@JohnRennie I saw you answered a question about the non-zero components of the Riemann tensor from the Schwarzschild metric. I also made a Mathematica notebook to compute the Riemann tensor, and am hoping to test my results against someone else's code. Would you be willing to share your notebook that computed the Riemann tensor?

@rob Good point. I just wanted to point out that it's not a simple matter of neutron : proton ratio. And on that note, there are isotopes that do both positive & negative beta decay, as we discussed a couple of months ago. physics.stackexchange.com/q/695466/123208

@tertius Go to swarchive.ratsauce.co.uk and grab AFT-Schwarzschild.nb

I have several other similar notebooks if you're interested. I didn't write any of them so I can't help explain how they work.

A quick question for the panel. Does this look like a second year undergraduate physics paper (link is to a pdf)?

A second year undergrad I tutor was set this as a practice paper, but it looks way too advanced for second year. I can't answer most of the questions!

"Robert Langlands did not actually introduce or use the term “dual group”. The original texts just speak of the “L-group”"

Calling it after your initial

What a lack of class

@JohnRennie It's not easy, but it's all questions I could've been plausibly expected to answer at the end of my fourth semester, so given that Indian physics seems generally a bit faster on the "learn this annoying computation" part, I find it equally plausible this is a (hard) problem set for a second year QM course in India.

Really? You'd covered time ordering and Dyson series by the fourth semester?

yeah - basics of QM in 3rd, and theoretical QM in 4th, which did include those things (but only at the very end - I don't think it was considered exam relevant). I was able to take QFT in my 5th semester, after all.

Wow. I no longer recall what the second year QM course covered at Cambridge (it was 40 years ago!) but I'm pretty sure it didn't cover topics like that.

the notes of the course I took are actually online here [pdf link, German], and probably even without speaking German you can see that there's a lot of perturbation theory (including Dyson series, even though the concept is introduced far earlier) and scattering theory of wave packets at the end in chapters 6 + 7

so yeah, it's a plausible if hard problem set to me, but it probably wouldn't be plausible to e.g. US second years

EM duality involves Galois theory

and me without any Galois theory

"Now, in all real physical systems, a particles position and momentum must remain bounded, and hence, for the remainder of the paper, we shall assume that M is compact."

Is that even reasonable?

I would guess that many realistic systems have unbounded position

@Slereah I think the point is that there are almost no systems where something escapes to infinity in finite time

True but that's not quite the same as the phase space being compact

unless you specify a domain of time

what I mean is that that means you can say "I'm only interested in the next 1000 years" and then choose your bounds accordingly

though I do hope that your $M$ there is a manifold with boundary

It's just "a symplectic manifold"

since e.g. saying "stuff never goes farther from the origin than X meters as far as I care" produces a solid ball, which isn't a manifold, but a manifold with boundary

@Slereah in that case I think whatever you're reading is cheating :P

I guess it's the closure of the subset where the phase space dwells

@Slereah but closures aren't manifolds!

close enough I guess?

like I said - the closure of "stuff never goes further than X" produces no manifold

It's about C^* algebras, so I guess the point is more that the observables are bounded

@Slereah It's really not "close enough": When I say "compact 1d manifold", there's only the circle, but when I say "compact 1d manifold with boundary", there's also the closed interval. These two are very different objects!

We shall see when the compactness of the phase space pops up I guess

math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Gleason.pdf

if you wish to give it a look

Step 2: Relegate the trivial work to the reader.

oh it's just the GNS construction

the compactness is indeed only there to avoid having to talk about unbounded $x$ and $p$ operators

Am I allowed to post here a link to my non-peer reviewed research paper on quantum speed limits? I am doing this because I seek professional perspective on it?

@GauravAnand Technically there aren't a lot of rules about what content you can post in chat - as long as you don't spam it, it's fine. But I wouldn't expect a lot of people to spend their time looking at a random paper.

Okay Sir, thanks for your response! I'll keep it in mind. I'll try posting the link all the same, since even if it reaches one interested person, it'll be a big help to me.

There is a simple way to find a quantum speed limit between two arbitrary pure states using just the Schrodinger equation. It's in a paper I have written whose link is preprints.org/manuscript/202201.0297/v2 . If you are interested, please look into it and comment on its validity. It's a small paper, and will be interesting because it contains a unique way to look into the Schrodinger equation. Thanks!

@GauravAnand If you really are interested in people looking at this: The typesetting is terrible (not the worst I've seen, but not good), which for a paper in 2022 is almost immediate grounds for me to stop reading.

$|x>$ instead of $\lvert x\rangle$ (perhaps not even produced with TeX?), inconsistent setting of equations (the spacing for minuses, for instance, varies all over the place), a lot of spacing errors around punctuation (English has a single space after periods and commas. No space before, not more than one space after, unless you are deliberately double-spacing in some cases), itemized lists but the items are not indented, etc.

This might sound like nit-picking to you, but really proper formatting is simply the first line of defense: If you didn't care enough when writing to adhere to standards in the field, why should I care enough to read? I usually only slog through terribly formatted papers when I have a very good reason to believe they contain something of value to me that I can't find anywhere else

I don't want to post any excuses, if it's terrible, I'm definitely at fault. I'll do my best to edit it and make it more presentable. However, I gather that you have read the paper, so I want to know if the speed limit I formed is correct and that such a limit hasn't been formed before?

@GauravAnand I stopped reading on the first page. But I've never heard of these "quantum speed limits" before (other than the often misused energy-time uncertainty), so I'm definitely not able to tell you whether what you did already exists in the literature somewhere.

Alright, thanks a lot for your helpful responses!

Hi all. I have a question about fluids.

I'm tasked with finding the density of an unknown liquid.

I have an object of known density (a dented wood block with density 0.92 kg/cm^3)

And I know the buoyant force on said block.

But how can I find $\rho_u$, the density of the unknown liquid, from here?

I've tried to employ Archimedes Principle $F_b=\rho V g$

The problem is, I don't know $V$, the volume of the dented object, and I assume that is not the intended solution.

I would greatly appreciate any suggestions.

Does anyone have any idea?

I got it.

"This is sometimes called the algebra of dual numbers, for no good reason."

@JohnRennie With reference to physics.stackexchange.com/a/709150/123208 isn't the positron field the same thing as the electron field?

@PM2Ring I didn't want to complicate things. As it is I'm taking desperate liberties with the truth :-)

Ah, ok. So it's a "lies to children". ;)

I like to think of it as "baby steps" :-)

Personally, I think it's potentially more confusing to pretend they're 2 separate fields. Explaining that it's the same field makes it a bit easier to justify / explain annihilation & pair production.

the language of different "fields" is not unambiguous

you could make an argument that the "electron field" is actually four different fields, since it's a 4-component spinor

To be honest that answer was something of an experiment to see if I could come up with a plausible reason for why mass of a bound state can change, and I'm not entirely convinced it was successful.

The basic argument is that a bound state is not the state we associate the rest mass with so there's no reason it should have the same mass.

@ACuriousMind But isnt that a bad argument?

I'm not a fan of the language in John's answer because it talks about "states of the field" (when the field in QFT is an operator, not a state), but that sort of phrase is used a lot, not just by John

@MoreAnonymous depends on why you're making it

see e.g. this answer of mine where I talk about conceptualizing the different chiralities of electron/positron as distinct particles

in some contexts, that's a neat thing to do, in others it's pointless

@ACuriousMind there are those of us who think the operator field is a description of some real object that we don't understand i.e. the real quantum field exists and has states.

@ACuriousMind Hmm, ok. But they're 4 very intimately related fields. ;) But I guess there's nothing wrong with that.

@JohnRennie that strikes me as an inconvenient choice of words (akin to people trying to redefine "virtual particle" to mean the actual intermediate state during an interaction) - sure, you can believe that, but why would you give that thing the same name as the operator field in QFT, that's just unnecessarily confusing

Hey, no-one asked me before they came up with the terminology :-)

@JohnRennie Sure. I think your answer is mostly ok, even though I think it'd be better if you said the electron & positron are excitations of the same field. But will it satisfy the OP?

Remember back in 1920 when Dirac started thinking about a quantised string and developed the operator approach to describe it. The result was a 1D quantum field, but there was still a real string hiding in the background.

@PM2Ring I'll have another look. I've made dozens of edits already so one more won't hurt :-)

Hey, people talk about the electromagnetic field of an individual electron, as if each electron had its own separate field, rather than there being a single electromagnetic field that fills space.

that is true

@PM2Ring I've talked about adding and removing energy from the electron field to create and annihilate electrons, but if I say positrons are states of the same field I need to explain why we can also add/remove energy to create/annihilate positrons as well.

@ACuriousMind I will argue that there is one field that's just the tensor product of every fields

i.e. explain why adding energy can create either an electron or a positron.

@Slereah sure, suit yourself

@Slereah if there are physical objects corresponding to the operator fields wouldn't the interactions between them entangle them so they ceased to be different fields anyway?

It depends what you consider to be "different fields" certainly

@JohnRennie all the talk about Fock spaces etc. only works for free fields anyway

True, all true.

You "flick" the field one way, you get an electron, if you flick it the opposite way you get a positron. Hopefully, nobody asks for the details of what "opposite" means. :)

What's an example of a differential equation that doesn't depend continuously on initial data, anyway

Does QM in non-trivial topological space have parastatistics?

@Slereah I got a copy of Steenrod

Hardcover

Seems it was Steenrod’s copy

Has some corrections penciled in

Practical