The h Bar

0celo7

00:00

To put it in context, when he writes the Sobolev norm he writes it in italics, like it's something new.

It's used in the context of topology and the Cauchy problem.

It's an inequality imposed on the initial data's evolution within the future domain of dependence (which is a cone in $\mathbb{R}^{3,1}$) IIRC.

Danu

yeah... I don't think I can muster the power of will to care about these type of things right now... sawry :P

good luck with it though

0celo7

It sucks that Math.SE is so saturated.

usukidoll

:/

sigh one of my matlab codes is giving trouble

0celo7

Lol, Arnold's Analytical Mechanics book has 200 pages of appendices.

Danu

Damn

0celo7

00:06

I have always wondered about the connection between Hamiltonian mechanics and symplectic geometry...this book seems to contain the answer.

Danu

Certainly does.

This is the go-to reference

0celo7

Cool. Can I skip the Newtonian review and Lagrangian stuff?

usukidoll

does anyone know matlab =(

0celo7

I'm taking a break from Wald to learn some more rigorous classical mechanics.

Reread Tong's Classical Dynamics lecture notes over the last couple hours, now I'm on an analytical mechanics kick!

@Danu: Are the symplectic diffeomorphisms Arnold speaks of canonical transformations?

ACuriousMind

@0celo7 Yes, canonical transformations are precisely the symplectomorphisms.

0celo7

00:13

"The phase space is the cotangent bundle of the configuration space"

That's a lot cooler than saying phase space = p & q

"Hamiltonian mechanics cannot be understood without differential forms"

ACuriousMind

@0celo7 It's also more precise, because it tells you that the phase space is the cotangent bundle, and not just any vector bundle of the same dimension over the configuration space.

0celo7

I swear people on this site quote him

@ACuriousMind I'm sure he explains this, right?

It's sure as hell not intuitive.

ACuriousMind

@0celo7 No idea because I haven't read him.

But he should

0celo7

@ACuriousMind I didn't want to read about Lagrangian mechanics for the 10th time, so I skipped to page 161. I hope I didn't miss anything.

@ACuriousMind @Danu Is $\mathbb{R}^n$ self-dual? If yes, is there anything meaningful to that statement? If no, what is an example of a self-dual space and what meaning does that have?

ACuriousMind

@0celo7 Every finite-dimensional vector space is self-dual. It's only in the infinite-dimensional case that things start to go wrong.

0celo7

00:20

@ACuriousMind So this doesn't imply anything special?

ACuriousMind

Not about $\mathbb{R}^n$, no.

0celo7

Are you implying that it could mean something special?

For another space?

ACuriousMind

I have the feeling that the self-dual spaces are those that can carry inner products

0celo7

Is $L^2$ self-dual?

ACuriousMind

Yes

0celo7

00:23

How does one prove that?

ACuriousMind

By noting that every element of $L^2$ defines a continuous linear functional on $L^2$ through the inner product, and that every continous linear functional defines an element of $L^2$ by summing up its evaluation on an orthonormal basis.

More generally, every inner product space is self-dual by this argument, if I am not mistaken.

0celo7

So is self-dual a tautology of having an inner product?

ACuriousMind

Yes, I think so, because given a self-dual space, you can define the inner product of $x,y$ just by the action of the dual of $y$ on $x$ (or vice versa)

Could be that this evaluation map fails some property of the inner product on some cases, though, I am not sure

0celo7

What's a space that doesn't have an inner product?

ACuriousMind

Hm. Well, the space of tempered distributions doesn't have one, I think.

user54412

00:35

@ACuriousMind That seems too broad.

Danu

@0celo7 It's self-dual yeah

ACuriousMind

@ChrisWhite Probably, that's why I phrased it as "a feeling" ;)

Danu

as are all Hilbert spaces, I think, right @ACuriousMind?

ACuriousMind

@Danu I believe that is correct.

user54412

In fact in these pdf notes above example 4.8: (L^p)^* = L^{p'}, where 1/p + 1/p' = 1

0celo7

00:37

Is knowledge of tempered distributions something I will need?

ACuriousMind

I think it is the completeness w.r.t. the inner product metric that forces self-duality

user54412

And in particular only for p = 2 is L^p its own dual

ACuriousMind

So perhaps the self-dual spaces are those that can carry inner products and be complete w.r.t them

Danu

@0celo7 For what?

user54412

I.e. Hilbert spaces

Danu

00:38

I've never had any practical use for them

0celo7

Just checking, since @ACuriousMind brought them up. (As a counterexample.)

ACuriousMind

@0celo7 Well, you don't need to, but technically you don't know what a quantum field is if you don't know distributions ;)

Danu

only technically

ACuriousMind

They are the dual space of the Schwartz function of rapid decrease, and are also needed to properly define unbound states in QM

0celo7

@ACuriousMind I'm fine with wavy thing in spacetime for now. You can keep your distribution theory.

user54412

00:39

@ACuriousMind I still don't know what a distribution is

ACuriousMind

But it's really technical, it doesn't lend much insight.

Danu

@ACuriousMind it's just like... good to know tempered distributions exist

nothing else really

as far as I can tell

0celo7

Wavy thing is full of insight, on the other hand.

Danu

even in my MQM course we never really needed it

ACuriousMind

@ChrisWhite You're not getting anything more precise than "linear thing that eats functions" from me

(which is basically how they are indeed defined)

Danu

00:41

also @0celo7 maybe it is a good idea to study some diffgeo before reading Arnold

0celo7

@Danu My diffgeo is good. (In my mind)

Danu

Ah cool

user54412

@ACuriousMind See, I do know what a dual space is. I've never been able to get a straight answer as to whether they're really the same thing.

Danu

just saying cause you asked about symplectomorphisms

(I know my diffgeo is mediocre at best, and I know what they are :P)

but maybe it was just my prof choosing random topics

0celo7

My question was more a "I'm pretty sure I'm right, pls confirm"

Danu

00:43

wouldn't be the first time, lol

Ah, yes

0celo7

I have Jost and Lee, which I've been meaning to read. No time though

So my DG is from lecture notes, Wald and Straumann

Danu

Lee is really nice

absolutely excellent

0celo7

Do I need to read his topology book before his smooth manifolds book?

Danu

no

I didnt

He has a very concise yet extensive survey of basic topology as an appendix

ACuriousMind

@ChrisWhite It depends, because the continuous and algebraic dual are two different things. (The latter is just the dual space without requiring them to be continuous). I think the distributions on a function space are indeed just the continuous duals.

Danu

00:46

Yeah I think so too ^

distributions are kind of a analysis/topology thing so they should be continuous

(yes, this is how I do math lol)

0celo7

Is the dirac delta technically a distribution?

Danu

yes

0celo7

Is it continuous?

Danu

ehm I think so

don't think about this handwavy physics way of thinking about it as an infinite spike

0celo7

Wow, does Arnold really need to prove that the wedge product is associative?

Danu

00:48

aren't almost all useful algebraic operations associative?

0celo7

Stuff like that I just YOLO

Danu

The only thing I can think of that isn't is the octonions

but then again... who cares about octonions

0celo7

Lol

Danu

awaits shitstorm

0celo7

Never heard of them

Danu

00:49

really?

0celo7

Doesn't mean much tho

Danu

c'mon you should know the chain

0celo7

@Danu I'm all self-taught

Chain?

ACuriousMind

@0celo7 It is in the sense that the prescription "multiply the dirac delta with a function and integrate" is a continuous linear functional on the real functions

Danu

$\mathbb{N}\to \mathbb{Z}\to \mathbb{Q}\to \mathbb{R}\to \mathbb{C}\to \mathbb{H}\to $... octonions

(idk the symbol for octonions)

H is the quaternions

0celo7

00:50

Was just about to ask what the symbol is

I know that

What does the chain say though

Danu

I just call it a chain

it's not a technical term

0celo7

wiki says $\mathbb{O}$

Danu

(well it is but I'm not using it as such)

ACuriousMind

@Danu Strictly speaking, the tensor product is not on the level of elements, for example

Danu

oh right makes sense

0celo7

00:51

lol what does it say

Danu

@ACuriousMind Shhhhh :P

ACuriousMind

@0celo7 The chain gains nice algebraic features up to $\mathbb{C}$, and then starts to lose features.

Danu

So @0celo7 you're not a student in the usual sense?

@ACuriousMind Yeah, it's beautiful

0celo7

At least Arnold uses $f^*$ for the pullback.

Danu

(H loses commutativity, O loses associativity too)

@0celo7 Is there anyone who doesn't?

0celo7

00:52

Wald got it into his mind that $f_*$ is the pullback.

Danu

hahaha, nice.

damn physicists!

0celo7

@Danu I'm in high school. So, yeah, a student.

Danu

Ah like that

ACuriousMind

I always forget where the star goes for what.

Danu

Also from your profile: You are reading Hawking & Ellis and Wald before Weinberg?

You do realize that Weinberg's book is not that advanced, right?

0celo7

00:54

Yes, but it's a different topic :)

More astrophysicsy

Danu

Gravitation & Cosmology will be hella boring

after Wald/H&E

0celo7

I read the first half already

Danu

right

0celo7

I learned GR from Zee and Weinberg

Danu

so just the cosmology left

I learned it from Carroll

0celo7

00:55

Then I reinforced with Straumann, Wald and H&E

Danu

I think Weinberg would be my number 2 for GR, but I HATE HIS TYPESETTING

0celo7

Oh yeah some Carroll in there too

Danu

(with a passion)

0celo7

I love Straumann, but I hate \var cap Greek letters with a passion.

That shit is not slanted.

ACuriousMind

@Danu Are you referring to the idiosyncratic notation, or the look of the formulae?

Danu

00:56

BOTH!!!

0celo7

@ACuriousMind there's nothing wrong with his GR notation

ACuriousMind

Hehe :D

Danu

urrrrgh :P

0celo7

Although using $I$ for action is weird.

Danu

@0celo7 Not as much as his QFT notation, but it's not perfect IIRC

Right, wtf is that!

ACuriousMind

00:57

@0celo7 There's nothing wrong with his QFT notation, either. It's just different from what everybody else uses :P

Danu

^No, it's wrong

Just... no. Not like that. No.

0celo7

What is wrong with his QFT notation?

Danu

Have you looked at his book? It's literally unreadable to me because of the terrible notation

0celo7

I have the first two!

Point something out that is wrong.

Danu

I mean not factually wrong

but morally wrong

0celo7

00:59

That's what I mean ;P

I don't see anything wrong

Danu

Have you read different QFT books?

0celo7

Zee, Tong's lecture notes

I have P&S on my list

Danu

Oh yeah, also his damn tables of content

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