Dense proper subgroups of SO(3)

Emilio Pisanty

Jun 15, 2017 07:56

@ACuriousMind @AccidentalFourierTransform @Slereah is this user of terminology correct?

0

Q: Reference request: Symmetry breaking from $SO(3)$ to its closed subgroups

As the title says, I'm looking for papers which discuss symmetry breaking from $SO(3)$ to some closed subgroup (e.g. the tetrahedral group, octahedral group, icosahedral group, etc.) This is in the context of looking at free-energy minima, where the free energy functional has the symmetry of $SO...

Aren't all SO(3) subgroups closed?

Slereah

Jun 15, 2017 08:16

They are all closed in $SO(3)$ if they're proper subgroups I think, yeah

Dawood ibn Kareem

Jun 15, 2017 08:27

@Kaumudi.H Not without a really good reason. I know someone who had a complete blood transfusion to a new blood type; but it doesn't just change for no reason.

Slereah

blood type is just markers on the red blood cells

those are determined by genetics

Kenshin

Jun 15, 2017 08:57

blood type never changes

Slereah

I guess maybe if you had all your bone marrow's replaced by a donor that might change it

But that's fairly radical

Kenshin

true

Qmechanic

2

Q: Open conjectures on the Fukaya category

This is ported from mathoverflow question https://mathoverflow.net/q/263693/. Can somebody give examples to open conjectures on the behavior of the $Fuk(M,ω)$ (that's the "mathematical A-model category of D-branes" in physics language) that come from string theory and can be phrased in a mathema...

string-theory mathematics topological-field-theory branes category-theory

To buy the community some extra time, I have temporarily locked this post, since it is currently just 2 vote short of a migration to Mathematics. Note that the Phys.SE community usually welcomes math questions relevant for physics. Is migration what we really want?... [June 15th, 2017: Unlocked.]

John Rennie

@Qmechanic I voted to leave that open. It's specifically asking for opinions from the string theory community so I consider it on topic here.

Slereah

Jun 15, 2017 09:15

This is a family website

I don't want no $Fuk$ there

Sanya

@Qmechanic I think it can stay here; I also think it would benefit from explaining its issue a bit more extensively

Slereah

Jun 15, 2017 09:40

@ACuriousMind why is the Regge slope written $\alpha'$

What is the original $\alpha$

Jannik Pitt

Jun 15, 2017 09:54

Quick question: The first law of thermodynamics is $dU=\delta W + \delta Q$. What does the lowercase delta mean? What's the difference to $\Delta$?

Slereah

Lower case delta in thermodynamic refers to inexact differentials

While $d$ is an exact differential

Inexact differential

An inexact differential or imperfect differential is a specific type of differential used in thermodynamics to express the path dependence of a particular differential. It is contrasted with the concept of the exact differential in calculus, which can be expressed as the gradient of another function and is therefore path independent. Consequently, an inexact differential cannot be expressed in terms of its antiderivative for the purpose of integral calculations; i.e. its value cannot be inferred just by looking at the initial and final states of a given system. It is primarily used in calculations...

Slereah

Jun 15, 2017 10:06

"Also the oscillator modes are promoted to operators - we will do without the little pretentious ^"

I like the little hats :(

ACuriousMind

Jun 15, 2017 10:24

@Slereah No idea

Qmechanic

@Slereah : Thanks for your effort to make this site family-friendly!

ACuriousMind

@EmilioPisanty I'm willing to believe that there are some U(1) you can put into it that aren't closed, but dense (and the embedding is then not continuous), but the user seems to mean discrete, not "closed", as Rod Vance comments.

@Slereah That's from Weigand's string theory script, right?

@Qmechanic I think migration to mathematics would be wrong, but I feel the question is a bit broad - it seems to be a list question, which MO does but we generally don't.

Jannik Pitt

Thanks @Slereah !

Slereah

Jun 15, 2017 10:40

@ACuriousMind it is

John Rennie

Jun 15, 2017 11:04

I find myself contemplating a canonical Q/A on why the expanding universe doesn't affect planetary orbits (much). For a de Sitter universe it turns out you can solve the equations analytically, and I think this would be a nice model calculation.

There are lots and lots of existing questions on this subject, but none of the answers provide more than a rather vague statement that gravity wins on small scales.

Kenshin

it's pretty obvious intuitively that gravity would win though

John Rennie

The point is that gravitational forces don't exist in GR. Objects move according to the spacetime curvature. The expanding universe doesn't produce a stretching force and the gravity of stars doesn't produce a compressing force. Discussing the situation in terms of opposing forces inevitably misleads.

Secret

I don't remember if I have asked this before, but as we all know, at the cosmological scale, energy conservation is a muddy topic and is commonly treated as not hold. But anyone checked whether given a spacetime manifold of cosmological scales, pick points A and B, determine the energy momentum density there (which obviously will conserve at those points because energy conservation holds locally), treat the tangent spaces of A and B as if they are two noninertial frames,
do some kind of parallel transport like operation to move from tangent space A to B, do after such transformations, the e…

More simply, when the curvature of spacetime at cosmological scales is taken account of, do energies at all points match up consistently even though it might seemed they are not conserved?

(A classical mechanics analogy will be that kinetic energy can take different numerical values depending on what reference frame you are in)

Slereah

Jun 15, 2017 11:30

Energy conservation isn't a muddy topic

Energy just isn't conserved in GR

Jim

@Slereah correction: it isn't necessarily conserved. There are many metrics in GR that would result in globally conserved energy, but for highly curved or non-de Sitter-like expansion, energy doesn't have a conservation law (asterisk)

John Rennie

Energy is conserved iff the metric has a timelike Killing vector

Jim

iif? Is that like "if and only not if"?

John Rennie

if and only if

Jim

that's iff

John Rennie

Jun 15, 2017 11:39

Oops :-)

Jim

well, now my comment makes no sense and makes me look like a jerk

win-win

@Secret the parallel transport may not be possible, but more generally, it'd still be on a local scale. It's only once you track energies at larger scales. Artificially moving local-scale energies across cosmological distances is not the same as considering global-scale energies

Secret

Ah I see, thanks

Now, a noobish sounding follow up question. Since energy conservation is violated in GR, yet the consensus is that there are no perpetual motion geerators, what in the metric that prevent a path in spacetime to be followed that lead to some net gain in energy?

(The question is noobish because I probably miss out the subtle details on how global scale energies is dealt with in GR)

Kenshin

there are perpertual motion generaters in GR

John Rennie

Jun 15, 2017 11:57

@Secret in a de Sitter universe just place any two (light) objects at rest relative to each other and they will accelerate away from each other.

Since our universe is asymptotically approaching the de Sitter geometry (or so we think) this will happen in our universe as well.

Secret

That's very strange. I suppose thermodynamics might have trouble describing the dynamics of moving bodies in such space

I suppose this is the first time someone told me that perpetual motion generators can be possible

GR is really much stranger than QFT

Slereah

It's really not

GR is fairly straightforward

John Rennie

@Secret trust me, QFT is faaaaaaaaaaaaaaar weirder than GR :-)

Slereah

Jun 15, 2017 12:14

It's not so much weird as poorly defined

Emilio Pisanty

@ACuriousMind I find that dense U(1) embedding pretty dubious

what's it mean, geometrically?

ACuriousMind

@EmilioPisanty Think of a line on a torus that's not a loop, but dense.

Emilio Pisanty

@ACuriousMind yeah, I know

but how do you fit that on SO(3)?

I mean, if you have a non-discrete non-$\{\mathrm{id}\}$ group, you're committed to a copy of $\rm U(1)$, right?

SBM

What is this discussion about?

Emilio Pisanty

@SBM whether SO(3) admits non-closed subgroups

bolbteppa

Jun 15, 2017 12:20

@ACuriousMind @Slereah this is the GR spinor question I tried to ask the other day in a very badly phrased way, does it make sense now: mathb.in/146369 ?

SBM

@EmilioPisanty topology?

Emilio Pisanty

@ACuriousMind hmmmm. Or maybe you can get a non-closed subgroup that's not continuous?

e.g. something like $\{R_z(n\sqrt{2}):n\in\mathbb Z\}$

or notch it up to 3D to something like $\left< R_x(\sqrt{2}) , R_y(\sqrt{3}) \right>$

so it generates a dense cloud of points

@SBM yeah, what about it?

SO(3) as a topological group

Secret

If I drew a strange attractor looking trajectory on a torus surface, will that be an example of a dense U(1) embedding since it never repeats exactly, hence not really a loop?

ACuriousMind

@EmilioPisanty I know you can for $U(1)\times SU(2)$, see physics.stackexchange.com/a/239276/50583

Emilio Pisanty

Chicken chicken chicken

►

oooooooh boy, that's fantastic

paper is here

isotropic.org/papers/chicken.pdf

ACuriousMind

Jun 15, 2017 12:53

I have seen that before in this room, I think

Slereah

it's an old joke

well, not that old

but a few years

Secret

Dvij

Jun 15, 2017 13:20

Hey. Can anyone explain what is the difference between a BPS blackhole and an extremal blackhole? I am not sure I have enough elaboration to this doubt so as to be able to form a question.

1010011010

Jun 15, 2017 13:44

Why is my question about a different derivation of the TKNN invariant closed as duplicate?

I'm not discussing completely new physics here but there are blanks to be filled in

0

Q: Derivation of the Kubo formula in Ashcroft & Mermin

I've been trying to rederive (13.37) from Ashcroft & Mermin: $$ \sigma_{xy}(\omega+i\eta) = \frac{-ie^2}{\hbar^2(\omega + i\eta)} \sum_{n\neq n'} (f) \left( \frac{\langle n\mathbf k|\nabla_x|n'\mathbf k \rangle \langle n'\mathbf k|\nabla_y|n\mathbf k \rangle}{\hbar\omega + i\eta + \varepsilon_{n...

quantum-mechanics condensed-matter

If you said you didn't "want" to fill in the blanks then that would be a different story, morally perhaps a bit controversial to mention

But this leans more towards untrue :\

Sanya

Jun 15, 2017 14:02

@1010011010 if I were in your shoes I would a) include in my text that I know there are this and this other question but point out in which way they do not solve my own question and b) make the question maybe a bit more accessible. Starting with a short version, giving a bit of structure makes it easier to read - because as it is, I think anyone not completely immersed in the topic will not attempt to follow all the steps

then you can dispute the duplicate flag

user685272

Jun 15, 2017 14:14

I'm looking forward to 0celou coming back in two days :P

Secret

This chat seemed to grew out of 0celo's. Activity is not known to be severely hindered due to his absence this time. However, it is always good to have him back as I need more guys to discuss about real analysis, math rigor and other things. Not to mention, he and Slereah and Acuriousmind often fuse to make interesting conversations

user685272

Agreed.

Slereah

Jun 15, 2017 14:36

@ACuriousMind math.stackexchange.com/questions/2323787/…

user685272

@0celouvskyopoulo7 nice to see you're still active :)

Beehumbole when or if you come back.

Balarka Sen

Jun 15, 2017 14:53

We're fighting with him in the comments.

user685272

I see, I see.

But if the comments go on for too long the system will suggest it be moved to a chat room, where he cannot talk :(

Balarka Sen

We settled it.

user685272

cool

vzn

Physicists Say They've Manipulated 'Pure Nothingness' And Observed The Fallout / sciencealert

Qmechanic

Jun 15, 2017 15:11

@EmilioPisanty : Example: Pick first a $U(1)$ subgroup of $SO(3)$. Identify $U(1)\cong (\mathbb{R}/\mathbb{Z},+)$. Consider dense subgroup $H:=\mathbb{Q}/\mathbb{Z}$.

Abcd

Can someone please help me with a basic question? I am stuck.

user685272

How are they defining "nothing"?

@vzn

Abcd

How to find the dimensional formula of coefficient of viscosity using the Poiseuielle's formula????

I know the formula.

DOnt know where I am going wrong.

Unable to obtain the right answer :((

$ \eta = ([ML^{-1} T^{-2}][L^4])/ ([LT^{-1}][L]) $

Is something wrong here?

I am getting the wrong dimensions of length.

Can someone help?

Sanya

@Abcd from what you have written, we do not know how you are trying to do your dimensional analysis

how could we hope to help you?

Abcd

@Sanya The formula is eta = (pi * p* r^4)/(8*v*l)

Where:

eta is coeff. of visvosity

p is pressure

r is radius

v is velocity

and l is length

@Sanya Now?

Sanya

Jun 15, 2017 15:25

and the unit of v is m/s for you?

Abcd

@Sanya yes

Sanya

I think v is a volumetric flow rate

so it's in m^3/s

would that change your result?

Abcd

@Sanya Oh. Didnt't know that.

@Sanya Yes it did

Thanks

Sanya

@Abcd you're welcome :)

user228700

Jun 15, 2017 15:40

@DawoodibnKareem Huh, I see. Well, that's never happened to me.

user228700

Perhaps they did it wrong when I was born, the one time I've had it tested (before now, i.e)

bolbteppa

"Note that it had always been assumed in the folklore of general relativity (and often written in texts) that GL(n,R) has no double-valued or spinorial representations;" pnas.org/content/74/10/4157.full.pdf who knows their GR folklore :p

Slereah

Well the largest spin group is the Clifford group I think

And that's just roughly O

Sha Vuklia

Guys, say we have a lamp like this:

According to Philips, the light is spread 30 degrees. However, when we shine the light on some surface, there is a brighter spot in the middle than on the the edges. I’m guessing part of the reason is that the lamp has a width, and therefore the intensity on the edges diminishes as I tried to show here:

We would like to give a theoretical value of the irradiance of the lamp at certain angles and distances from the lamp, but it seems to me can’t simply just the distance from the lamp in the formula $E=S/(4\pi r^2)$, because the light in front of the lamp has a greater irradiance in the middle than towards the fringes. So I’m not sure what $S$ to choose when I get off the middle. Any ideas on this?

Also, I just realised:

Even if the light bulb had a very small width, we would still have a vague "fringe" at the end, so I'm not even sure if the width of the bulb matters

Sha Vuklia

Jun 15, 2017 16:08

Maybe this is what I need

I don't see any units on the vertical axis, but it seems to me it's the angle from the lamp

Emilio Pisanty

@Qmechanic but that's dense on the embedded U(1), not on SO(3)

John Rennie

@ShaVuklia: the light intensity will depend on the exact shape of the reflector. In effect it's working like a crude lens so there will be an approximate image of the filament somewhere. The intensity will be a function both of angle and of distance from the bulb.

Emilio Pisanty

I meant that I found it dubious that you can get $f:\rm U(1)\to SO(3)$ such that $\mathrm{closure}(f(\mathrm{U}(1)))=\rm SO(3)$

John Rennie

@ShaVuklia: I would guess your only option is to measure the intensity and fit some convenient function of angle and distance.

Emilio Pisanty

or even just such that $f(\rm U(1))$ isn't already closed

John Rennie

Jun 15, 2017 16:17

@ShaVuklia: Or you could place a diffuser over the front of the bulb. That would even out the light a lot at the expense of losing a lot of the intensity.

Balarka Sen

Jun 15, 2017 16:36

@EmilioPisanty Huh, you can have those?

Emilio Pisanty

@BalarkaSen I dunno

I don't think so

@ACuriousMind seems to think you can

(cf. this comment chat.stackexchange.com/transcript/71?m=38130298#38130298)

Balarka Sen

I don't believe him either.

It feels like circle subgroups should always be closed, in any Lie group whatsoever.

user685272

:O

Balarka Sen

I mean, yeah, this is obvious. Lie subgroups are embedded manifolds.

ACuriousMind

@BalarkaSen It's not a Lie subgroup, just a subgroup

Balarka Sen

Jun 15, 2017 16:45

Oh.

user685272

It's all a "lie" :P

John Rennie

Have we got anyone present who'd like to answer a (hopefully) simple GR question?

(I ask because the current population seems to be mostly QFTers)

user685272

Shouldn't that be QFTists?

Bernardo Meurer

@Slereah I sent you the paper yesterday

John Rennie

:: tumbleweeds blow across the differentiable manifold ::

user685272

Jun 15, 2017 16:52

I enjoyed your AMA @BernardoMeurer

2

Bernardo Meurer

@JohnRennie I had an exam on those recently

@user685272 That's good to hear! Glad you liked it

John Rennie

@BernardoMeurer that's no good, I couldn't understand a word of the maths questions you were discussing here :-)

Bernardo Meurer

@JohnRennie Neither could I, hehehe

But whatever, I don't care anymore

Transferring to Santa Barbara, woohoo

John Rennie

@BernardoMeurer: while you're here, do you know of any USB DACs that (a) look like a pen drive, (b) are good, and (c) below $50? Kind of like a cheap DragonFly Black?

Balarka Sen

@ACuriousMind Well, that's an embedded circle in SO(3) in any case. That's a compact subspace of SO(3), so in particular closed, not?

user685272

Jun 15, 2017 16:55

Did you hear about the youngest wrangler? @JohnRennie

John Rennie

You're leaving Lisbon to go to Santa Barbara?

Bernardo Meurer

@JohnRennie Grrr

Let me see

And yes, I am

@JohnRennie Must it be like a pen drive? Can it not be dongle-like?

Then I have one suggestion

John Rennie

Sadly it must be pen drive like.

Bernardo Meurer

May I ask why?

John Rennie

Reasons ...

Bernardo Meurer

Jun 15, 2017 16:57

O.o

John Rennie

I'm thinking the only obvious option is a second hand DragonFly Black.

@user685272 have there been any record setting senior wranglers recently? I thoguht the last was a few years ago.

Bernardo Meurer

@JohnRennie I am afraid so, yes

My only competing solution is a dongle

schiit.com/products/fulla

John Rennie

I haven't been able to find any cheap good DragonFly-a-likes. There are loads of low end ones but they're all crap.

Bernardo Meurer

Yeah, DAC's are hard to make cheap because the IC's themselves are expensive

user685272

Yes, that's the one @JohnRennie Fernandez from a few years ago.

John Rennie

Jun 15, 2017 17:00

I agree with you that if it didn't have to be a pen drive there are lots of good options around the $50 mark.

Bernardo Meurer

@JohnRennie True that

John Rennie

I'll ask my GR question in the hope someone will see it. Even if they can't answer maybe they could firm it up into something that could be usefully asked on the main site.

Jaime Gallego

@Bernardo Are you going to be in UCSB by July 28? I got invited to a Caltech tour, Pasadena and SB are not that far away.

Bernardo Meurer

@bolbteppa I just realized I missed a question from you on the AMA. Yes, I do believe that Zizek, "actually says something". His recent book on the immigration crisis, as well as Violence are truly great reads. Trouble in Paradise and In Defense of Lost Causes as well.

@JaimeGallego SBCC, I'll be doing community college for one year to then transfer to UCSB

Because reasons

I should be moving at the end of August

dmckee

@JaimeGallego It's about a 2 hour drive if the traffic cooperates and as much as 3 hours if there is a problem on the route you choose. 118 through Simi valley is fastest most of the time, but 10 to El Calle Real is your backup route.

Bernardo Meurer

Jun 15, 2017 17:06

@dmckee When are you going to be in California?

Mithrandir24601

@JohnRennie I've been spending all day looking at the Kerr metric, so either I'll have some vague answer, or no clue whatsoever :P

dmckee

@BernardoMeurer No idea. Depends on possible job interviews and family stuff.

John Rennie

Notoriously energy isn't conserved in an expanding universe because there's no timelike Killing vector. But it's possible to write the de Sitter metric as:
$$ds^2 = -f(r)dt^2 + dr^2/f(r) + r^2d\Omega^2 $$
where
$$f(r) = 1 - \frac{\Lambda}{3}r^2$$
These coordinates are kind of Schwarzschild ish rather than the comoving ccordinates usually used. Anyhow this metric does have a timelike Killing vector and it therefore has a conserved energy. So written in this way energy is conserved in a de Sitter universe.

Bernardo Meurer

@dmckee You searching to move schools? You lectured at UCSB before didn't you?

Mithrandir24601

also, about the Dragonfly/DAC thing - I just gave in and got a dragonfly (back when my old laptop starting falling apart)

Bernardo Meurer

Jun 15, 2017 17:08

@Mithrandir24601 Dragonflies are damn good

John Rennie

@Mithrandir24601 yes, I think that's the obvious solution.

dmckee

@BernardoMeurer I did my undergrad at UCSB. But I am on the job market. Know anyone who wants to hire a mid-career physicsist or programmer analyst?

I looking in industry as much as academia. The money thing.

John Rennie

@BernardoMeurer Santa Barbara? Don't they make cartoons?

Bernardo Meurer

@dmckee Nice! Didn't know you did your undergrad there :)

Mithrandir24601

@BernardoMeurer You don't need to tell me that :) Mine gets a lot of use :)

Bernardo Meurer

Jun 15, 2017 17:12

@dmckee I know a couple people in the valley I can ask for you, can you email me a CV just in case?

Jaime Gallego

@dmckee Not that much by American standards. "Americans think 100 years is a long time, Europeans think 100 miles is a long way."

Bernardo Meurer

It's an absolute pleasure to try to help you :)

dmckee

@BernardoMeurer And my father's parents lived in La Canada, just three miles from Cal Tech, so I've made the drive @Jaime is proposing many times.

Bernardo Meurer

@Mithrandir24601 I need a good DAC soon, alas my analogue setup gets used the most

(That is thanks to @JohnRennie's hardware contribution :P)

Jaime Gallego

@BernardoMeurer Oh well, I can still meet with @DanielSank there :)

Bernardo Meurer

Jun 15, 2017 17:16

@JaimeGallego When do you go?

Jaime Gallego

July 28. I'm checking flight prices and they're all outrageous.

Bernardo Meurer

Oh yeah

They are :)

Jaime Gallego

The cheapest one seems to be an overnight Aeroflot flight for 2500 € with a layover in Moscow.

For my mother and I.

Haven't searched in depth though.

dmckee

@JaimeGallego You can often do better flying into Ontario than either LAX or Burbank, but that means a 2-3 hour drive from Riverside to get to Pasadena and it is in the opposite direction from Santa Barbara.

Watch our fro traffic at the I-10/210 interchange on that route: it backs up there something like 16 hours a day. Ugh.

user685272

Wow! 16 hours?

dmckee

Jun 15, 2017 17:21

If it is all the same otherwise Burbank is much more convenient to Pasadena than LAX.

Bernardo Meurer

@JaimeGallego That seems like way too much

I fly to Rio for 700€

Jaime Gallego

Checking Norwegian now, they seem to have a lower price.

Bernardo Meurer

Going to LA is more than that, but shouldn't be much more

@JaimeGallego Just check flights.google.com

Jaime Gallego

That's exactly what I did

theDoctor

@JohnRennie: Probably because holding a stable coordinate space consumes it.

There's no reason for space to be regular.

GOD must use energy to hold it... ;^)

Jaime Gallego

Jun 15, 2017 17:36

Dog backwards

ACuriousMind

@BalarkaSen Ah, its more an $\mathbb{R}$ than a $\mathrm{U}(1)$

user685272

goDog

John Rennie

Jun 15, 2017 18:02

@theDoctor you'll have to explain to me what holding a stable coordinate space consumes it means, and what regular space means. Neither of these are terms I recognise.

Qmechanic

@EmilioPisanty : Agree, but $H$ is also not closed in $SO(3)$, thereby answering your question here in the negative.

Slereah

@vzn I manipulate pure nothingness every day

Emilio Pisanty

whoa. $$\int_0^\infty J_0(\alpha x) \sin(x) \frac{\mathrm dx}{x} = \frac{\pi}{2}$$ for all $\alpha$ between 0 and 1.

that's crazy talk, right?

@Qmechanic sure.

6 hours ago, by Emilio Pisanty

or notch it up to 3D to something like $\left< R_x(\sqrt{2}) , R_y(\sqrt{3}) \right>$

that seems to also do it and it's likely to be dense in SO(3).

The h Bar

Participants