@ACuriousMind I might end up reading part IV of this book because it contains an introduction to intersection forms...you might get a lot of gauge fields questions from me in a week or two :P
@EmilioPisanty Okay thanks. Have you see a definition which basically defines it as a the eigenstates of a spin operator in some arbitrary direction. So the states $| \mathbf{S} \cdot \hat{n} \rangle$ in eignvalue equation: $$\mathbf{S} \cdot \hat{n}| \mathbf{S} \cdot \hat{n}; + \rangle = (\frac{\hbar}{2})| \mathbf{S} \cdot \hat{n}; + \rangle$$?
@ACuriousMind !!! ok thx for correction would have been hard to find that again for me... maybe there is more than one GSW in physics? found the wrong one :| ... am gonna fix it
@BenNiehoff sigh this false modesty is our worst enemy for the AMAs & maybe one of my main personal hurdles in getting/ "herding" guests... youve authored what over 7 papers on arxiv now? argh o_O
@BenNiehoff in case you hadnt noticed our AMAs work for both big fish in small ponds and small fish in big ponds & stuff in between too... anyway iirc at least 1 other endorsed idea of you as guest :)
@JohnDoe No really unique standard meaning. Common things it could denote would be a quantum state (if $\psi,\phi$ are already taken) or a Weyl spinor.
@ACuriousMind Okay thanks. Do you maybe know why it is that given the Hamiltonian $\hat{H} = \hbar \Omega_R \sigma_{x}$ for a two level system represented by a vector in the vloch sphere, why it follows that the time evolution operator $\hat{U}(t) = e^{(-u \Omega_{R} \sigma_x)t}$ implies a rotation of the vector by an angle $\Omega_R t$ around the $x$-axis?
@ACuriousMind Am I correct in stating that it doesn't make sense to talk about a vector in the Bloch sphere which is along the z-axis since that implies we know the coordinates of the other spin operators, it only makes sense to talk about the projection of some vector of a known magnitude?
@ACuriousMind Today the Elven Lord was saying that homology was known to Poincare. But how did those guys compute anything without Maclane/Mayer/Vietoris and their homological algebra?
@0celouvsky Because last time I asked you you threw a useless definition at me and didn't care to explain in a way that was accessible. With ACM at least I understood half the sentence
@ACuriousMind So I have this thing called the "index bundle" of a parametrized family $T:X\to\mathscr F(\mathscr H)$ of Fredholm operators which is a vector bundle $$\mathrm{index}(T)=\ker (T)-\mathrm{coker}( T),$$
@ACuriousMind All I was trying to figure out was whether the point is to think that every pure two level state is a point on the sphere or if there was some other info that is preserved.
but if you want to see the sphere more easily, you can write $z_1 = r \cos \frac{\theta}{2} e^{i(\psi + \phi)/2}$ and $z_2 = r \sin \frac{\theta}{2} e^{i(\psi - \phi)/2}$. Then the sphere is at $r = 1$ and $\psi = 0$.
@JohnDoe Well, there is some more information that you can see on the sphere, e.g. the Fubini-Study metric, and states that are "close" on the sphere are also "close" in the original state space (for a suitable notion of "close" for rays)
I once heard, not firsthand mind you, that my uncle was pissed at me because I didn't express enough gratitude for being allowed to stay at his house once
Here you mostly apply for scholarships and most of them give you the same amount of money as you'd be getting from the federal finanical aid. A little bit more and the main benefit is you don't have to pay that back later.
