6:00 AM
@ACuriousMind I'm not convinced that the algebraic properties of SO imply anything about the analytical equation $\nabla \epsilon=0$.

There we go
"Gaussian measures"
(Wiener measures are also called Gaussian measures)

post here
@JohnRennie Hello

You're back! :-)

Yes, and I'm here to teach you what a manifold is
because apparently you do not know

Didn't Cartan construct a covariant formulation of Newtonian mechanics?

6:03 AM
That doesn't even make sense.
Accelerated frames break Newtonian mechanics.

OK, non-relativistic mechanics

That's what I just said.
Covariant in which sense?

I'm sure he did. I'll have a quick hunt around. I think it's mentioned in Penrose's book Roads to Reality.

Penrose also thinks that brains are tensors or something
And don't think this Cartan thing will get you away from the balls!

6:06 AM
Jesus

I don't know what $\mathfrak{S}_2$ is

It says it's the $\sigma$ algebra generated by cylinder subsets of something
For the notion of cylinders in measure theory, see Halmos

Yeah I don't know that shit

Maybe complete Jost first
I feel like doing analysis exercises

hm, where was I in Jost

6:11 AM
Have you started the first section yet?

I think proving that $\sum \frac{1}{1 + a^n}$ converges
But this is $\leq \sum \frac{1}{a^n}$
and I already did that one I think

@JohnRennie Ok I found someone that talks about Cartan's formulation
But they give no details.

Eh, whatever
Enough convergent series
Let's go into chapter 1

found a book
"Topics in the Foundations of General Relativity and
Newtonian Gravitation"
Time to get it legally

I think Cartan did it just to show it could be done, but the approach is of no practical interest so it's faded into obscurity.

6:14 AM
I got the book

I think it's in MTW

Checking it out now

Since everything is in MTW

The argument about general covariance is a bit silly. Yes any theory can be made generally covariant, but metric theories are so naturally covariant that they are the obvious approach.

How can you make it generally covariant? What does that even mean

6:16 AM
Are you arguing with Einstein and the evidence, @JohnRennie

:-)
@0celo7 that was Kretschmann's objection to Einstein.

Yeah, well, you damn physicists need to tell me what it means
You will be happy to know that I managed to electrochemically polish tungsten ( also @Loong )
AND I AM ALIVE
I also did it to cerium

Cartan's theory is described in Penrose's Roads to Reality section 17.5.
However it's too long for me to easily reproduce here.
I'm sure there's a pdf version of RtR floating around the web somewhere though ...

that would be illegal
ðŸš¨ðŸš¨ðŸš¨ðŸš¨ðŸš¨ðŸš¨ðŸš¨ðŸš¨ðŸš¨ðŸš¨ðŸš¨ðŸš¨ðŸš¨

@Slereah amazon.com/…
we need this
@JohnRennie that sounds like something you would do

6:24 AM
oh hey, Malament
Of Malament's theorem fame
That guy just loves foundational theorems

In the unlikely event you want a copy of RtR see:

You have 60s before I delete that link ...

You're the room owner for pete's sake
that's illegal

Anarchy!

6:26 AM

yeah that's a bit of a risky gambit

I'm not going to defend book piracy, but if you want to read that one section of the book you'll find it there.

BREAKING NEWS JOHN RENNIE DEFENDS BOOK PIRACY

jesus

Alternative facts eh?

6:28 AM
Bringing politics in?
@Slereah Too bad the book is $69, it looks very interesting Oh wait,$42 used

Look dudes, I bought the book. I paid my own good money for the paperback version. That's how I knew it described Cartan's theory.

@JohnRennie not penrose
I refuse to buy anything by that nutjob

@0celo7 Ah, I was a bit surprised you'd be interested in Penrose's book especially since it's basically popular science not a textbook.

You bought his topology book

That was before he went crazy

6:33 AM
one day too you will be crazy
it is the fate of the scientist

@JohnRennie I got some GR books for Christmas
and @Slereah got some too

I got Hawking Israel
Which is nice

O'Neill

Yeah
I think for a good cross section of spacetime topology shenanigans, Hawking-Ellis, O'neill, Hawking Israel, Penrose and Sanchez is pretty good
That gathers most of the theorems

@0celo7: I'm sure the answer is no, but do you want to make a PDF of the bit of RtR that describes the Cartan formulation of non-relativistic mechanics. It's only a few pages so reproducing it is allowed by the fair use policy.

6:36 AM
also since this is for research purpose, it is fair use

I do not have it in my PDF folder
@JohnRennie do you want to know what a manifold is or not

@0celo7 in principle yes, but not right now.
I have to start work in a few minutes so I wouldn't be able to concentrate on it.

I should probably go to sleep then
@Slereah Sanchez has a paper book?

No
Just a big article online
I use it a fair bit though

6:41 AM
Hmm? I don't see equations
Nothing new
But I still don't know what "covariant" means

7:00 AM
Hey, everyone :-) @0celo7: Oh, hello, u're back!

Howdy

@Kaumudi.H Morning :-)
I got my new laptop!
My new new laptop that is.

:-) Didn't see u yesterday. Were u busy with ur new laptop?

I was a bit busy yesterday - partly work and partly other annoying but necessary stuff. Oh well, it happens.

I see. How is it, the laptop?
@JohnRennie x'D

7:07 AM
That's my new laptop on the right and a laptop like yours on the left for comparison.
The new one is biiiiiiiiiiiiiiig!

Yep, it's big :-) Is that something that u wanted?

Yes. Every now and then I visit my mother and stay for a week or so. When I'm down there I need a laptop to work on and it has to be big because I need a big screen for work.

Ah, OK. How is it? (Other than "biiiiiiiiiiiiiiiig", that is :-)

It's ... well, to be honest it's just another laptop.

:-P OK...

7:10 AM
It's nice. The screen is nice and clear and it's fast. But it's no different to all my other laptops except for the size.

Right, OK.

But it is exactly what I need for working away from home. So it wasn't bought as a toy - it really will be useful.

Yep, I know. What went around literally came back around :-)

You should see the little laptop I bought a few days ago. It's just as fast as yours but it's so cute!

Isn't that the one on the left?

7:13 AM
No, the one on the left is like yours. The little one is even smaller.

Wait, why dyou have THREE new laptops?!

Maybe ... :-)

You bought two on E-bay?

I might have done ... :-)
But actually I do have uses for them all. It wasn't just a random purchase.

Holy shit.

7:14 AM
I joke about buying laptops but in real life I don't buy them unless I have a use for them.

Still, it is astonishing that u have use for so many laptops!

I bought one like yours and the little cute one. The one like yours has a few faults, but I'm going to repair it and set it up for my own use with Windows 10.

Show me what the "little cute one" looks like, now you've made me curious!

Hang on, I'll take a picture. This will take a few moments because it's in a cupboard at the moment ...

I will wait...

7:18 AM
@JohnRennie I thought you had to work?

@JohnRennie BTW, dyou listen to music on your phone? If so, what music player app do you use? Neither Play Music (Google) nor VLC media player is any good...

@0celo7 I am working. Just ... not very hard :-)
I'm currently running a database reindex.
There. That's the baby laptop. Isn't it cute :-)

OMG! That is ridiculously small...and yes, very cute, I must admit :-)

:35227781 I prefer the Windows 7 classic mode desktop. I don't like the extra decoration that you get with Windows 10. But my new laptop has to run W10 because the 7th generation i7 CPUs aren't supported on W7.
So reluctantly I'm having to switch to Windows 10.
@Kaumudi.H I don't listen to music on the phone.

OK...

7:26 AM
Do you copy the music to the phone or do you stream it?

I'm still using the windows 95 theme

@JohnRennie SD card. So yes, I copy it.

OK so you're playing it locally. VLC is the obvious choice. What don't you like about VLC?

@Slereah XD

@Slereah on what OS?

7:28 AM
@JohnRennie It has a glitch when I close the app. It re-opens and I have to follow through with a few steps before it actually closes.

Windows 7
windows classic theme
Although the start menu isn't quite the same

@Slereah ah yes, that's I'm using.
@Kaumudi.H BTW this was lunch yesterday:

Those look like mini-samosas.
What are they, really?

They are samosas
Thirty samosas :-)

In India we have some big samosas :)

7:33 AM
Oh, really? That was lunch?! My mom would kick me if I asked to eat samosas for lunch.
Were they tasty?

They were OK. One of the supermarkets was selling them off cheap as a discontinued line so I bought them to see what they were like.

Oh, had u never tasted samosas before?

They were nice but not amazingly nice. The samosas my mum makes are far better.
I didn't mean to see what samosas are like, I meant to see what these ready made samosas are like

Ah, OK. Only occasionally do I find great samosas. A few months ago, I found that a restaurant near my house sells excellent samosas.

7:36 AM
...with Ketchup and some other dips, including chutney.
@JohnRennie I see. I don't like mini-samosas. They're a bit too oily and also crunchier than I like.

Ah, I like the crunchiness of the mini samosas.

Samosas are ideally a bit crunchy on the outside, soft and tasty on the inside :-) Mini samosas sometimes have a different filling, making them crunchier.

I baked the samosas instead of frying them, so they weren't too oily.

Nice. What was the filling?
Ah, potato. OK, then they must've been soft on the inside as well.

Three different fillings: vegetable, chicken and minced beef.
They were nice and soft on the inside.

7:40 AM
Ohh. The website only mentioned potatoes.
The mini-samosas here are often filled with fried onion and tomato and the former adds to the crunchiness, which I don't like.

Ah, OK...
No dip?

Tomato ketchup :-)

Hmm. Pudina chutney would've been nice too.

Mint?

7:43 AM
Yep.

We eat mint with lamb. I'm not sure about mint with chicken ...

@JohnRennie Ok, balls?

Oh, right, chicken. I'm not sure, then.

@JohnRennie When you use "baby" and "cute" to describe a laptop, it is proof that you are crazy (for laptops) :'D I believe you will inspire many people to take up laptop breeding as an occupation :D

@0celo7 I'm half working, half chatting. I certainly don't have enough brain cells left unused for point set topology :-)
In any case your conversation with Barry Carter is still on the chat if I wanted to go back and review it.

7:46 AM
@JohnRennie I'd probably want to redo it
And that conversation did not mention smooth manifolds
Just topological ones

I have this book (on paper not e-book) and I did have a go at learning about manifolds from that.

Just get a math book, @JohnRennie
I have some math books

The fun thing about GR causality theory is that it basically never writes down a metric

Or field equations!

yeah
I mean, you sometimes get one
But usually only when you get to the higher levels of causality
Like the metric $-dt^2 + g$

7:52 AM
@JohnRennie If you really want a wild ride, check out "Quantum Theory for Mathematicians" by Hall
He has Hermitian operators $A$ and $B$ for which $A+B$ is not Hermitian

Also, $X$ and $P$ cannot be defined on the whole Hilbert space

Did you ever find out if derivatives of unbounded operators made sense

ACM said it doesn't
And that one needs a whole formalism for the Heisenberg picture that is rigorous

Don't believe his lies

7:55 AM
But he didn't know details
He has no reason to lie!

yes
that's what makes his lies so damning
Once you know QM
mb u should look into AQFT on curved spacetime

What's the usual book for it?

Haag is the big one
I think there's another more modern one but I forget the name
Plus you already have a book on AQFT
Wald

That is true, I do have it
Is it any good?

Wald? I would say no

8:04 AM
I found it pretty incomprehensible two years ago

Yeah I'm still having troubles understanding it
Haag is probably a better way to start since it's in flat space

Unfortunately I want to read most of Hall

I am not reading all of the proofs of course
But I am getting a lot out of it
I will have to try hard not to correct my QM prof from now on

Uuuuh excuse me sir but that is not well defined

8:10 AM
if he integrates by parts I will get triggered

Oh odds are good he probably won't even do that
Usually hermitian operators in QM classes are just defined as $A^* = A$

Defining $A^*$ is nontrivial

Yeah but such is the way
I'm guessing the math definition just throws the word "automorphism" around a lot
I first learned the real definition when I did my master thesis
Because I saw a paper saying that in curved space, $\hat p$ wasn't hermitian
And I was like, whaaat?
Seems that $\hat p = \hat p^*$ to me!

@Slereah If $\mathscr H$ is a Hilbert space, and $\mathscr S\subset\mathscr H$ is a dense linear manifold, then $\mathscr S^\bot=\{0\}$.
True?

I don't knooow

8:18 AM
@Slereah Well, you want $\langle \psi,A\phi\rangle=\langle A^*\psi,\phi\rangle$

Yeah
That is when I learned that definition

But $A$ is usually only defined on $D(A)\subset\mathscr H$ (strict)

Although i had not yet learned that integration by part is tricky

So the question becomes, what is the domain of $A^*$?
And does such an operator even exist?
Is it unique?
but you only have $D(A^*)\subset D(A)$ usually
To be self-adjoint you need equality
@Slereah it is true, but sadly it's an exercise in Conway
Most the book is left as an exercise.

I wish there were more books of exercizes with solutions for hard stuff
I have one
Supersymmetry exercize book

8:23 AM
What does $\dot +$ mean?

$$\lim_{\epsilon \to 0} \frac{(+(t+\epsilon)) - (+(t))}{\epsilon}$$

It means $\oplus$ apparently
If $A$ is a symmetric op on $\mathscr H$, then $A$ is self-adjoint if and only if $$\mathrm{Ran}(A+i)=\mathrm{Ran}(A-i)=\mathscr H.$$

@JohnR: I'm off. Bye, have a nice day :-)

@Kaumudi.H Bye :-)

@Slereah will you read Wald with me when the time comes?

8:29 AM
I don't know, when is the time
I don't even like to look at it

maybe two weeks

it has very poor typesetting
i'm having a hard time seeing the difference between convergence and uniform convergence from the definitions
I've read what the difference is but it's hard to extract it from the definition

In uniform convergence, $\epsilon$ does not depend on $x$.

yeah, but
How do you see that

The difference is the order in which things are written

8:35 AM
$$\forall p \in D \forall \epsilon > 0 \exists \delta > 0 \forall x \in D, |x - p| < \delta \implies |f(x) - f(p) | < \epsilon$$
Versus
$$\forall \epsilon > 0 \exists \delta > 0 \forall p, x \in D, |x - p| < \delta \implies |f(x) - f(p) | < \epsilon$$

@JohnRennie (you should be paying attention)

Pretty subtle difference

That's continuity
Not convergence

I meant continuity

I'm currently checking the server backups have all completed successfully. The difference between that and physics is I get paid for checking the servers :-)
2

8:36 AM
PHYSICS
boi we out here analyzing

Oh wait
I think I get it
The important part is

That fact that in the first one, $p$ occurs first, means that every $\exists$ thing depends on $p$
and $\epsilon$ of course
In the second one, $p$ is not given anywhere

$\exists \delta \forall p$ versus $\forall p \exists \delta$

so nothing depends on the choice of $p$
@Slereah yes

in the second case the delta doesn't have to be the same

8:38 AM
correct

Alright I'm good

He has an op that is symmetric but not Hermitian
jesus
this is pretty savage honestly
dismantling physics

If you like that kind of bullshit
Get yourself a counterexample math book
Those are fun
i've got that one

thats in french

nothing but weird counterexamples for various intuitive notions

8:42 AM
Ok, something nice for a change
The position op $X_j$ is self-adjoint on the domain $$D(X_j)=\{\psi\in L^2(\Bbb R^n):x_j\psi (x)\in L^2(\Bbb R^n)\}.$$
@Slereah Ok, I have the precise result for integration by parts.

I'm amazed that they included hbar in it
Seems weird that such a mathy book would not pick $\hbar = 1$

Ok, and the self-adjointness of the Laplacian
So I'm reading this book because I want this stuff to work on manifolds
But it won't.
These proofs require Fourier transforms

I don't know if there's a lot of books about QM on curved manifolds
usually it's either flat QM or QFT on curved manifolds
Kleinert goes into QM on curved manifolds but it's using path integrals

@anonymous Where you at. I got you problems

for a smooth function, is there an easy criterion for uniform convergence?
like regarding its derivative or something

8:54 AM
Do you mean uniform continuity?

yes

Well, on compact sets, continuous implies uniformly continuous
@Slereah If the derivative is bounded, the function is uniformly continuous.

Ah yes
i suspected something like that

You see a proof?
You need $C^1$.