The h Bar

@Slereah I've gotta go, but just remember that if you vary the NG action, regardless of what connection you "think" you have, the critical points will be geodesics with respect to the unique Levi-Civita connection associated to the metric.

Slereah

I'm not sure

I know!

That is the issue

It's not like Nambu-Goto even depends on the connection

0celo7

So why would the other connection matter at all? Are you asking what action you need such that the critical points are geodesics of it?

@Slereah It (the NG action) depends on the metric, and the LC connection depends only on the metric.

Slereah

Either that or if there's a motivation for point particles to not follow geodesics in such a theory

Basically what is the proper action to use in affine gravity

0celo7

Why should point particles follow geodesics in GR? Equivalence principle?

@Slereah you need to reformulate your question and ask that, then.

Slereah

5:32 PM

Maybe!

Also if you add a term that depends on the connection in the action, the backreaction will change

and that may be an issue too

Point particles may not generate Schwarzschild solutions

(or the ultraboost garbage solution for null curves)

Hm, I wonder what's the metric generated by a spacelike curve with NG

0celo7

@Slereah what

bolbteppa

Point particles follow geodesics because GR is special relativity locally

Slereah

@0celo7 The NG action as a source terms gives the Schwarschild metric

@bolbteppa Is that true for a non-metric connection, though

I guess that may be an issue with the equivalence principle

there's that weird thing where particle rest mass can change if the connection isn't metric

bolbteppa

What does that mean, you have spacetime and you have a worldline, the action is the extremal of the arc length

Slereah

@bolbteppa But if the connection isn't metric, the extremal of the arc length isn't a geodesic

Mesentery

5:37 PM

@Blue chat.stackexchange.com/transcript/message/42179461#42179461

0celo7

@Slereah What is your motivation for affine gravity? You're throwing away the EP (else you'd get GR), so what is the actual guiding principle?

bolbteppa

How do people set up special relativity without being able to set up arc length

Slereah

@0celo7 I'm not sure

Anonymous

@Mesentery What ?

Slereah

I should look up some paper on affine gravity

See if there's a motivation behind it

other than "let's try to make the connection general"

bolbteppa

5:38 PM

I guess this is a motivation to try to read Kobayashi to see if SR can be set up with this stuff

Slereah

I'm guessing the answer is "maybe if we squeeze hard enough we'll get quantum gravity out of it"

it's the motivation of a lot of weird GR alternatives

Mesentery

@blue that's what they said in math.SE although I didn't understand any of it. So maybe the book is right

bolbteppa

You should probably set up weird SR first so that weird GR can reduce to it

Slereah

"Metric-affine gravitation theory straightforwardly comes from gauge gravitation theory where a general linear connection plays the role of a gauge field."

I guess that's probably the motivation

Emilio Pisanty

@Mesentery there is likely a typo in the second equation, it has one more factor of $\sin(\pi/n)$ than the bottom one

Slereah

5:42 PM

xxx.lanl.gov/pdf/gr-qc/9402012v1

Let's investigate

"The vain effort so far to quantize gravity is, perhaps, the strongest piece of evidence for going beyond a geometry which is dominated by the classical distance concept"

I KNEW IT

Emilio Pisanty

@Mesentery other than that, it's just the trivial identity $\displaystyle \tan(x)\equiv \frac{1}{\cot(x)}$, which comes directly from the definitions of tan and cot

Slereah

Let's squeeze that manifold for all the quantum

That paper seems a bit weird

Mesentery

@EmilioPisanty Ah! Thanks! Got it. You mean that there is one sin pi/n more in the last equation. Or maybe it should be cos pi/n instead of cot pi/n

Emilio Pisanty

@Mesentery precisely. as to how you might fix it - there's an infinite number of options, all of which depend on the context.

@hbar QFTers

0

Q: What is a gauge field exactly w.r.t. a gauge theory (QFT)

It is not really clear to me, what the term "gauge field" really encompasses. To illustrate this with an example, consider QED: I'm familiar with the canonical quantisation of the QED Lagrangian (Lorentz + Gupta-Bleuler formalism). I also know about the Gauge symmetry $A_\mu \rightarrow \ A_\mu+\...

duplicate?

@ACuriousMind @Qmechanic

Slereah

5:58 PM

xxx.lanl.gov/pdf/1110.5168v3

What the hell is this symbol

For the connection

Is it a Jesus fish

Emilio Pisanty

@Slereah the $(\!\!)$ thing?

definitely nonstandard

Slereah

yes

Emilio Pisanty

not a Jesus fish though

vertical, symmetric

it made it to the journal version, though, surprisingly enough

I agree that it's ridiculous

to quote Michael Berry,

> To emphasize the importance of notation, Robert Dingle in his graduate lectures in theoretical physics at the University of St. Andrews in Scotland would occasionally replace the letters representing variables by nameless invented squiggles, thereby inducing instant incomprehensibility.

Slereah

he should use emojis

Emilio Pisanty

(from Why are special functions special)

> Extending this one level higher, to the names of functions, just imagine how much confusion the physicist John Doe would cause if he insisted on replacing $\sin x$ by $\mathrm{doe}(x)$, even with a definition helpfully provided at the start of each paper.

I've been tempted to do this ever since

Slereah

6:05 PM

$\mathfrak{Pisanty}(x)$

Emilio Pisanty

@Slereah oh, hell, no

bolbteppa

$x(x)$

Emilio Pisanty

Fraktur can go burn in Hell

what's with that B-like $\mathfrak P$?

also, on that note

c'mon, Physics Today

what do we pay you for

you've typeset the $x$ in $\mathrm{doe}(x)$ correctly

was $\sin x$ so hard?

Slereah

6:28 PM

"Additionally, one can easily argue that within a metric-affine setting the Einstein–Hilbert form of the action is not necessarily well motivated anyway: under the assumption that the connection is the Christoffel symbol of the metric, the Einstein–Hilbert action is indeed the unique diffeomorphism invariant action which leads to second order field equations (modulo topological terms and total divergences).
However, this is not the case if the connection is allowed to be independent and it is not assumed to be symmetric: in this case there are other invariants one should in principle includ…

Balarka Sen

@EmilioPisanty dat typesetting doe

as the hip kids say nowadays

is that a hip kid

dat doe doe

6:39 PM

dat doe doe doe

Slereah

No, this is a hip kid

Emilio Pisanty

totally worth the time

ooolb

doe of x?

Emilio Pisanty

hmmmm

$$\LARGE 🦌(x)?$$

2

huh, not on this font, it seems

my phone does display it, though

but it's a deer, not a doe

ooolb

:o

lotta fake does in 2018

Slereah

6:48 PM

I prefer $$\large{\hat ☎^{-1}}$$

ooolb

ringring

Slereah

Hm, seems the telephone doesn't scale

Balarka Sen

$$\large{☎️^{+---}}$$

puts down headset harshly

0celo7

@Slereah Christian Geometry.

ooolb

$$\large{\textcolor{blue}{{{{}^o}^o}^o🐋}}$$

those were supposed to be water bubbles

Balarka Sen

6:53 PM

beautiful

ooolb

shit

i should have just used color

:'(

0celo7

@Slereah ugh what’s the chance people remember the definition of the Hodge star

ooolb

$$\large{\color{cyan}{{{{}^o}^o}^o🐋}}$$

2

Slereah

I'm not 100% sure I remember it off the top

Balarka Sen

@ooolb beautiful

10/10

ooolb

6:54 PM

thanks

Slereah

Except for "apply a big old Levi-civita tensor to it"

bc I'm a physicist

0celo7

@Slereah that’s basically it.

Slereah

Well I know how much you hate coordinates :p

Ah, there it is

$\alpha \wedge (\star \beta) = \langle \alpha, \beta \rangle \omega$

0celo7

Hmm, does the Hodge star require an orientation?

ooolb

$$\large{{}^{🐟}🐋}$$

the whale is eating a fishy

or about to

gtg bye guys

Balarka Sen

7:00 PM

@0celo7 it requires a volume form

so yes

volume form induces orientation

Emilio Pisanty

@ooolb cyan doesn't read very well

$$\Huge{\color{blue}{{{{}^o}^o}^o🐋}}$$

ooolb

$$\Huge{\color{blue}{{{{}^o}^o}^o🐋}}$$

those look pretty similar in size

$${}^{\tiny{🐋}} 🍽️ \Huge{🐟}$$

hmmmm

Slereah

@0celo7 How else do you get the big levi civitta

0celo7

7:50 PM

@Slereah write the symbols and pray no one is paying too much attention

(I’m learning the physicist tricks)

Slereah

I guess you can do the hodge star up to a sign if it's not orientable

Is there a generalization of line elements for tensors

Where you identify $T$ and $-T$

0celo7

8:13 PM

@Slereah have you found something on affine grav

@ACuriousMind what's the name for something satisfying $a^2=1$?

not idempotent

Slereah

@0celo7 found this : xxx.lanl.gov/pdf/1008.0171v3

DanielSank

9:18 PM

Hi, everybody.

CooperCape

ola

0celo7

@Slereah and what is the conclusion?

apt45

Why does a point particle break boost and translation invariance?

Captain Bohemian

9:34 PM

when the structure constants of a Lie algebra become nonconstants, what harm will be arisen?

0celo7

9:45 PM

@Slereah what's the significance of the DEC?

Slereah

@0celo7 To not have stupid matter propagation

There's the whole theorem on how DEC matter's influence doesn't leave the light cone or something

0celo7

hmm, right

isn't that in straumann

Slereah

@apt45 Here's how it breaks translation invariance : if you move away from the particle, the particle is at a different distance now

@0celo7 It's in HE at least

apt45

@Slereah So, why do we care about Lorentz invariance? Is just the vacuum Lorentz invariant? I am a bit confused

Slereah

what

Yes, only the vacuum is Lorentz invariant

It's the definition of the vacuum

QFT isn't Lorentz invariant, it's Lorentz covariant

The laws are Lorentz invariant, but initial conditions may not be

0celo7

9:50 PM

3.10 in Straumann

apt45

so, given a state $|\psi\rangle$ and a symmetry, this state transforms in a given representation of the symmetry. If I have a point particle state in $x^\mu$, let's say $|\psi(x)\rangle$, we said this breaks boosts (and then Lorentz). But, we know this states transforms accordingly to Lorentz, doesn't?

Slereah

yes

Well, to be precise

0celo7

ugh, the horror

Slereah

There's a unitary representation of the Lorentz group that acts on the state

apt45

But we said lorentz is explicitly broken;

0celo7

9:55 PM

trying to do GR without coordinates

$T^\mu{}_\nu \xi^\nu$ becomes $T(\cdot,\xi)^\sharp$

Slereah

The state isn't Lorentz invariant, yeah

On the other hand, momentum eigenvectors do have a symmetry group

Called the little group

Which is interesting to know about

0celo7

@Slereah where's the list of energy conditions

I know you have one

apt45

yes, I know

0celo7

wtf?

Slereah

@0celo7 There's a bunch of papers about them

I think Visser has one

How many do you need

0celo7

9:57 PM

I know I will be grilled on what DEC means

it's kind of a strange thing to demand

I wonder if something weaker gives me what I want

Slereah

The DEC implies like

Almost every other energy condition

It's a pretty strong one

Except the SEC and the SDEC

(Without going into the weird energy conditions, at least)

@0celo7 the hell kind of bundle is $VY$ for a manifold $Y$

0celo7

vertical subbundle of the tangent bundle, most likely

Slereah

Oh yeah

0celo7

er

of the double tangent bundle

geometry is too hard

Slereah

that it is

Let's just go into sociology

0celo7

10:17 PM

@Slereah I can't even remember how to define Lorentzian metrics

what the hell is a signature

Slereah

Eigenvalues of the metric?

0celo7

the signature of $g_{\mu\nu}(p)$ is num(pos evs) - num(neg evs)?

Slereah

yeah that's usually how it's defined

0celo7

I'm expecting at least two people to not know this stuff

most should though

Slereah

Hamiltonian is apparently applying the ~Legendre morphism~ to the Lagrangian density

$$\hat L : J^1 Y \to \bigwedge^n T^*X \otimes_Y TX \otimes_Y V^* Y$$

0celo7

10:21 PM

lol

so Christo Doulou only does nonconstrained stuff, which is...strange considering GR and EM are both constrained

Slereah

$$H = p^\lambda_i dy^i \wedge \omega_\lambda - p_i^\lambda \Gamma^i_\lambda \omega - \mathcal H_\Gamma \omega$$

What

0celo7

what's that supposed to be

Slereah

The Hamiltonian

First terp is $\pi \dot \phi$ I suppose

Also he never seems to talk about like a foliation or anything, which is weird

0celo7

what is $\omega$

@Slereah what are you reading

Slereah

cds.cern.ch/record/271943/files/9411089.pdf

$\omega$ is the volume form

0celo7

10:28 PM

oh, field theory of sections

this is in Christo

I think it's slightly more palatable

Slereah

I guess I'll download Christo

Question is, where the hell does the foliation come in

0celo7

"Given a Lagarngian density"

Slereah

What happens if you look at a Hamiltonian on some non-globally hyperbolic manifold

0celo7

spell check?

@Slereah that's an evil question

Slereah

I'm not even 100% sure if there's a time involved at all in this formulation

For all I know you could do it on any manifold

0celo7

10:31 PM

this is how Christo does things then

honestly I find this stuff very hard to read

maybe someone who lives and breathes bundles can parse it easier

Slereah

I'm already not even sure what the velocity space is and I'm on page 1

0celo7

@Slereah if you look on there seems to be way more details

I think the first bit is the goal of the lectures, it seems more sane later on

Slereah

Is it just the space of $n \times m$ matrices from the tangent space of $M$ to $N$?

Also what does that correspond to

0celo7

I'm familiar with the velocity space being Hom(TM,TN), but this is for bundles which I don't know about yet

oh are you talking about Christo?

Slereah

yes

What does that correspond to for like

0celo7

10:35 PM

an element of $\mathcal V$ is indeed a linear operator $T_pM\to T_qN$

Slereah

One particle Lagrangian

0celo7

ugh

so for a point particle you have $M=\Bbb R$ and $N=\Bbb R^3$

Slereah

Why $\Bbb R$?

0celo7

because you're looking at a theory of maps $u:M\to N$

Slereah

isn't the configuration space the position and velocity

Oh

So time and position

0celo7

10:37 PM

for instance

Slereah

So the velocity is just maps... $\Bbb R \to \Bbb R^3$?

which I guess is $\dot x(t)$

0celo7

well there's of course the meme that $T_x\Bbb R=\Bbb R$

Slereah

Is it not

0celo7

it is

Slereah

Phew

So far so good

0celo7

10:39 PM

so $\dot x(t)$ is really a differential operator at $t$

but it can just as easily be represented by a number

this choice is not canonical and thinking about it gives me a headache

Slereah

I guess for field theory it's gonna be $\mathcal M$ and $J^1 Y$

0celo7

no, the formalism for sections of fibrations is different

Slereah

then what's all this garbage for

0celo7

oh also I think Christo only does vector bundles which might be easier

Slereah

Isn't there already a perfectly fine tangent bundle fort that

0celo7

10:42 PM

@Slereah fluid mechanics is not a bundle thing

Slereah

Damn fluids

0celo7

damn GR too

Slereah

The problem with Legendre bundles is that only math people seem to do it

And as they are insane they don't seem to think time is important for Hamiltonian mechanics

0celo7

I think this is the part of math that only physicists do, actually.

What use would it have in mathematics?

Slereah

Who knows

Does any math have any use

Also is GR on $J^1 Y$ or $J^2 Y$

It has second derivatives

Should be like $J^2 O\mathcal M$

jom jom jom

0celo7

10:51 PM

I don't knoooow

Slereah

Except the configuration space isn't that because of the constraints and help why is physics so bad

0celo7

don't worry buddy I'm struggling too

I want to go back to math but I have this blasted physics talk

this is what I get for telling people I know physics

Slereah

Wouldn't $J^1 T^1_1 \mathcal M$ be the weird 56 dimensional bundle of what's his name

or close to it, anyway

0celo7

not so fast

Slereah

Hm, this paper says that matter fields are of the form $(P \times V) / G$

0celo7

10:57 PM

you want the subbundle of the symmetric 2-bundle of Lorentz metrics

@Slereah associated bundle?

Slereah

yeah

And the gauge fields are $J^1P/G$

Why does the gauge field have the jets but not the matter field

0celo7

ask ACM

Slereah

@ACuriousMind halp

0celo7

actually I bet Qmechanic would know

Transcript for