 7:41 AM
That's because all those diagrams are the same

2 hours later… 9:45 AM
Is there an equivalent of the Cauchy normal neighbourhood for non-Affine connections?
Seems trickier to define because they may have equivalence classes of curves with different parametrizations, but there are some invariants that are still there, like concurrence
So the boundary of the normal neighbourhood defined by caustics would remain the same

5 hours later… 3:03 PM
I am trying to articulate properly the different notations regarding angular momentum , so I'd appreciate if someone could help me here. If I have two particles 1 and 2, each has an angular momentum, in classical mechanics we have $\vec J_1$ and $\vec J_2$ the angular momentum vectors. In QM we have $\hat{\vec J_1}$,$\hat{\vec J_2}$ the angular momentum vector operators.
For particle 1, $j_1$ is called the main angular momentum quantum number, while $m_1$ is called the secondary angular momentum quantum number (analog for the 2 particle). Am I correct in the way i am describing this notations? 6 hours later… 9:55 PM
Are the Clebsch–Gordan coefficients used when one needs to add more then 2 angular momenta together? IMO it should be possible 10:14 PM
@imbAF Sure, in principle CG coefficients are all you need to add arbitarily many momenta. However, there's a whole zoo of "coefficients" for different use cases and number of momenta that are supposedly more efficient than starting from the CG coefficients every time again, e.g. Wigner's 3j/6j/9j symbols, Racah's W coefficients. I see
I am not familiar with the last part of what you wrote but at least
I know that adding a desirable number of momenta is possible with the CG coef. @Slereah I think a crucial question here is how did I recognize he was doing category theory :P
is he planting stalks of sheaves in his hermit garden? @ACuriousMind Can I ask you something about the Wigner eckart theorem? I am writing a thread right now but maybe you could give me an explanation, quick you'll only find out if I can help you if you actually ask me the question :P and ask you want confuses me
bear with me, I will have a poor terminology
I cannot figure out
$$\langle k,j,m|T^{(r)}_q|k',j',m'\rangle=\langle j',r; m',q|j,m\rangle \frac{\langle k,j||T^{(r)}||k',j'\rangle}{\sqrt{2j +1}}$$. This is the W.E-Theorem . the letter k is used for other quantum numbers, such as the principal quantum nr $n$ etc. I have no idea, why other quantum numbers are important here. Maybe they add the the degeneracy, but the interconnections and consequences of additional quantum numbers in the grand scheme of things
Regardless, this is an extra detail that I wish I could understand, but it's not the main issues here, right now 10:26 PM
they aren't important, that's why you'll often see WE written without the $k$s :P so they play no role
for this part Hallo I think I will make a thread, and maybe you can have a look at it when I finish @imbAF sure, if this is a question suitable for the main site you should post it there (but I've told you several times before that you don't need to come to chat to ask people to look at main site questions - people who are interested in that watch the main site anyways) Yeah sure
I am trying to use the right words
for this topic
man saying stuff like main total angular momentum quantum number etc etc
is taxing 10:38 PM
I don't think I've ever called anything "main total angular momentum quantum number" :P It's just related to the eigenvalue of $J^2$ as $j(j+1)$, just say that
personally, I think "quantum number" is a pretentious word that obscures we're just talking about eigenvalues yes but the eigenvalue of $J^2$ is $j(j+1)\hbar$
then naturally comes the question what is $j$? "total angular momentum" $\vec J$ right?
you have $\vec J$, $\hat {\vec J}$, $j$, $j(j+1)$ we just talk about spin-1/2 particles, too, even when $S^2 = 1/2(1/2 + 1)$. No one talks about "particles with main total spin angular momentum quantum number 1/2" does it make sense to add up spins? They act on different entities/objects
electrons for example
? 10:44 PM
sure, e.g. if you have two electrons, you want to talk about the spin of the two-particle system hmm you do the CG math for that exactly like for ordinary angular momentum, you find the combined system can have spin 0 or spin 1, it's exactly the same logic I find it odd
like I can understand when you do 2 orbital angular momentums
because they act on the same "axis"
of rotation
but two spins or spin/orbital (I am aware of LS coupling) act on different objects
well L doesn't act on anything really, but I don't know how to explain it otherwise I wouldn't think of anything here in such...material terms. You have a Hilbert space $H_1$ with an angular momentum operator $L_1$ (doesn't matter whether spin or kinematic angular momentum) and a Hilbert space $H_2$ with an angular momentum operator $L_2$, and they combine to a total Hilbert space $H = H_1 \otimes H_2$ with angular momentum operator $L = L_1 \otimes 1 + 1 \otimes L_2$ hello smart people, could someone explain to me whether upthrust and drag are the same thing? when travelling through a fluid 10:50 PM
What Clebsch-Gordan coefficients do is tell us how the common eigenbasis $\lvert jm\rangle$ of $L^2, L_z$ decomposes in terms of the tensor products $\lvert j_1m_1j_2m_2 \rangle = \lvert j_1m_1\rangle \otimes \lvert j_2m_2\rangle$ of eigenbases of $L_1^2, L_{1,z}, L_2^2, L_{2,z}$
for this math it doesn't matter at all what the "nature" of $L_1$ and $L_2$ is - the math works regardless of their physical interpretation I fully understand how the C.G coef arise
simply I thought in physical terms
how does it make sense to couple things acting on entire different things
But ok 11:08 PM
@ACuriousMind I have 3 questions in my thread, should I post them all together or 1 by 1 ? that always depends on how closely related they are: If one can answer one of the questions without answering the other two at least partially, it's usually better to post them separately all 3 have to do with the W.E-Theorem