/sci/ - Science & Math

Professor Emeritus edition
Previous thread: >>14607941

>what is /sqt/ for?
Questions regarding math and science. Also homework.
>where do I go for advice?
>>>/sci/scg or >>>/adv/
>where do I go for other questions and requests?
>>>/wsr/ >>>/g/sqt >>>/diy/sqt etc.
>how do I post math symbols (Latex)?
rentry.org/sci-latex-v1
>a plain google search didn't return anything, is there anything else I should try before asking the question here?
scholar.google.com
>where can I look up if the question has already been asked here?
>>/sci
https://eientei.xyz/
>how do I optimize an image losslessly?
trimage.org
pnggauntlet.com

>where can I get:
>books?
libgen.rs
z-lib.org
stitz-zeager.com
openstax.org
>articles?
sci-hub.st
>book recs?
sites.google.com/site/scienceandmathguide
4chan-science.fandom.com/wiki//sci/_Wiki
math.ucr.edu/home/baez/physics/Administrivia/booklist.html
>charts?
imgur.com/a/pHfMGwE
imgur.com/a/ZZDVNk1
>tables, properties and material selection?
www.engineeringtoolbox.com
www.matweb.com

Tips for asking questions here:
>attach an image (animal images are ideal. Grab them from >>>/an/)
>avoid replying to yourself
>ask anonymously
>recheck the Latex before posting
>ignore shitpost replies
>avoid getting into arguments
>do not tell us where is it you came from
>do not mention how [other place] didn't answer your question so you're reposting it here
>if you need to ask for clarification fifteen times in a row, try to make the sequence easy to read through
>I'm not reading your handwriting
>I'm not flipping that sideways picture
>I'm not google translating your spanish
>don't ask to ask
>don't ask for a hint if you want a solution
>xyproblem.info

>>11448310
Nope. Second quantization is named so inorder to distinguish it from first quantization. Given a symplectic space [math](\Omega,\omega)[/math], second quantization quantizes fields, namely promotes sections of vector bundles on the [math]\infty[/math]-jet bundle [math]J^\infty \Omega[/math] to operators, while first quantization quantizes coordinates, namely promotes the smooth Poisson algebra [math]C^\infty(\Omega)[/math] to operators.
>second countable
Did you mean separable? Second countable means a countable base for the metric topology, which is weaker than having a countable basis for the linear structure of [math]\mathcal{H}[/math]. Being separable is an assumption that allows us to leverage reflexibility and Riesz representation to study duals and preduals. This extremely useful in QM, since it in particular allows for a faithful infinite-matrix representation on [math]\mathcal{B}(\mathcal{H})[/math] and hence a resolution of unity.
>>11448716
Because it's just [math]SO(3)[/math].
>>11449108
It's looking up. Thanks for asking.

>>10227957
I'm doing a PhD

>>9481732
All research problems in this area were solved in Sakurai & Ballentine - An Intuitive Approach to Improper Integrals for Physicists.

>>9382523
Really interesting. I was actually studying topological superconductors before I switched to mathematical physics.
One of the most powerful tools for studying quantum cirticality is conformal field theory, since a condensed matter system has infinite correlation length at the critical point and hence achieves conformal symmetry. In 2D the symmetry group is the modular (Mobius) group of [math]\mathbb{C}[/math], which is so large that all observables in the theory can be described by an affine Lie algebra. This means that the conformal blocks satisfy a finite set of differential KZ equations and so the [math]n[/math]-point correlation functions of the CFT can be solved exactly at the critical point.
General mathematical structures of CFT have been outlined by Seiberg and Moore, and connections to category theory made by Nayak and others. The latter is manifestly important since this gives a way to frame CFTs in terms of certain types of TQFTs, and quantum criticality described by the CFT can be computed via topological methods. An example would be the fact that the non-trivial particle statistics arising from the fusion relations in a CFT can be computed via Vafa's theorem in the corresponding TQFT.
>>9382736
It was good. Did some oly lifts while flirting with my bf.

>>9132295
>What are you studying/researching?
Algebraic geometry on Riemann surfaces for gauge symmetry in CFTs and some quantization schemes. I couldn't find a book on TQFTs in the library of the new uni so my quest for a unifying picture of CFT/TQFT will have to wait.
For my actual job, I've already finished studying some specific aspects of topological superconductors that caught my PhD supervisor's eye. I'll be pitching ideas to her later this week.
>Any interesting problems/questions?
For an analytic family [math]\pi: R \rightarrow C[/math] of Riemann surfaces there is a map [math]\rho: \Theta_C \rightarrow R^1 \pi_* \Theta_{R/C}[/math] from the sheaf of germs of holomorphic functions on the Riemann surface [math]R[/math] to that of the Cech 1-classes, called the Kodaira-Spencer map. This map is isomorphic if and only if the family [math]\pi[/math] is "nice" under deformations (reparameterizations) of the base space [math]C[/math].
An interesting fact I've found is that the Schwinger-Keldysh contour in nonequilibrium quantum dynamics under physical assumptions can be thought of as the boundary of a Möbius band embedded within the Riemann sphere [math]S^2[/math], and can therefore be thought of as an elliptic curve on which the Dyson equations for the self-energy can be considered as a constraint in the corresponding Cech 1-cohomology class as long as we generalize the notion of sheafs of germs to include holomorphic distributions. This may allow us to use the Kodaira-Spencer map to investigate which class of Green functions behave nicely under reparameterizations of the contour, and will hopefully let us find anomalies relating to renormalizability.
>Textbook recommendations?
Ueno.