)),$$ ...

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The corresponding symmetry group is the Lorentz group and yes we can use Noether to derive conserved quantities: Invariance under translations $\rightarrow$ momentum conservation Invariance under rotations $\rightarrow$ spin and angular momentum conservation Invariance under boost $\rightarrow$ some strange, not really useful, conserved quantity

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My hint is too long for a comment, but maybe it is worth writing down. I'll not use symmetry arguments, but I'll try to get Newton's equations out of the non-degeneracy of a certain mathematical object. Let $M$ be manifold and $\omega$ a 2-form on it. If $\omega$ is algebraically closed and non-degenerate, then the dimension of $M$ must be even (let's say ...

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Answer posted by Lubos Motl in the comments; I reproduce most of it here. This answer was posted in order to remove this question from the "unanswered" list. Some (sketches of) answers to your questions, one by one: Physical states have to be invariant under gauge symmetries, so all of them are singlets and there are no nontrivial representations, (and 3....

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There's not reason to assume nature should treat everything symmetrically. There are many phenomena in nature that we actually know are asymmetric. For example the weak force violates parity symmetry (meaning the weak force has a preference for right or left handedness).

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I think we can pretty safely say this is not the case. The main reason is that we have a pretty good idea of where dark matter is--to some degree, it can be reconstructed from the gravitational influence it has on surrounding matter. The dark matter appears to be distributed evenly throughout the galaxy. Thing is, a galaxy is pretty "dirty," as far as space ...

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Like it was said before, there is no a priori reason why nature should treat everything symmetrically. Much to the contrary, we know several examples of P- and CP-violating processes. And in other cases we do not even know the reason why a process is "symmetric", when in principle it would be allowed to violate CP (see: the strong CP problem). I guess you ...

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An internal symmetry is a transformation acting only on the fields, therefore not transforming spacetime points, and leaving the lagrangian or the physical results invariant. Example of internal symmetries are gauge symmetries. These are local symmetries, which means the transformations are in general spacetime dependent in the sense they are, in general, ...

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It is a little different in General Relativity. Let's start with Special Relativity and all the 3 forces of the Standard Model in physics. Then we will talk about gravity and the universe. In The Standard Model spacetime is Minkowski, meaning flat in all 4 dimensions. If it is that way clearly any direction and position is equivalent. That's called ...

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The energy levels of the bound-states of a hydrogen atom only depend on the radial quantum number n . This is a special property of a (1/r) type of interaction potential . For a general central potential, V (r ) the quantized energy levels of a bound-state can depend on both n and l values. The property that the energy levels of a hydrogen atom ...

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The wave function of the hydrogen atom $\psi(r,\theta,\phi)$ is a product of the radial part, the angular part and azimuthal term $$\psi_{n,\ell,m}(r,\theta,\phi) = R(r)\Theta(\theta)\Phi(\phi).$$ The radial part $R(r)$ obeys Laguerre polynomials or, $$R_{n\ell}(r) = Ae^{-\rho/2}\rho^\ell L^{2\ell+1}_{n-\ell-1},~ \rho = \frac{2r}{na_0},$$ and $A$ is a ...

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The field inside an hollow infinite cylinder is $0$, just like the field inside an hollow sphere. This is because of Gauss' law: the flux of the electric field $\vec E$ through any closed surface $S$ is $$\Phi = \int_S \vec E \cdot d \vec S = \frac Q {\epsilon_0}$$ Where $Q$ is the charge inside the volume enclosed by the surface. Let $R$ be the radius ...

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