@ACuriousMind @DanielSank I would appreciate if you checked physics.stackexchange.com/questions/318497/… for correctness. This isn't my field.

@0celo7 I don't see how your answer links to the question he asked, e.g. why are you using $\phi$ notation, where is the resolution of the identity, seems like the question is about interchanging an operator with an integral...

@bolbteppa I had no interest in answering the question, see the discussion above.

Alright cool, I often find answers delving into tons of formality on this site that don't end up answering the question :p Reminds me of that saying when a paper says it will answer some question, it usually wont end up answering it :p

@bolbteppa The OP wanted to see what a fully rigorous proof of equation (1) is, so that's what I did.

after all, his question is asking about a step in the derivation of that

What is your opinion of characteristic classes and curvature as a characteristic class?

Opinion?

Curvature isn't a characteristic class afaik

It's some combination of curvature tensors

@DanielSank I'm a minor

That is an interesting pretty much direct way to motivate characteristic classes on manifolds, but I mean curvature is the only non-hand-wavey invariant I'd want to construct

What is $A$?

That's way too topological, I don't know what $BG$ is and I don't think I care either.

Commutative ring $A$, sorry

Oh, cohomology with coefficients.

$BG$ en.wikipedia.org/wiki/Classifying_space

math.ucr.edu/home/baez/calgary/BG.html

Ok, I know what BG is, but I don't think I care that I do.

poor person

poor, person

Maybe (s)he means something else?

Basically, who cares about characteristic classes of bundles, why would anyone think of such a crazy thing? The best I can do now is to note that the Riemann curvature tensor involves derivatives of connection coefficients, and you can use the Riemann curvature tensor to build scalars like Ricci and kretschmann, so characteristic classes are somehow fleshing that out into a general theory. Thoughts?

@bolbteppa I haven't read the sections in Bott and Tu on this, admittedly

but the characteristic classes I know of all come from polynomials

and there's an important theorem that says certain polynomials in the curvature give cohomology classes

Yeah, I wonder how the passage I quoted above leads to polynomials

do you know what theorem I'm talking about?

it's not obvious to me what the connection is, but perhaps it's in Milnor-Stasheff

I seen something like that in a physics book, don't really understand why polynomials even arise tbh

Right now I can vaguely see this is a (crazy but good) way of formalizing scalars associated to curvature as invariants, linking it to the general notion of invariants in algebraic topology

@0celo7 You get that $\mathrm{Tr}(F^k)$ is a cohomology class by writing down the Chern-Simons form and showing its derivative is $\mathrm{Tr}(F^k)$. The theorem you're thinking of is probably one that shows that things like Chern and Poyntriyagin classes must be integral cohomology classes.

(Don't expect me to write Pontryagin the same way twice in a row, ever :P)

@ACuriousMind Any homogenous invariant polynomial gives a characteristic class

@0celo7 Ah, you're thinking of the Chern-Weil homomorphism

Yes, that's the proper name.

What I don't know is how they dreamt it up

Nash Top Geom has it

what's that?

@0celo7 Well, if you want to know you either need to read the original papers or get the mathematicians to write their books more like the history overviews of physics :P

Either sounds bad. Did you read my translation operator post?

@0celo7 It looks fine to me, but I think it will not help Alex in learning quantum mechanics if they fall back on such rigorous formulations everytime something seems unclear.

That is, the post is probably correct but most practicing quantum physicists would just look at that with bewilderment

@ACuriousMind Alex said that he's a math student.

hmm... principal bundles involve lie groups, lie groups have lie algebras, on lie algebras, invariants like the casimirs are polynomial functions of the generators...

curvature form is lie algebra valued...

@ACuriousMind rare picture of dog

@0celo7 Picture is named "Cat". I'm confused

@ACuriousMind well the cat is obviously the star. Do you not see the dog?

I see the dog and half of the cat

The dog doesn't seem to be particularly interested in being photographed, though :P

@ACuriousMind it's impossible to photograph this dog

That's horrific

@ACuriousMind they're now saying one inch of snow and 3 inches of rain

oh boy, Maxwell's equations in curved space

@AccidentalFourierTransform oO

I dont like your new avatar

^

^

well, I don't hate it, but not a fan either

I'm running out of good characters :P

has anyone read the sections in MTW about how to generalize Maxwell's equations to curved spacetime?

I think it's one of the more horrible things I've read

Hmm? Is it not straightforward

dF=0 and d*F=0

oh, it's straightforward if you know differential forms

but instead, MTW drones on for like 5 pages agonizing over whether partial derivatives should go to covariant derivatives or not

What business do you have doing stuff in curved space if you don't know differential forms? :P

Ah, ok.

Well everything should be covariant, but then you can cancel christoffels.

@ACuriousMind oh please

That's like me saying what business do you have writing down PDEs if you don't know sobolev theory

ooh, I should be able to meet Misner this summer! Wheeler is dead, of course...and not sure if Thorne will be there

I was only half-serious

but I don't think I want to lug their book around for autographs :\

@BenNiehoff Rip out a page, let them sign it, then tape it back in ;)

In MTW they only discuss EM in curved spacetime on like 2 pages, where abouts are you referring to, didn't like the book that much tbh

I like the visualizing forms aspect of it, but that's about it

@ACuriousMind But there are so many pages I could rip out! How could I ever decide which one?

@bolbteppa I was exaggerating, no worries

I am not at all fond of that book

although it was my first introduction to GR

You have a paper copy?

Invariants of a lie algebra come from $\det(tI + a_i X_i)$, makes sense, you can get the Casimir, you can even get the electromagnetism invariant in the Lagrangian from this, but sticking the curvature form in there, $\det(tI + F)$ as in i.stack.imgur.com/U6V8H.png wtf

@ACuriousMind yeah well you could say "use Schauder estimates etc"

There's more than one way to do things

MTW is very heavy

I do, @0celo7, I use it to help weigh my bookshelf down

@BenNiehoff are you independently wealthy

It was on the windowsill in the library it is that out of place

no? Is it super expensive these days?

That book is like $500

university libraries...

no way! I think I paid no more than like $120 for it, maybe less

in a Barnes & Noble, over ten years ago

:(

The only book I like is Landau, the rest go to helping make sense of him when he does something weird

I'd never read it, but I do want it

@0celo7 But...why?

I need to wait until my PDE prof dies, then get it from his estate

@ACuriousMind we all have our vices

For me it's bourbon and books

@0celo7 Aaaand now we have the prime suspect in case of his premature death :P

@ACuriousMind he just has surgery. I'm sure there could be complications

@0celo7 Still don't know how you can prefer bourbon over scotch

one book I don't have a copy of is Wald

:o

oh, also Hawking & Ellis. Two books I don't have copies of!

Wald is a crappy book tbh. Paperback, yellow pages, awkward size.

I have an illicit copy of it :D

but I don't like it very much

imgur.com/gallery/6yfoh

^my GR books (most of them)

I also don't have Sean Carroll's book!

I do have Podolsky & Griffiths

Never even heard of that one

Exact Solutions

it's very nice

Oh, the brown one?

is it brown? I think mine is orange

Looks brown on my legal copy

but the most important thing in it is the Peblanski-Damianski metric (which I may have spelled wrong)

I've never needed those exact solution books

Books may change their colors between editions, no?

I could be mistaken about the color of mine, it is in a different room

@ACuriousMind yes

but my copy of Jackson is definitely the blue one

@BenNiehoff why do you want HE?

Or do you not

I do, actually

it's supposed to have useful stuff in it about singularity theorems, isn't it?

I'd take your Weinberg and add it to my collection

@BenNiehoff indeed

I think I actually don't have any real GR "textbooks" other than MTW

@bolbteppa I have two copies

I have Nakahara, and my PhD advisor hand-wrote a bunch of notes which he bound together and gave us

which is as big as MTW!

although I think it's like 400 pages, not 1100

I got his Cosmology, haven't looked in it yet, but the cover is amazing to look at

@bolbteppa cosmology is out of the frame, but it's there

Also Peskin is another with an amazing cover to just look at

I read the first two chapters a few years ago

Didn't work

The Susskind Cosmology videos seem to kind of follow it

I tried to learn some cosmology once

took a course which used Dodelson

I thought it was boring and illogical :\

although the illogical part was probably just my lack of understanding

Don't have the urgency to read it that I do with say a string book

I have shockingly few string books for a string theorist

GSW?

I have Becker, Becker, Schwarz because I took a course from Schwarz and we used it

BBS is the most horrible book ever

That is pretty good, very direct

I have only done the beginning

I do not have GSW, nor Polchinski, nor Kiritsis, nor Zwiebach

Zwiebach is kind of shockingly weird, no idea what he's doing in the middle

@ACuriousMind on a scale from paperweight to bad to good, where is BBS?

$n \cdot P$ ...

Schwarz is a very nice guy and was very helpful when I went to his office with confusions about the homework problems

@BenNiehoff All the string theory books seem very confusing if you don't already know what is going on

but as far as teaching, he basically read the book to us

@0celo7 I am not one to judge, but mine is only used as a paperweight

Alas, the only book I have as a paper copy and actually have had use for is still Henneaux/Teitelboim

Maybe Weinberg's QFT, very occasionally

what is Henneaux/Teitelboim?

Kaku is really good when you know what's going on

@BenNiehoff Quantization of gauge systems, which is exactly about what it says on the tin

@BenNiehoff best book on gauge theory. period.

Polchinski is good when you forget how to do spinor representations in d dimensions

::twitches::

@ACuriousMind I did not know the title :)

ah, yes, I know you still have that open question

oh, I also have a copy of Friedman & van Proeyen

oh, and I have Clifford Johnson's book

What are these books

@AccidentalFourierTransform you've read it?

My favorite book is probably my Whittaker

@0celo7 about 70% of it. And I understood about 50% of what I read :-P

but that was a year ago, I should read it again

@0celo7 Supersymmetry and D-branes, respectively

Hm, maybe the van Proeyen one is called supergravity, not -symmetry

Supergravity, yes

has apples wearing masks on the cover, because puns

Heh

I don't get it.

the cover of MTW is an apple with a magnifying glass showing you how its surface is a manifold

@0celo7 Think of Newton's iconic story about discovering gravity and superheroes

Oh. I thought it was about apples disguising themselves, i.e. Gravity emerging from the shadows

no, that will be Verlinde's book if he ever writes one :P

I have heard a rumor that almost everyone ends up writing a GR book

because everyone ends up with their own approach, and thinks all the existing books are crap

even Zee wrote a GR book!

Zee has some nice tricks

I haven't seen his GR book

his QFT book is pretty illuminating, though

Definitely, wish he did way more calculations

I can't say I've ever done a QFT calculation successfully :(

like a real calculation

Hmmm.. So Dieudonne explains how invariants relate to cohomology, then you have bundles with lie groups hence lie algebras and a way to generate invariants in a lie algebra, so what's the 'connection' between the two approaches? Curvature is lie algebra valued and links to cohomology, so you use that in your lie algebra to generate invariants, hmm

God, his QFT book is garbage

and this baby speak is formalised-by/called the Weil Homomorphism

GR book is amazing

I bet he writes a QM book next

whose QFT book is garbage?

Zee

@ACuriousMind needs to write a GR book

I thought it was great conceptually

He doesn't do BRST in his qft book :(

does a good job with renormalization and the Higgs mechanism

well, ok

@0celo7 I do?

He doesn't explain representation theory

That would require me to actually learn GR beyond the basics to begin with :P

Zee? I thought he had an appendix on it

Large parts of the book hinge on knowing that 5 x 3 = 10 +5 or some shit

@BenNiehoff no, he says "use Young tableaux"

His appendix is worthless

@0celo7 I think that's why we can remove them by surgery, yes :P

ok, I was pretty sure I learned from his book that rep theory is all about picking out the variously-symmetrized parts of tensor products, but maybe that was somewhere else

That is where you learned that

But nowhere does he explain what that actually means

And he doesn't give enough examples

Here's a question: for the classical Lie groups, are all representations obtained this way?

@BenNiehoff I'm pretty sure it's a special feature of $\mathrm{SU}(N)$ and $\mathrm{SO}(N)$.

I see

Because those are matrix Lie groups?

but they're all matrix groups

So how are you supposed to do representations in E_8

@BenNiehoff They are not - $E_8$ isn't.

what? I know I read an article several years back that, after some serious number crunching, some mathematicians had finally gotten a computer to spit out the 40k-odd dimensional matrices for E_8

What even is E8?

although one thing that puzzles me is that the lowest rep of E_8 is the adjoint rep...is that what you mean?

Don't even do that lattice thing

@BenNiehoff A matrix group is one that is isomorphic to a group of matrices, i.e. a subgroup of $\mathrm{GL}(N)$ for some $N$. $E_8$ is not - it has no faithful finite-dimensional representations

@BenNiehoff yeah, I guess there's no fundamental.

I have some vague notion of what E6 is after reading some heavy papers on reductions of IIB to 5d gauged SUGRA

The adjoint being the lowest-dimensional admissible representation is sort of a symptom of that, I think

Chrome freezes when I try to delete that.

That's weird.

Was that you or me?

That was me

That is shocking, how could it be that no one has ever mentioned to me that E8 has no finite-dimensional faithful representations?

Btw I only corrected because it sounded German ;)

how is it defined, then?

@BenNiehoff how is it defined in BBS?

it isn't

Garbage book.

this cursed program of mine

well, unless you think a Dynkin diagram is a satisfactory definition

@ACuriousMind, your explanation makes sense as to why my program is wrong, but I do not know how to make it right, even by rewriting it.

@BenNiehoff Ah, there is a subtlety where $E_8$ is used for different "forms", i.e. closely related Lie groups with a similar Lie algebra

lol

Only one of the non-compact forms is not a matrix group, there others are matrix groups, I think

and on top of that, I have a paper to finish, in a week, and I feel lazy as all get out.

thank goodness for spring break.

@heather so you don't want me to define Banach spaces for you tonight?

I honestly do not know which form the $E_8$ that appears in the heterotic string and similar places is

@0celo7, go ahead =) I'm very interested.

oh, in the heterotic string it will have to be the compact one

because it's a gauge group

and we wouldn't want ghosts

Right...that one should be a matrix group

Is there a homoerotic string?

yes

A gauge group has to be compact?

yes

because the kinetic term uses the Killing metric

and you don't want wrong-sign kinetic terms

@0celo7 Yes, you want a definite Killing form.

You never see the Killing form in books like Weinberg unless I'm mistaken.

What kind of link is that?

because in SU(2) and SU(3) it's kinda trivial

I'm not asking homework questions...

@0celo7 Sorry, fixed

@0celo7 The "trace" that's applied to the $F\wedge{\star}F$ is basically the Killing form up to some normalization, no?

yes

@ACuriousMind ah, ok

What is true is that people often forget to mention that the definiteness is needed to ensure unitarity.

Admittedly I don't recall why compactness is necessary for that to be positive definite.

@0celo7 It's an application of Bonnet-Myers

The real question is how does $E_8$ link to the dodecahedron/icosahedron

@ACuriousMind The diameter estimate?

Ah, ok.

@0celo7 Well, you don't need the estimate itself, just that the boundedness implies compactness

One assumes the killing form gives a Riemannian metric.

@bolbteppa Pretty sure you need a 24-dimensional cube. In split signature

And a finite diameter means compactness

Then computes the curvature.

aimath.org/E8/platonic.html

oh, yeah, I'd be curious to know that, too

@ACuriousMind Is one supposed to know that Lie groups are complete?

Says $E_8$ is the root system of the symmetry group of the dodecahedron/icosahedron