yes this is a computation that i needed to operationally know the fubini-study measure for
i was computing $S = 1 - \text{Tr}\left(\int_{\mathbb{C}P^n} d \Omega_n \rho^2\right)$ where $\rho$ is a time-evolved reduced density matrix with some $\mathbb{C}P^n$ dependence
the answer turns out to be nice and simple >:D which makes me wonder if there is a simpler way to do the computation
but i tried to ask on stack about coordinate free/other coordintizations of $\mathbb{C}P^n$ expectation values to no avail
@SillyGoose your cases are about $i=1$ and $i\neq1$, but some of the expressions themselves don't depend upon $i$ in the correct way. Must be some typoes. Also, these seem to be complete angular integrals. If they are, you can just use standard Gaußian integrals to extract the results directly.
@naturallyInconsistent the $i$ dependency comes from the fact that $n_i$ when $i = 2$ is different from $n_i$ for all other $i$. (i did mistype $i = 1$ instead of $i = 2$ though)
What Veritasium isn't telling you is that the Kerr geometry is (probably) unstable to perturbations and cannot exist outside of a mathematician's head. So the Penrose diagrams showing multiple universes linked through the Kerr geometries don't actually exist.
The same applies to the Reissner-Nordström geometry.
if that's enough to be a "multiverse", then if you accept that inflation may have causally separated patches of the early universe, it's actually not that far fetched that we live in a multiverse
you don't need white holes or weird Penrose diagrams to get causal disconnectedness
@SillyGoose Oh, that is an important mistake to catch. What is the standard result? What are $\mathrm d\Omega_n$? Usually that has trigonometric functions in there, but it seems to be missing. Are you sure $\theta_1$ does not have a wider integration region? In 3D polar we have an $\int_0^\pi$ and a $\int_0^{2\pi}$ to keep track of.
It seems though that alot of people who are interested in QFT seem to want to get past QED as quickly as possible to move onto QCD or something so Maxwells equations get relagated
My biggest gripe with QFT books is that they are too heavily reliant on literature. I mean, that's how it should be in science but QFT books push it to the limit
Every two lines there is a paper that proved something in a very specific setting
That's probably because as you say they want to get you into the business of Feynman diagrams quickly
> For a one-dimensional compact manifold, two topologies are possible - corresponding to the closed and open strings
Aren't all 1D compact manifolds homeomorphic to $S^1$?
