>coordinateless notation

You disgust me

what monster uses musical isomorphisms

are they more than a euphemism for index raising/lowering?

Nope, they're just the coordinate-free notion of raising/lowering

They are isomorphisms between the vector bundle and the dual vector bundle

which is usually the metric

There is always a relation $\omega(X) = \mathfrak{Some shit}$, and you have $g(X_1, X_2) = \mathfrak{Some shit}$

@ACuriousMind in the expression $\nabla\omega^\sharp$, where is the vector $X$ from $\nabla_XY$? What kind of object is $\nabla Y$?

It's a tensor of rank 2

well (1,1)

@Slereah ok, so it has one upper and one lower index, and the $\flat$ maps it to a (0,2)-tensor?

Yes

Some shit is very academic, @Slereah

Everything looks more serious once you put it in Fraktur

$\mathfrak{Slereah}$

Nope, still mostly funny ;P

$\mathfrak{JohnDuffield}$

Red alert

Why did you invoke he who shall not be named

$\mathfrak{Voldemort}$?

Jesus

You're gonna get us all killed you crazy Kraut

I don't fear the Dark Lord!

You're a wizard Bajoran

*not

But anyway, if you have a metric, then you can also express $g(X_1,X_2)$ as $X_1^\sharp (X_2)$

@0celo7 How would you know that, muggle?

And that is how the metric defines a musical isomorphism

@ACuriousMind wtf?

Literally anti-mudblood

@Slereah $X_1^\flat (X_2)$, you mean.

Can't raise a vector!

@Slereah of course, because $\nabla_XY$ is $C^\infty$-linear in $X$?

(If you know what I mean)

Master wizard master race confirmed

yeah w/e

@Bass did you ask above how to prove a coordinate statement without coordinates :P

@0celo7 yep :P

What do you mean by "quits" in that question

Translation glitch?

@Bass @ACuriousMind what's the German word for quits

Also for "cancel" in this context

@0celo7 in my solution of the exercise, I'm using the summation convention in the first line, but I didn't know how to write the 2nd and 3rd line with it, so I had to "quit" verlassen it

@0celo7 verlassen, he means he stops using the convention there

Wtf I would never use that word like that

@0celo7 any better suggestion?

Have to speak German in a few hours 😨

I wasn't sure, so I put "quit" in quotes

@Bass I think he primarily means he wouldn't use verlassen like that

^

@0celo7 "The terms cancel" is Die Terme heben sich auf.

ok.. maybe "leave the summation convention" would have been better... anyway

wat

wat wat

Does @Huy even know what a coordinate is

like $x$ and $y$ coordinates ???

I have some obscure math book that calls the tensor product "transvection"

I've seen those I swear

I have literally never seen that word anywhere else

@0celo7: I had to change basis of a conic section today and my students wanted me to do it explicitly without using quadratic forms

Well the coordinate notation isn't really about coordinates

the horror

It's more about a basis

you can use non-coordinate basis, whatever

basises

@Huy no clue what you're talking about

basii

@0celo7: dude did you never go to chinese elementary school

Never did conics

I have, I used to be chinese when I was a kid

well draw a cone and intersect it with a plane

But it was just a phase

simple as that

then I became white again

Constant time Hamiltonian evolution

Cheap Chinese Hamiltonian

@ACuriousMind this whole coordinate-free stuff is more like, sometimes you find an object that can be expressed without coordinates, but to convert one coordinate-free expression to another you still need coordinates, isn't it?

@Huy so what conic stuff were you talking about

@0celo7: ellipses, parabolas, hyperbolas are conic sections

What quadratic form thing

like the ACuriousMindian conjecture $\mathrm{d}\omega = (\nabla \omega^\sharp)^\flat$.. to check/prove this, you still need coordinates, right?

@Huy duh

if they're in $RP^2$ you can do a chance of basis to get from one conic section to any other

you can prove this

No

No need for coordinates

Wtf is RP^2

Real projective plane

@0celo7: projective plane???

Well duh?

What does that have to do with anything

Coordinatefull objects are just a special case of coordinatefree ones

you can solve fancy tangency problems

which I frankly don't really care about

Wtf are those

but it was my last geometry class anyways so who cares

$\omega(dx^a) = \omega_a$

@Bass Well, yes, at least some coordinate-free relations you have to prove in coordinates - after all, what defines your manifold is that it has coordinates, so it is not to be expected that you can prove everything without coordinates.

@0celo7: mathworld.wolfram.com/ApolloniusProblem.html

those are tangency problems

for example

Wtf how do fresmen do that

they stay fresh

B)

Is that not at least PhD level??

@Slereah is that coordinate-free?

@Bass Depends on your definition of coordinate-free ;)

@0celo7: you can also solve even less useful exercises

The point is that the components of a tensor are just the projection of it on a basis

@0celo7 for example given certain equation of an ellipse, you can find explicitly all rational points on it

no idea what that exercise was for

that is what coordinatey means

coordiner

coordinated

@ACuriousMind it's definitely not index-free.. is that the same as coordinate-free? :starting to get confused:

@Huy wtf that's insane

@0celo7: ikr

I could never do math at that level

@0celo7: one guy actually had a really impressive intuition of projective space

could just draw everything and see everything

@Bass I would not call it coordinate-free precisely because it has a "coordinate index", yeah.

@Slereah and the basis is mostly $\frac\partial{\partial_\mu}$, which comes from the coordinates.. I thought coordinate-free means something like $R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla{[X,Y]}Z$

It's sad I used to think I was smart

Well you can have non coordinate basis

@0celo7: wanna know Balarkas age? :P

but yes that is a coordinate free notation

@Huy I don't really care

ok

that's when I got frustrated :P

I'd rather be normal and shit at math.

Than...that.

so it makes more sense to say "index-free" instead of "coordinate-free".. because the manifold has coordinates in its definition

fukin indians with no life-perspective if they don't excel in something

There comes a point where someone is really good at something...but it's all-consuming.

so what are u working on these days

PDEs?

Not playing video games or reading the news is incomprehensible.

@Huy I was looking at a PDE proof of a geometry theorem.

geometrization theorem?

@Bass We-ell...if I take three forms and I want to sum then, I might label them $\omega_1,\omega_2,\omega_3$ and then write $\sum_i \omega_i$. It's coordinate-free, but there's an index

No, that all homotopy classes on a compact manifold have a geodesic

that is why there's an index on the basis

I'd say just don't get too worked up about any exact meaning of "coordinate-free" and "index free". You'll know it when you see it

the basis is just several vectors

just like pornography

@0celo7: ah ok

@Slereah I was trusting in someone to say that, thanks

I never did PDEs btw

just know some functional analysis

don't think I'd enjoy PDEs too much

functional analysis is hard

@ACuriousMind yep but that's not a coordinate index :)

regularity consists of way too many numbers for me

@ACuriousMind alright, I think my questions are answerd :not confused anymore:

thx

I only just figured out what a functional derivative, or first variation is

You basically beat the geodesic equation into the heat equation

and that is not even what functional analysis is lol

that's more of calculus of variations

@kevinTahN. cool, can you explain it?

lol

I don't know much about calculus of variations

And use some theorems on parabolic PDE to get the result. I'm not clear on the details.

just the lagrangian basic stuff

what theorems @0celo7

well yes. but this link does it in a little over 4 steps en.wikipedia.org/wiki/First_variation

@Huy fuck if I know I need to figure out the appendix

ah the cake derivative

seen that

hehehe

but why do you compare it to functional analysis

yes gateaux lol

while functional analysis is somewhat more general

it's not harder or anything?

Cake derivative??

gateaux = cake

I don't . . . funtional analysis is about sobolev spaces. . . etc and other things, and I tried reading a real math book about it an almost jumped off a window lol

yeah lol

close the window and read it again

which book did you use

It was a while ago, but it was pretty hardcore and my math background is very bad lol

well if you study something that's not analysis or (linear) algebra and your math background is bad it'll be hardcore :P

Sounds like an opportunity to learn more math!

Is there such a thing as a pedantic excursion through functional analysis for people like me lol

^

may be an easy book lol

you need topology and analysis at least

measure theory would be very useful

every other book that contains "functional analysis" won't teach you more than any basic QM book

which texts would be most accessible? With tangible description of the subject and easy language with examples carefully worked out?

do you know enough about topology? :P

@kevinTahN. did Cahill work out for you

Cahil was marvelous

I know just childish topology lol

good, just remember that wasn't rigorous

hehe

tbh the only English book (on FA) that I've been working through is my prof's

engineering needs to be more rigorous.

I want to take a course on calc of var

and while it has very good motivation it is very very very dense

has a lot of great exercises

it has engineering applications

not sure whether the examples are enough for you

@0537 not really

what is tbh

to be honest

oh lol

@0celo7 true though I want it to be.

if you want I can send you and you can tell me on which page you get lost completely

I tend to understand math texts written for engineers better. Is there something like that for things like topology, or functional analysis, etc?

no because they don't need to know the theory behind it

I see

@kevinTahN. amazon.com/Handbook-Mathematics-I-N-Bronshtein/dp/3540721215

>moves to laptop

>sees star

Wow $\mathfrak{Astronomer}$

@ACuriousMind why do you do this to me

I think you're smart.

...

you have crap standards

lol

@Huy lol

I have reasonable standards... and I consider your age as well.

wow lol

@0celo7 no more lies, only truth

- V

who the f is V

wow

oh man people speaking German

you don't watch a lot of Natalie's movies do you

Natalie?

reported

for troll

>trol accusation incoming

learn to type faster

no

zz

never seen V for Vendetta?

Never heard of it.

no comment

same.

tr0l

So quick question. Do you guys know about the book "fields" by Warren Siegel? One of my goals has been to read the whole thing and understand it. Do you think it is a good text to read? I have learned quite a bit of group theory from it so far.

It's a pretty good book

It has a lot of seldomly discussed topics

@0celo7 when are u watchin sta wars ?

@0celo7 oO

+1

@Huy after I get back to school??

what does school have to do with any of it

do you watch it in sci-fi-class?

I'm going with someone from school??

wtf

@ACuriousMind what

I'll spoiler everything on Thu

@Huy we've been over this

I'm going with a girl who goes to my uni

is this so strange?

I'll explain to you why the movie is sexist

not that one

and I will leave if that one shows up

that's what you think

@0celo7 Yeah, girls are icky, obviously

maybe she'll set you up with her

@Slereah yeah, I am looking forward to getting to some of those topics. Supersymmetry is the first topic I am looking forward to really grasping once I reach it. I have learned so much formal lingo and formalism from the text. I did not even know about indefinite orthogonal group, never understood chiral projections, even twistors and stuff. Pretty good text. I am going to continue reading

@Huy ...why would you say that

*continue reading it

@ACuriousMind they do have STDs cooties

@kevinTahN. what book?

@Huy must...resist...joke in poor taste

@0celo7 "Fields" by Warren Siegel

@0celo7 Have you read it? I am currently reading it. Any thoughts about the text?

@kevinTahN. it's on my list

I think my list is over 100 books long though

hehehe,

You seem pretty informed. What other cool goodness on your list, would you recommend for a starter like myself?

I'm not informed. Curse you $\mathfrak{Astronomer(s)}$.

One of Emilio's questions down, one to go:

lol

@Danu nice!

@kevinTahN. Thanks! Make sure to read it if you're interested ;)

brb 9 hour flight

cu

@0celo7 All the time in teh world to read my answer ;D

watch V for Vendetta on the airplane

and the Matrix

@Danu looking at it now. . . and proud to report that after two year I actually know what a "dual" in the case of the question is. . . "Fields" by warren Siegel has really been working wonders in my life lol

the dual is just the space of linear functions on the thing

@Slereah Not the Hodge dual.

Somedays I think I'm living in the dual space.

hehe

not very special in $L^2$

why is Z^000 called optical chirality? just curious

@kevinTahN. Because it's a pseudoscalar

@EmilioPisanty ahh yes . . . I see

@Danu laptop is out of reach

And I won't have interwebs during flight

Oh well... your loss ;)

@0celo7 Probably for the best. Otherwise you'll just end up spending $10 just to go on social media and say "hey look, I got the interwebz on a plane!"

^lol true

Or I'd read Danu's thing

@0celo7 I guess you can just do it later :)

Or forget ;)

@0celo7 I WILL NEVER ALLOW YOU TO

@Danu ok remind me on the 29th

@0celo7: just watch le moviez

Some dude has a PS4

Just watching these Germans and Austrians right now...they look funny.

I don't remember them looking like this

wtf

@Danu anyway, what makes you think I will understand your answer

a PS4 in a plane

@0celo7 Uh, what?

@0celo7 It's really quite nice and not too advanced, don't worry about it. You should easily be able to get it.

@ACuriousMind On mobile, what?

You need to be more specific

Why do they look funny?

For the same reason short people are smug.

@ACuriousMind There's just something...off about them.

They don't look like standard people.

Am I in a plane full of synths...

@0celo7: or maybe you Americans don't look like standard people

@Huy no.

We won the war.

you didn't win any

WW1, WW2

nah

that's just what your history teachers tell you

Is there anything that works like wolfram alpha, but in python? and that can may be do functional integral ? :)

I know these days there are all sorts of python packages out there, especially for QIT and Quantum optics

Also is there an alternative to wolfram alpha , that also shows the step by step solution to stuff? I really need a system to check my stuff lol

@kevin don't you have a subscription from uni for pro

and what exactly do you do on w/a that you need stepbystep for

@kevinTahN. That thing isn't even well-defined in most cases, and getting it to be in the cases where it can be well-defined is pretty hard math, you shouldn't expect there to be something that can compute that algorithmically.

@Huy you mean pro version of wolfram?

yes

@ACuriousMind yeah you are right. After I actually looked closely and understood how functional integrals and Feynman path integrals work in general, I finally understood why qft is hard lol

@Huy It can't do path integrals lol hehehe

Also, you mostly don't actually compute the integral, you just use its formal properties.

well that's why I'm asking what you exactly do on w/a where you'd need step by step

yeah it is tough stuff lol

hehehe

wat

do you english?

may be if something like G^2 could be computed for phi^4 in wolfram, that would be a cool start lol

@kevinTahN.: If you want to actually compute results with a computer, you'll have to do it numerically - put the theory on a lattice and do a Monte-Carlo simulation, for example.

@ACuriousMind : Can you give precise reference to where this argument is made?

@ACuriousMind is there a simple reference for say. . something simple like phi^4 . What is the recipe for putting it on a lattice, and how is basic (PI ?)Monte Carlo achieved? If a reference is provided , I would love to try it in C++ or python right now

@Qmechanic At least I think that's what is done in this paper, e.g. for the energy-momentum tensor - eq. (14) and eq. (18) are only arrived at under use of the e.o.m. eq. (9), and yet they "use Noether's theorem" for this.

Weirdly, they refer to Weinberg, which explicitly states that Noether's theorem is supposed to be for variations which are symmetries off-shell.

@kevinTahN. Uh...taking up programming such a simulation is something you should better take up guided by someone experienced in the field - lattice simulations can have many subtleties, and you should not jump right into QFT if you don't have experience with such numerical calculations, I'd say

@kevinTahN. itp.uni-hannover.de/saalburg/Lectures/wiese.pdf

@ACuriousMind I had a feeling it would be pretty tricky stuff

@Slereah checking it out now

Yeah, I think lattice field theory is something I should take up when I am more mature lol

So I finally understood feynman diagrams after going through Anthony Zee's book.

Is there some simple introduction to how penrose diagrams work. Usually when I encounter them in a text, there is already some background assumed. I have seen experts make predictions by looking at those pictures as if they were sorcerers.

I think my big problem is that I had an aweful math background, and if anything is beyond calculus and basic linear algebra I can't compute things right, unless someone holds my hand through them

Penrose diagrams?

Do you mean conformal diagrams?

Penrose diagrams

are they the same thing?

that kind?

My gr background is aweful. Most of the GR texts I attempted to read were hard. Eventually, I just settled on "Mathematics of Relativity" by George Yuri. Pretty much relativity for babies in diapers lol

It does not have penrose diagrams though

Conformal diagrams are basically just another spacetime conformally related to the first one

except that they are of finite extend

since conformal transformations preserve the causal structure, they are nice to see the causal structure of a spacetime

OMG it sounds like sorcery lol

I think I am lacking quite a bit of background to put those words in context

@ACuriousMind : On a trivial note, I wonder why second author Saletan has been left out in Ref. 10?

@Slereah this is the only relativity text I have managed to read through amazon.com/Mathematics-Relativity-Dover-Books-Physics/dp/…

@Slereah Penrose diagrams is also standard terminology.

@Qmechanic Hm...indeed a bit strange

It Conformal or Penrose diagrams is one of the things I would really love to know. Notice I have not used the word understand, as it might entail learning about some huge number of mathematical stuff. I just want to be able to know where they come from and be able to draw and interpret them. Most of the canonical gr

texts out there are a bit hard for me to self study with. Yuri's text which I mentioned was self contained, explained things rather pedantically (which I love) and assumed I the reader has almost no background. He omits the diagrams though :(

Hmm . . . actually just found this physik.uni-regensburg.de/forschung/wegscheider/gebhardt_files/… . . . . what do you guys think?