4:00 PM
When I see a correlation function like that, I think "path integral", but I don't think that's what's meant here.

@0celo7 Which correlation functions and what was the title of Weinberg's book again?

1. e.g. (A.125) 2. Cosmology
I'm judiciously skimming right now.
@JimdalftheGrey I want to interpret (A.125) as $$\xi_\mathcal{R}(r)=\int d\mathbf{x}\,\mathcal{R}(\mathbf{x})\mathcal{R}(\mathbf{x}+\mathbf{r})$$

@0celo7 Ah, those are the two point correlation functions of the power spectrum that help define the comoving curvature perturbation. You have to read the corresponding part of the main body, otherwise it'll make no sense
and no, I didn't read Cosmology

@glance , when did you do this edit? I think I saw your edit some an hour ago. Did you do something after that?

@ACuriousMind How does $\bar{\mathbb{C}}:=\mathbb{C}\cup\{\infty\}$ make sense without the Riemann sphere? Also why is this not parsing?

4:10 PM
@0celo7 That is the Riemann sphere.
(and you should write \bar{\mathbb{C}} )

@ACuriousMind Thanks. But on the Riemann sphere, is there no distinction between $z=\pm\infty$?

@0celo7 No, there is just one point at infinity.
(this is known as "one-point compactification", generally)

@ACuriousMind Do we then "lose information" when we go the Riemann sphere?

@0celo7 Define lose information. If you're asking if global functions on the sphere are more constrained than global functions on the complex plane, then yes.

Zeidler's theorem 4.4: Each holomorphic function from the closed complex plane into itself is constant.
What does he mean by "into itself"?
I know this is the Liouville theorem.
But I learned it with holomorphic bounded functions.

4:15 PM
@0celo7 "Closed complex plane" is $\bar{\mathbb{C}}$?

Yes.

(Stupid name, should be compact, because $\mathbb{C}$ is also closed in its own topology)
The theorem is indeed Liouville's theorem by observing that every holomorphic function that has a value at infinity is bounded on $\mathbb{C}$

Isn't $\bar{\mathbb{C}}$ just the closure of the complex plane?

@0celo7 Closure in what topology?

@ACuriousMind Oh not this question again...

4:18 PM
It's really a compactification, not a closure operation.

Lol why is $\mathbb{C}$ closed in its own topology? I know it's open. Is is perhaps clopen?

@0celo7 Yes, every space is clopen in its own topology :D

(The empty set is open, and the space is the complement of the empty set)

I REMEMBER

4:20 PM
@ACuriousMind clopen? both closed and open? I like this concept

It's like deja-vu
You told me this a week ago :D

@JimdalftheGrey Yeah, the topologists did a really good job with the names

@JimdalftheGrey Obligatory.
Actually, there is a mistake. In @ACuriousMind 's bad example, a closed interval was also open.
This is so dumb.

Not even the mathematicians are sure why they call these things closed and open, there's a funny thread on one (or both) of the math sites about that
The best they have is some weird analogy where the topology is like a set of rulers :D
136

I'm ashamed to admit it, but I don't think I've ever been able to genuinely motivate the definition of a topological space in an undergraduate course. Clearly, the definition distills the essence of many examples, but it's never been obvious to me how it came about, compared, for example, to the ...

@0celo7 That was histerical
And maybe a bit historical :P

4:28 PM
@JimdalftheGrey You've never studied topology?

I never studied physics until a bit over 3 years ago

I thought that graduate GR courses cover causal sets, which are defined partly in terms of their topological properties.

@0celo7 You self-studiers have soooo cute ideas about how university courses work

we did closed and open sets a bit, manifolds and the like
but topology like that? nope
Plus 3 years is not a lot of time. I'm understandably lacking in certain areas

@ACuriousMind Hilarious indeed

4:34 PM
@ACuriousMind I'm hoping you're not assuming I'm a self-studier, right?

@JimdalftheGrey Hmm funny how we physicists always seem to skip over essential details in mathematics ;0

@Danu I can't stop laughing about it

One of the main reasons I'm happy to be in a mathematical physics program now

@JimdalftheGrey No, I know you're not special :P

I'm not special in that way

4:36 PM
I mostly meant 0celo7 and JamalS with "you self-studiers"

ah, good. At least we agree I'd never be unlazy enough to self-study

In other news: How can I try to repair my sleep schedule after staying up till 7 AM last night?
(and waking up at 10:25 AM)

@ACuriousMind I'm in high school! I can't help it!

@Danu Lol. You don't. You just roll with it and spend the first few days you have to get up like normal people as a zombie on caffeine

4:38 PM
@Danu 3.25 hours of sleep? Sounds like all you have to do is stay awake until you want your schedule to normally start. That's it

@ACuriousMind Also why is $\bar{\mathbb{C}}$ compact whereas $\mathbb{C}$ isn't? Something something finite subcover, right?

Every open cover admits a finite subcover
that's the definition of compact lol

I know!

@0celo7 Compact means that every cover has a finite subcover. The plane is not compact because the cover of successively larger disks centered at the origin has no finite subcover

So the Riemann sphere is compact?

4:40 PM
...guess why it's called a sphere! :P

Yeah, because it is the one-point compactification. It's a sphere, topologically

@Danu I don't get it. Does "sphere" have some etymological meaning?

It is surprisingly annoying to show that spheres are compact, though

@ACuriousMind Yes, that's what I'm looking for.

The best way is to embed a sphere into $\mathbb{R}^n$ and use that the closed bounded subsets of $\mathbb{R}^n$ are precisely the compact ones

4:43 PM
@0celo7 If it's called a sphere, it's probably compact!
@ACuriousMind Meh this is cheap
I'd like to see it from the definition

@Danu Which one?

@Danu Spheres are defined by their embedding.

i.e. an explicit construction of a finite subcover from arbitrary cover

That is, how do you define the sphere in the first place?
@Danu I am not sure such a constructive proof exists

I mean from the definition of compactness, @0celo7

4:44 PM
@ACuriousMind what other definition is there? (of the sphere)

@ACuriousMind Would probably be too nice
@0celo7 You can have spheres in things that are not at all like $\mathbb{R}^n$

@0celo7 If you are a topologist, the sphere is best described as a disk glued to a point
And the disk, in turn, is just the unit interval times itself - a square

If you spend enough time to get comfortable with gluing recipes, and how polygons define 2D surfaces and so on, it makes a lot of things a lot clearer
You'll spend a lot of time being confused, though :D

@ACuriousMind I'm already confused enough as it is. Is Heine-Borel enough to show that spheres are compact?
Or does that only work in $\mathbb{R}^1$?

4:47 PM
Heine-Borel works in $\mathbb{R}^n$

@0celo7 Heine-Borel shows that the n-sphere $S^n$ is compact, because it is a closed and bounded subset

Cool.

...but as I said one can have totally different spheres

@Danu Like?

think about spheres in finite fields

4:48 PM
No clue what that means.

you know what mod is right?

@Danu Usually, when someone says "sphere", they mean $S^n$.

@Danu I'm going to say yes cautiously.

@ACuriousMind Yeah, but I'm just saying that my definition would be $x^2+y^2+\dots = 1$
which works in finite fields also
but gives very surprising things ;)

What is a finite field?

4:49 PM
a field with finitely many elements

Ok. What is one of these surprises?

a field is a commutative associative ring without zero divisors. You know the most famous examples: $\mathbb{R}$ and $\mathbb{C}$.

I know what a field is.

coolbeans

You mentioned some surprising results though.

4:50 PM

Care to share? Gotcha

basically one can do algebraic geometry
with this stuff

Uh-huh.
Is it at all useful?

@ACuriousMind, know anything about algebraic geometry over finite fields? ^^

@glance you asked me if I saw your last proof. Just now I looked again. But, you prove an anti-commutation rule in base of another anti-commutation rule. I repeat, it's not what the user needs. Not in whatever calculi he/she will have to do in the future, he/she will stay to check if there is some anti-commutator offended. This is why he/she wanted to know where is something wrong in his calculus, without altering with starting from vacuum. So, I repeat, Mark gave the solution.

4:51 PM
@0celo7 In what sense?

@Danu Oh, are you looking at the projective space over a finite field?

Physics

@Danu Yeah, I know something :D

@Danu About my reading speed: I've now hit topology and complex analysis that I don't know. I'm reading much more carefully, I assure you.

Woop found it
section 3.2
@ACuriousMind No... I think?
THis is what got me started last night, he mentions the 'size' of the spheres encapsulating very interesting information
of $\mathbb{F}_{p^m}$
viewed as a sequence ($m=1,2,\dots$)
sounds very interesting

4:56 PM
@Danu $\mathbb{F}_{p^m}$?

I'm doing like a billion things! Fine.

@Danu Yes, you are (almost) :) For example, the circle is the real projective line, and the sphere is the complex projective line.
I think your definition of "sphere" is the projective line over the finite field

@Danu Integers mod $p$?

@0celo7 to the $m$

4:59 PM
Yeah.
Wait it says something about polynomials...and I've lost interest.

Algebraic geometry is pretty much about polynomials
(I think)

@Danu That's the old-fashioned standpoint, but yes.
I'm not sure if Grothendieck even knew what a polynomial was ;)

I don't even want to try to pronounce his name.

The German way
...although he was French

@Danu No, was was originally born in Germany and at least his mother was German, too. (His father was Russian, I think)
So his name is fully German, he just happened to grow up in France

5:04 PM
It would like out like Grotendick in an American accent.
Or is it deek at the end
It sounds weird in a German accent too.

@ACuriousMind I know, I know quite a bit about his life

@Danu Alright, I'm not surprised to hear that from a mod of hsm ;)

@ACuriousMind He was French though, always lived there and finally acquired the nationality after waiting for a long time (he was nationless for many years)

Yeah, his soul was French, probably

The waiting was because he didn't want to serve in the army

5:06 PM
@ACuriousMind where you wander? Do you want a minus from me (which I would dislike). Why do you post an answer without reading the question? The question was in essence how can the e.m. waves travel without a supporting medium, i.e. which are the carriers. Stop wandering in 50 places at once.

@Danu Isn't the proper term "stateless"?

Probably

@ACuriousMind Someone just randomly upvoted this. I'll give you a check if you want to answer it.

@ACuriousMind please correct your answer. In essence, it is that e.m. waves are their own carriers. Or else, you didn't answer his real problem.

I get it now, but it should be closed after all this time.

5:09 PM
@Sofia EM waves have no carriers. Waves don't need carriers. They are solutions to a wave equation, nothing else, I stand by what I have written.
To ask "how can they travel without a medium" is to commit the fallacy of the aether. They can travel without a carrier because there is no reason they should need one.

@ACuriousMind what you say? Water waves have carriers the molecules of water. The energy is transmitted from molecule to molecule. Did you ever hear sound waves in vacuum?

Photons != phonons

@Sofia how is that relevant?

@Sofia I am not saying that there aren't waves that have carriers. I am saying that having a carrier is not necessary for being a wave.

@ACuriousMind you make philosophy with a beginner.

5:12 PM
@Sofia YES

@ACuriousMind do you indeed want to solve his problem?

I am not actually a friend of lies-for-children. I hate them with a passion, and I believe it is better to tell someone something they do not understand than to tell them something which will later turn out to be nonsense or false.

Eh. I like giving rep.

do it!

5:13 PM
@ACuriousMind I will see you when you have family and children.

I do have to say I agree with @ACuriousMind here
since I have been 'hurt' by these things myself, sometimes quite severely
for instance, special relativity was a huge mess in my mind for 2 full years

@Sofia I want to answer the question correctly. And the answer to the question "Why are EM waves called waves?" is *"Because they are solutions to a wave equation". The other stuff is all based on widely spread misconceptions, and I will not perpetuate them.

...all because people were too lame to tell me it's all about the invariant interval

@ACuriousMind do you mean to help, or not? A beginner asks for help and sometimes has even difficulty to express what troubles "them".

@Danu !!!!

5:15 PM
:(

It's like hearing from my own past self! ;)

Kept on getting those stupid 'intuitive' derivations of time dilation/length contraction and it all didn't make any sense

@Sofia Do you know what the socratic method is?
I like to give these harsh answers because I want people to think about their question

I want them to try and formulate what exactly it is that troubles them (i.e. why my answer is not a good answer)

5:16 PM
@ACuriousMind I am an engineer, and at home I am a mother. Beginners don't have the means that you expect from them.

That being said, I think @Sofia's answer is better
It has the same essential answer, but elaborates a bit more

@Danu I didn't say hers is bad, did I?

No, I was just observing

Good :)

Also, now that I think about it, who was it that first defined waves as 'solutions of a wave equation'?

5:18 PM
@ACuriousMind you learn to loose honorably (I just translate a saying from my country).

In particular, it doesn't seem obvious from 'just' the equation for a string or something that all waves should satisfy this

@ACuriousMind mature people tell you that you are wrong.

?

@Danu I think you found a good question for that little site of yours, eh?

okay, first sentence understood. Second question?

5:21 PM
@Sofia: Are you saying I should accept your position that my approach to answering "beginner's questions", as you call them, is wrong/bad because you are olden than me?

@ACuriousMind (I guess you do.) But, you'll follow me better when you'll have to explain to children.

@Sofia How do you know he doesn't already explain to children?

@ACuriousMind Oh! I am half joking. Just I am advising to take in consideration that they may not understand your answers.
@ACuriousMind nobody should accept my position.
@JimdalftheGrey he says that he is not a friend of lies-for-children.

@Sofia That's why multiple people can answer here. It's certainly not a good idea if everyone started just answering like me.

@Sofia That doesn't mean he doesn't explain things to children. Just that he doesn't simplify to the point of telling them things that are wrong. A risky strategy but maybe it works for him

5:27 PM
@ACuriousMind Pleeease! Never feel offended by my words. I am just saying that you do your best, give an answer, and the answer may pass by them with no help for them.

@JimdalftheGrey It, admittedly, works far better if someone told them the lies before.

@Sofia Alas, that is a reality every one of us must face

@ACuriousMind yes, I am older than you, that means more experience with people of different ages, including children.

@Sofia Not true, he could volunteer as a social worker, a teacher, or some other kind of people-oriented duty in his spare time. It's entirely possible he could have more experience than every one of us in dealing with children

@ACuriousMind You're not harsh towards my questions.
@Danu Is this acceptable?

5:31 PM
@0celo7 Hah, with "harsh" answers I mean those that simply retreat to definitions and facts without trying to be "intuitive". I don't tell stories, mostly.

@JimdalftheGrey , @ACuriousMind I saw once a movie (Ingrid Bergman, Liv Ulmann) if I recall correctly "The autumn sonata" (I am not sure). Bottom line, the daughter was coming with heavy accusations to her mother, e.g. "you gave me to read books that I didn't understand".

@ACuriousMind This is great
@JimdalftheGrey Stop playing devil's advocate, Jimbo
OH! Jimbo! How could I not have thought of that earlier!

I'm playing Devil's advocate. You've been making assumptions that aren't supported by available evidence. I'm just trying to point out how what you propose may not be the case. Not that it actually isn't the case
@Danu The devil is my advocate

@Sofia "You gave me books I did not understand" is not an accusation in my world. The realisation that there are many things I do not understand is very important

Dudes, this name is perfect
we have to call you Jimbo

5:35 PM
@Danu Hey, I'm a demon, not a devil.

@ACuriousMind I recall that someone told you "you are telling me that I am an idiot", and you replied "I should have been more compassionate". But again, we just talk, nobody should accept someone else's approach if the former doesn't subscribe to it.

@Sofia No worries, I'm not mad at you or anything.

@Sofia Can we please locate this exchange of words? :D

@Danu why not Jimmy? Or Tiny Jim? Or Jiminy Cricket? Or Jimbo Jones? Or Jimmy Jim Jim?

@Sofia Not if you subscribe to universally preferable behavior.

5:37 PM
@JimdalftheGrey There is only one Jimbo
Did you even look at my video clip?!

@Danu lol, it sounds as if it could have happened to me, but I don't recall it exactly

I think Jimbo Jones was around first

Jimosperm is the best

@ACuriousMind beware! Could it be that you heard of the (beautiful) movie "La beaute du diable?" (with Gerard Philippe). A daemon, or a devil, or whatever such thing, may be angelically beautiful.

Apparently 0celo7 is a big fan of conifers

5:39 PM
@Sofia No, I don't know that film. The thing with the demon and the devil was just an obsure joke that sadly, no one here got.

@ACuriousMind what is in your thought when you say a daemon not a devil?

Oh, great, now I get to explain my D&D joke :D

@ACuriousMind
0

In modern physics, a quantity is said to behave like a wave if it satisfies a wave equation. With some quick googling, it is quite simple to find out in what context the wave equation was first derived and applied (e.g. here). Furthermore, it is known that throughout the 18th and 19th century, ...

@Sofia My profile picture is a figure from a setting called Dungeons&Dragons, where all sorts of mystical creatures are real. In that universe, there are various creatures form worlds that are like hell, and they are the demons and devils. The devils are the rule-following sort, while the demons are rather free spirits
My profile picture is a creature from a demon world, hence my saying: "I'm a demon, not a devil"
It really gets even less funny when I explain it :D

5:44 PM
edited so now I reference you in it too, @ACuriousMind

@ACuriousMind "La beaute du diable" was a splendid scenography after Goethe's "Faust" with the celebrated actor Gerard Philippe.

@Danu Uh, you didn't need to :P

Just for the lols
Also, I noticed that the comment of yours I just starred is a nice example of German-type grammar :)

@Sofia If you say so, I'll believe you.

Es wurde allerdings...

5:47 PM
@Danu The positioning of the admittedly?
Ah, yes

:)

@ACuriousMind I wonder, do you do other things too than physics? Do you go to parties with friends, to movies, theatre, concertos, opera? (If you find my question too intrusive I withdraw it).

@Sofia OPERA :D hah

@Danu what hah? I am from Romania, we have opera in our DNA.

Just a fun thing to think about
Opera always appeared to me as the rich-mans musical

5:49 PM
@Sofia Yes, I do things other than physics - opera and concerts aren't my thing, but movies, parties, bars, playing games and all that, that I go to.
I also read a lot of fantasy books and play too many video games

It seems so cheesy compared to good ol' instrumental classical

@ACuriousMind @Danu What's the best notation for the volume element induced on the boundary of a region?

@Danu maybe you were exposed to inappropriate things. Opera is for us the "bell passato". Verdi, Puccini, Donizetti, Bellini.

Sometimes I wonder whether you are a time traveller from a bygone era that somehow got trapped on the internet, @Sofia :)

@Danu also Carusso, Beniamino Gilli, Mario del Monaco, Maria Callas, Renata Tebaldi.

5:53 PM
@ACuriousMind lmao
@Sofia I guess it's just not my thing
Also, only Italians?!

@KyleKanos That picture proof answer.... what do you think? not an answer flag?

It's a comment though, not an answer

@KyleKanos sometimes. Which one you talking about

@ACuriousMind our life is too much short. To miss opera is to miss unbordered enthousiasm.

5:55 PM
1

What is a tachyon? Most physicists think that faster-than-light particles cannot exist because they are not consistent with the known laws of physics. Are they right?

He wrote an 11 comment answer

respect

yeah. This is a no-harm-done thing

So no to the flag?

I think it should be converted into an answer

and people answer in the comments all the time for primarily-opinion closed questions

5:57 PM
despite the fact that it's closed

@KyleKanos What would you flag it for?

I will flag it with that suggestion, if you don't @KyleKanos

It's definitely on-topic commentary relevant to the post. It's nothing bad.

@ACuriousMind Probably personal message to mods, saying something like "answering to closed questions via comments"

@Danu that might be a good flag

5:57 PM
It should be flagged since an elaborate answer in the comments is not good in any case

It could fall under too chatty

@Danu Meh, I see it like Broadway musicals. A lot of them are terrible, repetitive and use uninspired music, but there are gems.

::puts on shiny, authoritative moderator badge::

I think to be chatty, there has to be a second person in a conversation.... at least

@DanielSank Hm, perhaps
@KyleKanos I think it's a good thing though in this case

5:59 PM
I disagree. Comments are meant to clarify the question/answer. It is not meant to be a work-around to answering closed questions

@Danu ::throws mud at shiny badge::

Take that @Danu!

@ACuriousMind It's possible @Sofia is actually a phenomenon which exists only within the net.

@KyleKanos I know, but it's a good contribution and it should be an answer.

@Danu Again, I disagree. It's a bad question and it does not deserve an answer