@SillyGoose Causal Perturbation theory is good if you can understand it

@Obliv That is the $\beta\to\infty$ and $\beta\to0$ limits. You know the Taylor expansion of exp, which applies to the zero limit. The infinite limit, convert to negative exponents, so that then the exp goes to zero.

@SillyGoose ok, nice; when you are doing these integrals, say the last one, is it $\int_0^{\pi/2}\cos^4\theta_{n+1}\,\mathrm d\theta_{n+1}$ alone, or are there a lot of $\int_0^{\pi/2}\mathrm d\theta_i$ to also integrate?

Anyway, I'm sure you will find the following identity useful $$\cos^2\theta=\frac{1+\cos2\theta}2\qquad\bigwedge\qquad\sin^2\theta=\frac{1-\cos2\theta}2$$

m i a o ~

thanks, I did have to give up and look at the solution. I'm definitely memorizing it I've seen it this many times

power series, geometric series, etc. I'm going to do a nice review of fundamentals of calculus after finals

wow it just occurred to me, as I was working this problem out that involved radio waves

gigahertz are really slow..?

er wait

why are radiowaves in the 20kHz to 300+GHz range

my familiarity with frequencies involve computer processors where people consider higher GHz to be "faster"

f =1/T

hmm ill have to recalculate then lol

yea nope i did it right.. I'm getting an associated radiowave of 841GHz for the absorption frequency of this paramagnetic solid

I'm buggin

or maybe this is normal and i wasn't aware cpus operated at the same frequency as radiowaves..

whatever CPU frequency even means

ye nvm that's too high to be radiowave. I suppose I did it wrong

@SillyGoose there's only one thing worse than QFT

Rigorous QFT

is q²/T a possible new username for you? (joke) @fqq :P

By the way, I've always wondered what "fqq" stands for

So are we considering complex transformations too?!

Infinitesimally, $D_n$ generates $\theta\mapsto\theta+\mathrm{e}^{in\theta}$

@Mr.Feynman hi. complex valued diffeomorphisms do not make sense on a circle, so theyre probably only using these as a basis for real diffeomorphisms

one can take linear combinations of these generators to get the $\sin n\theta$ and $\cos n\theta$ generators, which then only form a basis of real diffeomorphisms

Oh, like the usual Fourier basis

this is an example of a basis of an infinite dimensional lie algebra. i wonder if one cud also use the delta functions here

or maybe the Hermite polynomials for diffeomorphisms on the real line

really pretty how math ideas assemble

@RyderRude why would you?

idk. theyre probably too pathological for lie algebras

imagine $\delta (x-x_0)\frac{d}{dx}$ lol