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08:09

0

Q: Questions about physics teaching pedagogy

Are questions that relate to how physics is being presented in teaching on topic in this site?

skullpatrol

08:28

> a french mathematical text that does take this attitude (of rigour) to its extreme, by apologising at the beginning of the book for using the natural numbers to label the pages before they have been defined.

John Rennie

@skullpatrol Bourbaki?

skullpatrol

that was my guess also @JohnRennie but I couldn't find anything on the internet supporting it?

John Rennie

I couldn't find anything either. I suspect it's one of those modern myths :-)

skullpatrol

:-)

thanks for trying, sir

Slereah

09:09

morning

dsm

09:32

Hi there @Slereah. I'm trying to show that if two representations $(\Pi_1,V_1)$ and $(\Pi_2,V_2)$ are equivalent then there exists appropriate bases $\mathcal{B}_1$ and $\mathcal{B}_2$ such that $[\Pi_1(g)]_{\mathcal{B}_1}=[\Pi_2(g)]_{\mathcal{B}_2}$. I'm confusing myself. If they are equivalent, there exists and intertwiner $\phi:V_1\rightarrow V_2$ such that

$$\phi\hspace{1mm}\circ\hspace{1mm}\Pi_1(g)=\Pi_2(g)\hspace{1mm}\circ\hspace{1mm}\phi\hspace{3mm}\Rightarrow\hspace{3mm}[\Pi_2(g)]=[\phi][\Pi_1(g)][\phi]^{-1}.$$

To have $[\Pi_1(g)]_{\mathcal{B}_1}=[\Pi_2(g)]_{\mathcal{B}_2}$ doesn't this just require $[\phi]=1$, in which case the basis $\mathcal{B}_2$ should be the same basis used for $\mathcal{B}_1$ represented in the standard orthogonal basis of $V_2$? Is that correct?

Novalium Company

09:48

I just learned that the embryo at the beginning (during segmentation) divides into 8, 16, 32, 64 parts... is there any connection to why my computer can be 32 bits, 64 bits...? xD

Slereah

@NovaliumCompany The number two

Novalium Company

tf?

Slereah

Each cell divides in two

So one cell divides in two

Novalium Company

exponential division?

Slereah

Then each of these two cells divide in two, giving you four

Yes

At step $n$ you get $2^n$ cells

Novalium Company

09:50

Is there any connection to why my PC is 64 bits?

Slereah

And @dsm I don't know

@NovaliumCompany Yes, because your computer works in binary, so that an address must be in the $2^n$ range

Novalium Company

Yeah, I need to review my computer stuff knowledge.

dsm

10:18

@Slereah No worries, I imagine you're sick of me asking about representation theory XD

Slereah

Well I'm not super good at it

I mostly know it in the ways it relates to physics

dsm

I feel you. I'm looking forward to grounding everything back in physics here pretty soon. I've been roaming around the land of maths for too long, but telling myself it will be worth it.

Slereah

10:36

it is useful to know, yes

bolbteppa

@skullpatrol that's an incredible thing if true, apologizing for the page numbering :p

Slereah

If you wish I'm currently trying to rewrite what I remember of effective theories for mesons

samuel-lereah.com/articles/Physics/…

bolbteppa

10:51

What's a meson

Slereah

It's me, son

5

dsm

XD

Slereah

Mister string theory too busy looking for what a particle is to remember what two particles together are called!

dsm

That's cool to see some representation theory in practice, although I can't fully appreciate the physics quite yet.

Slereah

yeah it's a nice way of using it for something useful beyond the usual rep theory of the Lorentz group

You can do nuclear physics with it

With protons and neutrons as vectors in $SU(2) \times SU(2) \times SU(2)$

Well, in the associated vector space

It beats doing QCD

dsm

10:57

Effective theories sound fun, simplifying analysis by going to a general theory with the same symmetries. Sign me up

Slereah

Then you get the $p + n \to n + \pi^+ + n \to n + p$ diagrams and whatnot

dsm

I genuinely cannot fathom why anyone would willingly put their home under surveillance. Why are people purchasing Nest's and Ring's in droves? Blows my mind. Surely people have some underlying feeling of where this can lead, or at least I hope they do.

bolbteppa

Did you know that a Harmonic oscillaror $m\ddot{x} + k x = 0$ having solutions like $A \cos (\omega t)$ with $\omega = \sqrt{k/m}$ have the same frequency as systems whose new mass is $m = 1/k$ as long as the new $k$ is $k = 1/m$? The Hamiltonian however needs to be transformed by a canonical transformation of the form $x \to p$ and $p \to - x$, and that this is an example of T duality?

Slereah

I did not know!

bolbteppa

Very similar to EM being invariant under $E \to - B$ and $B \to E$!

dsm

11:04

Let me rephrase, surveillance that is not solely owned by the homeowner.

Slereah

Hm

How to write Young tableaux in HTML

I may have to draw them

IIRC for mesons it's like $$\Box \otimes \bar{\Box} = \Box \bar{\Box} \oplus \overset{\Box}{\bar{\Box}}$$

which is about the same as $$\psi_A \chi_{\dot{B}} = \psi_{[A} \chi_{\dot{B}]} + \psi_{(A} \chi_{\dot{B})}$$

bolbteppa

Is it the other way around

Slereah

pcteserver.mi.infn.it/~ferrera/teaching/young_tableaux.pdf

Ah, there's the rules

Conjugate rep is a column of $N$ boxes

So two box x one box

Slereah

12:56

Argh

The action of SU(2) on $\mathbb{C}^2 \times \mathbb{C}^2$ is a big ol' matrix

Johan Liebert

13:52

5

Q: Questions for Space Exploration Mod Nominees

If anyone has any questions for the Space Exploration mod nominees, please post them below in an answer and the candidates can edit in their responses.

discussion featured election

Similarly "As a representative of Physics.SE, what would you want the council to know about or take action on?"

Noah P

14:07

In general, is it possible for satellite galaxies to have satellite galaxies themselves?

Knight

14:22

@JohanLiebert I didn't get that

Slereah

just kill me

PM 2Ring

Wow. Look who's back:

3

A: "Gauge Freedom" in GR

Only diffeomorphism invariant quantities are measurable. The metric does not have this property. ''every observer has a local proper time that he/she can measure on their clock, and proper distances that they can measure with a meter stick much shorter than the radius of curvature. If I insist o...

Slereah

wasn't he banned for a million years

or am I thinking of someone else

JMac

14:37

@PM2Ring Looks like he never completely left. He seems to have fairly steady activity over the past few years, though perhaps slowed down in 2019.

Johan Liebert

Elections are going on on space.SE and one of the topic that was raised (asked to the nominees) was that "As a representative of Space.SE, what would you want the council to know about or take action on?"

So on a similar note I wanted to ask what our mods be representing there (though the question without context is a bit vague). I just wanted to see if anyone was willing to discuss about the working of the site and if there was something to improve that we might be willing to let our mods know to represent there (though I think it is better suited for meta).

@Knight

@Slereah I think you are talking of Ron Maimon!

PM 2Ring

@Slereah I don't think so, but it was before I was active here

JMac

@JohanLiebert You should try to post it as a meta question on physics as well. That way people can organize their opinions and we can actually see what the community thinks about it. Not to say we shouldn't talk about it here, just that it probably should be a meta question too.

PM 2Ring

I think Arnold just got sick of the fighting & ego clashes, etc, and reduced his activity before the suspensions on a few persistent trouble-makers came into effect.

JMac

14:54

@PM2Ring His profile says he's mostly active on PO. Given that PO seems dead, I assume he just checks PSE occasionally to answer questions, but doesn't participate too much besides that anymore.

Knight

15:05

@JohanLiebert Are you standing for Public Office on Space.SE?

Johan Liebert

@Knight no I am not (neither I am willing to nor do I have enough reps). But Yuvraj Singh... is.

Knight

@JohanLiebert Okay! How you doing these days ?

Johan Liebert

@Knight I am fine but my exams are going on.

Knight

@JohanLiebert What exams?

Johan Liebert

@Knight annual exam

Knight

15:17

@JohanLiebert Oh! How about Abhas?

Johan Liebert

@Knight he is preparing for jee so I don't chat with him much (FB is disturbing).

Knight

@JohanLiebert Oh! JEE would make him even more mad LOL (if going beyond that is possible)

Johan Liebert

15:39

@JMac

JMac

:53586216

Johan Liebert

I have made this post and if feel anything should be improved you can edit it.

0

Q: Moderator council- our concerns

Community working group updates: February 2020 In this post the Community Team says: Additionally, in a recent blog post, we mentioned that we are forming a moderator council. We have determined the structure of the council, but are still working out all of the details. We are going to kick ...

discussion

Knight

15:57

@JohanLiebert Aren't you preparing for JEE ?

Johan Liebert

@Knight yes but I am in 11th.

Knight

What's that?

Johan Liebert

@Knight ???

Knight

What is 11th?

Johan Liebert

Class

11th grade

PM 2Ring

16:00

@JohanLiebert As far as I understand, the purpose of the moderator council is to handle issues related to the community of moderators across the network, and how the moderators communicate with the company. It doesn't have much to do with regular members, apart from stuff relating to policies and practices of how mods interact with the regular members.

Johan Liebert

@PM2Ringp oh! But the space.SE community doing the same so it gave me the impression that it's for that purpose.

PM 2Ring

One of the main reasons behind the creation of this council is to deal with problems that have arisen in the moderator community. In particular, many mods have had bad experiences in the mod chat room, known as the Teacher's Lounge. As one mod put it, "when everyone in the room has mod powers, nobody has mod powers". So bad things can happen, and nobody in the room has powers to stop them.

@JohanLiebert Yes, there's also some confusion about that on Meta Stack Exchange. Hopefully, that will be clarified soon.

Mods have other ways to communicate with each other. There's a special Team for mods, and the mods on most sites have a chatroom or 2 where only mods can go. But hopefully, the new mod council will be able to act like "super-mods" when the need arises to stop bad things happening in moderator interactions.

I'm not a mod, so I don't know much about it. I'm just going off what I've read on MSE, and in the Tavern, the main MSE chatroom, where several mods hang out.

Oh. This looks significant:

21

Q: The company’s commitment to rebuilding the relationship with you, our community

My name is Teresa Dietrich. About a month ago, I joined Stack Overflow as Head of Product and Community. During my years as an engineer and technology leader I saw the impact this community and site has on people across the world and I am very excited to be here. While I have personally gotten a ...

discussion featured community policy announcements

Aaron Stevens

16:37

7

Q: Pepper shaker mystery

So I have this pepper shaker made of glass with a print on it: One fine dinner, it ran out of pepper, so I opened the lid to fill it up and noticed a peculiar thing – small particles of pepper dust were stuck to the inside glass walls but only in places where the print wasn’t present on the ot...

electrostatics home-experiment humidity glass adhesion

I have always felt odd about questions like these. I feel like most of the time all that is possible are speculative answers.

peterh - Reinstate Monica

Wow. There is no question tagged both by [general-relativity] and [electricity].

Sad. Afaik, it would be the General Electricity

JMac

@AaronStevens I'm usually on the fence about those ones. On the one hand, it's not really clear if there is a good non-guessing explanation for what is happening based on the limited information. On the other hand, if there is some clear physical reason why this happens, it can often be a really interesting result that could actually be helpful for site traffic; and I also usually find those answers quite interesting.

Aaron Stevens

@JMac Right, I always think the same thing. I guess as long as there aren't any speculative answers posted then it should be fine. I think an answer to the question needs to be sure that what they are proposing is what is actually happening. I just don't think the OP has given (or can even give) enough information to allow that to happen

PM 2Ring

16:55

I hope we get some more info from the OP. But I agree that even if we do get more info, most of the answers will be speculative.

But if it does start getting answers, it will surely hit the HNQ. Maybe we should edit the title to be less click-baity...

Aaron Stevens

@PM2Ring Already done, I think :)

I was thinking the same thing though. Everyone loves a good "mystery"

Well, the question was already tweeted. Only a matter of time before HNQ now

PM 2Ring

@AaronStevens Not yet, but it got Tweeted. See physics.stackexchange.com/posts/531884/timeline

peterh - Reinstate Monica

@PM2Ring Afaik de facto the SE has mod powers in that room. The problem arised partially from that the SE misused his power.

Aaron Stevens

@PM2Ring Sorry, I meant I already edited the title

PM 2Ring

@AaronStevens Ah, right. I had to refresh the page...

Loong

16:59

@PM2Ring Although I am still a mod, I haven't heard anything about that council except one strange announcement without any content "We'll be kicking off the council project soon".

So I am a bit surprised by these rumours on meta.

peterh - Reinstate Monica

Wow, it seems the company won't kill the SE. Probably they will kill some sites by merging them into the SO.

PM 2Ring

@peterh-ReinstateMonica Highly unlikely. I really don't know how rumors like that got started, but it seems that some people have misinterpreted various statements from SE Inc in the worst possible way.

Eg, SE Inc said stuff about creating better cross-fertilization channels between StackOverflow, SuperUser & ServerFault. Then some people thought that meant that SE Inc want to merge those sites.

peterh - Reinstate Monica

@PM2Ring 1) Large part of the community believes, SE is working quite closely to the always worst possible way, long ago. 2) In the numerous recent company statements, was the SE ever mentioned (except that they want to merge some sites into the SO)?

@PM2Ring There is a de facto migration blockade since the start. I mentioned on the metas, many, many times, that migrations should be so easy like re-tagging, I was always voted down (and closed/deleted). Now they want to change their mind? Well... I don't believe.

PM 2Ring

@peterh-ReinstateMonica What's migration got to do with it? There are good reasons why migration is discouraged. One proposal for better cross-fertilization is that when you post a question, not only do you get Related questions from the same site shown in the sidebar, you could get related questions from other sites too.

peterh - Reinstate Monica

@PM2Ring The problem of the SO is that it is too big. For example, I have zero experience with .net . If I would randomly pick 30 .net questions, and initiate their closure, most of them would be closed by robo-close-voters (and the rest would timeout). I would not ever do that, but I think, roughly this is the reason, why the reviewing does not really work on the SO.

@PM2Ring Uhm, so they will go further on the do-nothing way. Wonderful.

@PM2Ring Btw, what I fin quite surprising in the stats... many of them clearly show a close apocalypse. Like the probability that a question ends up with positive score, with a positively scored, accepted answer. Its probability is today below 5%.

PM 2Ring

17:17

Anyway, there's not much point talking about SO in this room. Unless it's about stuff that affects all sites, or at least has some impact on Physics.SE.

peterh - Reinstate Monica

@PM2Ring But other stats are still good and stable, even on the SO! Like the new user registrations. Or the count of the votes.

@PM2Ring It is more about the SE. And the PSE is an SE site.

@PM2Ring Btw, for example the ratio of the well-received questions is surprisingly high on the PSE.

PM 2Ring

@peterh-ReinstateMonica Lots of new members post bad questions. That's not a new phenomenon. Part of the problem is that Stack Exchange sites don't work like forums, the rules & culture are different, and it's hard for newcomers to learn all the details. But a huge part of the problem is that so many newcomers simply do not bother to even take the Help tour before posting, and make no effort to find out how to create an acceptable question.

Aaron Stevens

The question is if the "close apocalypse" is due to actual bad questions or just many "trigger happy" users

PM 2Ring

A few years ago, when newbies posted bad questions, some regulars would post comments telling them what's wrong with their question, and what they need to do to make the question suitable. But these days, far fewer people post such comments because they don't want to be flagged as unwelcoming. So they just downvote & close-vote instead, and go onto the next question.

Sure, some people may be too trigger-happy. And when you see a stream of bad questions it's easy to see a so-so question in a bad light, and give it harsher treatment than it might deserve.

Aaron Stevens

17:33

I don't have any experience with other SE sites, but I feel like PSE does a good job at closing posts that should be closed, and even reopening questions that shouldn't have been closed.

PM 2Ring

Eg, if you see 9 blatant homework dump questions, followed by a so-so homework question that could turn into an acceptable conceptual question with a little bit of work, there's a tendency to just close-vote that 10th question, rather than to help the OP massage it into shape.

@AaronStevens I agree. PSE probably has the strictest homework policy. Sometimes I think it's a little too strict. But despite that policy, every day there's a constant stream of homework dumps from newbies who don't bother to learn how to ask an acceptable question. And with some of those newbies, you could show them the homework policy, written in flaming letters 19 feet high, and they still wouldn't get it. ;)

Aaron Stevens

@PM2Ring Haha yeah, I agree... I am probably one of the more stricter users in terms of the homework policy though. Lately though I have felt like many users (not asking the qustions) skip over the "check-my-work" part. I feel like more questions like those slide on by. I think it is because a question like that shows effort from the OP, so it is harder to close them

Johan Liebert

Chemistry and biology SE community is stricter than us!

They are quite eager to downvote and we are quite eager to upvote!

PM 2Ring

@AaronStevens Yeah. It feels kind of mean when they go to the effort of writing a clear question, with equations in MathJax. And sometimes, such questions aren't actually asking to check for mistakes in the algebra or arithmetic. They want to know if they made a conceptual error that caused them to use the wrong equation, or misapply the right equation.

Aaron Stevens

I will probably sound like a jerk, but I think it is better to be strict. But only because it is clear that PSE is not intended to be a homework help site. I feel like a good "homework question" should be one where the actual homework question becomes irrelevant when compared to the more general question better suited for PSE.

There are many other resources students have for homework help, plus I do my own tutoring, so I don't feel too bad turning students away from PSE in terms of getting homework help

Johan Liebert

17:50

Usually I refer these OP to our meta with the following template

PM 2Ring

That's fair enough.

Johan Liebert

Welcome to Physics.SE. We are pleased when new members join us though we don't answer homework type questions and note that even if someone answers this the moderators or the community members might just delete them so why not try somewhere else or just ask a question which is related to the concept in the problem that you have trouble in understanding.

Aaron Stevens

The "we don't answer homework type questions" is a little misleading, because there are certainly "homework-based" questions that are actually on-topic for the site

Novalium Company

MORGEN

Johan Liebert

@AaronStevens so what should I say instead that might be more appropriate?

Aaron Stevens

17:54

@JohanLiebert I think the homework close banner does a good job at explaining it well.

For the most part that is why I just tend to VTC off-topic homework questions and move on. If the question gets closed then they will know why and have resources to do better next time. If it doesn't get closed then that information becomes somewhat irrelevant

Although I will still say something if I am feeling nice at the moment :)

Johan Liebert

@AaronStevens but the banner feature was removed and now it just says:"Closed. This question is off-topic. It is not currently accepting answers."

Aaron Stevens

@JohanLiebert The OP and high-rep users can see the entire banner

It is a new feature. The banner used to be visible to all

Johan Liebert

@AaronStevens ah! That's why I never saw that. Thanks.

Aaron Stevens

5

Q: New notice banners

From I have found that new notice banners have started appearing on questions. Some examples are: But what was the reason for this new look? Why aren't they question closed as homework and exercise type show the specific reason? Any other relevant information are welcomed here!

discussion support

JMac

I usually just say "Please note that homework and exercise questions are often off-topic. Please read the homework and exercise policy." Or something along those lines.

@JohanLiebert FYI, the timeline workaround works for that too. You should still be able to see who closed it if you check the timeline, it just doesn't appear on the banner now.

Johan Liebert

17:59

58

A: New Post Notices are live network-wide

status-deferred The old close notices had their own share of issues that made it desirable to redo them. But publicly displaying the close reason on a question provides an additional benefit of explaining to others why a question is unsuitable in its current form. Custom close reasons used to s...

It has been give status-deferred. Why? I mean if OP is able to see the close banner then shouldn't it be given status-completed?

ACuriousMind

18:15

@JohanLiebert That answer is asking to show something more informative to others, not to OP.

zabop

18:42

hey

my notes are telling me that

although wikipedia (and other sources) confirm that

so, what is sigma squared in f(x) is hbar lambda in psi

so delta x is sqrt(hbar lambda)

but it is supposed to be sqrt(hbar lambda over 2)

where is the 2 coming from?

Aaron Stevens

I think you need to check the normalization.

f(x) is normalized to 1, but we don't normalize psi to 1

i.e. the distribution that determines delta x is not just psi by itself

(I am assuming this is for QM)

When you look at psi* times psi, you will lose that 2 in the denominator of the exponent. Therefore, you pick up a 1/sqrt(2) in the standard deviation

Since psi* times psi is your probability distribution

@zabop I can give more detail if needed

zabop

ohhh thanks I think I see it now

because

psi* times psi is exp(-(x-<x>)^2/(hbar lambda) times exp(some function of p which we dont care right now)

Aaron Stevens

18:57

Well the part with p will go away, actually

zabop

which is equal to exp(-1/2(x-<x>)^2/(hbar lambda / 2)

Aaron Stevens

Right :)

zabop

@AaronStevens ohhh yep, it indeed does

and than we identify sigma ^ 2 as hbar lambda / 2, and we are done

great

Thanks!

Aaron Stevens

It all works out haha. You are not going crazy!

zabop

yep! :)

peterh - Reinstate Monica

19:21

@AaronStevens I regularly check the site stats on the SEDE. If you wish to I explain the details. My opinion about the results is that the site has stabilized itself in a far better state than the SO is now. For example, more than 10% of the questions ends up open, not deleted, positively scored, with a not deleted, positively scored, accepted answer.

@AaronStevens The same ratio on the SO is far below 5%, and quickly decreases. It seems, the community will completely vote out itself from its own site. On the PSE, it is stable (except the yearly 2 wave due to the academical year).

@PM2Ring They did it because they never heard from the site until now. They found it by google. And, if you see their algorithm from the effort/result view, they are doing it quite well: they see little chance for a result, so they invest the least possible effort. These people are working pretty rationally, despite that the major problem of many of them is not a physical one, but that they can't communicate in written form.

peterh - Reinstate Monica

19:41

@PM2Ring Who is not working rationally, that is the site. We. We should give them a playground, where they could ask each other their problems. If a question evolves with subsequent edits, then it should go to the main site. If not, it could left to rot, out of the view of the google. The result would be that a little part of these homeworkers would evolve, to produce replesnishment for the site community. Instead, the site expels them.

skullpatrol

Let the "playground" be created chatrooms?

JMac

@peterh-ReinstateMonica Who is going to monitor the playground? Who is going to decide when it goes to the main site? How do you know such a system would evolve into something good, and not a total confused mess?

skullpatrol

all i know is expelling people doesn't work

it sows the seeds of "sour grapes"

imho

JMac

@skullpatrol The only people who you could really say get "expelled" are people who get suspended for continually breaking rules or posting off-topic. In my opinion, often times the "sour grapes" happen as soon as the site doesn't react the way the new user wants everyone to react. Then they blame the site instead of just taking the time to understand the community and it's goals.

ACuriousMind

19:56

We're not "expelling people". We're rejecting posts that we consider off-topic. Unless they continually ignore our rules in blatant ways, everyone is welcome to come back when they have an on-topic contribution to make.

By this line of rhetoric, we'd also be "expelling" a lot of people who really want to ask about e.g. sports (because that's also off-topic here).

It is hard to see how that's not working out quite well :P

JMac

@ACuriousMind So I should probably stop writing the draft of my first PSE question "Hey guys, what are your favourite sports teams?"

ACuriousMind

@JMac If you make it "What were Einstein's favourite sports teams?", we could at least fight over whether or not to migrate that to History of Science and Mathematics :P

JMac

@ACuriousMind I was just going to start it by saying "We all know that Newton said 'F = ma'. I was thinking about that, and I started wondering, what are your favourite sports teams?" I mean as far as I can tell the question would be directly related to physics.

skullpatrol

this "moderator council" thing looks promising

perhaps,, community mangers can appoint site managers and site assistant managers?

pardon the extra comma :P

ACuriousMind

20:11

You have a very different view of SE's actions than I do if you think investing in more community-managing focused employees is a part of their current strategies.

In any case, there's by far too little to do to justify a full employee devoted to a site - that's the benefit of moderation and curation being community-run for them

skullpatrol

i meant to deal with the chatrooms that are full of mods

and "when everyone in the room has mod powers, nobody has mod powers".

ACuriousMind

What does a "site manager" have to do with that?

skullpatrol

aren't chatrooms part of the "site"?

ACuriousMind

Sure, but the typical site has half a dozen mods, tops (and on some sites, not even all mods use chat). How much "managing" to you expect that needs?

skullpatrol

well, if they are fighting uncontrollably in the Teacher's Lounge and "when everyone in the room has mod powers, nobody has mod powers" who can stop them

who can they flag?

ACuriousMind

20:22

I don't see how that's an answer to my question. The TL is a single room, not associated with any site (and "fighting uncontrollably" is a debatable paraphrase for anything that's ever happened there). What's this to do with your "site managers"?

Loong

@skullpatrol That has never happened.

dsm

Greetings @ACuriousMind. Do you have any input on a question I have?

skullpatrol

@bolbteppa they could have avoided the apology by using the words: "one," "two," "three,"... until they defined them :P

ACuriousMind

@dsm 1. I don't really understand the argument you're trying to make there. If you try to write the bases explicitly for the $[\cdot ]$ in the steps from the l.h.s. to the r.h.s. after your "such that", maybe you'll see why it doesn't work.

2. If you know that the representations are unitarizable, just diagonalize the matrices via the spectral theorem :P

3. In the general case, think about what kind of matrix the intertwiner is.

skullpatrol

@ACuriousMind what are your thoughts on Einstein's interpretation of "empty space" that I posted in chat?

dsm

20:38

Doesn't work as in that's not the correct $\mathcal{B}_2$ basis? Because this is a provable iff statement. One example in favor of my same bases is the equivalence of the adjoint and vector representation of $\mathfrak{so}(3)$, where the adjoint basis is the same $L_i$ generators as the vector rep: $[ad_{L_i}] = L_i$. Let me ponder what you're saying for a bit. I'm using the intuition of similarity transformations to guide me on this, but maybe there are subtleties with the intertwiner

ACuriousMind

After reading it again, it may be that you're actually saying the right thing but I didn't parse it that way

@dsm What exactly does $[\phi] = 1$ mean to you?

There is no "standard orthogonal basis" here - representations are not equipped with an inner product by default so there's no unique notion of "orthogonal" - so that part of your argument definitely doesn't work out of the box

It may be instructive to ponder the following example: One representation is the standard fundamental representation of $\mathrm{SO}(3)$ on $\mathbb{R}^3$ with the $L_i$ generators associated to rotations about the $i$-axis, the second is the representation where they're associated with a cyclic permutation of the axes (i.e. $L_x$ rotates around $y$, $L_y$ around $z$, $L_z$ around $x$)

(Or the one where the second representation is the first one, just scaled by a constant factor $c$, i.e. $\pi_2(g) = c\pi_1(g)$)

dsm

20:56

@ACuriousMind That right there's the crux of it, interpreting $[\phi]=1$. Since $\phi:V_1\rightarrow V_2$, and these reps are connected by a similarity transformation by it, I've been interpreting its matrix in the same way as a basis transformation: $f_i=\phi_i\hspace{1mm}^je_j$, where $\{f_i\}$ is the $\mathcal{B_2}$ basis and $\{e_i\}$ the $\mathcal{B_1}$ basis.

This would imply $\phi_i\hspace{1mm}^j=e^j(f_i)$, and in order to be the unit matrix we'd need $e^j(f_i)=\delta_i\hspace{1mm}^j$. So I suppose what I'm saying is the $\mathcal{B_2}$ basis must be dual to the $\mathcal{B_1}$ basis. Would that make more sense? Pondering your example

ACuriousMind

$e^j(f_i)$ as an expression isn't clear to me either - $e$ and $f$ are just bases of two vector spaces of the same dimension, there's no notion of applying one to other

dsm

Hm, can we endow both vector spaces with the same metric? Then $e^j(f_i)=L(e_j)(f_i)=(e_j|f_i)=\delta_i\hspace{1mm}^j$, which says to me that these bases need to be the same

ACuriousMind

What does "the same metric" mean?

You have no fixed isomorphisms to some $\mathbb{R}^n$, either.

dsm

let's say Euclidean metric on both $V_1$ and $V_2$ for the same metric. How should I interpret your no fixed isomorphisms? Kind of confused by that. $\phi$ is an isomorphism between $\mathbb{R}^n$ and $\mathbb{R}^n$, but is that not what you're talking about?

ACuriousMind

My point is that you abstractly know that $V_1 \cong V_2 \cong \mathbb{R}^n$ as vector spaces, but you have no natural choice which vectors in the $V_i$ to map to the standard basis of $\mathbb{R}^n$.

So you can't really pick "the Euclidean metric" on $V_i$, there's an infinity of possible choices for that

dsm

21:13

Aren't we kind of forced to use the same metric by it being an isometry, so it wouldn't matter which one we pick? Haha I might be going in circles

Actually not certain about it being an isometry, but I'm guessing an intertwiner is

ACuriousMind

There are no metrics here, the intertwiner isn't forced to be an isometry between two arbitrary metrics on the spaces

dsm

Ah, ok, I'm gonna ponder these things. Just read the chapter today so this intertwiner is new to me. Do you have a hint on how this $\mathcal{B}_2$ basis is connected to the $\mathcal{B}_1$ basis to have equivalent matrices for the two reps?

I wrote a lot of things in trying to do the proof, but was going in circles (like above)

ACuriousMind

Beyond what I've already said, I can't think of anything that doesn't straight-up tell you the answer

dsm

So what does $[\phi] = 1$ mean to you? ;)

heuristically, no math necessary

if you so choose

Haha you don't have to answer that, I'll ponder it while trying to correct my terrible current sleep schedule. Thanks for the chat, cheers

ACuriousMind

@dsm Nothing, really. My problem with that notation is the following: $[\pi_1(g)]_{B_1} = [\phi\pi_2(g)\phi^{-1}]_{B_1}$ is true. But you seem to have "distributed" the bracket on the r.h.s., i.e. $[\pi_1(g)]_{B_1} = [\phi]_{???}[\pi_2(g)]_{B_2}[\phi^{-1}]_{???}$

This is kind of the right approach, but there isn't a basis you can write in for the $???$ because $\phi$ is a map between two different vector spaces, it's not expressed in a single basis.

dsm

21:32

Exactly my hold up, you nailed it on the head. My text has

$$\phi\hspace{1mm}\circ\hspace{1mm}\pi_1=\pi_2\hspace{1mm}\circ\hspace{1mm}\phi\hspace{3mm}\Rightarrow\hspace{3mm}[\phi][\pi_1]_{\mathcal{B}_1}=[\pi_2]_{\mathcal{B}_2}[\phi]\hspace{3mm}\Rightarrow\hspace{3mm}[\pi_2]_{\mathcal{B}_2}=[\phi][\pi_1]_{\mathcal{B}_1}[\phi]^{-1}$$

leaving me scratching my head on those ???'s

It being a map between two different vector spaces is really confusing me for some reason, as far as its matrix representation goes

Ok, forcing myself to close the laptop. Gotta wake up in 3 hours and it's going to be a long day. Perhaps I'll get a chance to think more about this basis relationship and will ping you to pick your thoughts. Cheers

Aaron Stevens

22:01

@JMac That would require critical thinking that this site is not ready for or equipped to handle

JMac

@AaronStevens Because users like you spend all your time on the site trying to prevent scientific progress.

(this is a joke for anyone who isn't aware of me and Aaron's conversations about this)

Had to write a disclaimer there, I felt like I took the joke too far into Poe's law territory and that a random reader might get the wrong idea.

Aaron Stevens

I tried to make a joke at my lab meeting today, but no one realized it was a joke. So it made me look like an amateur physicist.

Someone had an equation on their slide involving RT, where R is the gas constant and T is the temperature. I said "you know, you might run into an issue because you are using R instead of k".

I thought it was a well-known "joke" that physicists use k and chemists use R

But people started explaining to me that they had to use R because their energies are "per mile", and how they are really the same thing

So I had to explain that I knew that and I was just kidding

JMac

22:17

Yeah that would have gone right over my head. Most of my textbooks used R.

Aaron Stevens

@JMac Oh well. Maybe next time

skullpatrol

22:46

@JMac those are valid points.

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