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15:01
@CaptainBohemian You could also study the homeomorphism, or the diffeomorphism group of the manifold
Which is hard
probably not of interest to all these geometers though. they are blinded by the metric
vzn
vzn
@Mithrandir24601 did you say new physics? music to my ears! see if you can squeeze this little bit )( into your "big chunk of daily time," just found it yesterday, think its very substantial, in line with a sparkling new/old inquiry of mine, really wanna share it, would really like to hear what you or anyone else thinks :) mdpi.com/1099-4300/18/1/34 oh and re "new science" try this vzn1.wordpress.com/quotes/#kuhn
inhale the r/vaporwaveaesthetics of topology
@BalarkaSen study the space M/D, quotient of the metric cone by the diff group
@vzn Any reason to link to your blog there instead of just posting the quote here?
@BalarkaSen homeomorphism or the diffeomorphism group are not in the domain of geometry? or not one kind of G-structure?
15:05
Diff(M) and Homeo(M) acts on M, sure.
It's just not the kind of blood and flesh geometry with Riemannian metrics
@Danu calls it geometry
vzn
vzn
@ACuriousMind uh, experimenting / still trying to find a link to anywhere posted by me you are not opposed to...? o_O
(I don't agree with him)
@vzn Sorry, what? I didn't object to any other links you've posted here recently.
I do object to you pointlessly linking to your blog when the content you link is surely available at many other places and could in this case even just be copy-pasted into a short chat message.
vzn
vzn
@ACuriousMind theres been some pileon re stuff posted by me. EP was complaining repeatedly about popsci refs. sigh life goes on :(
@ACuriousMind fine then just for you :) todayinsci.com/K/Kuhn_Thomas/KuhnThomas-Quotations.htm
@vzn What has Emilio's (justified, imo) complaint about pop science links to do with me? Stop moving the goalposts, you said you were "still trying to find a link to anywhere posted by me you are not opposed to", implying I consistently complain about your links, which is just untrue.
If you don't want to take my criticism into consideration, fine, but don't misrepresent others.
15:09
“Wir nehmen jetzt an, daß”
Wot
Since when is that spelled with a fancy s
Imma Daßßing at you in German
@0celo7 It was spelled that way before the spelling reform around 2000
💯
@ACuriousMind ah, so I have revealed what I’m reading
15:10
Old math :P
vzn
vzn
cannot win, there is no defense against the circled wagons, made mistake constructing any reply at all, SORRY ABOUT THAT, now fleeing again
@BalarkaSen Definitely geometry
Maybe it’s physics
Maybe it’s meybelline
@BalarkaSen So, if it's not flesh-and-blood geometry, is it...ghostly geometry?
Astral geometry?
Fadeev Popov geometry
Ah, of course the hypotheses are not the same! I’m not crazy after all.
15:19
A morphism of modules is just a group homomorphism such that the square involving the action of the ring commutes, right?
@Danu Yes
Right
So if I have a sheaf of rings $A$ and for each open set also a module $B(U)$ over the ring $A(U) $, I can actually talk about a sheaf $B$ of $A$-modules
if the restrictions play nice
In particular if I have a sheaf of super rings then the odd part is a sheaf of modules over the even part
Sure, you have to check that $B$ is actually a sheaf
15:25
because the restrictions play nice by construction and the sheaf axioms will follow from those for the super ring
I have to talk about supermanifolds in the seminar for PhD students on monday haha
My condolences ;P
Super- stuff is super-annoying
Everything's fine until you actually have to compute something, then it's sign issues ten times worse than anything before :P
I'm just going to make puns
super-definitions
super-remarks
super difficult proof
@vzn that link would have been potentially more relevant for me if it was looking at the Hamiltonian instead of the wavefunction... It's making me wonder if continuity equation might be helpful for me though, so you're not a million miles away from something that I could potentially find useful. The overall gist of it seems to just be an interpretation thing though
> The overall gist of it seems to just be an interpretation thing though
classic QM
@Danu We'll now carry out a super-proof
cue dramatic music
15:39
If you write it down twice, it becomes trivial.
Stupid basic question
Can I say that if I have an ideal $I$ inside a commutative ring $A$, then $I$ is maximal iff all elements in the complement are invertible? Should I assume $A$ is local? Should I assume the ideal is prime?
Really what I'm asking is: I learned that if $I$ is maximal and $A$ local, then all elements in the complement are invertible. What's the correct converse? How generally does something like this hold?
@Danu I think you don't need to assume anything - if all elements in the complement are invertible, then any ideal containing $\mathfrak{p}$ and one of these elements is the whole ring
If $A$ is not local, it can happen that not all elements in the complement of a maximal ideal are invertible, though
Although @vzn when I'm referring to 'new science', I don't mean it in that way - I spend the majority of my time looking at non-hermitian Hamiltonians, so it's nothing radical - just things like lossy systems
15:46
@ACuriousMind Right, if an ideal contains the multiplicative identity then it's everything
@Mithrandir24601 non-self-adjoint problems are fun
So if I now have a local ring and its maximal ideal, how do I prove the other things are invertible?
but one should usually not interpret them as new physics
l
oops
^unless one is requesting funding
@Semiclassical yes, but they're simultaneously a nightmare :P
15:48
lol
from a perspective of proving things, definitely
...and numerics, if I cast my mind back
it makes a lot of the assumptions/safeguards one has in such problems not applicable
@Semiclassical I have the tendency to refer to any research of unknown/undiscovered physics I.e. any research 'new', but that's probably just me :P
it also usually makes it a lot harder to find eigenvalues, since you can't just look at the real line anymore
@Danu Every ideal is contained in a maximal ideal. If there's a non-invertible element in the complement, its principal ideal is contained in a maximal ideal that's not the local ring's maximal ideal, which is a contradiction.
vzn
vzn
@Mithrandir24601 lol nicest thing anyones said to me all day, thx for poking at it. :) just thought it worth looking at by anyone interested in QM details. admittedly few are working in the particular area. maybe some mind expanding peripheral potential. "new science" can have nearly radically different meanings depending on context etc...
@Semiclassical nah, the eigenvalues just become complex :P
15:50
lol
@ACuriousMind Thanks
I mostly have in mind the problems I worked on a few years back where (for instance) $V(x)=e^{2ix}+2e^{-ix}$
@Danu np :)
Of course, when I say this, people get a tad confused. Including myself :P
So in a local ring I have an equivalence
15:52
it's a periodic potential and one generally still has a notion of 'band structure'
but now those bands can move off into the imaginary direction :)
And if I have an ideal such that the complement is invertible, then in fact $A$ is local, isn't it @ACuriousMind?
(though, because of the pt-symmetry, this can only happen once the endpoints of two bands on the real line collide.)
(blah blah blah)
@Danu Yes, I think so
@Mithrandir24601 one place I've seen non-self-adjoint stuff which I always meant to read on further was in hydrodynamics
@ACuriousMind By your argument
15:54
@Semiclassical joking aside, some of the point is that in certain regimes (I.e. when the Hamiltonian is PT symmetric), the eigenvalues are still real. Breaking the symmetry gives complex eigenvalues
oh, sure
broken vs. unbroken pt-symmetry
No wait, now I'm confused
the trouble is that the point at which pt-symmetry becomes broken (in the sense of eigenvalues becoming complex) needn't be easy to find
oh yes, actually it's easy
because any invertible element is only contained in the entire thing, no proper ideal
And in this sense, 20 years ago, this really was New physics as it questions the fundamental postulates of Copenhagen as a non-hermitian Hamiltonian can still be PT symmetric
15:56
hmm
@Mithrandir24601 postulates of Copenhagen?
If you mean the standard axioms of QM, these are not really tied to the Copenhagen interpretation
my sense was that, while PT-symmetric hamiltonians can still be interesting, one shouldn't take them as telling you anything new about QM
vzn
vzn
@Mithrandir24601 ps havent looked at paper in detail yet but surely theres a hamiltonian "mirror" to all the derivations. feel its tip of iceberg etc
Does anyone have a coherent statement of this evil "Copenhagen" program that everybody loves to criticize? :D
@ACuriousMind Copenhagen interpretation of QM - I'm on my phone, give me some chance :P
15:57
@Mithrandir24601 I understood you referred to that interpretation, but "the Hamiltonian is Hermitian" is not particular to any interpretation
vzn
vzn
@Danu lol yes but its in my blog & will get shot down by the link police if cfd by me :( ... anyway theres a nice paper analyzing community beliefs in interpretations...
@ACuriousMind Is it not? Hmm, fair enough then - I stand corrected
You just need to have spectral theorems and stuff to do the usual things @Mithrandir24601
To do things like functional calculus
That's one of the reasons why you need self-adjoint things
@Mithrandir24601 One can be operationalist or MWE or anything really (except for stuff like Bohm which uses a different formalism), the axioms of QM don't really change, just how you interpret phrases like "a measurement happens" or "probability".
15:59
@Semiclassical yeah, that's another thing in itself - looking at exceptional points
if you want to conserve total probability, you pretty much need a hermitian Hamiltonian
@vzn If your blog is genuinely the best source you know for something in particular, I have no objections at all if you link to it. Stop crying persecution.
@ACuriousMind ehhh, I think it's a bit wrong to say that Bohm uses a different formalism. You use the wavefunction and the Schrodinger equation in pilot-wave theory just as much as in ordinary QM.
@ACuriousMind yeah, fair enough
@Semiclassical Bohm certainly does not use the Hilbert space + operator formalism.
vzn
vzn
16:00
@ACuriousMind catch 22 double bind for me, thx for your ostensible flexibility
insofar as that's reflective of the wavefunction, yes it certainly does
what pilot-wave theory adds is that one should take the quantity $\mathbf{j}/\rho$ (probability current and probaiblity density) as not being just mathematically a velocity field but physically one as well.
but the probability current is still obtained from the wavefunction.
@Semiclassical The standard axiom of QM is "a state is a vector in a Hilbert space", it makes no reference to wavefunctions. Taking the wavefunction/pilot wave as a fundamental axiom is a departure from the standard formalism.
@Danu which is why it's hard when you can't use the spectral theorem :P there are physical systems that violate parity symmetry though, so it's not exactly unphysical by itself
We just use wavefunctions because that's a convenient thing to represent the states for systems with position operators, but the standard formalism can in principle do without any reference to "wavefunctions". Bohm cannot, which is why I claim the formalism is different, though the predictions are not.
on that, I"ll agree. the issue is whether Hilbert space is axiomatic or not. In regular QM, it is; in Bohm, it isn't.
vzn
vzn
16:04
@Danu anyway these are very nice papers/ surveys for anyone inside/ outside the field going over the essential semi controversy arxiv.org/abs/1306.4646 arxiv.org/abs/1301.1069
in that respect it does go back to the whole matrix mechanics vs. wave mechanics issue in the history of QM
@Semiclassical I'd argue that questioning one of the fundamental postulates counts, but it's not impossible there are ways round this
well, the thing is, when you actually look at the examples of pt-symmetric systems
vzn
vzn
@ACuriousMind was pondering this topic recently, theres this thing called the schroedinger wave eqn...
they're not systems which are 'exotic' from the quantum-mechanical perspective, as far as I remember. they're just systems which are conveniently represented by pt-symmetric hamiltonians
though it's been a while since I looked at this, to be fair.
so I"m forgetting what real-world problems have been successfully modeled using pt-symmetric systems. (optical lattices with balanced gain/loss, maybe?)
16:09
@Semiclassical they can still be non-hermitian though - e.g. there's a type of crystal that undergoes a transition at ~200K below which, it violates parity inversion symmetry
Though yeah, your optical lattice has been done before
I don't remember a lot of this stuff, but
my guess would be that it's not non-Hermitian in the sense that "zomg probability isn't conserved"
but rather that there's a representation of the system which, while physically not entirely sound, is nevertheless convenient and useful.
The other thing I know of is frustrated Mott insulators are the same. It seems plausible to use them to model loss as well
e.g. one is choosing to take the system to be such that one has particle transport out of it. but if one broadened the system to include the environment, then one really would have conservation of probability
in other words, I would suspect a failure of hermiticity to tell you more about how one has defined the system than anything about QM itself.
@Semiclassical yeah, including the environment can (in at least some/many cases) make things Hermitian again
question, of course, is whether "many" really means "all"
for my money I'd bet on all physically-relevant PT-symmetric problems being ultimately consistent with good old hermitian QM.
which again doesn't make pt-symmetry uninteresting. but I don't expect it to up-end our understanding of QM.
16:17
@Semiclassical well, I've got 2 physical examples that violate parity inversion symmetry. The eigenvalues are still real, so I wouldn't assume that it messes with probability off-hand
is the reality of eigenvalues in those cases a consequence of the pt-symmetric potential not being strong enough? i.e. is there a parameter you could tune so that some eigenvalues are complex?
@Semiclassical no, just that Hamiltonians must be PT-symmetric instead of 'hermitian'. This is all just 'maybe' and 'I don't know' as I don't exactly spend my time worrying about things like that and just look at them because they're interesting
@Semiclassical yeah, exactly that
in that case, the eigenvalues being real is contingent on having a sufficiently weak 'perturbation' to the original hermitian problem
and that doesn't entirely reassure me.
vzn
vzn
16:22
maybe qm (formalism) is an ultimate case of "if you have a hammer, everything looks like a nail"...
I also vaguely remember theorems that said stuff like: If pt-symmetry is unbroken, then the non-hermitian hamiltonian is unitarily equivalent to a hermitian Hamiltonian
@Semiclassical yeah. The point of what I was saying is that if someone asked me if I could guarantee hermiticity, my answer would be "I don't know", which to me is substantially different to the regular QM answer of "yes"
I think my general feeling re: pt-symmetry is rather similar to the situation in metamaterials
@Semiclassical pseudo-hermitian is the word you want, so maybe physically valid hamiltonians have to be pseudo-hermitian is a possible adjustment to the postulates?
it's not that the existence of metamaterials (with negative indices of reflection) somehow challenges the validity of maxwell's equations within their domain. It's that materials with interesting structures can behave at the macroscopic scale in ways one wouldn't expect.
negative indices of refraction are a statement about the macroscopic description of your system, not about the microscopic physics.
@Mithrandir24601 well, I'd take it in a different direction. if i've got a pt-symmetric hamiltonian, then this pt-symmetry is either exact or broken.
16:30
-3
Q: Riding in a bus with "HAUNTED WINDS"

KRISHNANAND JI believe that wind does no magic tricks. Why do I say that? Well, it just occurred to me that wind started flowing in the same direction as that of the vehicle. I was in a fast moving bus, when I saw the long hair of the lady in front of me getting thrown forwards due to the wind, instead ...

if it's exact, then it's equivalent to a hermitian hamiltonian. if it's broken, then i'll have to be careful about probability being conserved and it probably reflects how my system has been defined
@Abcd is he on crack?
seems like it xD
vzn
vzn
@Mithrandir24601 cant follow this debate exactly but find it intriguing & will pay more attn to it. but anyway the idea of collecting apparent "anomalies" wrt current theory and continually pondering them/ their nature is very kuhnian & do commend you for it... feynman also discussed a similar style wrt unsolved problems etc :)
16:40
@Phase I feel he's just spamming. The scene he has described is common in disgusting Indian horror stuff.
or maybe that "haunted winds" was just to attract attention.
"haunted wind" sounds like a euphemism for flatulence
vzn
vzn
@0celo7 ??? what is (mathematical?) "arm candy" anyway?
@Semiclassical except for the bit where the inner product is redefined :P
@Mithrandir24601 I suspect this is a good review article by the same author: arxiv.org/pdf/0810.5643.pdf
vzn
vzn
16:44
@Semiclassical aka emergence (in complex/ dynamical systems)...
I do think it comes down to the issue re: how the inner product works
16:56
Huh...
vzn
vzn
misc googling, this 2017 paper seems to relate to the issue, PT-symmetry, indefinite metric, and nonlinear quantum mechanics iopscience.iop.org/article/10.1088/1751-8121/aa91e2/meta
generalized nonlinear sch eqns linked to quantum fluid dynamics link.springer.com/chapter/10.1007/978-3-642-73193-8_24
@vzn one of that author's earlier papers, which he cites in the one you just linked, is this one: iopscience.iop.org/article/10.1088/1751-8113/49/10/10LT03
the abstract seems worth quoting here:
What the men not hot think I do: 2 + 2 - 1 = 3
In recent reports, suggestions have been put forward to the effect that parity and time-reversal (PT) symmetry in quantum mechanics is incompatible with causality. It is shown here, in contrast, that PT-symmetric quantum mechanics is fully consistent with standard quantum mechanics. This follows from the surprising fact that the much-discussed metric operator on Hilbert space is not physically observable.
In particular, for closed quantum systems in finite dimensions there is no statistical test that one can perform on the outcomes of measurements to determine whether the Hamiltonian is Her
that second-to-last line seems the key one to me
@SirCumference "what I actually do" First semester analysis?
17:09
I bet no-one can guess what today's lunch was:
that looks awful
what is it?
fries and beef+rice?
No other guesses?
it reminds me of the hamburger helper recipe for "rice oriental" i'd have as a kid a lot
It was ... dum, dum, DUM ...
Haggis is a savoury pudding containing sheep's pluck (heart, liver, and lungs); minced with onion, oatmeal, suet, spices, and salt, mixed with stock, traditionally encased in the animal's stomach though now often in an artificial casing instead. According to the 2001 English edition of the Larousse Gastronomique: "Although its description is not immediately appealing, haggis has an excellent nutty texture and delicious savoury flavour". It is believed that food similar to haggis (though not so named), perishable offal quickly cooked inside an animal's stomach, all conveniently available after a...
17:11
...
:S
@Semiclassical yeah, you do have to careful when it's broken - in this case, I think it's just a case of loss to environment where environment is traced out. I do disagree about it being hermitian due to the above issue about redefinition of inner product, but I do at least agree it's pseudo-hermitian, which is so close that it's effectively the same (hence 'pseudo') as none of the actual results that come out the other end actually disagree with any of QM
oh no
I was right
disgusting
"its description is not immediately appealing"
that's a bit of an understatement
@Mithrandir24601 yeah, see that abstract a quoted a few minutes ago
17:12
@0celo7 you been back at your PC yet?
It was actually an expensive gourmet haggis bought from the UK's poshest supermarket.
if it's a closed system, there's nothing that will experimentally differentiate Hermitian and pseudo-Hermitian QM
@JohnRennie Waitrose?
@Phase Indeed!
@Phase I accepted the request remotely
17:14
I always found waitrose overrated tbh
teamviewer
I feel the meat and snack variety you get from Tescos >>>> Waitrose
@0celo7 found any gaymes yet?
if it's an open system, differences are possible. (but if it's an open system, that reflects a certain choice of the experimenter on where one draws a line between system and environment.)
@Phase no, i'm at home
@Phase their haggis is very good
17:16
Fair. Do you normally stay in student accommodation or something @0celo7 ?
@JohnRennie Never had haggis, does it taste at all similar to black pudding?
@Phase no, completely different. It's more like faggots.
[flags]
Never had faggots
They're like sausage meat right?
No, completely different :-)
This isn't going well ...
Ah well I'll just try it some day
Actually, looking back at the image, I don't think I will
@ACuriousMind in retrospect, I suppose the point being discussed earlier could have been made like so. In standard QM, the story is ultimately about Hilbert space; position space is useful, but not fundamental. In pilot-wave theory the opposite is true.
(Standard pilot-wave theory, anyways. there's a way to write a version which takes momentum space as fundamental, but it's not the usual formulation.)
17:30
@Semiclassical oh yeah, thanks for this :)
@Slereah The term "indefinite integral" has never helped anybody
@JohnRennie ...
@SirCumference Do you prefer the term antiderivative
@Phase I have an apartment on campus
@Slereah primitive
Is it a nice one @0celo7 ?
17:31
@0celo7 probability of someone flagging John for that?
@Slereah I think something is either an integral or it's not
@JohnRennie haggis is so, so delicious :)
@SirCumference What if it's one, but
indefinitely
I abhor the term "indefinite integral", I don't care what you call it otherwise
@Mithrandir24601 yes! There is another physicist with good taste! :-)
17:32
@Blue Thanks for that playlist on LA. It was awesome! Have you got any other recommendations? ;p
@CooperCape not me. But he’s playing with Fire.
Faggots are a traditional dish in the UK, especially South and Mid Wales and the Midlands of England. It is made from meat off-cuts and offal, especially pork. A faggot is traditionally made from pig's heart, liver and fatty belly meat or bacon minced together, with herbs added for flavouring and sometimes bread crumbs. Faggots originated as a traditional cheap food of ordinary country people in Western England, particularly west Wiltshire and the West Midlands. Their popularity spread from there, especially to South Wales in the mid-nineteenth century, when many agricultural workers left the land...
The name is ridiculous
@JohnRennie Don't worry I'm not that out of the food loop ;p We used to have them at school for lunches... ewww
Someone really should start a movement to change it
17:33
@0celo7 Not really, faggots also refers to bundles of sticks
Oh i see its sarcasm nvm
@vzn and how did I miss this? - this is indeed interesting, thanks :)
@0celo7 What, the dish or the indefinite integral?
I’m not being sarcastic
isn't that fagotti? @Phase
@SirCumference the dish
17:33
That comes up a lot cause I'm doing the most useful A level of all - Music.
Indefinite integral is also stupid, it’s a primitive
@0celo7 I think the food predated the US slang by a few centuries...
You’re wrong @SirCumference
@JohnRennie on that note, see the second portion (number 6) of this list: cracked.com/…
17:34
That’s the high school terminology. When you learn analysis you’ll correct your thinking.
@Semiclassical I ain't clicking on a cracked link
@Semiclassical :-)
Tbh though whenever I say "antidifferentiate" in front of a physics grad student they laugh at me and tell me to say "integrate"
Not giving my hard earned internet dollars to those hacks
17:35
@JohnRennie the swastika predated the nazis by millennia but you don’t see me walking around with one on my jacket
@Semiclassical archive it
@0celo7 I've seen you
@SirCumference because that’s not a word
I'll just cite the relevant bit
@Semiclassical yeah, this makes total sense - slightly different starting points, but the results are at worst consistent and at best, the same
17:35
Goose stepping down the streets
@0celo7 Wait it's not?
@SirCumference no?
Antidifferentiation is a word, isn't it?
So my gf just sent me this With the message "Personally I'd rather inverse the matrix"
rip ;p
@0celo7 We should petition / start a movement to change the Name of geese, as the idea of a "goose walking" references the oppressive nazi regime
17:36
@SirCumference probably not
>rather invert a 4x4 matrix
lol ok then
hmmm yeah
:c
Though she did say "I'm not sure which is worse"
There's hope there ;p
"[The incidents] all stems from a Reddit rule that subreddits which haven't been moderated in 60 days are up for grabs for anyone who wants them, so a racist moderator who takes their eye off the ball for too long might find themselves replaced by someone whose mission isn't quite so nefarious...Other appropriated subreddits include ... r/faggots, which is now a community for discussions about the traditional British food item."
that's the topical bit :P
@CooperCape you have fb?
That I do
17:40
If so, Mathematical Mathematics memes is a good [sometimes] group
as is >implying we can discuss physics
Oooh physics... ;p
dont join >implying actually
Thanks for the interesting discussion @Semiclassical and @vzn I'm delighted that you're sending papers instead of pop-sci articles :)
It's just a worse version of the h bar with more nazis
17:41
Ah right...
ummm
okaaaay
so basically just join Mathematical Mathematics Memes
Okay
I was
noice
Really confused as to what was going on
can't guarantee it's quality lately but it used to be decent
It's just a meme group based around maths
Engineering memes seem to popular right now
Anonymous
17:52
@CooperCape Now start reading Artin :)
Is that a sort of "Artin is impossible lol you'll fail" joke or is that the 'next step' of sorts?
okay it seems legit
Anonymous
@CooperCape No, I'm serious.
if a little pricey... but meh
Okay, thanks :)
Anonymous
The first 4 chapters include Linear Algebra
Anonymous
17:54
Yeah, it's expensive. I read from the PDF
There's no way I'm buying it I'll just libgen it... (shhh) ;p
Anonymous
Need to revise it though
Anonymous
I forgot many things
Anonymous
@CooperCape Yep
Okay, nice one! Thanks a bunch
17:57
@ACuriousMind It is obscene geometry. The best kind.
@SirCumference I agree
Well, there's an interesting point to be made there actually
The indefinite integral, or the antiderivative, is just the "inverse operator" of the derivative operator. Whereas integral is just by definition area under the graph of a function over a fixed interval
It's a great magic that these two are intimately related: Fundamental theorem of calculus
There is literally like no reason to believe that on first glance

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