/sci/ - Science & Math

>>11376227
Because the kinetic part [math]\int_\Omega |\nabla \phi|^2 [/math] of the action [math]S[\phi][/math] describes the surface tension of a material defined by the scalar [math]\phi:\Omega \rightarrow \mathbb{R}[/math]; semiclassically we can treat [math]\phi[/math] as a "density" of the "electron probability material" over the shells [math]\Omega \cong S^2[/math] so that the strong EL [math]\Delta \phi = \hat{p}^2\phi = 0[/math] minimizes the surface tension by the maximum principle.
>>11376186
Unfilled shells have a non-zero spin [math]S_z \neq 0[/math], yes, but terms proportional to it do not enter the Hamiltonian unless something couples to it, such as spin-orbit coupling [math]S\cdot L[/math] or some external magnetic field [math]h \cdot S[/math], and these terms can actually reduce the energy when the ions are put on a lattice next to each other. So magnets defined as a collection of tightly-bound electrons [math]do[/math], in fact, like to exist.
>>11376095
Write [math]\max\{f,g\} = \frac{1}{2}(f+g+|f-g|)[/math].

>>9402035
>But just trying to interpret the cross terms by analogy to classical probability theory
Similar to the confusion caused by the Aharonov-Bohm effect, this is just an artifact of thinking about quantum mechanical system wrongly. If you interpret probabilities of mixed states (photons at the 2 slits) as that of pure states (photons at [math]each[/math] of the 2 slits) of course you're going to get the wrong results.
>Repill me on C*-algebra. I thought it was a generalization of the Fourier transform, how you can represent L^2 integrable functions with a bunch of different basis?
Von Neumann algebras, which is a special kind of unital associative [math]\mathbb{C}^*[/math]-algebra, underlies the foundations of quantum mechanics. The ladder operators (from SHO, say) are generators of the Heisenberg algebra, which is a specific kind of Von Neumann algebra on which a representation on [math]L^2[/math] exists, [math]then[/math] Fourier transforms can be used along with other results from harmonic analysis. [math]\mathbb}C^*[/math]-algebras aren't so much a generalization of Fourier transform, but rather a generalization of the Heisenberg algebra.
>What's the formalism for a Lorentz invariant probability density function btw, much less an invariant wavefunction?
It just require the existence of a representation of the spin group compatible with the Heisenberg algebra. The only problem is that there's only the trivial representation so only vacuua can be defined as a consequence of Ekstein's theorem.