wait i need to make an edit

Do I get to be Free!

yes

thnx pal

@Relativisticcucumber i don't understand bottom right lol

@qwerty usually sillygoose asks q1, acm answers, sillygoose still unsatisfied asks q2. this process repeats a few times and acm says 'what r u trying to figure out' then sillygoose reveals the true q, then acm is like 'that is not what you asked' then provides the answer to the real q

oh haha

@Relativisticcucumber lmao

@nickbros123 you will be pleased to know i am studying enm atm

@Relativisticcucumber this gave me a good laugh

Reminds me of the old memes a user here used to post

can't believe that was 7 years ago

@qwerty After it was shown to be in direct conflict with GR, it stopped being a thing to be concerned about.

@GroveRover ... why did you not just copypasta the Poincaré from Slereah's... wouldn't that be a lot less work...

@RyderRude You even knew that the bucket was treated by Newton; it is called Newton's rotating bucket argument, which you could have directly quoted from Wiki, and yet you still claim that it was Mach who imagined it. How many times do we have to tell you to stop making up fake histories?

@naturallyInconsistent hence my surprise at the mention of present day Machian people. i think it might be a slippery topic depending on the interpretation of what is meant by Machian anyway.

Well, there will always be people who only learnt bits and pieces of physics and decide that they know so much. There was recently a twitter meme about right wing nutjobs who think that Thomas Aquinous College's undergrad curriculum is how to seriously do an education. You might want to giggle at that picture too.

LOL what century are these people from

19th century, and rapidly going towards 18th and beyond

x.com/JeremyTate41/status/1861489359024791705

There, found it for yall

@naturallyInconsistent the bucket argument was treated by both Newton and Mach. Mach was trying to correct Newton on it. U can search for bucket on the Wikipedia en.m.wikipedia.org/wiki/Mach%27s_principle

@RyderRude dont quote the wiki on me when I just quoted it on you

@naturallyInconsistent ??

@naturallyInconsistent not sure what you mean

@GroveRover You spelt the name wrong

@naturallyInconsistent Lmao I spelt it like pancarrè which is a type of bread

om nom nom?

It's lively weather in the UK this morning

@naturallyInconsistent afaik RR is correct in this instance, as Newton did not reference distant matter/distant stars in his bucket argument (as he believed in some fundamental absolute space)? Mach's contribution was to err drag (pun intended) distant matter into the story

morning

@Relativisticcucumber haha nice one

@Relativisticcucumber nice. Im doing topology rn

@JohnRennie 😳 omg that looks very frisky, please keep us updated that you're safe 😟

I'm fine. Thanks :-)

It was the coastline that took the main battering. Chester is far enough inland that the wind speed didn't exceed 50 mph.

ok

@qwerty I have never assumed that. I checked Wiki before commenting.

@SirCumference thanks for the trip down memory lane, pal. I also found this along the way:

@JohnRennie do the cats have mysterious grins in Chester?

lol

@naturallyInconsistent sorry, I don't quite follow what you mean?

@qwerty :-)

@qwerty I meant that I was absolutely certain that Newton was the one talking about the rotating bucket, but did not mention distant stars, and Mach was the one with the distant stars, before making my first comment on the topic.

Because I had checked wiki before commenting

@naturallyInconsistent Newton believed in a theological absolute space but fixed stars were definitely a thing around Newton's time

@GroveRover in the context of defining inertial frames?

@qwerty If I am not mistaken

@naturallyInconsistent I see. I suppose I didn't think what RR said was quite objectionable other than not crediting Newton, which didn't seem imperative in the context

@qwerty "In Newton's time the fixed stars were invoked as a reference frame supposedly at rest relative to absolute space. In other reference frames either at rest with respect to the fixed stars or in uniform translation relative to these stars, Newton's laws of motion were supposed to hold."

interesting. As I was reading these lecture notes earlier.

>One of Newton’s early arguments for the existence of absolute space involves a universe which contains a single material object. On Newton’s view, even though there is no other matter in the universe, we can still determine whether the object is rotating relative to absolute space.

@qwerty he's saying that Mach imagined the rotating bucket rather than Newton. He has had ACM scold him for inventing history over and over

@GroveRover that I know; again, I checked before commenting

I had assumed (and indeed on wiki there's no mention) therefore the fixed stars were not really part of Newton's presentation or conception of his bucket argument; the fixed stars would be something of an observational rather than fundamental feature of the theory.

Mach objects to this argument since the universe is given to us only once ‘complete with fixed stars intact’.

@naturallyInconsistent I've read someone claiming that fixed stars were Mach's contribution to the problem, and I was pointing out that they are not

@GroveRover ?

He may have elaborated on them but certainly not introduced them

@GroveRover your quote doesn't seem to support this.

@GroveRover As mentioned earlier, Newton's argument was for an absolute space in which, even if you remove everything and left just the bucket, it would still curve. This is that which is contradicted by Mach. It is important to not be retroactively reinterpreting what historical figures meant, outside of what they actually said

@think_meaning_buildß Were you in the chat back then?

@qwerty What? I just told that using fixed stars as an inertial FOE was not introduced by Mach

@think_meaning_buildß Oh shoot you're skullpatrol

I had a suspicion when you said "pal" lol

@naturallyInconsistent If we're being careful, "curve" is probably ahistorical, too - I don't think the notion of "curvature", as we understand it today, existed for Newton, since intrinsic and extrinsic curvature really started with Gauß and Riemann

@naturallyInconsistent What am I reinterpreting? I think I may not have conveyed my point correctly, as I was only pointing out that fixed stars as an inertial FOE were not introduced by Mach. I am not arguing about any interpretation

And I was doing so because I believe I've read someone claiming so

@GroveRover that's not about the bucket argument though, nor what anyone has said about Mach's contribution. it's not about "using the fixed stars as an inertial frame".

@qwerty I'll try to find the message later, maybe I've read it wrong

@ACuriousMind a translation that is used on wiki is says "concave figure"; there is no incongruity nor contentiousness when using curves to discuss that. This is totally not the same situation as asserting whether Newton wanted to mean fixed stars or not. He only discussed in terms of absolute space or not, not fixed stars or not. In particular, removing all other material still gives us an absolute space in which the curving would have happened.

@naturallyInconsistent I did not assert Newton was meaning fixed stars please stop t.t

@GroveRover i never once asserted that you meant that

@qwerty I was correcting this

and so was miao miao

@naturallyInconsistent So what did you mean by "It is important not to be retroactively reinterpreting what historical figures meant, outside of what they actually said"?

@GroveRover oh okay

that makes more sense I thought you meant a different message later on

@naturallyInconsistent yeah sorry. I see what you mean a bit better now

@qwerty it was in reply to that... lol

I thought RR was saying the same things as we all were :P but on re-reading a third time I see he wasn't quite

@qwerty I also may have misinterpreted what @naturallyInconsistent meant by " I meant that I was absolutely certain that Newton was the one talking about the rotating bucket, but did not mention distant stars, and Mach was the one with the distant stars, before making my first comment on the topic." on the same line

isn't it nice when it turns out we're all in furious agreement lol

Mach's contribution asserts that if you remove the fixed stars, it could well be the case that the centrifugal effect curving the water level disappears

yeap

This is something that Newton would have disagreed, because Newton's argument is that there is an absolute space that God would use

the role of those fixed stars should be stated clearly when we're discussing.

As in, all these inventing histories and being sloppy with words is precisely why it is so frustrating to engage in discussions when RR is involved

we would not have needed to have all these extensive "furious agreement" if things didn't proceed this way

@naturallyInconsistent Not familiar with Mach's POV, but that sounds to me a lot like relationalism

How is it different?

@GroveRover hang on i will link some lecture slides

relationalism is Leibniz I believe

@qwerty Yes, while Newton was a substantivalist

@GroveRover Yes, because by the time Mach was on the scene, the absolute space viewpoint is on the way out. It was already quietly brewing that the æther would have to have extremely contradictory properties, e.g. needing to be extremely high tension so that speed of light is so big, yet so slack that things just pass by unhindered. Of course, without evidence, we cannot know what Mach himself consciously understood about these issues, at least not that I know of.

@GroveRover maths.lse.ac.uk/Personal/james/York I liked lectures 12 and 17

on this

@qwerty what do u mean?

i didn't say that Mach introduced these ideas

@RyderRude start with going back to your own comment and read all the arguments that arose because of you

i just said that Mach tried to make rotation relative to the star frame

stop making it sound like that is all you did

> ...allow us to develop a Machian theory. That is, a relationist theory in which inertial effects can be explained in terms of the relative accelerations of material bodies.

i have said many times that Mach thought the definition of an inertial frame was dependent on star configuration. General Relativity is a refinement of this idea

GR makes inertial frames dependent on local matter configuration instead of global star configuration

this is in contrast with what Newton thought. According to Newton, the inertial frame was a primitive concept

in other words, in Newton's universe, the stars just happened to be at rest wrt the God given absolute space. But in Mach's universe, the stars could dynamically determine the absolute space

@RyderRude You were just told you stop inventing history and reusing words in ways that were not intended by the original authors. Why wont you stop?

@naturallyInconsistent my statement is backed by evidence

u can see on the wiki that Mach and Newton disagreed

@RyderRude it is nonsense, and so it cannot be backed by evidence

"Das ist nicht nur nicht richtig; es ist nicht einmal falsch!"

nice

@RyderRude Mach was not about absolute space. I also think it's important to be more careful when speaking, just like the other day when you kept dropping "locally" off of Euclidean... it will stop nI getting riled up :/

@qwerty Precisely. It is so egregious

Also, it is not just meow meow getting riled up.

@qwerty I'll give them a read. Do you study there?

@qwerty My guess was right. But this was also an older theory, so what was Mach's novelty?

:66749609 what did I miss

well that was attempting to reply to the deleted post

@SirCumference you sure you want to ask him? lol

probably not lol

we just had about 100 messages of furious agreement

all thanks to a throwaway comment of his

so another saturday here then

you also missed a hilarious moment earlier where GroveRover cried

@GroveRover no, I finished my studies a while ago.

@qwerty To his defense, arguing by messages is hard and it also happened to me to misinterpret/misconvey in the past days

It is like someone never once got curious, nor even noticed, that some automation routinely being used by himself, added stuff to his comment box

@GroveRover if it happens once or twice, that is a honest mistake. This, however, is meme status by now

@naturallyInconsistent fwiw I think every frequenter of this chat has been in a heated argument with him over a throwaway comment

myself included tbh

@GroveRover well, I'm not extremely versed on it myself. relationism does not speak of how inertia arises, right? like what makes one frame inertial. my understanding is that Mach said that inertia arises from the global or distant distribution of matter.

@RyderRude like I don't want to sound too critical, but I really do think you gotta take these responses to heart at some point

whole place is getting riled up fairly regularly

@GroveRover the novelty is in realising that one cannot distinguish between substantialism and a relationalism with a specific distant distribution of matter. But don't worry about it too much, because we know it is in disagreement with GR.

i think it's a little slippery. like the lecture notes i linked described how in some ways you could argue GR is or isn't Machian, depending on what you mean, although overall it's "not very Machian". I have yet to understand what goes on in the case of rotation in GR, as there was a claim I read recently that surprised me - it has in some sense absolute rotation (wrt local geodesics).

@qwerty we should note that Mach's ideas are vague to a significant extent, so there is a no point in disagreeing over terminology. i just tried to put the idea into words the best way I can. but maybe i should say "inertial frame" instead of absolute space

@qwerty "not very Machian" is already very much a disagreement! It was originally a strict mathematical equivalence, and GR does not have that property at all.

@RyderRude do not confuse your own vagueness with people who had been clear in their exposition of complicated concepts.

@SirCumference i told naturallyinconsistent that their previous comment was an empty statement. Then a bot deleted my comment and suspended me

Mach's book was quite the nice read

@RyderRude "a bot"

@naturallyInconsistent note that there is a reason there is so much debate over what Mach meant. the ideas are not supposed to be precise

@naturallyInconsistent it is a bot named Feeds

You clearly have no idea how things work

a good reason why bots should not be trusted with such tasks

didn't Slereah say Mach's book fuelled Einstein on the general covariance thing too and not just Mach's principle, or did i misremember? I feel like that book has a lot to answer for lol

@RyderRude No, your comment was flagged and then deleted because the flag was validated (by human reviewers). No "bot" was involved.

@ACuriousMind but it said it was automatic and I clicked on the deleter. It was a bot named Feeds

@qwerty yes. When miao miao read Mach's book, it was quite convincing. It is no surprise that Einstein was hooked on the idea.

@RyderRude ...yes, because the flag review is supposed to be anonymous so of course they don't show you who flagged you or who validated the flag

@ACuriousMind it said "automatically". Idk how else to interpret that. If a human reviewer did this, they made an incorrect judgement because the comment wasn't inappropriate

if anything, many of naturallyinconsistent's insults have never been removed despite me flagging them

Unless it is a direct mod, it is usually a vote of many ppl.

@RyderRude Because you apparently don't understand the difference between a harsh criticism and an insult.

@RyderRude the implication is extremely clear, isn't it?

You have a consistent pattern of being vague or confidently wrong, and when you're getting criticized for it - with some justified exasperation by those who find themselves in that situation repeatedly - you try to "mirror" the criticism but it just turns into insults because you don't seem to really understand what you're being criticized for. Of course the things people say to you are sometimes somewhat harsh, but they're reactions to your behaviour.

And every time "the system" - be it in form of a flag or a manual suspension by a mod - intervenes you start another discussion about how unjust it is that you're being punished for your inability to listen. It's tiresome, and it has to stop.

...forgive me for asking if it's none of my business, but was the deleted comment really just "it's an empty statement"?

@qwerty No, of course not. It was a much more general insult: "u never make any actual comment. U just say empty statements "

@ACuriousMind and this is an inappropriate comment, as opposed to the several times naturallyinconsistent has insulted others?

and my comment is not even an insult

i will try to search if I can. naturallyinconsistent has actually said inappropriate things and he has never faced any consequences

@RyderRude No, you won't, I will not re-litigate the past here. There is an important difference between "this statement is empty" and "all your statements are always empty", and once again you are being just wrong when you said that "i told naturallyinconsistent that their previous comment was an empty statement". It's not what you said.

@ACuriousMind it's because I was obviously paraphrasing the situation. and in no world is this an insult. it is just a criticism of naturallyinconsistent's comments

and these comments are a general behavior. it is not just this instance

@RyderRude if I may be so bold to suggest something. the mods also care about correctness of physics in the chat. if any bias in moderation exists, they will naturally weigh in favour of those who are correct in physics. I'm not saying that there definitely is any bias to be clear/fair to acm and nI. but maybe being more mindful of your wording will also help in that direction.

@RyderRude Ah, now you're "obviously paraphrasing" to excuse you just saying things that aren't true? Another important difference between harsh criticism and an insult is, by the way, also that a criticism has to be true.

@RyderRude At this point, I think you're better off not being this defensive and just reflecting on what you're doing

repeated bans usually aren't a good sign

@qwerty nothing justifies tolerance of insults in the chat

I suggest you stop trying to litigate this as if it were a court of law and instead start reflecting on what the feedback you're getting means. Several people in this chat continuously express annoyance at you (although often enough in the form of subtext you seem to not understand), and it's because you keep being wrong about things and then never back down without an extended and rather pointless discussion. And you're doing the same thing now: Stop it, or you will be stopped.

Last time we had a guy rile up the chat repeatedly, he got several year long bans

After it became clear shorter bans weren't changing anything

@SirCumference i have reflected on it.. I've been suspended thrice including this time. the first instance was when I posted a free copy of a book, even tho other people had posted similar stuff before. the second instance was arguably about making incorrect statements. and then this instance where a non inappropriate comment was judged as suspension worthy

thrice is really not good

I mean I've been on here for 9 years without getting banned at all

i have reflected on it. And I just find huge biases

@RyderRude in an online chat, tone and so forth are not always going to be obvious. people may be less forgiving if theyre already annoyed by you unfortunately.

@RyderRude I don't know what to tell you dude

at the end of the day it's either the chat rules or just not being on the chat

this is all just my prediction/advice of course

@RyderRude Have a fourth time since I cannot interpret representing your first suspension for "posting a free copy of a book" when I explained it to you as anything other than bad faith and a continued refusal to abide by the rules you're told.

And no, before you come back complaining that this is an overreaction to another "obvious paraphrase" of yours: Of course this is not just for that. It's for the aggregate of all the problems you keep causing around here and your lack of interest in fixing any of it.

ACM are the flags for spam/offensive and flag for moderator both anonymous and do both go to mods for review?

@qwerty the flags are anonymous and all users with more than 10k rep and mods can vote on them, but a mod vote is binding, cf. meta.stackexchange.com/a/271268/263383

(but they are not fully anonymous, there is also moderator tooling to combat people who raise a lot of spurious flags)

@ACuriousMind oh I see. is this different to your comment about being a starkiller but not seeing who starred what?

> Violations of the Code of Conduct include, but aren't limited to: Subtle put-downs/unfriendly language

good to know, thanks.

@qwerty Yes, stars are different - there's no standard mod tooling to investigate those but if it gets particularly bad I could probably get an SE community manager to find out what's going on

I have a question regarding the invariance in QFT

I wrote this question

And the person who answered, he gave me thoughtful explanation, but there was one thing I couldn't understand

He said: "Gauge invariance occurs at the level of the action, not the Lagrangian. In the action, you can integrate by parts"

Why is that? Because this implies

that it can happen that the lagrangian under some transformation (are all transformations classified as gauge ones, or do we have other types as well)

is invariant

but the anction is not.

Which I don't believe is possible

And if it is not possible than the invariance should occur at the level of the lagrangian and not action, since the action is a function of the lagrangian/lagrangian density if one can say so

Am I misunderstanding something here?

@imbAF I left a comment, it's not really clear what this question is actually trying to show

ok

Maybe I don't get it

but I can't see how that answer my question about his commentary on the level of invariance

@imbAF however, the thing you ask about here is something different: Yes, Andrew is correct that symmetries (gauge or not) are to be considered at the level of the action. This means that Lagrangians can change up to total derivatives under a transformation and that symmetry is still a symmetry of the theory.

sometimes such symmetries are called "quasi-symmetries" to distinguish them from "symmetries" that leave the Lagrangian invarant, but they are symmetries in the sense that there is a Noether charge associated with them and that the equations of motion will also have that symmetry

But then if for some transformation the lagrangian changes

then in relation to it, the transformation is not symmetric

Because in one homework exercise I did, we were considering a list of different transformations perhpas gauge or not and we were asked to check whether these were symmetry transformations, which is when the lagrangian was left unchanged

So the exercise was considering the invariance to the level of the lagrangian

@ACuriousMind Ahaaa so there it is, the difference

And would it be accurate to say: One can consider gauge or not transformations, and a gauge or not trasformation can be a symmetry transformation or not

Or is it that every gauge transformation has to be a symmetry one

depends on what exactly you think "gauge" means, really

I usually would not call any transformation that's not a quasi-symmetry "gauge"

but sometimes by "gauge" people just mean any transformation that depends on spacetime

your other favourite topic!

lol

it is!

@ACuriousMind Which is what I consider it

I mean a transformation can be classified as a gauge or not one

But just because you classify one as gauge, that doesn't guarantee invariance on w/e level

let's call it action level

@ACuriousMind at this point maybe we could all pay you to write a scathing document on these topics and have it permanently pinned to the chat lol

@imbAF My point is that that is one version to use the term. It's not the one I prefer, since the special properties of "gauge symmetries" are not because they depend on spacetime but because they are related to a redundancy in the choice of variables for the given action/Lagrangian/Hamiltonian. See this answer of mine for an approach to "gauge transformations" in terms of first-class constraints

(which are related to the dynamics and hence you can't tell whether a transformation is generated by a first-class constraint without knowing the action you want to apply it to, making it impossible to look at any given transformation in isolation and "decide" if it's gauge or not!)

I wonder why no one ever mentions such distinctions

I don't expect you to understand this (as it is criminally under-taught almost everywhere), I'm just saying this to caution against the view "gauge = spacetime-dependence" being universal

To gauge, or not to gauge, that is the question.

they just give a half-assed transformation, slap the gauge term to it and that is it

@qwerty I mean, I usually link to the same handful of answers on the topic I've written - the write-up is already there

and I also never get tired of telling people to read the book Quantization of Gauge Systems by Hennaux and Bunster (formerly Teitelboim)

@ACuriousMind it's still on my list! but slipped down in the ranks I'm afraid.

What happened to Teitelbaum

did he get married and change his name

You are the Hamlet of gauge theories in this chat :P

@Slereah no, i looked it up back when acm told me about that book and i couldnt find the author lol

Dramatic

damnit acm beat me to a link

lightning fingers

Thanks for the infos

@Slereah you missed a lot of it

@ACuriousMind This is a nice use of the term "gauge".

I have just been told that there are solutions to Einstein's equation that are not covariant. I have no idea what that means

Gonna need more context here

There was no additional context

Well you're gonna have to figure it out on your own then

covariant by itself is kinda vague. they could mean anything...

I guess I won't

as i have been learning

I just had a conversation with a philosopher of physics trained in philosophy

Worst experience of my life

how come?

I also had a conversation with a philosopher last week on non-physics-related philosophy and was unable to tell if he was a crackpot. I don't understand philosophy one bit.

@imbAF Caution: I'm not saying I would call all quasi-symmetries that don't leave the Lagrangian invariant "gauge", I'm saying I would not call any transformation "gauge" that's not a quasi-symmetry (or symmetry), i.e. there are no "gauge transformations" that aren't (quasi-)symmetries - this relates to what I said after that about constraints, in my definition of "gauge" they are always automatically quasi-symmetries. But there are quasi-symmetries that are not gauge.

@GroveRover Good faith interpretation: This is a clumsy way to express that not all solutions of an equation possess the same symmetries as the equation itself.

@ACuriousMind I understood that. You are saying that gauge transformation = invariance/quasi invariance. Unless I am not reading this properly

@imbAF not, not "=" - there are plenty of (quasi-)symmetries that are not gauge

such as the "normal" symmetries you're used to - rotation, translation, etc

what I'm saying is that in my definition of "gauge", it makes no sense to apply the term to some transformation that isn't a (quasi-)symmetry, i.e. everything that's "gauge" is a (quasi-)symmetry, but not everything that's a (quasi-)symmetry is gauge

For it to be a gauge you also need it to be like extraneous information

ie the Hessian of the Lagrangian is not regular

@ACuriousMind Ok I get it. You are right there are many (quasi-)symmetries that are not gauge

But I get it how you define gauge

For it to be gauge you need to be able to quotient it out basically

@Slereah ooh is this in some textbook or reference?

@qwerty Can you guess which one it's gonna be

ahhh

when it comes to digital books and files I keep forgetting they exist, even when organised on my zotero

I think Gotay is better than Henneaux tbh but on the other hand he never wrote a book

He just has a million little articles on gauge theory

any one of those million you would recommend in particular?

you can try this idk

pubs.aip.org/aip/jmp/article-abstract/19/11/2388/459192/…

It has the benefit of being a proper geometric theory

thanks

@ACuriousMind seems like you found a way to solve the last name thing

@naturallyInconsistent That's a very interesting argument, much more reasonable than Newton's

Ofc it's not relevant anymore after GR but I believe that the knowledge of these cases can quite often be useful when reflecting on modern physics

@GroveRover yes, which was why miao miao decided to look it up in the first place. A theorist should have some... philosophy... miehehehe

@ACuriousMind But that's obvious even for the dumbest of problems. Do you think they are referring to something more specific in GR?

nope :P

@ACuriousMind All right I'll assume it's nonsense

It's like Epoché^{-1}

@Slereah I don't like when there's the word "infinite-dimensional" next to the word "manifold"

Would you prefer "Frechet space"

@Slereah A bit better but still scary

@GroveRover what about “sufficiently nice [insert object]”

@SillyGoose I prefer infinite-dimensional manifolds

At least I know what to be scared of

I wonder if I’ll finally see an application of the sylow theorems in physics

the basic idea really is that your presymplectic space is bad because it is presymplectic (the symplectic form is degenerate), but we want a good one

So we need a submanifold of this space that is symplectic

and the choice of this submanifold is the constraint

@SillyGoose Don't give ideas to stringists

More like in quantum information. I wonder if they can be used conveniently to count abelian subgroups or something

@SillyGoose maybe not directly, but they obviously have applications to physics as part of other theorems being used. For example, equilateral triangle symmetries being a subset of hexagonal symmetries?

The symplectic form / Hessian is, at a given point, a matrix that isn't of full rank, hence not invertible and bad to do unique evolutions of your system. This means that you have a subspace of your system that is degenerate. You need to reduce this space to a new space that only has the full rank part of the matrix and drops all the degenerate directions basically

That's what the gauge does

is there a reason that i should expect a priori that in e&m the electromagnetic energy current is proportional to the electromagnetic momentum density? and furthermore that these things are proportional to $E \times B$. I did a derivation of both $S$ and $\Pi$ to get this result but im a bit perplexed about it conceptually

@Relativisticcucumber photons are massless, such that its energy current density is the same thing as its momentum density, up to choice of units causing an artificial constant of proportionality

Well in terms of relativistic EM there aren't that many simple ways you could write elements like that

@naturallyInconsistent hm i guess four vectors would imply this

You have your field A, by gauge invariance its first derivatives are gonna be of the form F = dA

how about the fact that these quantities are given by $E \times B$ in particular?

And there aren't a whole lot of maps from a 2-form to a 4-form

Like the terms are either $F \wedge * F$ or uuuuuh

@ACuriousMind what's the other EM term that you can have

I forget

The one that is forbidden due to breaking P symmetry

uh, do you mean $F\wedge F$ :P

Maybe?

that's the "topological" term because it's a total derivative (of the Chern-Simons current)

Why isn't $T^{0i} = \frac{1}{c^2}S_i$ (momentum density of a EM field in vacuum)?

Shouldn't the $0i$ components be momentum density

The units in a matrix have to be consistent

It is much easier to just work in systems of measurement where $c=1$ so that we can stop worrying about all these nonsense

But shouldn't they have done $T^{0i} = \frac{1}{c^2}S_i$ and put appropriate units on $T^{00}$?

^ This is seemingly the usual convention for a four-vector. (e.g. $E/c, p_i$)

well, it is not clear if they have a mistake or if you have a misunderstanding, and it is quite unimportant and silly. If we really need to make it good, there will be some way to get things to work properly.

But it is correct that the natural choice for what to pick as the standard between energy, momentum, and mass, is momentum.

hmm

L&L vol. 3 also have this

okay I will give Jackson the final word (pg. 606 in 3rd ed): $\int d^3 x T^{0i} = c P^i_\text{field}$. Hence, $T^{0i}/c = \frac{1}{c^2} S_i$ is the momentum energy density and I am unsure of the origin of this $c$ convention for $T$.

@SillyGoose good on ya

I suppose the convention for T is that it has units of energy density

inconvenient, but it is tolerable as long as it is consistent

@ACuriousMind Why is it allowable to change the metric to $\Omega(x) \eta_{\mu \nu}$ when it comes to conformal field theory? Shouldn't the interval be kept at $ds^2 = \eta_{\mu \nu}x^\mu x^{\nu}$ but just allow the fields to be invariant under conformal transformations which seems to be a different thing.

E.g. For example, in non-rel, one can choose to have a theory that is invariant under some weird symmetry but that doesn't change the length interval from $s^2 = x^2 + y^2 +z^2$

For example, I could force the fields to have the same value along the $z$ axis

But that doesn't change the metric

That just makes the fields invariant under transformations along the $z$ axis

Anyone?

Is ACM like an oracle here?

I suppose in the example of fields being invariant under $z$, we could just consider ourselves to be living in a world that is 2 dimensional i.e. $s^2=x^2+y^2$

But this is a subspace of the 3d world

Unlike the conformal metric we had before

I suppose we could then write the interval as $ds^2=f(z) \delta_{ij} \mathrm{d}x^i \mathrm{d}x^j$

Maybe not actually

I don't know

@DIRAC1930 if you do the same with conformal transformations you end up with constant fields

Oh it seems all my definitions are wrong

We just require $g_{ij}' = \Omega(x) g_{ij}$

@DIRAC1930 That's the very definition of conformal map

Is this a subset of all Lorentz transformations for the case of Minkowski space?

@DIRAC1930 o_O

What's the action of a Lorentz transformation on a Lorentzian metric?

$\Lambda^T g\Lambda=g$

So you got your answer

It's the other way around

Why is this a good thing to consider?

Because it's a symmetry arising in a variety of physical situations

And symmetries help you solve your theories

But it doesn't fall under a Lorentz transformation

@DIRAC1930 What do you mean? It's just a kind of symmetry a theory can have, like being rotationally or translationally invariant. We don't divide the world into "good" and "bad" symmetries.

Why do we have to change the metric though

We are essentially saying $ds^2 = \Omega(x) \eta_{\mu \nu}x^\mu x^\nu$

I don't understand the question.

My point is the metric of the world we live in is just $\eta$

@DIRAC1930 I suggest you take a serious look at symmetries in some textbook, also remembering yesterday's discussion, since their role is ubiquitous in physics

It's like saying that is we have a field invariant under the $z$ axis, then we are living on a flat sheet

But this is different since it is the other way around

@DIRAC1930 Have you ever solved the particle in a box in QM?

Yes, we add the symmetry of parity invariance

It's not a feature of the underlying space

@DIRAC1930 you don't "add" it, the problem naturally has it

The same with CFTs

@DIRAC1930 That's like saying "the world we live in" is not the same as the world where everything is translated by 1m so we shouldn't ever consider translations as symmetries.

@GroveRover ^^^

@naturallyInconsistent be nice

only pointing out an unfortunateness. Just unluckiness