The h Bar

Slereah

08:21

"Bram then additionally says that $|\uparrow ,k \rangle$ is disallowed, because its BRST-momentum is non-vanishing, and its pairing with a genuine physical state results in a delta function. This, for whatever reason, “violates kinematic law.” I don’t understand this."

Not reassuring

Slereah

08:44

It bothers me that so many sources say that the constraint $p^2 + m^2 = 0$ is natural

It is very much not, that's a choice of gauge!!!

It's the proper time parametrization certainly but that's not the only one possible

That's why you add it to the action, to force the gauge onto the system

Physics Meta

0

Q: Resource recommendation for Physics

Like this recent post on Chemistry SE which is the collection for different types of resources for learning Chemistry ( which I find quite useful) do we have any such thing over here at Physics SE? If not, then why not? Is it possible do it( if it isn't already available)? What I think is that ...

bolbteppa

09:35

@Slereah but the action is naturally just minimizing proper time, so the proper time parametrization is natural

Slereah

@bolbteppa I mean sure, but it's certainly not stemming naturally from it

You could decide $$p^2 + m^2 = f(\tau)$$

It is entirely fine

There's no mechanical process which produces $p^2 + m^2 = 0$ from the action

is my point

You have to know in advance about proper time parametrization

Is there even a proper way to compute whether a constraint is correct from the get go?

Or do you just have to check the determinant of the hessian of the constrained action

bolbteppa

Where does $p^2 + m^2 = f(\tau)$ come from

Slereah

Well the proper one is $$p^2 + m^2 = [f'(\tau)]^2$$

A change of the parametrization of $x$ produces such a constraint

Or something like that?

I forget

bolbteppa

The action is invariant under reparametrizations so idk how this extra factor comes in

Slereah

The constraint is here to fix the parametrization

Therefore different constraints fix different parametrizations

The proper time parametrization basically fixes $\dot{x}^2 = -1$

But if you switch parametrization, it becomes $$\dot{x}^2 = [f'(\tau)]^2$$

Oh wait

Hm

Am I wrong

I may be

bolbteppa

09:44

$S = - m \int \sqrt{-g_{\mu \nu} \frac{dx^{\mu}}{d \tau} \frac{dx^{\nu}}{d \tau}} d \tau = - m \int \sqrt{-g_{\mu \nu} \frac{dx^{\mu}}{d \tau'} \frac{dx^{\nu}}{d \tau'} (\frac{d \tau'}{d \tau})^2 } \frac{d \tau}{d \tau'} d \tau' = m \int \sqrt{-g_{\mu \nu} \frac{dx^{\mu}}{d \tau'} \frac{dx^{\nu}}{d \tau'}} d \tau'$ both will have $p^2 + m^2 = 0$

Hmm

Slereah

\begin{eqnarray}
p^2 &=& p^\mu p_\mu\\
&=& m^2 \frac{\dot{X}^2}{(\sqrt{\dot{X}^2})^2}\\
&=& m^2 \frac{[f'(\tau)]^2}{ \sqrt{[f'(\tau)]^2}}\\
&=& m^2
\end{eqnarray}

Dang it :V

Nevermind that

How do you fix the gauge

I've seen it done for Polyakov

$$S_G = \int d\tau \lambda(\tau) \ln(e(\tau))$$

bolbteppa

Well

I think because $S = - m \int ds$ possesses constraints you have to write it as a constrained action first, and then fixing the Lagrange multiplier is equivalent to fixing a gauge

Slereah

Hm wait

Let's see

bolbteppa

That book with the picture you posted also says setting $\dot{x}^0 = 1$ is a choice of gauge so it's not just fixing the Lagrange multiplier

Slereah

Well yes that is the obvious gauge

bolbteppa

09:51

hmm

Slereah

the Coulomb gauge of the point particle :p

not very covariant

bolbteppa

So fixing a gauge is literally choosing a parametrization, one way to do it is just to set $\dot{x}^0 = 1$, another way is to force the Lagrange multiplier to have a specific form, because it is a function of $\tau$ which changes under parametrizations in a certain way so fixing it fixes the parametrization

Slereah

Apparently the proper gauge fixation is $$\dot{\lambda} = \xi(x, p, \lambda)$$

Fairly wide gauge

Hm wait, what's the EoM for $\lambda$ anyway

You have $$S = \int \dot{x} p + \lambda (p^2 + m^2)$$

bolbteppa

The EOM are $p^2 + m^2 = 0$

Slereah

So wrt $p$ that's $$\dot{x} + \lambda p = 0$$

bolbteppa

10:03

If you find the EOM for $p$ from that action and use them to eliminate $p$ you get the einbein action

Slereah

That is indeed the parametrization

But it doesn't fix it

I guess it is basically the einbein, yes

So... $$S_G = \int d\tau \alpha(\tau) \ln(\lambda)$$

Maybe???

bolbteppa

But now you set $p = - \dot{x}/\lambda$ and plug it into $S = \int \dot{x} p + \lambda(p^2 + m^2)$ to get $S = \frac{1}{2} \int d \tau e (e^{-2} \dot{x}^2 - m^2)$ i.e. a 1D GR action with metric $e$

Slereah

I'd need to recheck the EoM for $\lambda$ tho

Wait no

That ain't working

bolbteppa

This is how you get Polyakov from NG btw if you do all this in 2D but it's pretty complicated writing out the metric

Slereah

Yeah that seems obvious enough

bolbteppa

10:07

So string books usually just say assume the 2D version of $S$ takes that form and use the EM tensor to show you get the NG action back

Slereah

Just trying to think about how you fix the gauge

Do you just decide it by fiat?

Or can you write down a sufficiently bespoke Lagrangian

You can write down a gauge fixing term for EM

bolbteppa

In EM you still have freedom though

Slereah

True

I hate gauge theory so much

bolbteppa

All of physics is basically gauge theory :p

The game is ignoring the subtleties just enough but not too much lest you descend into nlab madness

Slereah

It's too late

I have caught gauge fever

bolbteppa

10:21

Still need to work backwards from EM Gupta-Bleuler to BRST language to see it as natural

Slereah

BRST seems natural enough using the proof with path integrals

it's a bit trickier for canonical quantization tho

I shudder to think about BRST quantization of the Einstein Hilbert action

spooky metric ghosts

sci-hub.tw/https://link.springer.com/chapter/10.1007/…

aaaaaaah

Wait I think the gauge fixing is just the einbein constraint

$\alpha(\tau) p_e(\tau)$

bolbteppa

10:36

From $S = - m \int ds$ we have $p^2 + m^2 = 0$ is a primary constraint and so $S = \int d \tau [\dot{x} p - H + \frac{1}{2} e(\tau)(p^2 + m^2)] = \int d \tau [\dot{x} p + \frac{1}{2}e(\tau)(p^2 + m^2)]$ is the constrained action, this action must be parametrization invariant which implies $e(\tau)$ has to change in a certain way under reparametrizations. Thus fixing $e(\tau)$ fixes the parametrization i.e. fixes the gauge.

Solving the $p$ EOM for $p$ gives $S = \frac{1}{2} \int d \tau e(\frac{1}{2} e^{-2} \dot{x}^2 - m^2)$ and again fixing $e$ fixes the parametrization.

Slereah

10:49

I should do something simple to relax, rly

compute apples falling from trees

You know what I haven't done in forever

Optics

Also thermodynamics

Let's do a thermo page on my site

make a little engine

choo choo

Slereah

11:22

though to be fair

doing it properly requires contact manifolds

yuvraj singh

11:51

Any body can answer my question which I posted on the site, astronomy.stackexchange.com/questions/33748/…

ChoMedit

12:13

What is Dirac Quantization Condition? Roughly, we can say that it is $q_m q_e \in 2\pi \Z$ for eletric charge $q_e$ and magnetic charge $q_m$. But why someone said the existence of monopole implies quantization of electric charge, without any assumption, for instance, "If there exists a quantized monopole.."?

Did I miss something?

Slereah

Bad news, it's fiber bundles

The magnetic charge is related to the structure group of the EM principal bundle

ChoMedit

Then, Is magnetic charge always quantized?

if exists

Slereah

If it stems from an EM gauge theory, yes

ChoMedit

Okay thanks. I have seen most people omit that kind of hard stuff and just summarize the result without any caution about this. I think.. they need to mention it..

Slereah

You can do it without bundles by doing the whole Dirac string derivation

Though it's not quite as satisfying

ChoMedit

12:24

So, do you mean that the quantization of monopole can be shown from Dirac's original argument?

If yes,,, then I miss something.

Slereah

Well it's a bit of a fairly specific case

if you want a more general argument it is best to do it via gauge theory yes

amslaurea.unibo.it/16776/1/tesi_mazzoni_finale_1.pdf

stuff like that

Slereah

12:48

I've never actually worked out how geometric optics stems from EM

The ~Eikonal approximation~

bolbteppa

13:01

Landau gets it in 2 ways pretty fast

Abhas Kumar Sinha

14:50

@bolbteppa hello

Is there any easy and quick way to convert Laplace operator to spherical coordinates

?

bolbteppa

@AbhasKumarSinha yes see equation 16 and 24 here jfoadi.me.uk/documents/lecture_mathphys2_05.pdf

Abhas Kumar Sinha

@bolbteppa Gradient, Curl and Divergence, which one is laplace?

Slereah

@AbhasKumarSinha Laplace Beltrami operator

bolbteppa

Laplace is 'divergence of a gradient' as explained above 16 where they then state 16 (which is Laplace)

Abhas Kumar Sinha

@Slereah is that related?

Slereah

14:54

$$\Box = g^{\mu\nu} \nabla_\mu \partial_\nu$$

just gonna use the metric tensor of spherical coordinates

Abhas Kumar Sinha

@Slereah Tensor is verrrrrrry high-level thing.

bolbteppa

@AbhasKumarSinha read that pdf, it sets everything up from the beginning to make it easier and will help prep you for tensors

Abhas Kumar Sinha

@bolbteppa Is tensor required? What can I do with it?

bolbteppa

When you know tensors very well you can get the formula even faster than as explained in the pdf I sent you

Abhas Kumar Sinha

@bolbteppa Oh okay... Can it help me in high school mechanics or mathematics (like matrices or vectors)?

bolbteppa

14:58

Vectors are 'rank 1' tensors, matrices are 'rank 2 tensors', you will gain experience as you study more and more advanced things one step at a time

Abhas Kumar Sinha

@bolbteppa Okay.... nth rank vectors are multidimensional, so how do you represent them on paper?

bolbteppa

A vector can be represented as $v_i$, a rank 1 tensor (one index), a matrix can be represented as $M_{ij}$, this is a rank 2 tensor (2 indices), you can guess how to represent higher rank tensors

Abhas Kumar Sinha

@bolbteppa are tensors hard?

bolbteppa

It depends on what you want to do with them

Abhas Kumar Sinha

@bolbteppa My teacher defined moment of inertia as 7th component of tensor, does that make sense?

bolbteppa

15:06

Moment of inertia is a 2nd rank tensor, and it's components give the MOI so I guess that's what they meant

Abhas Kumar Sinha

@bolbteppa Well that pdf is well written... I finally understood what is curl

@bolbteppa Probably something like that...

Slereah

If you want some very bad ways to get the Laplacian I can also do that

using

BAD MATH

Dun dun duuun

bolbteppa

What's a bad math way

Abhas Kumar Sinha

@Slereah Show your bad math way.

Slereah

Pretending $\partial x$ is a quantity

Abhas Kumar Sinha

15:08

dun duun duuuun....

@Slereah okay

Slereah

I mean it's basically the chain rule

Abhas Kumar Sinha

@Slereah Those stunts are meant to be performed by the experts only, don't try at home...

@bolbteppa Thanks sir... That method was helpful.

Slereah

if you have $x$ as a function of $r, \theta, \phi$, then
\begin{equation}
\frac{\partial}{\partial x} = \frac{\partial r}{\partial x} \frac{\partial}{\partial r} + \frac{\partial \phi}{\partial x} \frac{\partial}{\partial \phi} + \frac{\partial \theta}{\partial x}\frac{\partial}{\partial \theta}
\end{equation}

This is just the chain rule

Do this for all coordinates and work it out

Abhas Kumar Sinha

@Slereah I know that, that's messy...

Slereah

It certainly is

Abhas Kumar Sinha

15:11

you'll end up doing silly mistakes.

Slereah

You can also do it the physicist way, where you present some argument of what the derivative will be

doing infinitesimal variations of the coordinates and argue that this is an infinitesimal arc and such

Abhas Kumar Sinha

okay...

that's variety of ways.

@Slereah see that, for example, most elegant proof, I've seen today

I dun know how postdocs always manage to think one step ahead and this level of proof is insane.

he's literally controlling maths.

@Slereah here's the pdf link: projecteuclid.org/download/pdf_1/euclid.bams/1183510788

@Slereah What's the most elegant mathematical proof you've come across?

Slereah

16:08

@AbhasKumarSinha here's a silly one

en.m.wikipedia.org/wiki/Eilenberg–Mazur_swindle

Aaron Stevens

16:38

@ACuriousMind 9th iteration of yourself? physics.stackexchange.com/users/245302/curiousmind9

ACuriousMind

@AaronStevens Sure, we call them ACuriousNine ;)

bolbteppa

@Slereah that's about 5 minutes working out the first derivatives, now do it again to get second derivatives

Aaron Stevens

16:58

@ACuriousMind There it is

Slereah

17:22

@bolbteppa that's why you use the tensors

bolbteppa

Is $g^{\mu \nu} \nabla_{\mu} \partial_{\nu}$ right?

Well it is, but you need to do simplifications on the Christoffels right

There's a direct formula $\nabla^2 f = \frac{1}{\sqrt{g}} \partial_{\mu} (\sqrt{g} g^{\mu \nu} \partial_{\nu} f)$

Using the scale factors you can simplify it in ortho coordinates

Slereah

that too

bolbteppa

17:37

Where does $A_J(s,t) = -\frac{g^2(-s)^J}{t-m^2}$ come from

Slereah

Is that the propagator thing

bolbteppa

It's a part of a scattering amplitude in Mandelstam variables that motivates going to strings

Slereah

I vaguely recall it in Schwarz et al

It's the "this is why we can't have spin 2.5" bullshit isn't it

Sir Cumference

18:10

already seeing posts with things like "my pronouns are var/let". sounds like javascript joke but i guess there's no way to tell if it's legitimate

JMac

@SirCumference I saw someone saying their friend used "var/let"; but they also claimed to use "grimble/gromble" as their own personal pronoun, and googling their pronouns + "pronoun" linked right to the SE question, so it seemed like a possible troll scifi.stackexchange.com/posts/221861/revisions

Sir Cumference

yeah that was the post

maybe i'm overthinking it, though var and let are the only assignment operators in JS. but who knows

JMac

@SirCumference Yeah I have severe doubts about that authenticity, the majority of the pronouns they chose plus the word "pronoun" literally gave that question as the top hit in google. Most of the examples they gave are grammatical nightmares, and "grimble/gromble" fits really well with "troll pronouns". The whole post just stinks to me, though proving it's trolling would be difficult.

Loong

@JMac clearly a troll; did a similar thing on another site

Sir Cumference

The CoC requires we refer to the OP as grimble though

ACuriousMind

18:19

@JMac Fortunately we don't need to "prove" trolls are trolls to simply not engage with the trolls.

JMac

@SirCumference I think in this case it would still be within the CoC to just completely disengage instead, and flag as probably trolling

Sir Cumference

Would it though?

The CoC itself seems really not thought-out

ACuriousMind

@SirCumference You were neither required to bring up that post nor its OP here, you could've simply chosen to ignore it. Don't overthink it and don't feed the trolls.

Sir Cumference

Even putting aside the whole coerced speech aspect, if the troll had been more subtle it's not clear what the course of action would be

@ACuriousMind I'm not feeding the troll, the fact is the rules laid out are not written well

JMac

@SirCumference Sure. Not addressing the user would make your use of their pronouns irrelevant. I agree that the CoC isn't well thought out; but I think "try to be nice" is still the ultimate goal. If someone has a weird request like that, and you feel like it's trolling and don't want to oblige, just don't engage at all so that things don't get sour if they actually were genuine (or are trying to wind people up).

Sir Cumference

18:24

It's inevitably going to become a problem elsewhere

@ACuriousMind Not to mention what I brought up was the (probable) JS joke, not the post itself

Emilio Pisanty

any astronomers or astrophysicists around?

I'd like some external input on the reliability of a spectroscopic observation from the 70s

Slereah

18:40

not my forté I'm afraid

Sir Cumference

never mind

bolbteppa

19:09

@Slereah equation 6.13 here (p.12)

Emilio Pisanty

21:24

@DanielSank atomic physics, of course ;-)

Jake Rose

22:22

What does integrating over solid angle mean?

Having some troubles understanding why we need to do it

For example, when calculating total luminosities?

bolbteppa

140

A: Why do we not have spin greater than 2?

Higher spin particles have to be coupled to conserved currents, and there are no conserved currents of high spin in quantum field theories. The only conserved currents are vector currents associated with internal symmetries, the stress-energy tensor current, the angular momentum tensor current, a...

Oh look

Again the person banned on this site is the only one who has ever posted about $s^j/(s-M^2)$ propagators on this whole site

Luckily once March '92 comes I could ask him on this site

Sir Cumference

@JakeRose Solid angle is the 3D generalization of angle

Generally the situations in which you'd want to integrate over solid angle are analogous to 2D situations where you'd integrate over angle

Solid angle

In geometry, a solid angle (symbol: Ω) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The point from which the object is viewed is called the apex of the solid angle, and the object is said to subtend its solid angle from that point. In the International System of Units (SI), a solid angle is expressed in a dimensionless unit called a steradian (symbol: sr). One steradian corresponds to one unit of area on the unit sphere surrounding the apex,...

bolbteppa

22:44

0

Q: Spin J $A_J(s,t) = - \frac{g^2(-s)^J}{t-M^2}$ Amplitude?

In GSW equation (1.1.2) they define the scattering amplitude for a spin $J$ particle at high energies as $$A_J(s,t) = - \frac{g^2(-s)^J}{t-M^2}$$ mentioning it is an asymptotic approximation to a formula involving Legendre polynomials, and in this stackexchange post it is also defined, and in Wil...

quantum-field-theory

@JakeRose see that picture

Note the $\sin \theta d \phi$ part is that red line which comes from noting that the arc length of the circle is found from $s = r \theta$ (not the $\theta$ in the picture, just the way this formula is usually written), but here $r = \sin \theta$, and the angle is $d \phi$, so you have $\sin \theta d \phi$, and the vertical line is $d \theta$ so that the area is $\sin \theta d \theta d \phi$.

$user image$

Sir Cumference

23:42

@Slereah no

NO

Transcript for