@ACuriousMind It follows from the fact on a flat Riemannian manifold a p-form is parallel iff it is harmonic

and apparently the space of parallel k-forms is of dimension $\le{n\choose k}$

jeez

@0celo7 Semi-related to your earlier question:

@DanielSank @alarge Would any of you by any chance be able to recommend a good read on Regular Expressions?

@HDE226868 Ok, Penrose

But Ricci invented the component notation, right?

I think so, but I'm not sure.

@BernardMeurer I'm not, but I think there are some standard books on the subject. Probably listed somewhere over on Stack Overflow. Why are you interested in them?

@HDE226868 working title: "How is abstract index notation different from Ricci calculus notation? "

@ACuriousMind perhaps a dumb question, but can one always find a basis for a given Hilbert space

@0celo7 Could work.

@HDE226868 I'll throw in "conceptually"

Should please the anti-homework crowd...

@0celo7 If you believe in the axiom of choice, yes.

@ACuriousMind ok, I knew that

@ACuriousMind What exactly does an element of the Fock space represent

(this is all relevant to index notation)

@alarge Its being foisted upon me to write shell scripts, and it is the worst thing ever. I'm doing a lot of things in a stupid way and I think I could save both my sanity and my thumbs by learning Regex.

@0celo7 why are you asking me, then?

@ACuriousMind to remind me!

Also I had to confirm that it's an AoC thing

@0celo7 It's a state vector.

@ACuriousMind ayy, what does that mean

Wave function?

@ACuriousMind Are the different parts of it the $n$-particle probability amplitudes?

@BernardMeurer To my understanding regexp and regular expressions are not the same thing per se (although often used to mean the same thing). From what I remember the latter is a limited subset and has all sortsa fancy math to it whereas the former is uglier, but more useful. Don't remember the details.

@0celo7 It's a quantum mechanical state vector, what do you think it means?

@HDE226868 Can your modness answer this: what textbooks use abstract index notation?

@0celo7 What "parts"?

I know only the basics to be able to use sed and have not really needed more.

@0celo7 I'm not an encyclopedia!

@ACuriousMind the fock space is $\oplus\otimes H$

so there's a "part" in $H$, in $H\otimes H$, etc.

@HDE226868 That's too bad.

@0celo7 If you cared to formulate that properly, those "parts" are the projections onto the $n$-particle spaces, which should answer your question

@ACuriousMind lol, how do you want me to formulate it properly

@alarge I was using Regex as the lazy version of Regular Expression. They seem so complex, I was trying to learn some but it was making me cry blood

@0celo7 Dunno, I find that the extra brain space is nice.

@ACuriousMind The $n$-particle space being $H^{\otimes n}$?

@0celo7 Yes

@ACuriousMind well it doesn't answer my question..

@BernardMeurer Just do some tutorials online. There was at least this one site where it had problems to filter some stuff and you had to come up with a regexp that would do the job. It was all interactive and nice. Can't remember the name, but I'm sure you can google it.

if I write an element of the Fock space as $(\psi,\psi_1,\psi_2,...)$, what exactly does $\psi_n$ tell me

i.e. what physical information can I obtain from it

@BernardMeurer I think "learn the hard way" might also have had regexp.

@0celo7 If I give you a quantum state $\lvert \psi \rangle$ (say of the harmonic oscillator), what does $\langle \psi \vert n \rangle$ tell you?

I've seen that site recommended for people wanting to learn C quite often.

@ACuriousMind Probability amplitude for the system to be in the state $|n\rangle$

@0celo7 Exactly.

@ACuriousMind ???

@0celo7 Do you not see that this is the exact same question as "what do the $\psi_n$ mean?"?

@ACuriousMind No

@alarge I found this one that seems quite nice.

Unless you're telling me that the space of states of a QHO is a fock space

If this is the case, then what is $H$?

@0celo7 The $\psi_n$ are exactly $\sum_i \langle \psi\vert \phi_{i,n}\rangle$ for $\phi_{i,n}$ a basis of the $n$-particle space.

@alarge And I believe you are refering to this book?

@ACuriousMind Ok, what is $H$

@0celo7 Huh?

What $H$?

@ACuriousMind What is the space of states for a quantum harmonic oscillator

@0celo7 What type of answer do you want? It's a seperable Hilbert space, and those are all isomorphic.

@BernardMeurer Yeah that looks about right.

@ACuriousMind I want an answer that's not that!

Is it a fock space?

@alarge I'll read it through the week, thanks!

@0celo7 Technically, yes, it is the Fock space for a "single particle space" $\mathbb{C}$.

@ACuriousMind Ok, so $H=\mathbb{C}$

jeez

I have no idea why we're suddenly talking about Fock spaces, btw

@ACuriousMind Because there's an abstract index notation for Fock spaces

wat

@ACuriousMind Well I want to include a little thing on it in my Q&A but I realized I don't remember anything about it

@ACuriousMind $\Psi=(\psi,\psi^a,\psi^{ab},...)$ is an element of the Fock space.

If those are "abstract indices", that's a stupid notation for $\psi_n$ :P

what

also, why does one take the symmetric/antisymmetric Fock space? Boson/Fermion stuff?

If I want to express that something is the 5-particle part, writing $\psi^{abcde}$ is more inconvenient than $\psi_5$ (or $\psi^{(5)}$)

@0celo7 Yep

@ACuriousMind Well Geroch and Wald say it's useful so...it's useful

QED

@BernardMeurer Not that regexps are not useful, but I guess if you're into physics and are not sure where to spend your time, you could probably learn to code some physics stuff as well. If you haven't already done so, that is.

@Slereah Will love this: Wald's source for saying abstract indices on Hilbert spaces is useful is an unpublished thing by Geroch

They were probably smoking some weed and decided it was a good idea

Being stoned doesn't make you like abstract indices, I think.

@alarge I'd love to, I just have absolutely no idea what to do in order to do so. I learn based on the challenge of doing something mostly, and since my knowledge of physics isn't exactly great I find it difficult to begin projects that mix programming and physics.

@BernardMeurer Well, stuff like box2d, right, the principles are super easy (Euler integration). So start with solving with a computer the kind of stuff you've done in class: how things fly in a parabola. Then maybe change the force to not be constant or add friction and see what happens. You might also pick up an introductory book on computational physics.

I didn't mean that you should use box2d, but rather you should implement its basic equations from scratch.

@alarge I got what you meant, loved the suggestion. Do you have any books on computational physics in mind?

@ACuriousMind It's Chicago Weed

You don't know what that stuff does to a man

@BernardMeurer Giordano & Nakanishi. I have it somewhere on my shelf and I think you might be able to find it somewhere online. From what I remember, it is quite basic (and I used it in a 1st/2nd year course). Once you have some more physics under your belt, you might go with Thijssen, which still covers a wide range of topics (after which you'll probably start to specialize in one field or another and pick a book accordingly).

@ACuriousMind thoughts?

@0celo7 Apart from the typo "does it related" in the last line, fine

@alarge I found a pdf of the book you mentioned but I'll try to get it on paper here in Brazil because I can't stand reading on a computer screen. Thanks a lot for the recommendation, I'm already going through the first chapter.

@ACuriousMind o.o did I mean "how does it relate" or "how is it related"

@ACuriousMind so it's far enough in the "physics math" category to not be closed?

@0celo7 I'd think so, yes

@BernardMeurer No problem. If you have problems understanding something (be it the mathematical notation, the physics, or the algorithms), do ask the chat here, most everything is covered during a standard undergrad curriculum (from what remember, you're not in uni yet, so obviously you may find some parts a bit puzzling).

@alarge Will do alarge, I have found people here to be far more welcoming and comprehensive than in mostly any other online chat I've been to. And you're correct, I've just finished high school and am waiting for my college apps to return; applying to study in the US.

@ACuriousMind What are the rules for draft deletion again?

Damn $145 USD

@BernardMeurer Abusive.

Don't reply, this should not become a discussion.

Abusive indeed @0celo7!

Howdy

Did you learn calculus?

Figured out integration and differentiation, head to toe

So yes?

I really don't know any other calc stuff besides those

Didn't you watch the lectures?

I can't, my internet is too crappy

dl them.

I sorta used them while also reading from sites on integrals and derivatives

you should be ok, move on to linear algebra and multivariable calculus.

I tried. "6 days remaining"

Wait really?

I haven't learned hyperbolic functions, taylor series, etc.

the details will come to you as you progress.

yeah go for it.

Wait, I got just one question

for calc

A derivative is the slope of the tangent line for a function or point on a function. The integral is the area between the x-axis and the function, right? So how are the two related?

I get how they can be found similarly, but that's all I see.

Completely different concepts to me

well do you know the fundamental theorem of calculus?

@SirCumference Uh, by the fundamental theorem of calculus?

Nope...

Oh. "The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function with the concept of the function's integral."

so like the definite integral definition.

Er, I never really looked into the definition. I just assumed that when no domain is given, I'm looking for the definite integral. Otherwise, I'd be looking for the indefinite integral...

$\int_a^b f(x) \, dx = F(b) - F(a)$

Yeah, I know that

then you know the relationship between integration and differentiation...

I'm reading the definition. I know that finding the integral is essentially doing backwards differentiation.

right the antiderivative is defined such that $F'(x) = f(x)$

But I still don't get how the two seemingly foreign concepts bear any resemblance. Shouldn't there be an obvious similarity between the slope of a tangent line and the area under the curve?

I can't find one