what is a math subject that ya'll wish you learned sooner and why

Almost all of them?

Say you have a state in Hilbert space $\Psi \in \mathcal{H}$ and some unitary operator $U \in SU(2)$. Then we have that $U \Psi$ sends $\Psi$ to some other state in hilbert space $\Psi'$. Is there a way to deal with unitary transformations at the algebra level? Like there is some state in a corresponding space $ln\Psi \in ln(\mathcal{H})$ such that for some map $\exp$ we have

for $u \in \mathfrak{su}(2)$ we have $uln\Psi \mapsto ln\Psi'$ such that $\exp(uln\Psi) = \Psi'$

In other words I want a way to enact unitary transformations on vectors in Hilbert space by doing some analogous action with hermitian transformations (in the Lie algebra of the Lie group of the unitary transformations) on some object analogous to vectors in Hilbert space such that there is a mapping going between the two descriptions

@SillyGoose This cannot work. Just think of the mismatch in dimensions: consider that the elements of the lie group are matrices, the lie algebra are also matrices, but once you apply them to a quantum state vector, you are left with a vector. The exponentiation can only be done on the lie algebra level, not on the state.

The underlying idea that you want to do, however, is quite perfectly captured by Lie algebra analysis. At most, it would be those, acting on certain states, akin to Dirac's suggestion to always work in Heisenberg picture in order for the quantities in QFT to stay sensible.

Here's something that's confusing me. When I write $E^2 = p^2 + m^4$ should the mass term be interpreted as a potential for a turning point?

@MoreAnonymous Has been done before, and will continue to be done. What is the point of doing it, though?

@naturallyInconsistent sorry?

Why would you be wanting to treat the mass term as a potential / turning point?

@naturallyInconsistent cause it's the only term which is unaffected by a boost

I put a question on bounty

With a similar confusion

Oh, I will not be commenting much more on that because I think your confusion is just physically trivial, nothing much to really get out of. Sorry.

The current rewrite project of my book is even worse than the previous one

I have learned too many GR things by now

yay? progress!

Well mostly I decided to start anew

By now I have a different feel about how you should construct spacetime

I don't think we should be so ambitious as to construct spacetime

sounds like something we should leave for the theologians to do

/s

@naturallyInconsistent sure

@Slereah You have a book on GR?

Where can I buy it from?

I have an attempt at writing a book on GR

Well if you want to finance it you are certainly welcome to it

I could probably do it faster without having to work

Maybe I should have a kickstarter

Big spenders get a spacetime named after them

I'm writing a text on quantum. Starting with QFT directly. So much work to write...

May I suggest writing a blog that eventually may grow into a book.

Already doing that

✅

The hardest part of doing that book is doing consistent notation that will not overlap

Boy how many things are called f

@Slereah Impossible to solve. Too many quantities, too few symbols to play with.

I mean I could technically do it like some authors have done, by doing really long and ugly names

But I also try to keep the notation mostly similar to standard notation

I'm not doing the nlab thing of calling things SimpSmoothCartSp

@Slereah I would actually consider this. For what level is it for?

I mean consider funding

Can funders get a sample chapter?

you can look at the old version as it is :

samuel-lereah.com/Spacetime.pdf

thanks

Im at office

But I'll definitely look at it

Would u recommend that book for someone who wants to refresh and extend their GR reach?

@naturallyInconsistent Personally I had found QFT super confusing until I learned how to go back and forth between QFT and QM

I mostly wanted to try to include as much demonstrations and properties of GR as possible, if possible in an order that makes sense

Without skipping on proofs

and trying to motivate things properly

Although this is probably not a good introduction

@Slereah I would be up for that

The current split is basically a big section on geometry (just spacetime as a manifold with no dynamics), then dynamics (in general and then applied to GR), how matter behaves on spacetime, the Cauchy problem, quantum theory in GR, quantum gravity, measurement related stuff, experimental stuff, and history

What would you suggest as prerequisite required reading?

Probably some diff. geom. and GR

I try to introduce all concepts myself but I doubt it would be good pedagogically

I try to introduce things in a "logical" order so bundle connections are defined before the covariant derivative

Skimmed a bit

Seems quite mathematical

Would have liked some more examples. Also no exercises ?

It's still in early stages :p

And probably no exercizes

I do it all :V

lol

I don't think the conversion rate translates to much. But I'd still like to fund. Seems like a worthy cause.

Lemme know when u start a kickstarter @Slereah

I'll probably wait until there's at least one presentable chapter :p

@Slereah good call :P

First chapter is supposed to be a motivation of why spacetime can be modelled by set theory

and the alternatives

Looks like your book will be about as thick as MTW

@naturallyInconsistent This is the best book of QFT i've read so far. Quantum Field Theory: The Why, What and How

@naturallyInconsistent I bet the final version will be thicker

@Slereah is there gonna be a section on Mach's princple?

Thanu's Book is motivated for someone like me :P

I do have a section planned on philosophy-related issues

So probably

not looking forward to doing the section on algebraic topology😔

My understanding of Mach's principle is as follows: kinematically you can similar situations. But dynamically different. It still confuses me till date.

Mach's principle is unfortunately not gonna be not confusing because there's like ten different interpretations of it

Most of which don't really apply to GR anyway

why does wikipedia say $SU(2)$ is used as the group for quantum spin due to being the double cover of $SO(3)$

as in why is being the double cover of $SO(3)$ important for a group to be identified as the spin group

@MoreAnonymous Just looking at the page counts and it just seems too short.

@SillyGoose I mean what do you think the spin group is

well that is the confusion i am trying to resolve :P

im not sure what the criteria are for defining a group to be the spin group

@naturallyInconsistent it's not an introductory text but more for someone who has had a course but is still confused like me :p

@MoreAnonymous Still unlikely to be helpful. Just covering purely the conceptual issues alone would take a tome.

@naturallyInconsistent i mean solely relying on one qft book will take one to a time for qft.

*tomb

Im saying that the book Ive been reading, to cover conceptual issues, took about 200 pages just to cover the properties of QFT so that Lorentz, locality, clustering, etc, are incorporated, and why second quantisation becomes the natural way to get working stuff, i.e. LSZ and stuff. No part of that is wasted on silly "free propagator" textbook stuff. And that is before we get into interactions.

That is why I am sceptical that a good text would be able to squeeze conceptual issues down to so few pages and still be useful.

what other group theoretic topics (at the level of undergraduate and for the purpose of understanding textbook non-rel quantum mechanics) would you include here?

Seems fine?

Okay I added Wigner's theorem

@SillyGoose What is the intended audience? In particular, are they coming from maths or from physics background?

the intended audience is coming from a physics background (working through latter parts of sakurai) and who is mathematically inclined (better than me :P) but just has never had the chance to take an intro group theory course, rep course, lie theory course, etc.

i want to make a very straight path towards understanding the business surrounding the irreducible representations of SU(2) since i think that would be quite cool to know; also i need to learn more about this topic

more precisely i think it is cool because my impression is that one understands irred reps => one understands a method of how to characterize states w.r.t. certain properties => gateway to understanding things like the standard model. and the spin case is a very easy way to see this?

but also to throw in additional group theory related topics (e.g. Wigner's theorem) that are pretty pervasive and essential and that i can still understand :P

it is meant to be 8 1.5 hour lectures worth of material (more informal than a lecture of course)

Well, if that is what you want to do, then you should be doing Wigner and Weyl approaches to characterising all the possible Poincaré invariant wave equations, and that is only natural in the relativistic case. Non-rel just does not cover these.

And you would also be in a good position to cover spin-stats if that is the case

Are you allowed to assume that the maths peeps would have already done Sturm-Liouville?

Ah hm then probably would not go that route; but is using this example of irred reps too much of a stretch to compare it to what is done with the standard model?

well i have not done Sturm-Liouville hehe so that would probably be off the table

Well, you do want Sturm-Liouville, because that is the connection between Schrödinger and Hilbert spaces, what maths peeps care quite deeply about.

@SillyGoose I think it is an ok thing to connect, but I can tell you that maths peeps would be "I understand this, I understand that, but I do not understand how these two come together". That is what they have always said when we physicists try to motivate things. Communication across the chasm is always a difficult thing.

I should really do a LaTeX shortcut for \emph{x}\index{x}

I do it a lot

Well, I define \schroedinger, \coordinate, \op, \bra, \ket, \braket, \extd, etc, and the front two naturally take one argument to use as suffix, so that my code is nicer to read. My obsession with niceness in the code borders upon OCD, though there is not much that is actually compulsive in that.

hm I see okay then maybe I will try to look into S-L

One very important shortcut I do is \int_\infty^\infty as \i

That one is quite a time saver

Also, is it accurate to say that spin, orbital angular momentum, and angular momentum (as in spin + orbital) are all representations of $SU(2)$, just different representations

@Slereah I use cool package. Look it up on CTAN

because their lie algebra commutators are all that of SU(2)

@SillyGoose No, because often it is not a SU(2) representation.

They all transform under some rep of SU(2) yes

Well you can do it with SO(3) yeah

But they all have a spinor representation

I have been confused about this point for a while. Let's say I am doing a quantum mechanics problem involving a spin-1/2 particle with some orbibtal angular momentum, so we are dealing with $J = S + L$ angular momentum. And I want to represent the $J$ operator; what do I write? Is it really $\hat{J} = \hat{S} + \hat{L}$?

The basic orbital angular momentum representation is the spherical harmonics, and that is a U(1) representation. Of course, you use that and the basic SU(2) pure spin fundamentals to generate the SU(2) spin spherical harmonics. When you combine more stuff, you get SU(3) and above. You should not be thinking that they are all just SU(2). The physics is easier to understand as separable total angular momentum subspaces, than as combinations of SU(2).

hm is there a text which talks about how to do this more in depth?

@SillyGoose Actually it is $\vec{\hat J}=\vec{\hat L}\otimes\hat{\mathbb I}+\hat{\mathbb I}\otimes\vec{\hat S}$

Hm now why do we separate the degrees of freedom?

@SillyGoose What do you mean by this question?

Well the spin and orbital angular momentum of a system should be describable using one hilbert space, that of the system

or at least i would imagine

and?

well the tensor product of operators you wrote acts on a composite of two hilbert spaces

so then $J = S + L : \mathcal{H} \rightarrow \mathcal{H}$ (exactly as written) is wrong? What is always meant is that implicitly separate spin from orbital angular momentum and define the operator as you did above?

and to recover what i would call the "actual system space" (not separated into two tensor factors), could we then find irreducible representations of $J$ and look at the resulting direct sum of invariant subspaces?

It is probably not natural to split the spin and space parts; the issue is that physically we can only start with experimentally and theoretically studying them initially as separate things. And as we have seen, space part is at minimum a U(1) representation, whereas the minimum non-trivial spin representation is SU(2), which is thus naturally separate topics to cover.

It is more logically coherent to treat them separately and then combine them. Students would likely be completely lost if we do not start with simple things and then combine them.

Hm but that shouldn't change the answer to how things really are. Since I am wondering what $J$ looks like as defined over solely the system Hilbert space. Although the above still cleared up a confusion I hadn't thought about

the wikipedia seems quite dubious :P

since i think it means to say that the spin rotation acts on spin degrees of freedom and the spatial acts on spatial

where in reality it looks like you are normally multiplying the operators together

@SillyGoose But that is immediately going to mire you in what you consider as the system. Do you do just the orbital and spin parts of the electron? Which electron if you have more than one? Do you include the nucleus? How are you going to expect to teach students if you deal with them all right off the bat?

okay i see that; but let's focus on the case in which we want to focus on the spin and orbital AM of the electron and just those degrees of freedom

Sure, then one has to ask, why only those? What do you mean this is the system Hilbert space? In particular, why are you allowed to just ignore the nucleus in the system Hilbert space?

Questions of this sort can go on forever

well this question is a personal question of mine heh

one learns to exercise judgment

@SillyGoose applies judgement eyes on the goose

Trying to check what the theorem guaranteeing that manifolds are smoothable is, and wikipedia just says "some theorem due to Whitney"

XD

and it's not in the bibliography >:|

@Slereah cosmic iron theorem?

there is plenty indeed left unclear in these tomes of quantum mechanics still heh

although perhaps that is a sign of learning :P recognition of insufficiencies

@SillyGoose Less than a century old, do not even attempt to present the theory as understood.

well not even that there are questions left unanswered in that sense. take for example sakurai's chapter on identical particles. it seems a notational trick that pops out tautological results that there exist bosons and fermions

I would not think of it that way. It is a tremendous discovery that particle indistinguishability is even a possible thing at all.

the concept is interesting, but sakurai's presentation of identical states seems purely pedagogical and not accurate; though i of course am more probable to be wrong than sakurai

i don't mean to say the results stated are wrong.

It might also be that when we try to be glib and make things appear smooth to students, the resulting treatment becomes impossible for mortals to understand.

here is a critical response to using indices in the way sakurai does, at least to my understanding

from this paper ifi.unicamp.br/~cabrera/teaching/referencia.pdf

oh oops i suppsoe i cut off part of t he criticism

@naturallyInconsistent at what level are u?

@naturallyInconsistent or to keep all their secrets to themselves ;)

@MoreAnonymous Your question lacks detail or clarity. Where is the flagging when we need them?

@SillyGoose I think that particle indistinguishability is a thing that should only be mentioned in QM text and really is a stat therm topic. There, and only there, can you really absorb the shockingly profound consequences.

does an UG stat mech course cover such material?

@SillyGoose Strongly dependent upon location.

But in general, yes

i am excited at the least to work through weinberg's derivation of (in the usual scenario) the only possibilities of particles being bosons or fermions :). my professor made it out to sound far more satisfying than the dubious indices of sakurai

oh interesting i shall look forward to that as well

it seems identical particles in the classical case is interesting in stat mech as well but i shall see i suppose

@SillyGoose That is why I talked about spin stats thm earlier. You need that for that.

hm I suppose I should work up to that then

@SillyGoose Failure to take into account that particles are indistinguishable will result in pV=NkT losing N. Since N has Avogadro's const in it, if you miss that, then you cannot even breathe.

Lol

And realise how fucked it is: We are talking about ideal gas

When we say that our universe is quantum, we really mean that a classical description is impossible to be correct even in principle. We may only model random bits and pieces classically if we ignore the inconsistencies that are always threatening to appear.

what will the next physical theory be named

@naturallyInconsistent ah i meant in the academic ladder

@MoreAnonymous worm. wormmm

it seems like stat mech kind of really blows classical up :P

@naturallyInconsistent ???

isn't the bohr vanleevan (I cannot remember her name) another instance where classical physics just simply cannot be true if stat mech is true

Van Leeuwen

@MoreAnonymous What are you asking that for?

@naturallyInconsistent like have you done a undergraduate,postgraduate, PhD, postdoc,etc?

@MoreAnonymous I know you are asking about that. It is very clear that I am postgrad and above; I'm asking why you are asking about this.

is there a way to show that invariant subspaces induced by irreducible representations are disjoint using an equivalence relation

i know that the orbit of an element is an equivalence class, but an invariant subspace is not just an orbit

though perhaps one can work off of this more basic structure

(i would like to prove that invariant subspaces are disjoint by proving that an invariant subspace is an equivalence class; and the fact that an equivalence relation on a set induces a partition of a set will yield disjointness)

@Slereah Sweet!

hm the orbit business will not work. it needs to be an equiv relation defined over the whole Hilbert space

@naturallyInconsistent ah i was just curious. I was wondering at what level of the academic level does one feel confident enough to write a book

Duffield wrote a book and he has no qualifications

The sky's the limit

@Slereah fair point :P

@Amit : Thanks.

Let see how many of u react to it it is just 4 min will get u to divine knowledge of mathematics

Ans phy please do have watch for 20 sec and share ur thought

I already read ancient greek math books, I'm all set for divine mathematics

@Slereah hi please do watch the video

Amazing India 🇮🇳👍

all these videos online about coffee machines and best ways to brew, etc. but no where near as many for tea

Actually writing a book is a great way to learn a subject

Lecture notes are not so different than a really diligent student notebook. Expanded lecture notes are basically a textbook. But a good textbook is more than that of course :) But many books don't go past the level of expanded lecture notes

True.

trying to type up notes meant for someone else really shows me holes in my knwoeldge :P

@Amit qft or gr?

I don't understand..?

@Amit ignore me. I misread your sentence

Hello

Hi

Hello everyone, If one asks forcefully, what is the partition function for microc. ensemble, is it ok to say Ω ( the no. of microstates)?

Because partition function in canonical and grandcanonical is defined as the normalization factor for the probability distribution function, and in microc. , Probability distribution function is 1/Ω . So in a sense, Ω is the normalization factor. How do you people see this?

@Slereah Another net split?

Do u think the quantum framework will not hold for a Quantum gravity theory

Hilbert space and Hamiltonian and Born Rule and stuff

Currently writing a it about the alternatives for the local structure of GR

Here's a terrible idea

Is there a p-adic version of GR

u need a p-adic spacetime topology?

Idk how it cud b relevant 4 GR

The geometry being based on p-adic numbers instead of real numbers

Well you can replace Rn with various other fields for GR

I've seen Q and finite fields being used

Yeah. So u usually hav a manifold and u hav the usual topology on given by euclidean metric

The other obvious candidate would be p-adic numbers

But u need the manifold 2 hav the p adic topology

Write better

Sorry

Ok so if you attribute a p-adic topology to the manifold, then points with far distant co-ordinates in the usual sense would be identified as being close

arxiv.org/pdf/1705.04758.pdf

oh god it's real

It may have some applications in infinite sums

one of the papers propose that the type of number field involved could be subject to quantum fluctuation

Academic publishing is ridiculous in many ways

I'm very uncomfortable with these ideas. They seem like too nerdy mathematics

I hope the real world doesnt hav this stuff

if these ideas come true, it would be like re-inventing the wheel. Very uncomfortable.

So far, the math of physics has been intuitive and attached to reality

@Slereah got any good resources? The p-adic thing broke my brain, but I'm curious about something other than reals being used as the underlying numbers

@DanielUnderwood hal.science/hal-00833213/document

Oh wait wrong paper

philsci-archive.pitt.edu/17234/1/structure.pdf

that's the one

3.1 has all the weird alternatives

p-adic numbers also re-define addition. But we shud note that the usual notion of addition is not arbitrary. In the usual notion, 2+2 is defined to be 4, etc. This definition doesnt allow 2+4+8+16... to converge

The usual notion is designed to model computations. It does not have some arbitrary dependence on topology.

Natural numbers and addition are much more primitive than topology ideas. They r fundamental to computation. This is y physics uses them. It's not an arbitrary choice

I saw someone saying that 1+2+3+..=-1/12 has applications in physics because nature is implementing some "other" definition of addition

But it's not true. Any applications of sums like this in physics is always justified by usual addition. We do not re-define addition in an ad-hoc way

@RyderRude Feyeraband said "anything goes" :) I interpreted it to mean, in this example say, if you model nature with a weird addition rule and it agrees with experiment, cool...

I won't apply the same principle to ethics, that's relativism, buh :)

@Slereah this is the second time I've looked at a philosophy paper this week...it feels dirty

@Amit yes but that model shud have the weird addition in its axioms. If we re-define addition whenever we get a wrong prediction, we cud get any answer we like. So ad-hoc stuff shudnt be allowed

in QFT, the use of these sums is justified by regularisation and renormalisation. There is no ad-hoc business. The theories use the usual definition of addition in its axioms

I think there's no restriction in putting weird addition into the axioms of a model. But we shud also note that the usual notion of addition is not arbitrary. It's as primitive as math stuff can get

The problem is that usually ad hoc stuff won't generalize, it's patch work by definition. There's no problem however to begin from a ad hoc proposal if you take care to see how far it generalizes and what are its limitations, and include it in your paper if you gonna publish it :)

That paper doesn't include the finite field spacetime otoh

what's the lagrangian of a cat

A furrier transformation?

Oh come on that's an insane ability :P

lol, thanks

Funny, I googled "furrier transformation" and got that xkcd one as the first result, where he transforms his cat, but i don't think he came up with that phrase

Yeah -- just verified he didn't. See I was on my phone, and couldn't see the tooltip you get when you hover over this so I just verified that it wasn't in the tooltip and I just subliminally remembered it :)

@Amit To see the xkcd title text on mobile, use the mobile site: m.xkcd.com/26

@PM2Ring Sweet! Thanks

No worries. :)

There's also a JSON interface: xkcd.com/json.html

Ugh ive been thinking about this since morning and getting nowhere

What is the minimum potential energy required (to behave as a turning point) for an elastic collision between $2$ point particles $A$ and $B$ with velocity $v_A^{\mu}$ and $v_B^{\mu}$ in a relativistic classical mechanics?

any thoughts would be welcome

@JackRod Most of this is blatant misinformation or at least not substantiated by actual historical evidence. A nonexhaustive list of errors: 1. The use of 360° degrees in the Western tradition is generally thought to come from the Babylonians, not the Indians, cf. hsm.stackexchange.com/a/1885/3797.

2. There is no generally accepted etymology for the Arabic term al-jabr, but "al" is just the definite article in Arabic languages and does not "stand for anything holy", and the only attempt at an etymology for "jabr" I could find posits it comes from an older Assyrian word (much more plausible since Assyrian is also a Semitic language).

3. Hindi "jyamiti" is a borrow from the English word geometry (cf. en.wiktionary.org/wiki/…), which comes from a Greek compound "earth" + "measure"

although I cannot find a source that this actually came via English and it might just come from Sanskrit via the same compound formation (since Sanskrit is Indo-European like Greek, there is no need for one language to borrow this term from another there to explain the similar sound)

Linguistics in general and etymology in particular is so fascinating and it really annoys me when it's done poorly, especially to "support" dubious historical claims

It is unfortunate that so many of our big physics fundamental principles were made with point particles in mind

can be tough to generalize

When did you write that book? @Slereah

few years back

I want to write a book too :P

Something about differential geometry in Physics

More specifically in gauge theories

@SillyGoose This holds for stuff that I've learned, I'm learning or I'm planning to learn. Differential geometry, Lie theory, Complex Analysis, Category Theory, Topology and Differential Topology, Algebraic topology, Functional Analysis and I think that finishes my list

do we really need more differential geometry books for physics

@Slereah I think this is the most ambitious GR book.. I didn't read much of it, but I liked the overall idea of writing such a book and it did seem good from what I read... it's just that OT1H I wanted to read it all, OT2H it takes so long to get to where it isn't too elementary for me

we don't need another introduction to EM or whatever

lol

I disagree... as I said I think writing can be for yourself, for studying :) who cares if someone reads it... (it's nice if anyone does... but that ain't necessarily the goal)

even better

I want to read something wild, too

read ST textbooks?? :hides:

find the worst formalism you've heard of and write a book about it

Maybe you can read about Newton's adventures in Alchemy, sounds right up your alley

@Slereah I would write it for myself mostly :P

In SR, do you need to know about the Lorentz transformation to calculate coordinate time? Or is a time synchronization method between you and whatever object coordinates you're measuring sufficient? I think the latter is true...

You can just use a single set of coordinates

@Slereah I prefer to find the best formalism with books written in a way I don't like and "fix" it

The good ones will usually have like 40 books about it

Also, writing Physics books isn't something that appeals to me as much as writing Math books

@Slereah Right, I think I just got confused. It doesn't require the LT at all

Physics books are chaotic

And coming up with problems is a wild challenge

I only come up with solutions

@Slereah I come up with neither

@Mr.Feynman You happen to know this guy? It's a long shot, but he's your compatriot so I thought why not ask :) and these lectures are great, anyway

Wait, I'm not sure I'm right about that, lol

I am probably awful at identifying accents :)

Anyway nvm, forget the question, pretty sure I'm wrong :) But I can't regret linking these lectures so I suppose no harm done :D

@Amit that's my uncle

lmao

Just kidding

good one

I googled a bit and that guy is not Italian

Judging from his name he's from a Spanish speaking country

yep lol

could you tell from his accent too?

Origins of Mathematical Words: A Comprehensive Dictionary of Latin, Greek, and Arabic Roots

Honk

in Lie theory lingo, what are the concepts surrounding “the irreps of SU(2) over complex VS H are invariant subspaces characterized by an eigenvalue?”

Is it roots and weights

As in is spin-1/2 irrep associated with a weight? Or something

@SillyGoose Roots and weights are indeed the generalization of what happens in the representation theory of $\mathfrak{su}(2)$. A Cartan subalgebra here is just the one-dimensional algebra $\mathbb{C}$ spanned by e.g. $J_z$, and so a weight is a linear map $\mathbb{C}\to\mathbb{C}$, which is equivalent to just a number $s\in\mathbb{C}, x\mapsto sx$. If you work out what the definition of "dominant algebraically integral weight" means concretely here, you end up with $s\in \frac{1}{2}\mathbb{Z}$

then the theorem of the highest weight reproduces the usual classification that irreps of $\mathfrak{su}(2)$ are given by half-integer spins.

if you unpack what the definitions mean, the definition of a weight is essentially just "a choice of eigenvalues for each basis element of a Cartan algebra"

ah okay cool :DD