The h Bar

Charlie

01:22

I've just seen something that has confused me, $SU(n)$ is the set of unitary matrices with $det|U|=1,U\in SO(n)$. But on wikipedia it is claimed that any unitary matrix has $det|U|=1$, so why is $SU(n)$ different from $U(n)$?

that should say $SU(n)$ not $SO(n)$

Slereah

unitary matrices have $|\det(U)| = 1$

So $\det(U) = e^{i\theta}$

Charlie

so $U(3)$ and $SU(3)$ are isomorphic?

Slereah

Well no, not quite

as an example

$U(1)$ is the circle

$SU(1)$ is $I$

Charlie

hmm

is there another property that $SU(1)$ has that $U(1)$ doesn't that distinguishes them? because I was under the impression that it was just the unitary determinant

but apparently they both have that property

Slereah

There might be something like $U(N) \approx S \times SU(N)$?

Charlie

01:27

is that the homeomorphism symbol?

Slereah

since for any element of $U(N)$ on a circle there corresponds one on $SU(N)$

No that's just me going "Eeeeh, maybe?"

Charlie

ah ok

Slereah

Lemme check the book of lie stuff

Unitary group

In mathematics, the unitary group of degree n, denoted U(n), is the group of n × n unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group GL(n, C). Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. For the group of unitary matrices with determinant 1, see Special unitary group. In the simple case n = 1, the group U(1) corresponds to the circle group, consisting of all complex numbers with absolute value 1 under multiplication. All the unitary groups contain copies of this group...

ah yes, there is a connection

Charlie

in this context do you mean connection as in there is a link or connection as in the connection on a manifold

I haven't really dug into lie groups yet, this was just something I saw offhandedly that surprised me

Slereah

$SU(N)$ is a subgroup of $U(N)$

so it's not terribly surprising that they both obey that property

Semiclassical

01:36

U(N)/SU(N) ~ S, maybe?

Charlie

what is S?

Semiclassical

circle group

Charlie

ah

Semiclassical

ah, yes:

1

Q: Quotient of unitary group by special unitary group

Can someone help me, I don't understand the following question. "Using without proof, the homomorphism theorem, or otherwise, show that $U(n)/SU(n)$ is isomorphic to $U(1)$." Here, $U(n)$ is the unitary group while $SU(n)$ is the special unitary group.

group-theory

Slereah

$S$ is, as the letter indicates, the sircle

4

Slereah

02:03

So I have trained a neural network on TeX formulas

It's a bit early in the training, but so far, here's a sample :

\begin{eqnarray}
m (\lambda_p, \lambda_0) &=& \lambda_0 \ldots &=& \mu_g(\lambda_p) \ldots\\
&=& \lambda_g(\lambda_p)
\end{eqnarray}
\begin{eqnarray}
m (\lambda_0, \lambda_g) &=& \lambda_g(\lambda_0) \ldots &=& \lambda_g(\lambda_0)
\end{eqnarray}
\begin{eqnarray}
\sigma^{-1}(\lambda_p, \lambda_0) &=& \lambda_0 \ldots &=& \sigma^{-1}(\lambda_p, \lambda_0)
\end{eqnarray}

Mysterious formulas

ZeroTheHero

@Charlie best seen using U(2) and SU(2)..

Take SU(2)... you can write such a matrix easily in the standard parametrization...

multiply all entries by some constant phase factor...

Slereah

\begin{eqnarray}
[\mu_{\mu\nu} (\varepsilon + \frac{1}{2}(a^a (x_2, e_1))] &=& |(\kappa^a|^2) \sigma^a (x_2, e_1)\\
\end{eqnarray}
\begin{eqnarray}
[\mu_{\mu\nu} (\varepsilon + \frac{1}{2}(\kappa^a (x_2, e_1))] &=& \varepsilon \sigma^a (x_2, e_1)\\
\end{eqnarray}
\begin{eqnarray}
[\mu_{\mu\nu} (\varepsilon + \frac{1}{2}(\kappa^a (x_2, e_1))] &=& \varepsilon + |\sigma^{a}|^2 (x_2, e_1)\\
\end{eqnarray}
\begin{eqnarray}
[\mu_{\mu\nu} (\varepsilon, \frac{1}{2}(\kappa^a (x_2, e_1))] &=& \varepsilon + |\sigma^{a}|^2 e_1 (- \varepsilon + \frac{1}{2}(\kappa^a (x_2, e_1))\\

ZeroTheHero

the unitarity property remains $M^\dagger M=$unit...

Slereah

Not bad for just a hundred cycles!

ZeroTheHero

but now det(M) is multiplied by square of that phase factor.

Charlie

02:09

but the det(M) is still 1 right?

oh wait

ZeroTheHero

Well if M is initially in SU(2) then Det(M)=1...

but if you multiply all entries by a phase then Det[phase M] = phase^2

Charlie

is that not no longer an element of SU(2) though?

ZeroTheHero

correct... it will an element of U(2) now.

it is still unitary...

although the determinant is no longer 1.

Charlie

oh

ZeroTheHero

in fact you can take any SU(2) matrix...

and premultiply it by a diagonal matrix containing phases...

it remains unitary but the determinant is no longer necessarily 1.

often one writes the previous comment U(2)=U(1)xSU(2)

where the U(1) is this diagonal matrix of phases.

I’m being quite loose here...

Charlie

02:13

But unitary matrices must have det(U)=1, is that not a property they have?

ZeroTheHero

NO...

they have Abs[Det[U]]=1

not Det[U]=1

Charlie

is this wrong then? en.wikipedia.org/wiki/Unitary_matrix#Properties

the fourth point

oh wait

the abs

ahhhhhhhhhhhhhhhhhhh

ok i see

ZeroTheHero

that’s easy enough to show:
Det[M^\dagger M]= Det[unit] = Det[M] Det[M^\dagger]

so it’s really Abs[Det[U]]=1 for unitary but strict Det[SU ]=1 for the special unitary

Charlie

thanks for clearing that up

ZeroTheHero

no worries.

Slereah

02:53

More neural network math

danielunderwood

You've just automated the jobs of the people who have to fill up blackboards in TV shows/movies!

Slereah

I mean so far it has some troubles doing it correctly

The syntax isn't great!

It does look sufficiently bespoke, though

danielunderwood

So what kind of model did you make that's doing that?

Slereah

I'm just using the standard GPT-2

I have to correct it a little bit by hand because so far, it's not perfect as far as syntax goes

a lot of it isn't proper TeX syntax

Slereah

03:20

knzhou

04:11

I've been through this!

Crammed my 500 pages of QFT notes into a Markov chain.

(with syntax minimally fixed)

Student404Mus

08:05

I couldn't understand why they are asking energy is frame dependent? physics.stackexchange.com/questions/538791/…

ACuriousMind

09:42

@Slereah Well, this is really hard for it! I mean, the HTML and other code it can generate usually looks convincing to a non-programmer but won't compile, no reason to expect it to do better with TeX

Slereah

@ACuriousMind You'd think it would be easier to learn than human speech, since it's a rigorous grammar!

ACuriousMind

Well, but if we think of a human, how many humans could learn to produce correct C code just from a bunch of examples without access to a compiler?

Especially if the examples themselves are sometimes full programs, and sometimes snippets that wouldn't compile on their own anyway

Slereah

All my examples compile!

Johan Liebert

10:40

Simulating an epidemic

►

Slereah

10:59

Perhaps a more interesting use of the neural network would be

Input a theorem, output a proof

And see what kind of proofs it writes up!

but doing a big dataset of proofs would be tedious

Plus they would need to be ideally all in the same style, notation and formalism

If I just dump a bunch of unrelated sources it wouldnt' be great

Using Metamath for this might be useful, actually, if I can manage to download all the proofs

Hm

I can probably do a siterip of just the proofs without too much troubles

Slereah

11:26

More mysterious robot math

MadSpaceMemer

Hello. how can i imagine the relationship between the Sin form of a wave which is to me fimiliar and the so called Wavefront drawn to be as circles of balls? is there an article or a video or a topic you can link me to to understand the Connection?

My understanding is that a Wavefront is the place where all the particles have the same phase. but it is always draw in distance from the next wave front. However should not the whole plane be a wavefront because if you consider a surface of water and a source generating waves then in distance R from this source always on a circle all the particles have the same phase right? because the waves spread in each

direction with the same speed. So what is the point of drawing these (wavefronts) in distanced intervals? it seems to indicate that the distance between the one wavefront and the other does not have particles that are in phase. But they are. or am i confusing something?

it would make more sense if the wavefronts are indicating to the place of the peaks

ZeroTheHero

13:27

@JohnRennie Don’t you mean: “The geeks shall inherit the Earth”. There was a typo in the original.

Novalium Company

13:41

Why in the f-ing world are people running out to buy toilet paper. Like, I see pictures of people running with full carts of toilet paper... is this the most important thing... Americans are weird sometimes

Loong

13:52

3

ACuriousMind

twitter.com/Nick_Craver/status/1243900025232601088

(if anyone's wondering about read-only mode)

John Rennie

14:54

@NovaliumCompany to be fair, defecting is one of the few human activities that is not optional and wiping your bottom afterwards is generally considered civil.

Loong

The nice part about this mess is that fuel is relatively cheap now and that there isn't much traffic when I drive to work.

John Rennie

Oops, just spotted the typo - defecting :-)

@ClaraDiazSanchez you're a science fiction fan as well! :-)

Aaron Stevens

2

A: Why are some energies dependent on reference frame, and some are not? And why is transfer between them possible?

What puzzles me, is: 1 - why are some energies relative to reference frame and some are not? 2 - why the "absolute" energy from gasoline can be changed into kinetic energy of car, and therefore change into "relative" energy? The reason is there are two types of energy of an object or s...

What is the distinction between "internal" and "external" frames of reference?

And what is the difference between "microscopic" and "macroscopic" kinetic energy?

I would ask on the answer directly, but I ask questions of this user a lot, and lately they either ignore my comments or seem upset by them

ACuriousMind

@AaronStevens Microscopic kinetic energy is the one responsible for temperature in the classical thermodynamic picture - Brownian motion.

Macroscopic kinetic energy is everything else kinetic

As for internal and external frames of reference, there is of course a natural preferred frame of reference in which to measure things like Brownian motion - the frame in which the thermodynamic system as a whole is at rest.

I'd assume this is the "internal" frame of reference, external frames are everything else

Aaron Stevens

@ACuriousMind I guess I don't like the distinction because kinetic energy depends on velocity, which doesn't have a micro/macroscopic distinction. It makes more sense to just say you are in a specific reference frame in my opinion

"internal kinetic energy" is just the kinetic energy as seen in the center of mass frame

ACuriousMind

15:07

@AaronStevens No, it's not

Aaron Stevens

@ACuriousMind I mean it is whatever frame you use to look at things like, say the temperature

I guess I feel like saying internal kinetic energy is reference frame independent is like saying time intervals in SR are relative except for the proper time interval.

ACuriousMind

yes, it's similar

Aaron Stevens

You have to assume a particular reference frame to say it is reference frame independent, which seems like an odd way to say things

ACuriousMind

and temperature/internal kinetic energy is even a bit worse, c f. physics.stackexchange.com/q/262382/50583

Aaron Stevens

Right, I know we get questions about temperature in different frames (with SR or not) every once and a while

ACuriousMind

15:10

I'm not saying the answer is perfect, but I think it's firmly within "standard" thermodynamics

In the end, all the different energy forms are just silly categories humans stick on things. The Lagrangian doesn't care.

Aaron Stevens

I suppose. But you could then say that about anything: "The kinetic energy I measure in this specific frame of reference is reference frame independent"

Really it is not unique at that point

Also, is there such thing as "external potential energy"?

If a conservative force is considered to be external, then technically isn't it not associated with a potential energy for that system anymore?

ACuriousMind

It's again the distinction between the potential energy the constituents of the system have w.r.t. to each other and the potential energy the system as a whole might have w.r.t. to external fields, much like with the kinetic energy

I think - and I sympathize! - that you just don't like thermodynamics :P

Aaron Stevens

@ACuriousMind Yeah I used to think that. But then I have seen a lot of questions/answers on PSE saying that it has to be part of the system

Yeah, it must be the isolation making me super nitpicky :P

Clara Diaz Sanchez

@JohnRennie guilty as charged

John Rennie

@ClaraDiazSanchez there are a lot of us around :-)

skullpatrol

16:05

our world is stranger than science fiction

John Rennie

16:27

@skullpatrol if you think the world is stranger than science fiction then you haven't read anything by Jeff Vandermeer! :-)

skullpatrol

:-)

Student404Mus

I need help understanding the Wigner-Eckart theorem

stating the lemma,

$F_{\kappa}^{(k)}=\int T_{\kappa}^{(k)}\left(\mathbf{r}_{1}, \mathbf{r}_{2}, \ldots\right) d \tau_{1} d \tau_{2} \ldots=0 \quad$ if $\quad k \neq 0$

$T_{\kappa}^{(k)}\left(\mathbf{r}_{1}, \mathbf{r}_{2}, \ldots\right)$ is an irreducible tensor of degree $k \neq 0$ which is a physical scalar field of the coordinates only.

If $T$ is of degree different from zero what makes sense of the above lemma?

ACuriousMind

I have no idea how a tensor operator is supposed to be a "physical scalar field" (or what that even means) or what the $\tau_i$ are you're integrating over...

Student404Mus

16:47

they are space variables

I know this is somewhat confusing even from the book I am reading from, the author introduces some comments to overcome ambiguities, but I didn't get it properly.

Definitions of irreducible tensors and tensor fields are striking

@ACuriousMind Could you tell me what to do to avoid ask ban?

Novalium Company

@JohnRennie You know what else is civil? Eating food, not toilet paper.

I'll develop the first edible toilet paper so you never get hungry while pooping.

ACuriousMind

@Student404Mus See meta.stackexchange.com/q/86997/263383

ZeroTheHero

17:04

@Loong source please?

Loong

@ZeroTheHero starecat.com/fortress-made-of-toilet-paper-coronavirus-drawing

ZeroTheHero

@Loong priceless. Thanks.

@Student404Mus avoid asking questions...

Novalium Company

@JohnRennie I understand your point but I feel like food is much more important than toilet paper so that's why I'm surprised to see carts full of toilet paper and not food.

Student404Mus

@ZeroTheHero Why?

ACuriousMind

@NovaliumCompany It's a bit of a volume thing - someone hoarding food can just buy like a dozen packages of flour, and that doesn't make a cart even remotely full. But the same hoarder buys more than one or two packages of toilet paper and then their carts is "full of toilet paper" :P

Novalium Company

17:11

I guess that's right

Student404Mus

I cannot understand why I am banned. All my questions are greened

ACuriousMind

I can assure you that e.g. noodles and flour are also being hoarded right now judging from the postings in the supermarket telling us to only buy one or two packets of each

Student404Mus

no down votes either.

ACuriousMind

@Student404Mus You have plenty of closed and downvoted questions that have been deleted.

Student404Mus

"deleted"

they banned me after deletion

skullpatrol

17:14

Go back and rewrite them pal.

ACuriousMind

If you read the post I linked, you will find that it quite clearly states deleted posts still count for the atuomatic blocks.

@Student404Mus I'm not sure who you think "they" are, but all the blocks on your account are purely automatic due to your question record, no human activity involved.

Charlie

is there a way to turn off the "new feed items" thing in the top left corner?

skullpatrol

no

Charlie

aw

Student404Mus

Okay. If they are deleted what I would do to them, it's too late?

skullpatrol

17:17

Show what you have tried and ask for an explanation about why it doesn't work @Student404Mus

Student404Mus

I asked a question after the first ban, and I tried to add new question just after that one, the message of question limit reached pops-up

I checked my questions and try to edit them

and I didn't find what to modify

ACuriousMind

That's likely because even your undeleted questions are not being upvoted, your highest score is a +1

Student404Mus

Ah! Now it makes sense. The answer should be up-voted. The only attempt I see is, to wait 6 months to ask a question that will may get up-votes, very sad.

ZeroTheHero

17:32

@Student404Mus why not take this occasion to contribute through answers and do some constructive lurking to understand what makes a well-received question?

Student404Mus

Actually, I'm on it from now on.

Charlie

No one gets banned for no reason

Student404Mus

What kind of reason?

Charlie

?

Student404Mus

Johnrennie got -2 for asking about some personal theory

Charlie

17:35

ok

ACuriousMind

...John doesn't have a single negatively-scored question.

Student404Mus

since then

I know that. But there are some restrictions about asking

ACuriousMind

sorry, I'm not following you

ZeroTheHero

I don't know what you mean...

Charlie

then don't ask questions that are restricted

ZeroTheHero

17:37

by "since then".

Student404Mus

I misunderstood ACuriousMind

it doesn't matter.

someone had up-voted my post?

skullpatrol

isn't that a good sign?

Student404Mus

lol

you took it seriously?

Charlie

???

Student404Mus

Yes. Thank you!

ACuriousMind

17:44

@Student404Mus It's pretty hard to understand what you're talking about. I know language barriers can be tough, but this might be another reason your posts aren't getting many upvotes. Try to express yourself as clearly and correctly as you can.

Student404Mus

yes that's right.

I miss a lot with English.

But I 'm still struggling with it.

Physics Meta

0

Q: Transfer answer from duplicate unregistered user account

I answered a question as not registered user, after that I registered, is it possible to transfer the registered answer to my registered account? The answer gave is here: Is there a difference in the infrared absorption spectrum of a greenhouse gas when pure and when mixed with non-greenhouse ga...

support user-accounts new-users

Student404Mus

Next time when I ask a question I seek an advice from you

I mean here in h-bar

How do you recommend academia.edu?

Charlie

If I have an operator that maps $|V\rangle$ onto the zero vector, is it an eigenvector of the operator with eigenvalue $0$, or does it not count if it nulls the vector?

ACuriousMind

@Charlie it counts

Charlie

17:51

rt

ty

skullpatrol

18:24

^ extreme brevity

:-)

bolbteppa

19:12

5

A: What is the usefulness of the Wigner-Eckart theorem?

The Wigner-Eckart Theorem http://en.wikipedia.org/wiki/Wigner_Eckart_theorem is a formula that tells us about all "simple constraints" that group theory - the mathematical incarnation of the wisdom about symmetries, especially in the $SO(3)\approx SU(2)$ case (rotations in a three-dimension...

'1) Remember that a tensor operator is a collection (i.e. a set of 2j+1) operators which transform amongst themselves under group transformation (let's stick to SU(2) or SO(3)) or commutation with the generators of su(2) or su(3) (if we use infinitesimal transformation).

2) The Wigner-Eckart theorem ultimately states that some operators (the components of the tensor operator) can be written in terms of the spherical coordinates as a common function of the radial variable r multiplied by a spherical harmonic specified by the component.'

@JohnDuffield string theory is lord

Student404Mus

@bolbteppa did you mean me? :) I was looking for this earlier.

bolbteppa

19:31

My bad incomplete understanding of W-E is that an irreducible tensor rep of $SO(3)$ has $2l+1$ components and transforms as the $2l+1$ spherical harmonics $Y_{lm}$ and the application of any of these $2l+1$ operators to a wave function in spherical coordinates is a linear combination of the stationary states of the system where the coefficients are the projection of states with all possible commutation relations onto the wave function,

but this involved adding the angular momenta of the states so the coefficients are expressed in terms of addition of angular momenta, then blah need to finish this stuff

Semiclassical

W-E is a black hole in my brain

i remember it being in Shankar

and I remember not understanding it

that's about it

bolbteppa

I think that physical idea makes sense I just need to get why the coefficients are fixed by addition of angular momenta so uniquely

Projecting a stationary state in a spherical basis onto the overall state clearly involves two different angular momenta which can add according to angular momenta addition rules so in this basis of course they'll involve these addition rules which seems reasonable

Semiclassical

what weirds me out, looking at the statement of W-E, is that it breaks the symmetry of the matrix element $\langle jm|T_q^{(k)}|j'm'\rangle$

i.e. you choose to couple either $j'$ with $k$ or $j$ with $k$

that's presumably a very useful thing to do

but it looks weird

I guess the C-G coefficients already have that asymmetry tho

Avyansh Katiyar

21:04

Hello

So I finally wanted to start coding

Because ai and ml interest me

Python or c++

I have no prior coding experience

alarge

@AvyanshKatiyar Probably start with Python then especially if AI/ML is what you're after.

skullpatrol

21:22

@AvyanshKatiyar Why is Python such a popular language in the AI field?

danielunderwood

21:42

+1 on python! When I was in high school, I was looking to learn a bit of programming. Many people (including from physics backgrounds) recommended python but I instead went for FORTRAN because I thought that's what real physicists used...I don't even know if I fully made it through hello world

ACuriousMind

22:33

Python is a nice language to get started on coding stuff but I'd recommend to learn at least one language with a less dynamic type system if you want to learn more about programming in general

corona patrol

22:44

@ACuriousMind what would you recommend for the complete novice?

2 hours ago, by Avyansh Katiyar

I have no prior coding experience

Charlie

23:13

c++ is quite a monsterous language to start with if you have no prior experience

2 hours late to the party but still

corona patrol

not to mention trying to learn two languages at once

Charlie

Java is a nice language, if a bit outdated if you asked some people, python is also among the easier ones to pick up and very widely used

ACuriousMind

@coronapatrol Depends much more on the quality of the instruction than the language

danielunderwood

IMO, being dynamic and interpreted makes python really easy to get started with. Just install the interpreter and you're off. The whole compiling/linking process can be a bit much to learn on top of the programming basics

Charlie

^ that's true, from what I remember getting c++ even setup in the first place can be challenging if you don't know what you're doing, a million things can go wrong

fond memories of spending hours trying to figure out what binaries are and why I don't have them come to mind

corona patrol

23:23

danielunderwood

Yeah that's kind of the wall that I hit with fortran back then. Then I used matlab for some stuff a couple years later and thought it was wonderful. Now I've come full circle and spend most of my coding time staring at python anyway

c++ also has pointers which seem to be a major sticking point for some people, though I always found that more intuitive than having things implicitly be a value or reference

Charlie

iirc if you learn c++ badly and don't take advantage of how powerful the language is you might as well learn something simpler and save yourself an hourly headache while learning

ACuriousMind

1. If you just want to write stuff that works it's fine to just learn a single language and never do anything else

2. If you want to learn programming - as a craft and not just as a means to an end - you need to learn multiple languages, preferably with as many different paradigms as possible

3. Yes, some languages are more "difficult" than others. C++ is a monstrosity whose full specification very few people fully understand (I'm not one of them). But understanding what a compiler does or what linker errors are is not esoteric knowledge, they are fundamental to understanding what a program is and how it is executed, and I would expect any "programmer" to understand them at least in principle

Charlie

23:40

I haven't touched it in ages, does c++ lend much advantage in programming in physics over say python?

In terms of solving problems, not necessarily programming entire pieces of software

ACuriousMind

Like, yes, you can just use physics formulae without any knowledge about their derivation, or knowing anything about the theory behind them. Knowing how to apply these formulae to solve specific problems doesn't make you a physicist (at best it makes you an engineer ;P)

Charlie

D:

ACuriousMind

@Charlie Depends on what you want to do, but you simply can't write things like hardware drivers or operating systems in Python.

Charlie

fair enough

ACuriousMind

And for things where performance is crucial, Python still is suboptimal because the nice abstractions it provides make it impossible to apply the optimizations e.g. a C compiler can apply

danielunderwood

23:49

@ACuriousMind that's absolutely true. I just remember a time when it took a bit to really learn variables, functions, returns, etc where I think something like python would clear up some of the other difficulties. I think plain C is a good one as well since it's a fairly simple language that forces you to do a lot, but I don't know that I'd recommend it to someone learning for the first time

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