is there generically a duality between a physical theory and a geometric theory?

you'll have to be more specific :P

Intuitively, what are called "dynamics" in a physical theory can be encoded in some trajectory(ies) in some space, which can then presumably be framed formally in terms of, say, geodesics on an effective spacetime.

I guess more specifically, I am wondering if generically an interacting system in spacetime can be described in terms of a non-interacting system in a generically distinct, effective spacetime within let's say the framework of quantum mechanics.

@SillyGoose The trajectories of Hamiltonian mechanics are not geodesics.

There is no metric on phase space

@SillyGoose I have no idea what this sentence is supposed to mean. Do you have an example?

For example, I imagine a huge blob of positive charge is concentrated at $x = (t,0,0,0)$. This point in space is then effectively inaccessible to a negative charge (assuming nothing dynamically happens to the positive charge blob). So why can't I model the motion of a negative charge then as living on the space $\mathbb{R}^3 \backslash \{0\}$

You can, and I don't see how that relates to the other stuff you've said :P

it is encoding a physical interaction into some modification of the underlying spacetime. that is the idea i am asking about

no, it's not

the dynamics is in the electric field on $\mathbb{R}^3\setminus 0$

@Allie mew

there are infinitely many different Hamiltonians/Lagrangians/equations of motion you might write down on any particular configuration space

the configuration space alone does not specify the dynamics at all

well part of the question is if the dynamics encoded in the charge and electric field can be instead encoded in some effective spacetime

Is infinite more than three?

@naturallyInconsistent do you know of mark tuckerman?

no, why?

@HerrFeinmann usually yes :P

assuming +q is fixed, the dynamics of -q is essentially acceleration rightwards in the interval $[-R, 0)$

oh hes just someone i might be working with for my phd

weve been in contact his stuff seems really interesting and im going to meet him

i just need to get accepted....

nice

@SillyGoose I mean, this is true in some sense, but I'm not really seeing the point

well actually maybe i should phrase it like this

what does this have to do with an "effective spacetime"

well i thought spacetime includes a notion of "natural motion" (i.e. motion of a system when no interactions involving that system are present)

so i am asking if there is a way to map the motion of -q in $\mathbb{R}^3$ due to an electric field into another space in which the free particle motion of -q now is the acceleration that we observe in the original formulation of the system.

and since we don't have interactions at our disposal to fit the motion, i was thinking that we would need some sort of effective spacetime which carries a free particle motion that matches the actual motion of -q in actual spacetime.

I'm not really following the train of thought here

Spacetime is a pseudo-Riemannian manifold and the usual assumption is that test particles that have no other interaction than gravity follow geodesics in that spacetime

It is very specifically gravity that is encoded in the metric (and hence geodesics) of spacetime, not arbitrary forces or motions

in galilean spacetime, there is a well-defined notion of a free particle (that is how a free particle moves), concretely essentially defined by the laplacian

^

The analogous construction for the electromagnetic field is to take the U(1) principal bundle and demand that particles follow the "geodesics" of the connection on that bundle, i.e. curves where the covariant derivative of the tangent vector vanishes

yes but i do not want to preserve the electromagnetic field. i want to remove all traces of it and encode it in another object.

@SillyGoose look, how are you going to have the E and B fields correspond to free particle motion in the first place? How does that make sense? What happens if you have a non-zero charge density everywhere in spacetime?

@SillyGoose you'll have to make this your question right from the start

@naturallyInconsistent well that's not very physical anyway

@naturallyInconsistent Be nice, and if you want to be critical, be constructive

and you would have to assert precisely which entity you want to model E&M with

@naturallyInconsistent non-zero charge density everywhere in spacetime

@SillyGoose how is that unphysical? I can literally choose an interpretation whereby the electron is the wavefunction, say of an Hydrogen atom, extended throughout spacetime, and there is nothing that is unphysical about that.

@SillyGoose I don't know what you mean by "preserve the electromagnetic field". I've just told you what the analogous construction of geodesics is for the EM case. Attempting to phrase this in terms of some higher dimensional metric and actual geodesics is famously the Kaluza-Klein construction that predicts an infinite tower of ever more massive unphysical states.

@SillyGoose this has ABSOLUTELY nothing to do with telling us that you want to eliminate the standard E&M fields and redoing the description some way else

@naturallyInconsistent i am not sure what you think describing an interacting system in terms of a non-interacting system means

@SillyGoose Well, the standard go-to example is electron quasiparticles in Hartree-Fock or something like that

meow

M E O W ~

am i one of the good cats

@SillyGoose no matter what, it has literally nothing to do with gutting the usual description of E&M fields and redoing the description. There is no evocation of any semblance of the concept. How can you expect that to be what we obtain from your description?

@Allie ... mew? ...

correct answer: yes

Like, miao miao could have asked if you have floofffed around in bed today, or miao miao could have asked if you have pushed a billionaire off a very very tall table, and both would have been appropriate

@SillyGoose the problem here is that this is an extremely vague description when it comes to interpreting it in actual technical terms and each of your clarifications seems to be to go in a completely different direction; I have not yet gotten any coherent picture of what the starting point and what the goal here is

both

@ACuriousMind a simple example of what i mean is that the two charge motion drawn in the image can be dually described by two masses and newtonian gravity so long as $m_1 m_2 = -q^2k/g$ where $k$ is the constant in the Coulomb force and $g$ is the gravitational constant.

@Allie then obviously yes, very much yes

studying hartree fock is also a cat-like activity

dogs would definitely do biology or some shit

@SillyGoose are you trying to introduce the old Newtonian action-at-a-distance back into E&M at the quantum level?

@SillyGoose Yes, that's because there essentially is only one "good" force law for spherical symmetry. The similarity between the two situations is due to the simplicity and the high symmetry of the example, not due to some deep general map between EM and gravity

@Allie now, now, there are plenty of doggos doing hartree fock too

this stuff is so lovely ilinear algebra is just neat

@ACuriousMind yes which is why i didn't use this example in the first place. but it illustrates somewhat what i mean. encoding an EM interaction into something completely not EM related.

but again: The formal analogy between EM (gauge theories generally) and GR is precisely that the gauge theories can be phrased in terms of connections and covariant derivatives exactly like GR; the difference between the two is that the Einstein-Hilbert action and the Yang-Mills action are rather different and the GR connection is soldered to the spacetime

also i don't really care about EM and gravity in particular. it is just the two most common forces at an everyday level

@SillyGoose but then you need to define "completely not EM related"

@SillyGoose Where is there an "encoding" happening here?

We simply observe that both force laws for two masses and two charges are "of the same form"

@ACuriousMind the motion is equivalent and Coulomb's law was not used

Even in old classical Maxwell's equations, the EM force law is not merely Coulomb's law.

and if you are aware of how special this example is and have no better example to offer, I'm deeply mystified as to why you would ask the question in such general terms and expect anyone to understand what you're talking about

And in GR, there are also corrections

And those are classical

we haven't even added the quantum corrections

Like, the moment you consider anything more concrete than the extremely basic, it becomes immediately obvious that the foundations completely collapse. If you want to make speculations in these directions, you'll need to be a lot more descriptive than your vague notions

@ACuriousMind now imagine this happens to you everyday with every topic possible

Oh Lorentzian vegetable, help your friend make sense

@naturallyInconsistent I am the major victim of this offense

miao knows

it is so bizarre, and im sure it is a daily occurrence

i mean, it clearly isn't a one-off

meow

i mean i have not read the reference, but there is a paper describing newton-cartan gravity as being the correct framework for describing fractional hall effect arxiv.org/pdf/2111.08052

Look, there is a TONNE of literature covering the analogies and melding of E&M with gravity in so many ways; if you are going that direction, have you at least cursorily read the wiki? Even there you will find a list of ways things miraculously work and another list of ways that they fail

i already said that i do not really care about EM and gravity in particular

Then what are you on about?

You haven't managed to make anything clear to 3 people trying to engage you in conversation

And when you raise questions about your wording of the situation, I gave you the standard literature that maps clearly to your particular wording

How were you expecting a sensible conversation along the lines that you were hoping it would have gone down?

i dont know whats going on but its intense

@Allie none of us knows what's going on

And that includes the questioner

The conversation had been seriously confused and painful to read, like, even earlier than the 3-component wavefunction thingy

@SillyGoose So your two examples are 1. Two charged particles/two masses and 2. A paper about the fractional Hall effect (also an EM effect!) you haven't read. But you maintain your question a) isn't really about EM when both your examples are about EM and b) should be clear to someone to the extent that they could answer it?

please consider for a moment how this looks from the side that doesn't have all the context that presumably exists in your head

Lorentzian vegetable, you really ought to buy a Fast Fourier Transform. Otherwise, the deconvolution alone would bring about global warming.

you crazy little physicists on my. screen

You'll need to continually observe the wacky fowl in appropriately deconvolved states so that the entropy is kept low

aww, gone

and so will meow; way too late for medical appointment

ok by

bye

@ACuriousMind usually? 💀

Slereah waiting to rock his new weird ass topology

iim out here focking my hartree

@naturallyInconsistent good. So it seems you have some conjecture that you feel is causing the spike and you want to investigate that. 👍🏻

If it were a statistical problem and I would have to investigate whether that one agent is influencing the spike or not, I would employ Kruskal-Wallis, without further context. Seems interesting.

@Allie lol

@Allie lol

@User1865345 yes we have. It isnt even myow own idea, but an idea of a colleague.

but he is like, "this is then numerically solved" and miao miao iz like, HOW PRECISELYYYY??? I NEED TO KNOW IF YOUR SCHEME EVEN CONVERGESSSS

i.e. miao miao will dot the i and cross the t and make sure that every step is perfect, and then maybe put his name on the paper

@naturallyInconsistent yeh. Sounds good to me. 😺👏🏻

I see an upload button in chat now :3

Ahh! Yes!!

Wait. Wasn't it there earlier too? Or am I hallucinating? Lol.

You need 100 rep

I see. So you crossed 100!! 👍🏻🥳

just in time for new year in 1 minute here :D

🎇

hehe nice

and happy new year to you ;) we have to wait some more time here

Best wishes for the upcoming year to all.

Happy new year everyone!

even if it's a bit early for most parts of the globe :)

This is circulating after President Carter's passing away:

'A community of galactic civilizations': Jimmy Carter's decades-old message for aliens

interesting

Can someone help me understand one thing. When we talk about continuous/ discrete transformations of the configuration of a system, the transformation is called symmetry or symmetry transformation, if a physical quantity is invariant.

And in all the cases considered, we start by checking how the transformation, of any kind, affects the lagrangian and the action

So, my question is, is the term configuration of a system the same as the lagrangian density or action of it ?

@imbAF i don't think so. i mean, the lagrangian isn't unique. acm wrote an answer about that recently.

I see

But how would you explain the following then:

The red lined sentence, first one

directly imples that if transformation is labeled as symmetric -- > invariant or put to a constant or smth lagrangian

but also symmetric transformation is one, that transforms the configuration of the system and leaves a physical measurable quantity invariant

So some sort of connection must exist here

between the configuration of a system and the lagrangian density

mhmm isn't it just that the Lagrangian or action describes the system? i'm not confident to put it into more precise language off the top of my head. but also, it's different to say there exists a correspondence between two things and it being "the same".

That's what I also suspect

But I am not sure

since the configuration of the system, which perhaps it's the generalized coordinates, change under the transformation, but a physical quantity is invariant. And apparently this is = invariant action/langrangian under the transformation, then the tranformation is labeled as symmetric

i think it says something more/differently than the "configuration of the system", but it's past 3am here and I'm sleepy... i'll catch you later :)

Ok

Happy New year

In rel qm, one does not require the representation of the Lorentz group acting on the spinor indicies to be unitary since the probability density $\rho = \psi^\dagger \psi$ is the time component of a 4-vector which is not a scalar

How does one formalise the argument requiring the other part of the representation of the Lorentz group to be unitary?

How do the energy of the singlet and middle triplet not depend on magnetic fields? There is another paper that shows rephrasing of the singlet in response to the coupling to the nuclear spin magnetic field (I think) but here it’s saying the sub space of this system is a DFS?

@qwerty when I was in China my little cousin in the US was convinced I could time travel. Every time I’d try to explain to him, he’d say “you are in my tomorrow, right?” and I’d say yes and he’d say “exactly” and all hope at changing his mind was lost.

He isn’t even that little, just not the sharpest tool in the shed

@Relativisticcucumber probably because $S_z \lvert S \rangle = 0 = S_z \lvert T \rangle$

@Relativisticcucumber so you were in China 🤔

Last time I play dumb this year :P

Oh God, is the thumb not called a finger in English?!

This language is so messed up, having the toe-finger distinction was already enough :P

So the word is digit, I see

@HerrFeinmann um yes it is?

Some disagree D:

But I feel like usually you’d just say thumb if referring to just thumb. Like I guarantee if I go outside and hold up my right hand and say “how many fingers am I holding up” everyone will answer 5

I’ve never heard a nonmedical person say “digit” for this purpose in my life

@Relativisticcucumber informally, maybe. Technically it seems to be controversial because the thumb has a different anatomy

Wait wait

wdym toe finger distinction ??

@HerrFeinmann yeah well indeed medical personnel do say digit

@Relativisticcucumber yeah, only in more technical context that word arises. I started thinking about it because in Italian we just have the word "dito" which means "digit" and you use only that

the rest of us plebs call it like we see it

DITTO

begins flailing

You say "dito della mano/del piede" (literally "digit of the hand/foot"), which made me think about the tendency I have to conflate fingers and digits

KMS

@Relativisticcucumber but you would never call your toes fingers!

Sorry I’m struggling w the mobile version

And in this context I learned the sad truth about thumbs

@HerrFeinmann ah ok makes sense. Actually that’s v reasonable

Hm so I wonder why the big toe is called big toe but the thumb is called thumb

shouldn’t thumb be big finger

i hate the mobile version but sadly my laptop is in the shop

NO

@Relativisticcucumber probably because the big toe is just the same as the other toes but bigger, while the thumb is different

Also, the thumb is not bigger than the other fingers, it's just thicker and shorter

so the big toe has no anatomical difference ??

I do remember learning in bio that thumbs are OP

I think it doesn't but my word in biology is not so reliable

@HerrFeinmann true it’s the girth really

Yes, it has more girth

Wait no

See both the thumb and the big toe are missing the middle bone, no?

can a squid transmitter leave a trace?

@Relativisticcucumber damn it, this is a game changer

Happy new year

Happy new year, nerds!