03:00 - 12:0012:00 - 00:00

3:56 AM
@user1271772 Commented on your answer, but I didn't want to add more to that comment in case it makes the OP think he is validated: The extra dimensions nonsense is actually quite common to quantum physics in the following sense: while we are used to 1st quantised systems to be talking about one single quantum particle hence living on 3+1 spacetime, those wavefunctions actually are in configuration space, and to mesh well with SR, it is common to have each quantum particle in an entangled mess
to have their own time directions. This is the headache that leads to the multiple time form of the Bethe-Salpeter equation, which is still a headache in the 2nd quantised form. Everybody who are working on those systems fully understand that the internal relative-time coördinates is completely meaningless and ought to be excised from the theory, but only in specific scenarios do we have some way to perform the excision. No general method exists to reliably remove the unwanted time coördinates.

4:16 AM
is there a reason that in mech we start with lagrangian but in qft (it seems) we start with lagrangian density? will we ever use the lagrangian density but not the lagrangian ? even the canonical E-L equations are in terms of the lagrangian?

@Relativisticcucumber In basic mechanics we work with point particles. In that scheme, it is far nicer, and less confusing, less mathematically sophisticated, to use Lagrangians. But if you start doing fluids, then even in classical mechanics, you would use Lagrangian densities. QFT has no choice, because, literally, fields. It is even in the name!

@naturallyInconsistent thanks for these elaborations here :)
By the way I had a -2 notification from somewhere, then it became a +8, and now I can't find where it originated. Is it possible that you downvoted that answer before you upvoted it? This would explain the -2 notification and then the -2 +10 = +8 notification that I got.

@user1271772 it is not; The +8 is from me giving you the +10, and you have a -2 from somewhere else. I think you can check your scoring in the activity log, IIRC.

@naturallyInconsistent Yea I can't find the -2 anywhere though, so I though it might have been a -2 that was then turned into a +10.
It now says +10, but it said +8 before (and -2 before that):

@naturallyInconsistent but everything ive seen so far cares only about E-L equations, action, and converting to Hamiltonian formalism. but all of these can be done with lagrangians, right? so it seems strange to be given the lagrangian density instead of the lagrangian but i think im just missing something

4:29 AM
@Relativisticcucumber No, if you have fields, you have to use densities. If you want to use the Lagrangians, you would be doing 1st quantisation.
@user1271772 Did you downvote something and then remove it? Or check the day before; there might be a timezone traversal. Or it can be that you downvoted something and later that got deleted / the downvoted user deleted the account. So many possibilities

and another thing that i think i keep conflating. tong says that qft is invariant under lorentz transformation, specifically saying if we make a coordinate transformation $$x^{\mu} \rightarrow (x')^{\mu} = \Lambda^{\mu}_{\nu}x^{\nu}$$ then none of the physics changes. i am confused because i thought the point of relativity is that physics is invariant of choice of coordinates, so what does this mean? why are the lorentz transformations special? i think im missing the point once again.

@naturallyInconsistent as a personal policy, I do not downvote :)
6244 up, 0 down.

@user1271772 why

4:35 AM
acm loves downvoting
@user1271772 ah thats nice

@Relativisticcucumber If you look up the links we were going on, there is a link to an early downvote of his and it turned somewhat bad

If it's a VLQ (very low quality) post, then I'll flag it.

i often feel physics stack main site is a ruthless place

I don't contribute to the ruthlessness, haha.

I am very much into downvoting stuff when they are, say, homework questions, etc.

4:36 AM
@naturallyInconsistent u seem v no nonsense but in a good way

@Relativisticcucumber invariant of choice of coördinates is kinda closer to GR than SR. Lorentz transformations are expressing the "speed of light is constant" part of that.

@naturallyInconsistent yes i think whats confusing me is my relativity framework is solely based on gr

@Relativisticcucumber which is very funny because I'm quite the source of nonsense even in the classroom, both as student and as teacher...

so i feel like any coordinate transformation should work but that doesnt seem to be right here or it seems to be missing the point

@Relativisticcucumber did uni not cover SR separately from GR?

4:39 AM
@naturallyInconsistent they did but i couldnt understand it.

@Relativisticcucumber Well, I can give you a direct argument for why we do otherwise: Nobody has ever succeeded in making a self-consistent quantum gravity theory. This means that when we teach QFT, we are only doing QM+SR but not QM+GR. In this way, coördinate invariance is not going to be ok in QFT, at least for now

Physics.SE also gets a lot of questions and when a site is that popular, it tends to get a lot of homework questions and questions from all over the place. I can imagine that if I spent as much time at Physics.SE as I do on MMSE, I would also get tired of trying to leave helpful comments all the time as an alternative to downvoting. MMSE only gets 3 questions/day and they're usually research-level questions rather than homework questions.
The question that we were discussing earlier was from a new user that hadn't ever posted on MMSE before, and we don't usually get questions about "quantum foams" and such.

but if coordinates only describe a space how can a theory not work for any coordinates and be right? i feel like im not getting how coordinates work in this context because i think of them as like how we "lay down a grid on a manifold" or really anything. like in gen phys we can choose whatever coordinates to label $\mathbb{R}^2$

@Relativisticcucumber At least they did. There are unis who fail to even attempt. But yes, I will totally understand that students might be confused by extremely bad coverage of SR. I think it is nicer to cover SR first from Minkowski diagrams, because it is great for answering spacetime questions. But then there is also a need to convert to mathematical treatments, not least because the energy-momentum parts are awkward to cover in Minkowski diagrams.
@user1271772 still, it is nice to see someone being so wholesome
2

At 3 questions/day, I can follow everything each day and not get too frustrated. Maybe 1 or 2 times per week it can be frustrating having to edit/fix some extremely poorly formatted questions, but it's not as common as it would be on Physics.SE, so I can have a lot more patience!

4:48 AM
@Relativisticcucumber I am kinda sympathetic to the viewpoint you are espousing here, but I think you should take note that the standard QFT treatments we work with, we write down, tends to only be in Cartesian coördinates. Some of the formulæ are so egregiously specialised that they would not even work in spherical polar coördinates. You don't have to go so deep and think of the GR way of thinking on it.

hm this makes me very uncomfortable

@Relativisticcucumber anyway, I'm more interested in the quality of discourse. I get extremely annoyed by people who just want to hear their own voice, or maybe a few bros chatting about their big brain ideas, when in fact they are just wrong from the very beginning.

well there is somewhat of a dichotomy here that is unclear to me. iirc, in my sr course, we have this idea of literally moving everything. so like if we "move everything to the left physics stays the same" is the canonical statement, but this isn't really a coordinate transformation come to think of it?

@Relativisticcucumber But it is a straight up consequence of the fact that we don't have quantum gravity yet. What is the point of being general when the general thing fails?

@naturallyInconsistent YES

4:52 AM
@Relativisticcucumber Oh, that is kinda scary. Yes, Lorentz boosts can be seen as a coördinate transform, but we really think of things as being actually moved.

okay then lets reduce this problem to relativity and not qft since that might isolate the issue. even in sr i dont understand this idea because we deal with mink space, right? so presumably there are tons of coordinate transformations that describe flat space. what is special about the poincare group?

mink space

I think I should put it this way: The interesting bit of SR is NOT the coördinate transforms. We don't so much care about coördinate transform in SR. The interesting part is that spacetime behaves in so-and-so a way such that Lorentz boosts are a coördinate transform, as opposed to the Galilean transforms.
That is, we are deriving the appropriate coördinate transforms that are natural to spacetime. This is a totally different thing from just allowing any coördinate transform we choose.
Poincaré group is special because it is the group of symmetries of spacetime. The translational symmetries, thanks to Noether, immediately lead to the conservation of energy and of momentum. Rotational subgroup lead to conservation of angular momentum, and concept of centre of mass, etc. The Lorentz boosts are the correct subgroup for characterising uniform motion. Combined, these characterise all possible quantum fields of interest via rep theory.
You can simply try to apply Galilean transformation onto Poincaré generated QFT fields, and show to yourself that weird nonsense will have to appear. Those are thus not of any interest in the space of theories.
wait, @Relativisticcucumber you know that the Lorentz subgroup of Poincaré group is the unique group for boosting that respects N1L and constancy of speed of light, yes?

5:13 AM
@naturallyInconsistent what is N1L? what i know so far is part of what you said: that these coordinate transforms describe "moving things" in a way that can be reflected by a coordinate transformation. i also have seen that the poincare group consists of all of the isometries of mink space and, as you said, these are all associated by noether's theorem with a conserved entity. the part where i start to go awry is that i have no idea what implications this has for anything [...]
[...] in field theory or physics. sure we have established there are a group of isometries, so when we move things along the isometries, the metric (and the physics) wont change, but what are the implications of this for field theory? i guess i dont get what it means to move anything along the isometries here. in gr this has a very natural notion because parallel transport and geodesics and whatnot

@Relativisticcucumber Newton's 1st law
I'm confused, because Lorentz boosts has the whole set of standard SR insanities that we learn about. Relativity of simultaneity, time dilation, length contraction, twin paradox, etc, are all standard points of confusion that come straight from there. The interesting bits for QFT would be things like being able to restrict our attention to representations, the spin-statistics theorem, that "macroscopic causality condition that space-like separations commute" lead to microscopic causality,
and stuff along that lines.
The example I always love to go for when teaching Minkowski diagrams, is that "going faster than the speed of light" necessarily implies "antiparticles exist" (but not necessarily the converse!)

2 hours later…
7:00 AM
wait i think i have narrowed it down to a new question
okay so our beloved tongy boy says that we want our field theory to be relativistic. then he says that we want the theory to be LI. my qualm is that i do not see why ensuring that our theory is LI satisfies our desire for a relativistic theory.

@Relativisticcucumber Lorentz invariance is SR. These are not two different statements. They are different versions of the same statement.

7:18 AM
Hi everyone, I would need help with the following problem. Consider a point charge q placed at a point (0,0, b \geq 0) on the positive
half-space of the z (horizontal) in the presence of a very thin isolated and overall neutral
metal sheet arranged along half of the spherical surface of radius a centered in the
origin and placed in the negative half-space of the z. Discuss qualitatively the force to
which the q charge is subjected and determine whether or not for some positive value
of b it can be repulsive, that is, directed as +\vec k the versor of the z axis.

7:33 AM
how much have you learnt to work with questions similar to this sort?

@naturallyInconsistent I know a little about Electromagnetism, but I cannot understand this question...

This is not the first question of this type to be thrown at students.

@naturallyInconsistent In fact it is not a homework assignment, it is my personal curiosity. I did some Electromagnetism problems myself but I got stuck on this one, but it's not homework.

@naturallyInconsistent oh i have grossly misunderstood sr. because i thought sr should be a complete theory of relativity for flat space? is this so?

7:48 AM
@Bml If z=0 outside of the hemisphere is also a flat metallic sheet, maybe there is a simple solution. As it is, it might be a complicated sum.
@Relativisticcucumber We typically think of SR as a complete theory of relativity for flat space, but if you are thinking in terms of GR, that is not the case. SR is a highly restricted subset of GR, so highly restricted, that its merger with QM worked.

@naturallyInconsistent OK. I had thought: In the b>>a case, can we think of a simple charge arrangement that might approximate that of the hemisphere?

@Bml I am not sure why you would want to do with the hemisphere? Why not a whole sphere? Or a flat mirror? Why would you want to make your own life difficult?

@Relativisticcucumber What's the misunderstanding here? "Relativity in flat space" -> The metric is the Minkowksi metric -> The isometries are the Lorentz (or rather Poincaré) transformations -> The physics should be invariant under Lorentz transformations

@naturallyInconsistent Sorry. What and how should I do with flat mirrors or a whole sphere?

@ACuriousMind that is fine to me but the issue is going from "physics is invariant under Lorentz transformations" > "Relativity in flat space is satisfied/accounted for fully"

7:54 AM
@Bml Then standard textbooks cover the specific trick that happens to work extremely well for them.

@naturallyInconsistent note that flat space implies the existence of globally Riemann normal co ordinates

@RyderRude What does this fact provide as a positive influence on the preceding discussion?

so SR with lorentz invariance is a complete theory of flat space

Is anybody claiming otherwise?

@naturallyInconsistent u said it is not the case

7:56 AM
@Relativisticcucumber Who is claiming this implication?

becuz of GR. but GR has no effect on flat space theories

Your QFT is not "just Lorentz invariant", you're also using the Minkowksi metric to raise and lower indices on stuff

@naturallyInconsistent Can you recommend one? In Italy, where I live, these tricks are not used. Thanks.

@ACuriousMind i thought that's what tong is saying. because what im trying to understand is why we are so fixated on the poincare group and lorentz invariance. he seems to say "we want qft to be relativistic" and then seems to think that showing its lorentz invariant is sufficient to show we have made it a relativistic theory and im just not understanding this logic

@Relativisticcucumber The translational and rotational invariances are already covered beforehand, namely with (even if they hide the fact) the use of tensor fields, differential geometry, etc, and so the moment you handle Lorentz (and Minkowski metric) parts properly, the result is automatically going to be SR.

7:59 AM
@naturallyInconsistent but there is no geometry in qft i thought? sorry im missing a lot here it seems

there is the geometry of flat space
v. geometric setting

@Bml The standard textbooks for EM are things like Griffiths, Jackson. I heard there are people using Purcell, and a list of others.

@Slereah right but i thought in qft there is no geometry

There's no dynamic geometry
What does "no geometry" mean

u can have non translationally invariant theories in flat space. translation invariance is not a requirement for SR

8:00 AM
Classical mechanics has geometry

@Slereah it's something I said recently to get Silly Goose to stop trying to do QFT on general manifolds to make it "easier" to see the structure of QFT... :P

it's literally the first thing you study in school
@ACuriousMind You fool

of course I didn't mean that you are literally forbidden from ever using geometrical concepts in QFT

but the other day acm said "there's no geometry here"

@Relativisticcucumber Well, it can be extremely subtle. For example, we never write $\vec r$ alone, only having $\vec r-\vec r\,^\prime$, and this immediately implies translational invariance. It is not being emphasised, but things like that come into play.

8:01 AM
@ACuriousMind kms

@Slereah it means a topological theory
in this context

@ACuriousMind yeah this is what i was referencing :P
so should i develop geometric intuition for qft or not D:

@naturallyInconsistent I had heard of these books. How is this trick you speak of marked in these books? How is it made recognisable?

@RyderRude I literally said nothing of that sort.
@Bml The specific trick is called method of images

@Relativisticcucumber What I meant in completely unambiguous terms is: Standard QFT is a theory formulated on Minkowski space. It uses a lot of tools that are specific to both flat space and spacetime being $\mathbb{R}^{4}$ topologically instead of some other topology. You need to first understand this formulation of QFT before it is any use to try to generalize it to more general notions of spacetime

8:04 AM
@ACuriousMind okay that seems sensible

@Relativisticcucumber I think one will always be in trouble in physics without some generally used geometric concepts. Things like tensors unify a whole host of concepts in physics, and should simply not be avoided. But that would not be considered as too much geometry.

@Relativisticcucumber i think this is still representative of where im at before this whole geometry tangent

note that traditional QFT cannot even be formulated in spacetimes lacking space-like slicings

Anyway, ACM, Slereah, I'll leave you two to take over, before RR makes my blood pressure go unhealthy again. Not to mention FFS the Lorentzian vegetable would not be reading anything from him anyway since there is a publicly announced blocking.

i blocked too so it's symmetric

8:08 AM
lmfao

arxiv.org/abs/hep-th/0012084 This paper is more clear about the De Donder quantisation

If you blocked too, i.e. symmetric blocking, why do you think it is worthwhile to chime in a conversation whereby you cannot even see what the conversation is about? It is not like what the whole set of us talking would be able to give you enough information as to what the original question is about, since we have all assumed that the Lorentzian vegetable understands enough physics to not need covering all the basics?

15 mins ago, by ACuriousMind
Your QFT is not "just Lorentz invariant", you're also using the Minkowksi metric to raise and lower indices on stuff
that's still my response
I don't think anyone is making the strong claim here that just any theory invariant under Lorentz transformation is "special relativistic" (for one, a Euclidean conformal theory in 2+0 dimensions would have the 3+1 Lorentz group as its conformal group, but is not a 3+1 special relativistic theory)

so you mean LI coupled with the mink metric means the theory is special relativistic ?

I mean the Minkowski metric is more-or-less the one thing you need to consider for a special relativistic theory

8:16 AM
@naturallyInconsistent hmm i was just replying to ur comment that SR is not a complete theory of flat space if one thinks in terms o GR

@RyderRude And that was NOT what you think you read it about. It was a reply in itself, and I specifically worded it the way I did that does NOT mean what you think it means. The opposite, in fact.

what did u mean by it??

@ACuriousMind But then the conformal theory has the Minkowski metric [on the ambient space]
It has Lorentz invariant tractors

oh no not the tractors

i am overwhelmed

8:20 AM
@Slereah okay, let's see if you can quibble with this example: I could write down a Yang-Mills type gauge theory with gauge group the Lorentz group. That would be "invariant under the Lorentz group" but have nothing to do with special relativity.

@RyderRude I am under no obligation to explain to you, especially since that would entail explaining the context, and that is long, and no other commenter, who are also here, is interested in that particular detail. Are you trying to insinuate that ACM and Slereah would have failed to catch a serious mistake of mine while you would?

@Relativisticcucumber please don't feel like you have to engage with the argument between Slereah and me :P

@ACuriousMind now, now, do not ask for the impossible

@ACuriousMind i just see nI screaming at himself

It is not about engage. It is about witnessing the clash and being scarred for life
@Relativisticcucumber screaming noises

8:22 AM
@ACuriousMind can i just clarify what it means for a theory to be fully special relativistic ? i thought it means like a full theory of flat spacetime but this is broad

@ACuriousMind i am wondering if the use of the metric to raise and lower indicies is distinct from the use of the symplectic form to induce a canonical isomorphism between tangent and cotangent spaces at a point. because previously my interest in putting the basics of textbook qft into this bundle language is to understand it in terms of what i know happens in classical mechanics

I'm still curious about mink theory - it sounds bit hairy to me.

Nov 7 at 21:32, by ACuriousMind

@JohnRennie It just means Minkowski. We love mink

8:23 AM
so cute

combs fur

@Relativisticcucumber In this context I'd say it's just a theory where you use the Minkowski metric as the metric on your spacetime (in particular to raise and lower indices)

No-one loves mink - not even other mink. The mustelidae are all rapacious carnivores!

i love mink

@SillyGoose As I had repeatedly stated to you, the symplectic form is NOT used to induce a canonical isomorphism between tangent and cotangent spaces at a point. You would need the metric to raise or lower indices before you can compare tangent and cotangent spaces. The very act of raising and lowering indices is literally for this thing.

8:25 AM
@SillyGoose They are the same in that both metrics and symplectic forms provide an isomorphism between tangent and cotangent space (i.e. allow raising and lowering indices). They are distinct in that a metric is not a symplectic form and vice versa.

LOL YO

@naturallyInconsistent u can use the symplectic form too as it's a (0,2) type tensor

@naturallyInconsistent see. u sometimes say things that r incorrect. thats y i replied!

@naturallyInconsistent You can use any non-degenerate bilinear form to define an isomorphism between a vector space and its dual (and hence between tangent and cotangent spaces). You may not think this is the "point" of the symplectic form (and I'd agree to some extent) but Silly Goose is correct that you can phrase it in that language

8:28 AM
Every non-degenerate bilinear form defines some isomorphism between vectors and dual vectors

jinx

Since it's just changing columns to rows
Though course semantically metric tensors and symplectic forms act on different spaces in physics
At least typically

Sigh, I stand corrected. The point I am making is that he is already plentifully confused by things here, and trying to be technically correct is not a smart way to go about this.

@ACuriousMind so then for the theory to be special relativistic, we just need to invoke $\eta$, right? why does LI come into the picture at all as something we need to check
but this is also confusing to me because if all we need to do is invoke a metric, why cant i invoke a curved spacetime metric?

8:32 AM
@Relativisticcucumber Well, the Lagrangian is supposed to be a scalar, right?

@Slereah It is quite annoying to have to teach to students that, technically, tensor indices do not actually map to columns or rows, but only that for two-index stuff, the horizontal placement of the index can be uniquely mapped to column v.s. row.

@Relativisticcucumber As I said, at some point standard QFT will invoke special properties of flat space to proceed
you cannot see why this restriction is necessary when just writing down the classical field theory you're starting to quantize

any saw Killers of the Flower moon? is it good

@RyderRude There are two physics discussions actively happening in this physics chatroom, why do you think this is a good time to bring in something completely unrelated?

@naturallyInconsistent two discussions happen at the same time. u just reply to the msgs u like

8:36 AM
@RyderRude Will it kill you to be considerate of other people?

@ACuriousMind for scalar fields, right? or this is true always?

@Relativisticcucumber No, the Lagrangian is always a number!
You want to integrate it to get an action, which is also a number, right?

@Relativisticcucumber Even for Dirac Lagrangian, its Lagrangian density is a scalar

Still, I think you're still misinterpreting what's going on: What did you mean by LI as something "we need to check"?
there's really no need to check this

when the fields are operator-valued, how does this work if $L(\phi,\dot{\phi})$? i think you are saying to get from $L$ to $S$, we integrate over $dx^4$ so it must be a function and thus, a number, right?
@ACuriousMind let me send the reference one sec

8:39 AM
@ACuriousMind hm wait what do you consider the point of the symplectic form. my current impression is that it allows one to define Hamilton's equations of motion. but to do so you need to define the isomorphism I sent above first (at least to my understanding)

@ACuriousMind damtp.cam.ac.uk/user/tong/qft/one.pdf the page starting with 1.23 is what im referring to

Err... I guess it is not necessary maybe? It seems convenient though as it allows one to write a Hamiltonian vector field explicitly

@SillyGoose This is confusing. $\eta$ and $\zeta$ are clearly vectors; $\omega(\eta,\zeta)\in\mathbb R$, so it would be better to say that $\omega_\eta$ is defined as $\omega_\eta(\zeta)=\omega(\eta,\zeta)$

@naturallyInconsistent yes i agree...i gots to change the notation

@Relativisticcucumber Again, I don't think this is making the strong claim that any Lorentz invariant theory is necessarily automatically a correct special relativistic theory
but any special relativistic theory will be invariant under Lorentz transformation (you agreed to this earlier!)

8:43 AM
oh god. wait so why is LI even being brought up

yet another case of someone overanalyzing a throwaway line in some lecture notes :P
@Relativisticcucumber because symmetries of theories are important?
why do we talk about rotations in non-relativistic mechanics?

well tbf i have seen this a lot and talked with @SillyGoose extensively about this whole poincare business so i did think it was worth figuring out

The Lorentz transformations are literally the analogue to rotations

@ACuriousMind To be fair in classical mech classes they don't talk about rotational symmetry that much

Wacky Fowl, before ACM chimes in on this, I would like to point out to you that a lot of the stuff we talk about symplectic forms, originally had nothing to do with Hamiltonian mechanics. Physicists worked out the properties of Hamiltonian mechanics, noted the behaviour of Poisson brackets, and then noted that all those Poisson brackets stuff is just nice the maths of symplectic forms

8:45 AM
@Slereah we don't?

@naturallyInconsistent I don't see how it could be done with charge images. I think the force is repulsive for b near 0, attractive for b very large, and I don't know if it can be calculated analytically in closed form. Any hints?

@ACuriousMind very much don't

@ACuriousMind we dont in Newtonian mech. we do in Lagrangian mech

@ACuriousMind Not in what I remember anyway

8:46 AM
I mean maybe in a few classes, but that's not usually a big focus

@Bml Again, if you want to cook up horrendously difficult problems for yourself, do not expect anybody else to help you. At least begin with the simple stuff that textbooks cover, so that people would know how to help you because they are familiar with standard examples.

I guess I already knew I had weird classical mechanics lectures, I do remember learning about infinitesimal rotations pretty early on :P

@ACuriousMind Particularly weird.

In Newtonian mech, angular momentum conservation doesnt even follow from Newon's three laws
conservation of momentum does follow from the third law, but we still dont talk about translational invariance
the crucial reason for this is that, in Newtonian mechanics, conservation need not follow from symmetry

@ACuriousMind man this brings us to our recent discussion where i was confused about what it means to rotate a field

8:49 AM
@ACuriousMind my c. mech course did not cover rotations in this language, but if i ever was able to teach c. mech i think i would. a first exposure to lie algebras :D

0

Suppose there is a particle whose position, velocity and acceleration are $\vec{r(t)}$, $\vec{v(t) }$, $\vec{a(t) }$. $\vec{a(t) }$ is always directed along $\vec{r(t) }$ but $|a|$ is not just a function of $|r|$, but explicitly depends on the co-ordinates. This system is not rotationally invaria...

is it that we care about the poincare group because "quantum fields transform in a rep of the Poincare group"? @ACuriousMind

@SillyGoose Really the thing you want is to define the Hamiltonian vector field $X_f$ corresponding to an observable $f$. That we have to use this (co)tangent isomorphism in the process is correct, but to me it really doesn't have the same ubiquitious significance that the musical isomorphisms have in Riemannian geometry, where the difference between 1-forms and vectors is even notationally erased by us just raising/lowering indices everywhere

In our case, we only had one compulsory module of classical mechanics, and so it was quite rushed to end at derivation of Coriolis effect. People could follow because, in general, we happened to be better equipped at the maths for classical mechanics, and so people could survive giving up on Coriolis.

@Relativisticcucumber I mean, very generally the operators and states of a quantum theory transform under the symmetry group of the theory (if they didn't transform under it what would it mean to call it a symmetry group?)
so the statement is true but somewhat vacuously so

8:52 AM
i think my problem w qft is that i need to understand why things are needed so they have a place in my framework to live. qft feels like a ton of random facts that i cannot figure out their purpose. sad.

@Relativisticcucumber This is an extremely common feeling. hugs

okay i will keep the LI in mind for now and move on. BAHHHHH

@Relativisticcucumber Really I think the problem is that you're trying to learn field theory and quantum theory at the same time!
Nothing about "the fields have to transform under Lorentz transformations and our theory should be Lorentz invariant" is quantum
the Lagrangian formulation of classical EM, for instance, works exactly the same way

You could even just learn the same principles for fluid mechanics

8:54 AM
@naturallyInconsistent which thing are you referring to? i think i have indeed read every message that you sent. many concepts are muddled to me but i am piecing them together via this chat

this first chapter of Tong is correctly titled "classical field theory" - nothing here is quantum

just with a different group

@ACuriousMind Actually, I think you are the one making an unlikely assumption: I think it is far more common that people got taught a bunch of different things, e.g. this classical EM field, beforehand, and then suddenly get thrown to QFT and are expected to know that all the earlier stuff have to gel together.

hm i now see that nothing tong said is against anything i knew. oh no i have misconstrued him oncemore

so trying to focus on understanding why "QFT" requires these things is the wrong starting point - it's just the FT part, not the Q part, that's at play here and so trying to simltaneously think about operators or states from the quantum theory is just needlessly confusing

8:56 AM
@Relativisticcucumber there had been so many big discussions about this that I am not even sure if I had written it down for you! But if I did, I would have explained what "rotating the field" means, with a concrete example.

@naturallyInconsistent hm i actually am not sure if i saw this

@Relativisticcucumber Consider an electric field, somewhat similar to that inside a capacitor, that is everywhere in the universe, uniform in magnitude, and always pointing towards the +x direction. This is a very simple field, a field that does not matter which input position you give it, a constant field. Now, if you rotate around the z axis, the position dependence does absolutely nothing, but the field now needs to rotate to point somewhat in the y direction too. This is what is being meant.

ah i am scrolling through some CFT notes and it indeed seems i have very much missed a lot of this content
or at least posed in the way it seems to be

We say that an electric field is a vector field, because when we rotate, it appears as if it is a vector.

@naturallyInconsistent It's time for us to make a new way to teach physics [makes teaching program even worse than all the previous ones]

9:02 AM
do u think classes r helpful

@naturallyInconsistent what do you mean "the field now needs to rotate"?

@Relativisticcucumber If you put a test charge somewhere, the electric field maps onto a force vector. If you rotate the universe around the z-axis, then this force vector will rotate.
insert obligatory XKCD yourself.

But I vehemently object to that characterisation. If we do not attempt new ways to do anything, then there will never be progress. The very fact of the matter is that we already have evidence that improvements in teaching are very much possible.

Oh we should
But that idea backfired quite often
ie. the New Math
or whatever they had going on in California

We should teach CFT to first year students instead of Newtonian mechanics

@Relativisticcucumber If you are only looking at scalar fields, then the numerical value of $\phi(x)$ is just moved and mapped by (active) coordinate transforms from position $x$ to position $x^\prime$ in $\phi(x^\prime)$. But there are vector fields, fields that, other than position dependence, also point some direction. Then when you rotate spacetime, not only do you have to change the position dependence, you have to rotate the vector too
In particular, if it were $\vec E(\vec r)=E_x(\vec r)$ and you do a rotation, it can not just be $\vec E\,^\prime=E_x(R^{-1}\vec r)$, but is $E_y(R^{-1}\vec r)$
that is, an E field originally pointing in the x direction rotated to become pointing in y direction
No amount of fiddling with the position dependence alone will be able to give you the correct behaviour; you have to actually rotate the vector of the vector field itself.
And it is this that we are saying are representations of the Lorentz group.
(continuing from capacitor) Imagine we now swap for a beam of electrons that are meant to fly into a Stern-Gerlach apparatus. The spin half character of electrons means that when we rotate along the z axis, the +x spinors rotate with half the angle. This is a totally different representation of the Poincaré group, even though the position dependence is, again, assumed to be trivial.

9:14 AM
@Mr.Feynman It's very basic "field theory" but we did do Lagrangian formulations of continuous systems (e.g. waves on a string) in our first year (2nd semester, "theoretical physics 2" lecture)

@Mr.Feynman If you wish for most of the students to fail, sure.
@ACuriousMind envious
There is somewhat of a chicken-and-egg issue going on there, actually. It is, of course, wonderful to let students see Lagrangians early, but that would require the students to understand E-L equations insanely early, way earlier than they could be expected to 1) learn fast enough to yet not hinder other education goals 2) be appreciative of the mathematical toolset and be motivated in learning
As such, most treatments would teach classical fields in an ad hoc manner, and often just omit the Lagrangian density description. If they are really good, they might treat this omitted part later on.
@Slereah I was very shocked when some interlocuters vehemently defended New Math, with statistics on their side. I had, prior to that point, assumed that it was universally derided.

I mean it was popular enough to get started and widespread
also IIRC there wasn't really that much hard data for or against
Probably didn't stick around long enough to get much

@Mr.Feynman why not newtonian mech

@Slereah The general impression was that it was hated, and forced top down as a response to Soviets getting a headstart on the space race.

Pretty much
There were also various other reasons
Apparenty the Bourbaki are part of why?
Just the notion that all math is just based on set theory therefore we need to teach set theory
Jean Piaget, né le 9 août 1896 à Neuchâtel et mort le 16 septembre 1980 à Genève, est un biologiste, psychologue et épistémologue suisse connu pour ses travaux en psychologie du développement et en épistémologie à travers ce qu'il a appelé l'épistémologie génétique (ou structuralisme génétique). Ses travaux apportent un éclairage sur l'« intelligence », comprise comme une forme spécifique de l'adaptation du vivant à son milieu, sur les stades d'évolution de celle-ci chez l'enfant et sa théorie de l'apprentissage. Cet éclairage exercera une influence notable sur la pédagogie et les méthodes...
also that guy

9:25 AM
If that is the case, we would have to refer not to Bourbaki, but rather to Russell and Whitehead

Bourbaki certainly did a lot of damage to people who thought their books were how you were meant to teach math

@naturallyInconsistent Bourbaki were the people who really popularized the idea apparently
Like idk how many mathematicians actually read Russell back then

Not Serge Lang?

Who knows
Tracking trends is a complicated topic
Currently reading the book on the spread of the popularity of higher dimensional geometry in the 19th century and boy does it take some weird turns

I thought Bourbaki is kinda acceptable as uni textbooks?

9:28 AM
Many twists and turns in that book
There are a lot more wizards involved than I thought there would be

@Slereah ah, literally showcasing the essence of high-dimensional geometry?

also Lenin makes an appearance?

@naturallyInconsistent they're fine for some people, but not for everyone. Cartier said: "The misunderstanding was that it should be a textbook for everybody. That was the big disaster."

@Slereah politics refuses to leave anything alone

and: "The misunderstanding was that many people thought it should be taught the way it was written in the books. You can think of the first books of Bourbaki as an encyclopedia of mathematics... If you consider it as a textbook, it's a disaster"

9:31 AM
@ACuriousMind ah, everybody. No wonder

I mean you have to remember that Lenin was a big philosophy nerd
He wrote a lot about various philosophical topics
He was just a big fan of Marx in particular

@ACuriousMind Again, that is where I'm not particularly convinced? Isn't it common for 1st two years of maths undergrad to basically follow the Bourbaki-style introductions?

@naturallyInconsistent It is certainly done in some settings, although I don't know if it's a good idea
The sink or swim kind of classes

@naturallyInconsistent I wouldn't say so. Yes, we do a little bit of set theory and logic at the start, but we generally are much more hands-on with linear algebra and analysis and other intro topics than the true abstract Bourbaki-style would be

@Slereah I'd disagree with this too: Profs can really just come in and make their support beloved...

9:33 AM
Like, the thing about Bourbaki isn't just that they founded everything in set theory at the start, it's the extremely abstract presentation style throughout

^the new math style

@ACuriousMind ah, extremely abstraction: That is what I would have referred to as French school, rather than as of Bourbaki

Well they were French :p
Although it is true that apparently the French mathematicians thought themselves as more rigorous than the rest before the Bourbakis even

@Slereah This is slightly confusing? I don't think the other schools are necessarily any less rigorous (with heaps of salt on the Russian side), but they are definitely less dogmatically abstract
Still, it is most enlightening being in a classroom with people from all the different schools of thought and hearing everybody tackle ideas from completely wacky directions.

More recent schools of mathematical pedagogy tried to put more emphasis on intuition and examples
@naturallyInconsistent I mean they're not hiring 6 teachers per class :p
And usually they will use a common program

9:39 AM
@Slereah recent? Isn't it just somewhat returning to the historical norm?

Mathematical education is just trends rly
The methods used varied a lot through history
Although I think it was pretty common in ancient times to just make you read the Elements of Euclid until you got it
But you always have people trying to reform education in various directions

Well, at least that is a great textbook. Not easy to write a textbook so good that it could last > 2000 years

ie
Ramism was a collection of theories on rhetoric, logic, and pedagogy based on the teachings of Petrus Ramus, a French academic, philosopher, and Huguenot convert, who was murdered during the St. Bartholomew's Day massacre in August 1572.According to British historian Jonathan Israel: "[Ramism], despite its crudity, enjoyed vast popularity in late sixteenth-century Europe, and at the outset of the seventeenth, providing as it did a method of systematizing all branches of knowledge, emphasizing the relevance of theory to practical applications [...]" == Development == Ramus was a cleric an...
@naturallyInconsistent It is shockingly good by the standards of the era
Reading most "math" texts in ancient greece it is barely understandable
Hence why it is so popular I guess

Of course. Euclid's Elements somewhat look like they are divinely inspired

From what I hear he used the old Lobachevski trick

9:43 AM
???

It is a common theory that Euclid's elements is just a compilation of various other texts
For mathematical pedagogy you can also look at Oliver Byrne's version of the Elements, too :
I'm not sure it's as readable as he hoped, but it is a nice idea

@Slereah loved it enough to write my own geometry proofs in that form.

@ACuriousMind Heidelberg is a place of its own, my friend. At my place you learn about lagrangian formulation of classical mechanics during the 2nd year, so what you mention would be between the 2nd and 3rd year
Then the fact that there is no such course is another business :P
@RyderRude @naturallyInconsistent did you really take me seriously?! :P

The OG educational reformer for math being Socrates I guess
Teachers love talking about the socratic method

@Mr.Feynman not meow

9:58 AM
Instead of the previous method in common use [beating children when they get things wrong]

@Mr.Feynman Yeah, I know it's a bit unusual but I think the reasoning is sound: The principle of least action is such a fundamental part of modern physics that you should introduce it as soon as possible, even if it will be challenging. That theoretical physics 2 course was whirlwind of concepts: Lagrangian mechanics, Hamiltonian mechanics, fields/fluid dynamics, thermodynamics.
None of them in great depth, but enough so that you knew where to look if you ever needed it again, and I think it really set us up extremely well to engage with the lectures that followed.

@Slereah Am not sure Socrates did any Socratic method for maths; He completely just let Aristotelian physics nonsense alone, as if Socratic method was not able to pick out the contradictions inherent in Aristotelian physics (even giving it more slack than the Medieval physics)

@naturallyInconsistent It's in the Meno

@Mr.Feynman i did becuz i dont like Newtonian mech

10:04 AM
@ACuriousMind Of course! Maybe I didn't make it clear enough but I think that's how I think things should be done.

@Slereah is that so much Socratic? It seems like Socrates simply gave the slave boy the answer.

That is also the reason I became mostly a self-learner (not saying that it worked, because it's like doubling the number of classes and thus minimize efficiency)

@naturallyInconsistent A cynical man might say that's what the socratic method is

let's discuss the unreasonable effectiveness of mathematics

@Slereah ah, the cynic comes to the rescue. Woe be to those who think that cynics never bring anything positive to the table

10:09 AM
is the universe mathematical or is the mind mathematical

The next attack on the neusis came when, from the fourth century BC, Plato's idealism gained ground. Under its influence a hierarchy of three classes of geometrical constructions was developed. Descending from the "abstract and noble" to the "mechanical and earthly", the three classes were:

constructions with straight lines and circles only (compass and straightedge);
constructions that in addition to this use conic sections (ellipses, parabolas, hyperbolas);
constructions that needed yet other means of construction, for example neuseis.
It was a strange era

10:25 AM
@Slereah will learning it give meow meow a neurosis?

10:38 AM
I haven't read Plato's argument so idk what are the sinister implications
I'm guessing you are straying away from the Truth

It won't be for much. History has proven that the Truth holds the reins to my slave collar

"Using neusis where other construction methods might have been used was branded by the late Greek mathematician Pappus of Alexandria (ca. 325 AD) as "a not inconsiderable error". "

before long I'll be yanked back in whatever direction

"neusis" is a fancy word for "using a ruler where you can mark distances"
It's essentially the use of a pencil
but it is a Sin

Such appeals continue to live today in the form of coördinate free geometry?

10:44 AM
I mean marking a ruler is coordinate free

i suppose

I don't know why neusis is better than a regular straightedge and compass
Can't you also transfer a distance using a compass?

@Slereah I think the trick is the pole of the neusis - you can transfer a length with a compass, but the neusis construction allows you to transfer a length anchored at a point $P$ somewhere in the middle of the length. The neusis construction becomes the compass construction when you choose one of the ends of the length as the pole of the neusis

Oh I see
You can just put your point anywhere on the uuuuh neusomter
I'm still not sure if that is impossible to do with compass and straightedge tho
Neusis seems to be kind of an uncommon topic for some reason

11:13 AM
@Slereah probably because Plato hated it? :P

I think math people stopped caring that much about Plato and his ilk by the 19th century
Bring Neusis back I say

11:26 AM
arxiv.org/abs/hep-th/0012084 see page 8 for the equivalence of De Donder quantisation and QFT
there is indeed the "degree of freedom" problem. they assume the qft wavefunctional factorises into $\psi[\phi]= \Pi _{x} \psi (q, x)$
they say they r assuming no correlations. but qft experiments have space-like correlations
so i think this. is less general than qft unlike what the previous paper said
they r basically looking at a subset of qft wavefunctionals for which they r able to write a covariant Schrodinger eqn
it's not a surprise that space and time r on an equal footing for wavefunctionals that can be factorised like this. these wavefunctionals can be derived from wavefunctions of the form $\psi (q, x,y,z,t)$
the method also replaces the wavefunctional Schrodinger eqn with the Dirac eqn which is covariant

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