The h Bar

\mathcal is great for saying "here's some kinda abstract thing that I don't quite understand myself"

$$\mathcal{LOVE}$$

@BenNiehoff in which role it is second only to \mathfrak, of course

or it says "oops, ran out of letters"

\mathfrak strikes me as a bit more concrete than \mathcal

since we use it only for Lie algebras

and when we run out of letters

@BenNiehoff there's no such thing as "ran out of letters"

7:15 PM

haha

there is "did not plan correctly for how many symbols would come up in this thing", though

there are only so many accented letters you can write before it starts to look ridiculous

@BenNiehoff that's why God gave us subscripts

oh, no, when you use indices belonging to a bunch of different vector spaces, you run out of letters right quick

@BenNiehoff number your vector spaces and then you don't need different letters for vectors from different spaces

7:18 PM

no, I mean indices, for index notation

@BenNiehoff subscripts on your indices

ew

$T_{i_1\cdots i_n}^{j_1\cdots j_n}$

yeah, that works for that, fine

but now say you have indices belonging to 6 different vector spaces

$T_{i_1\cdots i_{n_2}}^{j_1\cdots j_{n_1}}$

@BenNiehoff example?

7:20 PM

well

say I want to do an uplift from 5d gauged SUGRA into 10d IIB

@BenNiehoff no idea what any of that means, but sure

well, that's at least two distinct index sets you need already

@BenNiehoff no, but how does this look like in practice

symbols-wise

if you take type IIB supergravity and reduce it on S^5, you get a 5d supergravity theory which is "gauged"

Latin upper & lower, Greek upper & lower, then Hebrew

7:22 PM

hold on, that's not even the half of it

cyrillic

in the 5d gauged SUGRA, there are 42 scalars that sit in an SL(6)/SO(6) coset inside of E_6(6)

Until you show me examples, imma stick with indices can totes solve it

@BenNiehoff what is $E_6$?

so, you need E_6(6) indices

the Lie group E_6

E_6(6) is a different real form with 6 non-compact directions (I think)

7:23 PM

@BenNiehoff what's the definition?

it's faster to link you to a paper, hold on

sorry, it's an old paper, and I think it's behind a paywall

inspirehep.net/record/219727

@yuggib I'm having trouble with Plancherel's theorem. I understand the proof in e.g. Conway where he extends $\mathscr F:\mathscr S\to L^2$ using the bounded linear operator theorem. Yosida defines the extension via TF of distributions $\mathscr S'$. He needs the estimate $||\hat f||\le ||f|$, and says it follows from the estimate $\int f\hat \phi \le ||f||||\phi||$ for $\phi\in\mathscr S$ using density of $\mathscr S$ in $L^2$.

but anyway, to write the E_6 Lie algebra, they use an SL(6) basis, so there are branching rules and they involve gamma matrices

so you have SL(6) indices and Spin(6) indices

I think

@yuggib What I think actually works is to take $\phi\in\mathscr S$ with $||f-\phi||<\epsilon$ and $||\phi||=||f||$. This works because $\mathscr S\cap B_R$ is dense in $B_R$, where $B_R$ is the $R$-sphere in $L^2$. Actually that doesn't work. I wanted to do $||\hat f||\le ||\hat f-\hat\phi||+||\hat\phi||=||\hat f-\hat \phi||+||f||$, but I don't have an estimate for $||\hat f-\hat \phi||$.

actually, USp(8) is in there somehow, now I'm actually forgetting!

the point is, there are a shit-ton of different portions of the alphabet that have to be used for different things

and "running out of alphabets" is not to uncommon of an occurence!

7:32 PM

what is the beta function of the alphabet?

Since the beta function is an inverse binomial coefficient in disguise, $C$?

ah, now I remember!

there are 42 scalars (from the 5d perspective) that live inside $E_{6(6)}/USp(8)$

so, you need to construct that coset

and it's a pain

($USp(8)$ is the maximal compact subgroup of $E_{6(6)}$, of course!)

How do people know this crazy lie stuff

I have no idea

from a bunch of ancient papers in the 80s, I guess

but how did they know?

This is my favorite example of lie theory

motls.blogspot.co.uk/2007/11/…

"Unfortunately, Garrett Lisi, painting himself as an E8 expert, already doesn't know that E8 has a SU(5) x SU(5) subgroup. It's kind of amazing to be ignorant about these elementary facts for a person who is pictured as an expert."

7:40 PM

lol, Lubos Motl...

I don't blame him, because the subgroups are crazy, but it's still hilarious

Only realised why Engel's theorem even arises in Lie theory like 2 days ago :(

@BenNiehoff one of the great minds in string theory, no?

I think he did a handful of moderately important things before becoming a bit of a pariah

A pariah?

I think he had a permanent position at Harvard? And then either left or was made to leave

I don't actually know, because nobody talks about it

all I know is his personality which shines through his blog :)

7:45 PM

I like his personality

A rational thinker in science which is shockingly rare

oh, he's certainly rational and brilliant

no question there

just...abrasive

I'm being slightly facetious

alas, I literally know no facts about his situation!

this summer I'm looking forward to meeting one of the other prominent physics bloggers, Sabine Hossenfelder

@BenNiehoff same

I hate not knowing, it keeps me up at night

and knowing is half the battle!

7:49 PM

Show her this and ask her to write a proper response to it motls.blogspot.ie/2004/10/…

that's from 2004, I'm sure she already knows about it

I don't think she's very pro-LQG, aside from just generically thinking we should keep our options open

lumo has been shitposting for over a decade :O

I'm actually very little aware of what she does exactly

but I think it's a variety of quantum gravity things

as for me, I'm pretty anti-LQG

I got to meet Rovelli once, he was invited to give a math seminar on LQG

As of now, that article has convinced me to ignore it as a subject, and any time I look into it I see people basically validating that article, so I stay away

I went in with an open mind, but after seeing him present it, it seemed like it was full of holes :shrug:

7:54 PM

You've met every famous physicist of the past half century

That's a little crazy

maybe it means the wrong ones are getting famous

none of these people has a Nobel

Witten has a Field's medal, I guess

Because they haven't done anything real lol

I'm going to name my next cat Witten

oh, and then there's that Milner prize, invented by some Russian oligarch more or less specifically to give out to string theorists because he likes string theory

Either that or Sobolev

^ that's some joyful use of notation there

7:56 PM

ye gods

what does it mean?

that's the Jacobi identity

are you sure? don't you need a third thing in there?

you define ${:}f{:}$ as the operator that takes $({:}f{:})(g) = [f,g]$

oh

that is totally not how I've ever seen $:f:$ used before

Lol

What the hell is that notation for

@EmilioPisanty that's sad

7:58 PM

some monkey business with $[]$ and $\{\}$, but whatever

@0celou7 well, it lets you write things like $e^{\alpha \, {:}f{:}}$

usually it means some sort of canonical ordering

say f is a function of x and p (position and momentum)

whats wrong with $e^{\alpha\{f,\cdot\}}$

toddsatogata.net/2011-USPAS/lieAlgebras.pdf there it is if you're that angry about it =P

quantum mechanically, x and p get promoted to operators, and now you have ordering ambiguities if you expand f in some series in x and p

@AccidentalFourierTransform dunno, ask Todd

7:59 PM

:...: tells you that the operators inside go in a standard order

@EmilioPisanty is that even convergent :P

but if it puts ${:}{=}{:}$ on the page then I'm on board

dammit, LaTeX, behave

@0celou7 yes

proof by because we're physicists

You're going to say that every time I raise the mathematician flag, aren't you

@0celou7 probably, yeah

though in this specific example asking about the "convergence" of $e^{\alpha \, {:}f{:}}$ is kinda pointless because it's not really defined as a series (which you probably already know), so there

@EmilioPisanty I don't really know how you want to define it.

Functional calculus?

8:04 PM

@0celou7 no, its action on $g$ is the solution of the differential equation $g' = \{f,g\}$, roughly speaking

Ohhh, I wouldn't have guessed that.

Seems like a physicist thing, is that differential equation even solvable?

it's a first order ODE

It's a PDE assuming that's a Poisson bracket

Exponential map (Riemannian geometry)

@0celou7 it's this kinda thing

In Riemannian geometry, an exponential map is a map from a subset of a tangent space TpM of a Riemannian manifold (or pseudo-Riemannian manifold) M to M itself. The (pseudo) Riemannian metric determines a canonical affine connection, and the exponential map of the (pseudo) Riemannian manifold is given by the exponential map of this connection. == Definition == Let M be a differentiable manifold and p a point of M. An affine connection on M allows one to define the notion of a geodesic through the point p. Let v ∈ TpM be a tangent vector to the manifold at p. Then there is a unique geodesi...

I think

Doubtful.

8:06 PM

true...the prime is differentiation w.r.t. time, which is a parameter in Hamiltonian mechanics

pretty standard object

What's the manifold?

What's the metric?

the manifold is phase space

What are the geodesics?

I don't think we care whether phase space has a metric, although we do care that it has a Poisson structure

8:07 PM

@BenNiehoff he's trying to exponentiate an operator on functions

In physics you treat them like the weak topology, whatever works you just use that

It's not an exponential map in the riemannian sense

@EmilioPisanty it's a physicist expression, I'm happy with that

I don't expect it to be correct or well defined

@0celou7 You've got so much joyful geometric analytical mechanics in front of you

I take it you haven't gone through VI Arnold's mechanics

@EmilioPisanty what?

I have, what about it?

Arnold is just Landau where he tries to define the domains properly

and some frills

which are hard

Plug of Arnold's teacher:

8:12 PM

I can't remember whether the Arnold book I have his mechanics or just differential geometry

and I'm too lazy to get up and check

Maybe there's a way of understanding it in terms of Hamiltonian vector fields. Not sure

yes there is

@0celou7 of course there is

f is a moment map

I think

Well, what is it?

8:13 PM

in $g' = \{f,g\}$, f is the "Hamiltonian" that generates a flow

this is also in Goldstein, it's pretty standard

If you think I don't know that, then I might as well stop asking

I'm asking why that PDE should have a solution

it's not a PDE

well, kinda

Is it not? You've got a time derivative on one side and other derivatives on the other

Maybe you can flow along f and get a solution. I don't remember how exactly to do that

@0celou7 not really

you're solving for the flow, not for $g$

i.e. you have some manifold $M$ and you're solving for $\Phi: M\times \mathbb R\to M$

its derivative at $t=0$ is $X= \frac{d\Phi}{dt}$

I don't need a lecture on geometry

8:19 PM

you want this to be equal to the hamiltonian under the symplectic identification

so you're solving for $d\Phi/dt$, i.e. not a PDE

@EmilioPisanty ok it make it sound like you were trying to solve for g

Maybe you can do that with phi, I'd have to check Arnold

@0celou7 yeah, well, solving for the flow is messy

I'm not going to argue there

and you're mostly interested on the effect on observables $g:M\to \mathbb R$

which handily enough is probably enough information to reconstruct $\Phi$ if you really want to

Ok, I read Arnold, we can talk about something else

8:21 PM

(I think)

Like atoms

Or Fourier transforms

what about 'em?

better not talk about Fourier transforms

@EmilioPisanty I'm working through the details of the Plancherel theorem and I need an estimate

@BenNiehoff why not?

OK, question

consider the operator $$\mathbf S = \sum_{\mathbf k} \frac{\hbar \mathbf k}{k}(a_{\mathbf k,+}^\dagger a_{\mathbf k,+}^\phantom{\dagger} - a_{\mathbf k,-}^\dagger a_{\mathbf k,-}^\phantom{\dagger})$$

paper claims that

> for an arbitrary unit vector $\mathbf u$, the operator $R(\alpha) = \exp(-i\alpha \mathbf u \cdot \mathbf S/\hbar)$ rotates the polarization of each $\mathbf k$-component of the field around its wave-vector $\mathbf k$ over an angle $\alpha \cos(\mathbf u\cdot \mathbf k/k)$

surely that's a typo and the angle in question is $\alpha \times \mathbf u{\cdot} \mathbf k/k$, right?

8:44 PM

strange, why would $\cos$ of something be the angle

So I want to show that $||\hat f||\le ||f||$. I can take a sequence $(\phi_n)\subset\mathscr S$ with $\phi_n\to f$ in $L^2$. I claim that $\hat f=\lim \hat \phi_n$.

@0celou7 well, there's a cos in the inner product, right?

but that would make the angle $\alpha\cos(\cos(\theta))$ for $\theta$ the angle between $\mathbf u$ and $\mathbf k$

For $\psi\in\mathscr D$, I have $(\hat f,\psi)=(f,\hat\psi)=(\lim \phi_n,\hat\psi)=\lim (\phi_n,\hat\psi)=\lim (\hat\phi_n,\psi)$

I guess I don't know that $\lim \hat\phi_n$ exists :(

it's a shame, too, it's a really important paper

tandfonline.com/doi/abs/10.1080/09500349414550911

I don't know what any of those symbols mean off the top of my head

$\mathscr S$ probably

$\mathscr D$ no idea

@EmilioPisanty that's $a\cdot b=||a||||b||\cos\theta$

the angle isn't inside of the cosine

@0celou7 no, the angle itself is $\alpha\,(\mathbf u\cdot \hat{ \mathbf k})$

8:51 PM

@EmilioPisanty $\mathscr S$ are the rapidly decreasing functions, $\mathscr D$ are the test functions

because geometry

@EmilioPisanty why that $\times$?

just to indicate a product

the angle should be $\boldsymbol u$-linear, so yeah, the $\cos$ makes no sense

Maybe I should show that $\hat\phi_n$ is a Cauchy sequence

@EmilioPisanty Ahhhh

One uses the fact that norms in $L^2$ can be computed as operator norms on the dual, which is $L^2$

And that $\mathscr S$ is dense in $L^2$, so one can compute $L^2$ norms using $\mathscr S$ functions

$$||\hat f||=\sup_{||\phi||=1,\phi\in\mathscr S}\int \hat f\phi\, dx$$

Q: Under which conditions are static links displayed by their title?

9:17 PM

@EmilioPisanty you'd love Yosida's notation

$\mathfrak D\subset\mathfrak S\subset\mathfrak O\subset\mathfrak E$

$\subset \mathfrak C$

heather

9:33 PM

@HDE226868 no need to apologize - I'm honestly just glad I didn't unload my sarcastic response =)

Slereah

9:50 PM

that is too much Fraktur

peterh

I am thinking on tachyons. With positive real mass. They would live in a Universe with 3 timelike and 1 spacelike dimensions

They would interact with ordinary matter only on nondeterministic ways. It would solve the problem of the tachyonic antitelephone.

Physics Meta

10:23 PM

0

Whenever I link a related/relevant question in a comment, I use the "share" static link of the question or answer I want to refer to. I've seen some comments where the link is replaced by the question title, but I'm not able to reproduce that behavior (without doing the [URL](link description) in...