guys I wanna express the states $|qlm\rangle = |01m\rangle, N = q+2l, N = n_x+n_y+n_z$ in terms of the 3 energy eigenstates corresponding to the first excited level for the 3d isotropic QHO $|n_x n_y n_z\rangle = |100\rangle, |010\rangle, |001\rangle$

I've tried solving it but couldn't do it, therefore I checked the solutions

but there's a passage I can't understand

I don't understand how to get the expansion coefficients.

what I mean is: the system of three equations has solution $$\langle 100|01m\rangle = \langle010|01m \rangle = 0 $$

also how is it possible that for $|010\rangle$ the expansion coefficient $\langle001 |010\rangle = 1$

when just above it is has been found to be equal zero

Shwank

I didn’t see a discussion of micro to macro fields in Wald’s enm text

@Mr.Feynman You did not manage to state the task you want help on...

@SillyGoose shwank

Einstein was opposing group theory in his 50s

this means that it's normal to be afraid of new math

How are u @naturallyInconsistent

Soooo tired

what about you?

About to start new semester and find out abt grad school results in coming months :P

do u like lectures

I don’t like most lectures

Because i’d rather read texts but sometimes they can be good especially if i only want to learn a bit of the subject

Or if the subject is particularly challenging

some professors can be fun

most lectures are a waste, yeah

@naturallyInconsistent very nasty stuff happened yesterday

I'll take care of mathematica later on

contrary to what almost every human wud believe, humans r awful at generating lists of random numbers

I am interested to see how the lectures are for a quantum information course i am taking this semester—i hope they are good:D

@Mr.Feynman oh nose! what hap?

@Mr.Feynman Do you just want to compute the Christoffel symbols?

@naturallyInconsistent that, Riemann and Ricci

@naturallyInconsistent such a big gaffe that I'm not ready to tell yet :P

@Mr.Feynman One of the guys who hangs out here gave me a notebook for calculating all the common GR tensors. You're welcome you a copy if you want, though I guess you've already got something along these lines.

@ClaudioMenchinelli no

@ClaudioMenchinelli $$\begin{align}\tag1\pm\left<001|01\pm\right>&=0\\\tag2\pm\left<100|01\pm\right>&=-i\left<010|01\pm\right>\\\tag3\pm\left<010|01\pm\right>&=+i\left<100|01\pm\right>\end {align}$$

Equation (1) here explicitly sets two expansion terms to zero. You are supposed to combine Equations (2) and (3) here to get $$\tag4\left|01\pm\right>=\frac1{\sqrt2}\left|100\right>\pm\frac i{\sqrt2}\left|010\right>$$

And then you can find the specific case for $\left|010\right>_s=\left|001\right>_c$

what is time according to u

@SillyGoose thing with lectures is it only gives a short buffer time to be stuck on something in the lecture. A random side train of thought- generally cannot entertain, atleast not at the cost of losing track of wats going on in the lecture

It doesn't help if u hv a notion to prove everything yourself, then lectures are basically 1.5 hrs of spoilers

@nickbros123 That is precisely where the lecturer could save everybody's time if they would express something rigorously, and then the students would be much less likely to fall into random interpretation errors.

i feel books always do a much better job

sometimes lectures go too slow over basic topics

@Mr.Feynman I'm rusty on this but I think you should have $\Gamma^\varphi_{\vartheta\varphi}=\cot\vartheta,\ \Gamma^\varphi_{r\varphi}=1/r=\Gamma^\vartheta_{r\vartheta},\ \Gamma^\vartheta_{\varphi\varphi}=-\sin\vartheta\cos\vartheta$ remembering that the lower symbols are symmetric, and the others are all zero. The $\Gamma^r$ and $\Gamma^t$ will follow shortly

$\Gamma^r_{\varphi\varphi}=-r\sin^2\vartheta f,\ \Gamma^r_{\vartheta\vartheta}=-rf,\ \Gamma^r_{rr}=-\frac12\partial_r\ln f,\ \Gamma^r_{rt}=-\partial_t\ln f,\ \Gamma^r_{tt}=\left(\frac12f\partial_r f+f^2\partial_r\Psi\right)e^{2\Psi}$

apparently Martin Schwarzschild, Karl Schwarzschild's son, won the Karl Schwarzschild medal

Seems a bit fishy

$\Gamma^t_{tt}=\frac12\partial_t\left(\ln f+\Psi\right),\ \Gamma^t_{rr}=-\frac{\partial_t f}{2f^3}e^{-2\Psi},\ \Gamma^t_{rt}=\frac12\partial_r\left(\ln f+2\Psi\right)$

That's it

what do u gain from learning standard model

At 3:10, Carroll has a proof by contradiction that recurrence theorem doesnt exist for our universe

does weinberg really not mention bargmann's theorem?

@JohnRennie I'm an absolute noob, so anything will be useful :)

hm well it seems the notion of central charges, and so central extensions, are integral to proving bargmann's theorem, so I guess weinberg just talks about the actual maths and does not name the end result theorem: mathematik.uni-muenchen.de/~schotten/LNP-cft-pdf/…

@Mr.Feynman I've put it on my server. The file is AFT-Schwarzschild.nb. Chrome will probably complain the download is unsafe because the site doesn't use https.

Written by AccidentalFourierTransform, who doesn't seem to hang out here these days.

I've read a couple of answers of theirs!

@naturallyInconsistent it seems to agree with my calculation

@Mr.Feynman yay

@naturallyInconsistent I see that was my plan from the start as well I just wasn't dealing with the right quantities: my solution is: $$ |011\rangle =c_x^{+} |100 \rangle_c + c_y^{+}| 010\rangle_c + c_z^{+} | 001\rangle$$

moreover $$\sum_{i=1}^{3} |c_i^{+}|^2 = 1$$

but $c_z = 0$

I can therefore combine 2 and 3 to get $c_x^{+}$ and $c_y^{+}$ up to a phase factor

setting the phase factor of $c_x^{+} \in \mathbb{R}^{+}$

I get from the system of equation that $c_y^{+} = +ic_x^{+}$

Is this what you meant to say

@ClaudioMenchinelli correct, and then you normalise

one problem remains: you said that I can now find the expansion coefficients for $ |010 \rangle$

but I get all zeros hahaha

No, m=0 helps create a new possibility

don't you get three $0 =0$ eq's

You do

where's the new possibility then?

But the new version of Equation (2) and (3) assert that two expansion terms are zero, however, the new version of Equation (1) is trivially true, and then you have a degree of freedom to freely choose.

oh so all the probability is contained in the last exp.coeff

Now I see it

yeah

$$ |010\rangle_s = e^{i\phi}|001\rangle_c $$ and then I set the coefficient to be real and positive and I've done. That's kind of illuminating @naturallyInconsistent you might be goated for this one

Thanks a lot for the suggestions @naturallyInconsistent

This is totally not worth goating

I teach semi-professionally.

the important thing is that you've been helpful. That's always praiseworthy

meowth

in thermodynamics why do we not care much about gravity

The sizes of our engines are too small for gravity to make a considerable impact

Feynman Lectures covered the isothermal atmosphere, where gravity makes an explicit appearance.

@nickbros123 no one cares about gravity except astropeople and crazy QG fellas

" a subgroup of a group has elements m such that m divides n"

this proof is easy but i cant see this result as an obvious thing

i cant see it "without the proof"

this also has a consequence that prime groups are always cyclic

shud these results b obvious

i think these results never showed up in physics so far

in physics, u just hav Lie groups

the Poincaire grp, the internal symmetry groups and the gauge groups

is this the only part of grp theory relevant to physics

@RyderRude you really need to stop making general claims about "physics"

condensed matter, in particular crystallography, is full of discrete point groups

oh

so it shows up in cond matter

@ACuriousMind Those are all Lie groups :p

everything shows up in cond matter. maybe even number theory

i feel the local transform grp $\phi ---> \phi e^{i \alpha (x)}$ is in some sense the same grp as $A_ {\mu} ----> A_{\mu} - \partial _{\mu}$ alpha$

so gauge grp and local internal symmetry grps r the same

Groups that aren't gonna be Lie groups in some form are basically going to be infinite groups that don't have a proper Lie group structure and those are unlikely to pop up

Except for like automorphism groups of functions I guess

But even then they are sort of Lie groups

They'd basically have to fail to be manifolds in some sense

u cud call the full thing as "the gauge grp". $\phi ---> \phi e^{i\alpha (x)$, $A---> A - d\alpha $

or have non-smooth actions

so Yang Mills gauge grps r "made up of" finite dimensional lie grps like U(1), SU(2), SU(3)

im calling the local transform thing "the gauge grp" and U(1) the "internal symmetry grp"

@Slereah oh

u can also hav discrete infinite grps. these wont b Lie grps

Why not, they're just non-connected zero-dimensional manifolds

but the Cyclic grp as cycle tends to infinity becomes the U(1) grp which is a lie grp

@Slereah oh

but none of the important Lie theory wud apply there

because no generators or vector fields or diffeomorphisms

without vector fields, u dont even hav the Lie bracket

The Lie bracket is the Lie bracket of the tangent bundle of the point

Since there is only one vector it is the trivial bracket $[0,0] = 0$

v. much a Lie group

This is the vector given by the curve $\gamma(\lambda) = e$

every grp is a lie grp then :P

i think grp theory is the most important thing in physics

what about field theory, etc

i think all u need to know is that vector spaces need fields

not much field theory, right?

we mostly just use R or C fields

@RyderRude Well no, this only works because a discrete group is given the discrete topology

Topological groups can be not-Lie groups if their topology doesn't match that of a locally Euclidian space

oh

But physics is mostly made by considering things with a somewhat Euclidian topology

or discrete things also

It is not common to have too weird topologies

p adic fields hav a different topology

but theyre not in physics mostly

so p-adic numbers r not a lie grp under addition

@Slereah No one thinks of discrete groups when someone says "Lie groups" (even if I grant you that technically they are 0-dimensional ones)

Is the empty set a Lie groupoid, more importantly

Pair groupoid of the empty manifolf

@Slereah I think the definition of a groupoid should involve the underlying set being non-empty

did Andrew Wiles.make use of p adics

and I think you don't really want the empty set to be a manifold, either (what would be its dimension, for instance?)

@ACuriousMind all of them

seems like these numbers r cutting edge these days in number theory

But if the empty set isn't a manifold, what's the boundary of a manifold without boundaries

well a manifold without boundaries isn't a manifold, either :P

I think you probably want the empty set to be a manifold with boundary but not a manifold

hav u read Wiles's proof

he converted it to a geometry problm

prob algebraic geometry stuff

only one link mentions p adics and Fermat's theorem

and that's an article on p adics

it should b called Wiles's theorem tbf

it's only Fermat's conjecture

people say that the math machines that Wiles used werent available in Fermat's time

crazy tht he proved it. maybe he even has intuition for why the theorem is true which is crazier

Veritasium has the best exposition on p adics youtu.be/tRaq4aYPzCc?si=lZTO7RoT5UN5AdIG

ok so p-adics basically form a field when p is prime.thats y theyre important

@ACuriousMind How do you feel about the covariance field arxiv.org/pdf/1008.3170.pdf

I hadn't heard of it but I've read stuff from Gotay before

He is the big name in geometric dynamics

at a quick glance I think this is just the careful field theory version of the "make any theory generally covariant" from QoGS

Looks like Dirac's time reparametrization invariant mechanics is an example

"Consequently the covariance field has no physical import. We are free to suppose η is dynamic, and so we have accomplished goal (II): we have constructed a new field theory in which all fields are dynamic."

Oh no

So far I am getting the impression that absolute objects are mostly vibes based

it's like a made-up concept to remove diffeomorphism invariance from SR

it's all definitions really. define the metric to not transform

mathematically, it does make sense to talk about symmetries of only dynamical objects. and it's a useful concept sometimes. e.g. it gives Poincaire transforms in SR

Are central charges (in the context of lie algebras/rep theory) called so in both physics and math contexts?

@SillyGoose not always, you might find math texts that present the same concept talking about 2-cocycles on the algebra etc. but never call anything a "central charge"

In the Poincare patch of AdS spacetime, translation along the radial $z$ direction correspond to dilatation(in the CFT sense) according to this pg. 12 first line, but according to this eqn. 15 pg 15, a scaling along $z$ and other coordinates is dilatation.

But scaling and translation is different...How can I reconcile the two?

AdS spacetime has like two things here

Like every spacetime, you can add some conformal transformations to it, but also its isometry group is uuuuh SO(3,2) or something

Which is isometric to the conformal group

Meaning that there is some motions on it that can be interpreted as dilations idk

> The connected component SO(d−1,2) of the anti de Sitter group is isomorphic to the connected component of the conformal group of ℝ d−2,1. This is the basis of the AdS-CFT correspondence.

neat