Any experts in qft in the chat?

has anyone read the book Geometric Phases in Classical and Quantum Mechanics by Chruściński and Jamiołkowski?

what makes the linear algebraic approach to quantum mechanics so popular over the geometric approach?

@SillyGoose what is "the geometric approach"?

it seems like it is taking the perspective: projective hilbert space is a kähler manifold; and writing all the usual quantum results in the language of differential geometry

but im not sure precisely yet

have fun defining that for the usual infinite-dimensional Hilbert spaces of QM

all the beauty of differential geometry in its standard formulation exists only in finitely many dimensions; you have to do a lot of work to establish similar things in infinitely many dimensions

so it is not as "natural" a language is the arugment against the geometric formulation?

I mean, what even is your "geometric formulation" here? Sure, the states of QM live in the projective Hilbert space, and sometimes it is useful to make that explicit

but the core of QM is that states/expectation values are linear functions on observables

"superposition" is just this linearity

i have not read much of it yet, so i am not all sure what this formalism buys us. but it seems like they are using this language to write quantum dynamics as one does in symplectic mechanics, which is just interesting i guess. i am interested in looking more into this stuff because surely if you are using geometry instead of algebra you might encounter new stuff not obvious or sensible in the algebraic formalism but that is still physically relevant

e.g. eigenvalues of observables are significant, and those make much more sense in an "algebraic" formulation, i.e. functional analysis on Hilbert spaces

@SillyGoose why the "surely"?

i tempered the surely by the might :P

but i guess i would expect two different approaches to not just spit out exactly the same stuff

and i would expect people to pursue this geometric formalism not just to capture usual QM but to do something more with it

don't get me wrong, I think it is valuable to look at stuff from various angles, but I really don't like this tendency of "selling" unconventional approaches with a bunch of handwaving

@SillyGoose To be blunt, I think ordinary QM is perfectly well-understood in terms of functional analysis. You can re-cast its finite-dimensional parts in terms of projective Hilbert spaces, and sometimes this is illuminating, but it will most likely not reveal some mysteries that remained unsolved in the traditional approach :P

it's fine to look into that, but if you do that before you've understood the usual approach, all you get is that you now can't communicate with anyone else about it because you're speaking a non-standard language

hm just generally do you think this geometry business is really only of discussion in hep theory related matters? like symplectic geometry, complex geometry, and so on

I'd say it's particularly for non-hep-th matters

oh what sort of matters

since it's simple, finite-dimensional quantum systems where you might apply this without worrying about all the terrible subtleties of "infinite-dimensional manifolds"

oh

hm but isn't that worrying if, say, people in condensed matter theory want to use this formalism that really only works for finite-dimensional systems when we would really like a formalism that is natural for both finite and infinite dimensional systems

well maybe it is just reality :P

do you have an example of "people in CMT" using this formalism?

caveat: you can talk about Berry phases etc. without all this baggage

hm not this geometric formalism of QM, but it seems like such stuff is used in this preprint: chapmanlabs.gatech.edu/papers/…. this is where i got the reference for the book i mention above also

also: "people in CMT" might just be looking at systems with finite-dimensional Hilbert spaces! :P

there is a pretty large gulf between the applications of "QFT" in cond-mat and of "QFT" in hep-th

there's a lot of shared ideas, but neither of those subsumes the other

nvm actually duhem eventually drops the metaphysics and says it's probably a bad idea

Duhem says : A theory of physics is not an explanation. It is a system of mathematical propositions, deduced from a small number of principles, which have for their aim to represent as simply, as completely and as exactly as possible, a group of experimental laws

I know, I am reading it

explainxkcd.com/wiki/index.php/1132:_Frequentists_vs._Bayesians

is frequentist probability bad for this

this problm is really weird becuz u can technically say "the machine's answer is 98% likely to be correct"

as in, this holds for any answer to any question that this machine gives

but it doesnt hold for questions like "has the sun exploded" becuz we have additional information here about the sun's exploding probability

under lack of additional information, it wud hold

so we can say that the machine's answers r 98% likely to be correct over a random sample of questions, but not for specific samples

even for "has the sun exploded?", this machine gives the right answer 98% of the time

but in the sub-sample where it answers "yes" , u cant conclude that it's 98% likely to be correct

this is all resolved by Bayes theorem

under a lack of additional info, its "yes" wud be 98% likely to be correct

@ClaudioMenchinelli It is seldom useful to brute-force through a particular method when a much cleaner and nicer way had already been presented. However, if you did more digging, you would have obtained what you wanted: math.stackexchange.com/a/711309/1175760

@Supersymmetry not specific messages - we can only anonymize all your messages at once, see meta.stackexchange.com/a/157511/263383

@ACuriousMind can't you just delete some specific messages?

I could, but we generally don't delete things here just because people ask us to, neither on the site nor in chat. Everything on SE is public and eternal by default, if you don't want something on the record, don't post it.

@Supersymmetry Yes, if you're sure I'll press the button. It should happen instantly (modulo caching), since you have fewer than 1000 total messages (please confirm that this is what you want, since it is irreversible)

@ACuriousMind so will SE servers outlive the universe?

@Mr.Feynman I'll take a backup with me into the next one

just realized that Lovelock's theorem was actually made using the theory of concomitant objects

@Supersymmetry r u gonna delete the account

some people say that everything u do is going to be effectively irrelevant in a hundred years, so it doesnt matter what u do

Kurzgesagt often say this in the existential crisis videos

i guess people mean it in a positive way

it's done (reload the chat if you don't notice any changes)

Great but we might also have to remove the conversation about anoynmization, because the messages tagged my username. Or it could also work if you could untag me without deleting the message at all

unfortunately the @ in the replies are never removed or changed by any of these systems; if you look into the transcript you'll still see people replying with your username to the anonymized messages. The point of anonymization is that it's no longer possible to directly find those messages by searching for messages from your user

as Shog commented on the meta post I linked:

@ACuriousMind damn you're even immortal

gravitymodification.com

The Heim theory people are still at it

You'd think they'd drop such a stupid theory by a nazi scientist

@naturallyInconsistent I already looked into that answer but forgot to mention it here

Ok~

@naturallyInconsistent anyways, thanks

Apparently as Klein geometries, the Galilean space is characterized by a one-form $\tau$ (the "time metric") and a bilinear form $\gamma$ (the space metric), but otoh the Carrolian space is characterized by a vector $\xi$ and a rank $(0,2)$ tensor $h$, which are basically the same thing

That's a pretty subtle nuance between the two

not sure why the duality gives them the change in behaviour, although I guess I should work out what happens to dual vectors in a Galilean transform (if that notion makes sense since the metric is degenerate)

You do get some weird duality between the two but it's not entirely obvious how it works

Can someone provide a feedback for my derivation of Ward identity here, if possible?

I guess I can bring this up next time someone asks what's the difference between a vector and a dual vector

@Slereah oh, cool. Where did you learn about that?

ACM reading my message and wondering how could I fall again into Slereah's trap

arxiv.org/pdf/2210.13924.pdf

You may find it here

why is classical mechanics so complicated

I neglected to post a plot for the recent solstice, but here's one for the upcoming perihelion. Times are in UTC, with a 6 hour timestep. My plotting script is at astronomy.stackexchange.com/a/49823/16685 & I have some info about the perihelion at astronomy.stackexchange.com/a/49546/16685