The h Bar

12:39 AM

@DanielSank not invert, diagonalize

and yeah, your Pauli-matrix formalism is almost certainly the way to go

write your matrix as $\vec n \cdot \vec \sigma$ and you'll be able to read off the components of the eigenvectors from the spherical coordinates of $\vec n$.

I dunno whether that'll be helpful, but it's guaranteed to work (though it can give ugly expressions)

Emilio Pisanty

12:59 AM

@penelope: Now, I acknowledge that to some extent we are patronising the asker here by giving them a protection they did not necessarily ask for, and in an ideal world I would let the asker make this decision. However, this requires the asker to be present at the right moment, to know what is about to be unleashed unto them, and a moderator to be present to enact their decision, once made. In my experience, this is rarely the case. We may leave a comment along the lines of: “If you do not say otherwise, I will de-HNQ in fifteen minutes.”, but that’s about it. — Wrzlprmft ♦ Mar 12 at 16:53

"de-HNQ"

good verb

we should keep it

(CC the discussion leader, @rob)

user351417

2:07 AM

@dmckee I read a very nice bit about that and pedagogy in an excerpt of a book called The Upside of Down. It was in the context of an author searching for inspiration. Unfortunately, there appear to be two books by the same name ([Wikipedia](en.wikipedia.org/wiki/The_Upside_of_Down ), and I can't tell which one it was :(.

dmckee

Haven't read either so I'm afraid I can't help.

user351417

@dmckee Ah, I searched for some phrases and I found that I'm talking about the one by mcardle. It was a very well-written section, if you're interested.

user351417

It was related to the way some people (gifted authors, in this case) receive praise in high school, and then wonder if they aren't actually praise-worthy because they start relying on talent instead of hard work, and so on.

DanielSank

3:18 AM

@EmilioPisanty Just say dehnq. I dehnq, you dehnq, he dehnqs. Yesterday we dehnqed, etc.

vzn

3:42 AM

Feb 7 at 22:43, by DanielSank

...and I think it's important to understand why classical is classical and quantum is quantum.

← still working on earthshaking theory linking classical to quantum harmonic oscillators/ qubits, trying to connect very many scattered dots, very slow going, still no cohorts/ collaborators... :( o_O

user351417

4:00 AM

I've been saying "un-hnq" until now; I pronounce it as "unhunk"

Physics Meta

4:17 AM

0

Q: Is this a bad question?

One of my question that I asked a week ago is not yet answered. I just wanted to ask if there was something wrong in that question. Here’s the link Can flames be Frozen?

user351417

5:01 AM

@Blue I just saw this one too... perhaps that v1 stuff is relevant only when the badge holder has edited the post?

dm__

5:39 AM

for complex gurus out there: under what conditions must $f(z)$ satisfy such that $\int_{-\infty}^{\infty}f(z) dz = \int_{-\infty+i\alpha}^{\infty+i\alpha}f(z) dz$?

for a real $\alpha$

trying to get an intuitive feel for this

and not a homework problem XD. usually see these integrals with time dependent perturbation theory, and has always just been equal and I didn't give it much thought

Anonymous

7:51 AM

@Chair I see; didn't know that! :)

pZombie

8:15 AM

If i wanted to predict the speed of sound in air based on first principles which in turn could be in theory derived from first principles of the standard model (QM/GR), then how deep would i have to go?

pZombie

8:26 AM

I assume that it is electromagnetism/photons which transmit the initial vibrations of sound generated at a source. electromagnetic waves however travel near c through air, so how does sound end up being so slow then? Could one write a simulation down to the molecular level and below, with photons transmitting the vibrations and end up with as simulation which results in the simulated soundwaves moving at the known speed of sound through air?

I guess this would be related to asking why a compressed spring would de-compress at a given velocity, given that we would want to solve this only by looking at its particle composition and run a quantum simulation based on its particle structure rather than solve it with the known newtonian formulas

Captain Bohemian

9:34 AM

what is the definition of scattering amplitude in quantum field theory? I have only learnt the definition of scattering amplitude in nonrelativistic quantum mechanics as the squared wave function, but have not seen anywhere a definition of scattering amplitude in quantum field theory.

Captain Bohemian

9:45 AM

I have only seen scattering amplitude in quantum field theory is some limit of coorelrator or Wilson loop in some cases, but I wonder if there is a generic definition of scattering amplitude in quantum field theory.

user351417

10:11 AM

@Blue Unfortunately, I can't find any posts on meta SE which talk about that behavior. I'm actually starting to wonder if it's a bug.

Slereah

10:38 AM

hello folks

You wouldn't really need GR for the speed of sound

what you need to do is statistical mechanics for QFT

Which is unpleasant

I mean if you go "from first principles" showing protons exist will take a few years already

Emilio Pisanty

@Blue @Chair Dupehammer only requires that the close-voter didn't edit in the tag themselves

which would obviously be an open invitation for abuse

2

Q: Dupe hammer - "original tags only" limitation

The question was asked and answered earlier: Is there anything to stop a Gold Tag person editing a tag to close any question they want? and the answer is "yes - only original tags on a question count for the dupe hammer". I have a problem with that. A question such as this one is absolutely abou...

discussion dupehammer

for one

user351417

Thanks! That's what the behavior of the system looked like. =)

Emilio Pisanty

5

Q: Is there anything to stop a Gold Tag person editing a tag in to close any question they want?

Someone with a gold badge in a tag can Dupe Hammer, and close it in one go, like a moderator. Is there anything to stop a Gold Tag user from editing in that tag to close it? Or is that allowed, like edit to unvote?

discussion dupehammer

user351417

I'll suggest an edit for the faq post about the dupehammer on meta se.

Emilio Pisanty

which one are you thinking of?

user351417

10:49 AM

Apparently whatever it was, it doesn't exist.

user351417

meta.stackexchange.com/questions/tagged/dupehammer+faq

user351417

I'll look at Tim Post's answer to the original feature request.

user351417

Or actually, could you edit meta.stackexchange.com/a/231212, @EmilioPisanty? You seem to be more familiar with how it works. Perhaps the bullet list could use the update.

Emilio Pisanty

ah, here it is

157

Q: Only prohibit those who edited the tags from using the dupe hammer

I have a gold c++ badge but this question wasn't closed when I marked it as a duplicate. The OP had left out the language tag, and it was only added later by an edit. Looking closer, no less than three "golden badgers" voted to close the question, yet it still waited for 5 votes and attracted a...

feature-request status-completed dupehammer

originally proposed on MSO

43

Q: It looks like the duplicate banner changed. How does it work now?

I noticed this morning that posts that were closed with a gold badge dupe hammer have a new banner. A couple of examples of the new look are below. Is this a new change or just testing as I can't find a post on meta post about this? What happens if the user doing the closing has a gold bad...

support duplicate-questions dupehammer

the behaviour then documented on MSE

@Chair It feels to me that barging in with an update into that post wouldn't be appropriate, I don't think

user351417

@EmilioPisanty Okee, sure. I'll read through the comment thread, and if nobody's talked about the matter, I'll ask Tim post if that bit could be updated.

user351417

10:54 AM

@EmilioPisanty That question looks kinda badly written because I don't see a picture of the old banner in it.

user351417

Writing normal stuff in code is an supremely effective way to communicate.

user351417

@NathanOliver: What BoltClock is requesting: if(hasHammer()){ showBadge(); } else { showDiamond(); } — Cerbrus Oct 7 '15 at 13:32

Emilio Pisanty

@Tim Is it possible to change the language in the bullet points (particularly the "originally asked with a tag") to take into account the behaviour documented here? It's confusingly different to the final behaviour of the system (any question with the tag, so long as you didn't edit it in). This seems to be the canonical resource for this mechanism, and it'd be good to have it correspond exactly with the system behaviour. And it'd also be good to have it as faq, while we're here. — E.P. 9 secs ago

user351417

@EmilioPisanty Thanks =)

Emilio Pisanty

@Chair it's just the addition of the badge in the banner

@Chair no worries

it's good that you point these things out

user351417

10:57 AM

@EmilioPisanty Fun.

user351417

@EmilioPisanty Huh, what things?

user351417

Also, why does someone want to add the homework tag here?

user351417

For a regular editor, that's a pretty bizarre edit suggestion.

Emilio Pisanty

@Chair places where the documentation is unclear and confusing and needs to be streamlined

user351417

Ah okee :)

Emilio Pisanty

11:06 AM

@JohnRennie can you give a second-opinion look at this closure?

hi ~~@Red~~ ~~@Green~~ @Blue

4

user351417

Hehehe

Anonymous

Lol

Anonymous

Just passing by. :P

Anonymous

Thanks for the Meta SE reference on the gold badge closing; I was looking for it. :)

user351417

I just realized that facebook messenger does the lamest markdown-like formatting ever. You use ` for code, ~ for strikethroughs, * for bold, and _ for italics, but it only shows up on desktop browsers. The phone app apparently just shows the symbols, which makes communication pretty difficult.

user351417

11:11 AM

And there's some nonsense running around the internet about how it periodically supports TeX.

Anonymous

@Chair Ya know...that's their implicit way of telling people to use Whatsapp instead. :P

Anonymous

Whatsapp has the same markdown format.

Anonymous

Though, I like some of the features the FB Messenger has (I think it even supports MathJax these days). And at least it stores the messages on the server side; so I wouldn't lose important messages even if my phone drowns (unlike Whatsapp).

user351417

@Blue Eeek I was last seen on whatsapp approximately a year and three months back, and I don't care at all. It's a pretty convoluted system if it wants both a google account and a phone number. Plus my email id is outlook.com, which makes the google account crabby.

user351417

@Blue It supports tex/mathjax periodically. It doesn't let me paste nice pretty links, and the oneboxing-equivalent is way too big.

Anonymous

11:14 AM

OTOH I absolutely love Telegram (it has a nice web interface too), but I couldn't convince my classmates to join. ;_;

user351417

@Blue I've never heard of that one; I'll check it out.

Anonymous

@Chair "periodically" XD

Emilio Pisanty

@Blue I can't see any functional difference between whatsapp and telegram

the only real difference that I can see is that sci-hub has a bot on telegram but not on whatsapp

user351417

@EmilioPisanty People use whatsapp in spain? I was given to believe that though it was created by some people from the US, it was a big deal only in India.

user351417

People use it so much here, which makes it easy for me to ignore everything because I don't have a functional phone :P

John Rennie

11:18 AM

@EmilioPisanty that's a tricky one. On the one hand the OP has no idea about Lorentz contraction as they explicitly say they think it's just due to light time of travel, so they need to read the question you linked to find out what is actually going on. On the other hand, the duplicate doesn't really address the specific question.

user351417

My school internet blocks scihub as 'immoral or unethical', which is questionable. But it sometimes blocks stack overflow as 'science and technology', so it's not exactly easy to understand.

Anonymous

@EmilioPisanty IMO one of the downsides of Whatsapp is that everything is stored on the user's end (there's a possibility of cloud backup but that's a hassle). And another is that I can associate a Whatsapp account with only one device. The seamless sync between devices just doesn't exist! Additionally, there's no way to retrieve accidentally lost data/media. Also, I hate typing much on mobile; would prefer to use a nice web interface instead (which Telegram has).

Emilio Pisanty

@Chair from the communities that I've seen, whatsapp has been the primary messaging application in Mexico, the UK, Germany and Spain

> would prefer to use a nice web interface instead

whatsapp too, right?

Anonymous

WhatsWeb?

Emilio Pisanty

or what difference do you see between the two web interfaces?

Anonymous

11:22 AM

That sucks. :P

Emilio Pisanty

@Blue web.whatsapp.com

Anonymous

Ugh, I need to scan QR code and keep the mobile phone switched on.

Anonymous

Also the mobile needs to be connected to the Internet.

Anonymous

Too much hassle.

Emilio Pisanty

@Blue "need to scan QR code once at set-up time" is hardly a high bar for hassle

Anonymous

11:23 AM

I'm not much of a mobile guy...so mine lies with 0% charge most of the time.

Emilio Pisanty

@Blue but this is true

@JohnRennie =|

anyways

let's see how OP responds

user351417

@Blue Exactly. I think mine dies in a moment anyways, so I barely care.

Anonymous

> Unlike WhatsApp, Telegram is a cloud-based messenger with seamless sync. As a result, you can access your messages from several devices at once, including tablets and computers, and share an unlimited number of photos, videos and files (doc, zip, mp3, etc.) of up to 1,5 GB each. And if you don't want to store all that data on your device, you can always keep it in the cloud.

Anonymous

These are nice stuff too.

Anonymous

Oh, and the lovely bots. :P

Anonymous

11:30 AM

@Chair You might want to read through this faq page. :)

Captain Bohemian

12:17 PM

I may be wrong above. I think what I know as squared wave function in nonrelativistic quantum mechanics is called probability amplitude, not scattering amplitude, which may mean the inner product between two states.

PranshuKhandal

12:44 PM

Will someone help me out with this physics.stackexchange.com/q/466776/221932 question

Avnish Kabaj

Real intellectuals use slack for social media

Captain Bohemian

1:09 PM

Slack (software), a team communication tool

user351417

@AvnishKabaj I remember trying to convince people to move here from FB messenger. I got them convinced that it's worth trying to earn 20 rep, but they lost it when I mentioned that there's no privacy, for unknown reasons.

Captain Bohemian

I think I have got tired of that kind of social media in which people tend to add you as friend arbitrarily and send frivilous message to you arbitrarily, such as Facebook.

Avnish Kabaj

@Chair I would have been bullied the shit out if I asked them to use SE instead of insta

You guys use Facebook¿

user351417

Yeah. I refused to do instagram. Interestingly, searching online for "rishi acharya facebook" doesn't get anything related to me.

user351417

I need to make that account look less abandoned. Or maybe I don't.

Captain Bohemian

1:17 PM

twitter and linkedin are better; nobody I don't know there has send message to me.

user351417

@CaptainBohemian Linked in shows you who viewed your profile, right? Isn't that scary>

Captain Bohemian

they may follow me, but I don't care who follows me.

Avnish Kabaj

@Chair ha ha

user351417

I remember making one a few years back because I was bored, and I got emails of my friends' parents viewing my linkedin profile, which was scary.

Avnish Kabaj

I will hax

Student404Mus

1:18 PM

Hello students and professors!

user351417

@AvnishKabaj You won't findz.

Anonymous

Hello, 404. I can't find you.

Student404Mus

I need help in research concerning "concurrence" which is an entanglement measure

Captain Bohemian

@Chair no, it doesn't show me who view my profile. It only shows someone (without specifying who) in xxx (an affiliation) has viewed my profile.

Avnish Kabaj

Found yer GitHub

user351417

1:20 PM

@CaptainBohemian Interesting. Perhaps it's time for me to make another empty linkedin profile for experimentation

Student404Mus

I want to understand this quantity in more detail

user351417

@AvnishKabaj Meh I abandoned that one ages back too. Nothing interesting on it, and I wasn't even using github properly.

user351417

And that name's too corny too :P I used to have it on my SE profile, but I decided that it's kinda lame.

user351417

(if you found the right account)

Student404Mus

If someone has good knowledge in this domain of entanglement?

Captain Bohemian

1:21 PM

I don't feel scary because I have never shown real name in any social media.

Avnish Kabaj

Gimme some time I found out 2 users from their name

Captain Bohemian

I also never show personal pictures there.

user351417

@AvnishKabaj Nah, they're probably wrong.

Anonymous

@Chair Found it (I think) :P

Student404Mus

@blue I need help in entanglement measure "concurrence", how to understand it? I explored some articles but without self-confidence!

Anonymous

1:26 PM

9 mins ago, by Chair

Yeah. I refused to do instagram. Interestingly, searching online for "rishi acharya facebook" doesn't get anything related to me.

Anonymous

BTW I guess there's a missing "i" there?

user351417

@Blue Eh that one's easy. It used to be on my SE profile, and I removed it because it's lame.

user351417

@Blue Heh, no. The Achairya bit was because I thought it was funny a while back.

PolarBear

Chair is a high school student?!

user351417

No U

user351417

1:27 PM

But yeah :P

Anonymous

@Chair Oh, LOL! :P

Anonymous

Missed that pun.

user351417

@Blue It's a reasonably common last name.

Avnish Kabaj

Ah

Nice

Anonymous

Yeah, I have a few friends with Acharya as their last name.

Anonymous

1:29 PM

Not Achairya, though. :P

Avnish Kabaj

is this you

That was a logger link

Anonymous

4 mins ago, by Blue

@Chair Found it (I think) :P

Anonymous

Ninja'd, by 4 mins. ;)

Avnish Kabaj

@Blue shit

I felt that was unethical

user351417

I'm really regretting not using my real name on SE. I remember I created it with my real name, and then I changed it before I became moderately active.

Avnish Kabaj

1:31 PM

And I used it wrong I guess

PolarBear

@AvnishKabaj When did you start using logger?

Avnish Kabaj

@PolarBear When I realized you used one on me an year ago

And made me feel bad for calling you a hacker

user351417

What is logger even?

Avnish Kabaj

Ip adress

Tracker

Bad stuff

PolarBear

@AvnishKabaj I was going to say I don't remember but now I do. That was funny xD

You were like, ahha no one can find me

Avnish Kabaj

1:33 PM

How do you use it though

PolarBear

But, I found out your real name before that

Captain Bohemian

i really hate to use real name in any social media.

actually I also don't like to use real name in my email showing name.

PolarBear

@AvnishKabaj Just for location and scaring you

Anonymous

@CaptainBohemian Is there any place where you like using your real name? :P

Captain Bohemian

@Blue in real world interaction.

user351417

1:35 PM

(Please don't say quora)

PolarBear

I got locked out of my Quora account and couldn't log in because my name wasn't real .-.

Captain Bohemian

if a social media only has contacts of my real world friends, I may not mind using real name.

Anonymous

@PolarBear Send them a passport with Polar Bear written on it...along with a nice picture.

Anonymous

Anonymous

Just found this. :P

PolarBear

1:43 PM

@Blue Yeah, totally :P

Captain Bohemian

@Blue he is a Taurus.

@PolarBear really? Quora requires a name which looks real to log in?

I think Quora has become more degraded. It keeps sending irrelevant notices to me.

Avnish Kabaj

very difficult

Who lives in greece

Yo

@Chair you clicked

Hahahha

I has ip

user351417

Fine.

user351417

Wait, have you been spending all this time subjecting my name to search engines?

Avnish Kabaj

1:59 PM

Yes

ACuriousMind

2:44 PM

@Chair you know you can change it again if you want to, right?

user351417

@ACuriousMind Lol yeah, but it becomes a mess with other answers which mention mine and comments which are directed towards me. I've seen other people with changed usernames, and it's a mess for future readers.

Anonymous

4:46 PM

On teaching mathematics is a lovely essay by Arnold.

ZeroTheHero

7:07 PM

is there an easy way to make a global change to some posts? I have in mind that the name "Bernoulli" is incorrectly spelled in a number of questions and answers as "Bernouilli", with an extra "i" between the "u" and "l".

ACuriousMind

@ZeroTheHero nope, there isn't

ZeroTheHero

@ACuriousMind Hmmm... While I don't think it's worth going through all the posts and kicking them back up the queue just for this, the spelling bug is not good for searching.

ACuriousMind

Search with Google or some other search engine instead of the inbuilt search, they usually are fuzzy enough to not care about the spelling mistake

ZeroTheHero

good point.

ZeroTheHero

7:27 PM

@ACuriousMind I meant to tell you: the term " covariant representations of the Poincare group" is likely related to finite dimensional representations. Something very very similar is used in : "General principles of quantum field theory" by Bogolubov et al, Section 7.2.D, and in "Mathews, P. M. "Covariant fields: Poincaré group representations and metric structure in the space of quantum states." Pramana 3.4 (1974): 261-276."

ACuriousMind

I find it strange they wouldn't just say "finite-dimensional representation" but I guess making up new terminology is fun :P

ZeroTheHero

Agreed. Maybe there's a subtlety I have not quite found about the nomenclature but I think it more likely fell into disuse. Mathews' paper is 1974 and although Bogolubov is "more recent" (1990) it is based on older material. I suppose this kind of nomenclature just didn't "stick".

Captain Bohemian

7:42 PM

what on earth is a pomeron? Wikipedia says pomeron is a Regge trajectory. I don't exactly know what it means. Are pomerons scattered particles themselves or (force mediating) particles exchanging between scattered particles?

Sir Cumference

8:58 PM

@Blue Sadly it seems like time and time again, essays urging a reform of math education go unnoticed :/

Understandably though. The only ones who honestly should be teaching math are those who have a passion and understanding of it. Obviously there are not a sufficient number of them to teach classes, which causes fewer students to develop passions, etc.

Seems like there's a cycle going on

DanielSank

9:21 PM

Is there a trick for finding eigenvalues of things like $$\sigma_z \otimes (\sigma_x + \mathbb{I}) \, ?$$

@EmilioPisanty @ACuriousMind

ACuriousMind

@DanielSank The eigenvalues of a tensor product are all possible products of eigenvalues of the factors

DanielSank

Well heck, that's convenient.

That is absolutely convenient.

ACuriousMind

wait

might be too convenient in that it's not true :P

DanielSank

I think you're right

Kronecker product

In mathematics, the Kronecker product, denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors to matrices, and gives the matrix of the tensor product with respect to a standard choice of basis. The Kronecker product should not be confused with the usual matrix multiplication, which is an entirely different operation. The Kronecker product is named after Leopold Kronecker, even though there is little evidence that he was the first to define and use it. Indeed, in the...

ACuriousMind

Ah, no it's true :)

DanielSank

9:25 PM

I'm trying to find the modes of $$\left( begin{array}{cccc} -i \omega_1 & 0 & -i \kappa & 0 \\ 0 & i \omega_1 & 0 & i \kappa \\ -i \kappa & 0 & -i \omega_2 & 0 \\ 0 & i \kappa & 0 & i \omega_2 \end{array} \right) \, . $$

I'm trying to find the eigenvalues of $$\left( \begin{array}{cccc} -i \omega_1 & 0 & -i \kappa & 0 \\ 0 & i \omega_1 & 0 & i \kappa \\ -i \kappa & 0 & -i \omega_2 & 0 \\ 0 & i \kappa & 0 & i \omega_2 \end{array} \right) \, .$$

dmckee

@CaptainBohemian They are one of the theoretical object that prodded me to coin the term notaton for any partilce-like theoretical object not believed to represent an actual physical particle.

DanielSank

I think that's $$-i \kappa (\sigma_z \otimes \sigma_x) -i \omega_1 (\sigma_z \otimes P_0) - i \omega_2 (\sigma_z \otimes P_1)$$ where $P$ means a projection.

So I guess $-i \kappa (\sigma_z \otimes \sigma_x) -i \omega \sigma_z \otimes (\omega_1 P_0 + \omega_2 P_1)$.

I guess that's not just one tensor product. Darn.

If If anyone wants to throw that matrix into Mathematica I will send you appreciative thoughts.

Oh wow, Wolfram Alpha can do it.

rob

10:30 PM

I'm listening to this podcast and the last segment is an interview with an "Anders Sandberg." I thought, "why is that name so familiar?" It's because they were interviewing Anders Sandberg about his arxiv publication in response to this deleted question here.

Anonymous

@rob Hah. :)

Anonymous

I actually like those what-if questions. If only we could protect them from spam and the HNQ effect...

dm__

perhaps obvious question, but I'm confusing myself. if a hamiltonian of two spin particles consists of a spin-spin interaction, $\vec{S_1} \dotp \vec{S_2}$, how do I immediately see that this is invariant under rotations? It seems like the generator of rotations would be $\vec{S} = \vec{S_1}+\vec{S_2}$, which doesn't commute with that spin-spin interaction, does it?

DanielSank

Wouldn't the rotation generator be more like $S_1 \otimes S_2$?

dm__

yes, sorry, I'm using sloppy notation. $\vec{S} = $\vec{S_1}\otimes 1 + 1\otimes\vec{S_2}$

rob

10:42 PM

@Blue I kind of feel the same way. Maybe now that we have some control over the HNQ effect we can sometimes allow them to stay.

DanielSank

Hahahaha, look at this user's profile page

Jesus Christ

1

entropy statistical-mechanics thermodynamics

"Jesus Christ - Apparently, this user prefers to keep an air of mystery about them"

LOLOLOLOLOL

@dm__ Oh, yes, that. Not what I wrote.

$$G = S_1 \otimes 1 + 1 \otimes S_2$$

Hmm, I guess it is reasonable to write that as $S_1 + S_2$...

rob

@DanielSank That's just the default profile text if you don't enter anything. If it amuses you, perhaps make an account with the name "Agatha Christie".

dm__

mitigates the clutter, but can sometimes be confusing

Anonymous

@DanielSank Jesus' answer was downvoted? Shame on us. O:)

Anonymous

>

If in a free-fall or when not acted upon, all gas molecules travel in one direction in uniformity just like everything else does. When free-fall ends Gravity arranges matter in a known order with an order using Bubble sort method and that's how invisible Lord (Gravity) creates stars by an order. By the way, Holy books (Quran and Bibles) are correct about how everything obeys the orders of an invisible Lord who created the stars with an order. Only Gravity calls itself Al-Lah and has been trying to educate humans by sending these messages.

dm__

10:49 PM

with the canonical commutations for $S_1$ and $S_2$, correct me if I'm mistaken, don't we have $[\vec{S},\vec{S_1} \dotp \vec{S_2}] \neq 0$. meaning, the hamiltonian is not invariant under rotations?

sorry, just got my render to work, seeing now the above errors.

dm__

11:05 PM

I suppose just rewrite $S_1\cdot S_2 = \frac{1}{2}(S^2-S_1^2-S_2^2)$, and then it commutes with all of the rotation ops.

Transcript for