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3:00 PM
@0celo7 what?
@0celo7 I might find it interesting if I didn't have the suspicion you're making that up :P
 
@0celo7 I do not see how that's particular to Lefschetz fixed points tho.
 
@0celo7 Symbol. A Generic element. I could have used the element of surprise: $\mathrm{Ah}$
 
@ACuriousMind If you have a map $f$ with one fixed point, you can homotope it to a map that has (possibly many) Lefschetz fixed points
and, independent of the homotopy, the Lefschetz number remains the same
 
I can, like, take a smooth map $f$ of equidimensional manifolds, and take a value $x$, pullback by $f$. Pick $y \in f^{-1}(x)$.
Perturb $f$ to make it transverse to $x$. Then $y$ spits into bunch o' regular points.
Also, the sum of orientation no's is independent of how I perturb
 
@ACuriousMind My point was that for you to have a serious issue as to the validity of my question on qualitatively predicting why $a(k) = \int d^3 x e^{ik_{\mu} x^{\mu}} (\omega_{\vec{k}} \psi + i \pi)$ should be the answer you get for the free scalar KG field before doing any calculations, where I said I could calculate it 3 different ways, buttressed by the fact people are also voting to close a fantastic question in stage 3,
indicates a pattern of stage 2 mentality, and it should be troubling, and I want to know how you justify ignoring such a question in a way that doesn't admit being outside of stage 3
 
3:03 PM
@BalarkaSen $f^{-1}(x)$ might be some fucked up shape though
if $f$ does not have $y$ as a regular value, who knows what $f^{-1}(x)$ is
 
Um, sure, so can be the fixed point set of an endomorphism of a smooth manifold.
 
@bolbteppa I'm not troubled at all that you think I have a "stage 2 mentality".
 
@ACuriousMind If a good question is not on-topic here then the site is stage 2 level, not good enough for experts, is that really what this site is supposed to be and why lots of them left or rarely post like they used to?
 
@0celo7 y is just an element of f^{-1}(x)
 
@bolbteppa: ACM does not have to justify ignoring your question. That's his choice. You are not paying him to answer your question.
 
3:04 PM
A point out of there.
 
@BalarkaSen oh, well we're talking about isolated fixed points here
 
@JohnRennie I'm not ignoring it, I voted to close it as unclear. That's why he's so salty.
 
@JohnRennie obviously, but that doesn't mean I can't ask him directly
 
I can assume every point of $f^{-1}(x)$ is isolated too :D
 
@bolbteppa your persistence on this matter is already well beyond what many of us would consider polite.
FWIW I would answer your question if I could, but I can't.
 
3:08 PM
@BalarkaSen so you think that any preimage point is just a combination of regular points of some homotoped map?
 
user116211
Well, sooner or later we have to think about who would be our next AMA guest after @yuggib.
 
@JohnRennie I have mainly responded to statements directed at me, that is an absolutely amazing statement to make towards me given that, you don't consider it polite because I am calling out an issue that is uncomfortable, but I find it extremely rude to close and block good questions on this site because of some ridiculous reasoning
 
Yes.
 
perhaps
 
user116211
What do you say @JohnRennie, would you like to be the next guest?
 
3:08 PM
but GP think this is more interesting and deserving of recognition :P
 
My intuition comes from branched covers though.
@0celo7 Which is weirding me out, but ok.
 
the heck
@BalarkaSen I think they're trying to find an algorithm for calculating $L(f)$ for general maps
it's easy if $f$ is Lefschetz, but in general $I(\Delta,\Gamma(f))$ might be terrible
 
@MAFIA36790 I've been asked before, but honestly I don't have anything interesting to talk about. The only thing I was even remotely expert in was colloid science, and I stopped doing that 19 years ago so I've forgotten most of it.
 
@ACuriousMind I did not think you had a stage 2 mentality, you have great answers sometimes, I am asking about your comment on my question and where your thinking comes from on it
 
so we homotopy $f$ a bit to make it Lefschetz, count the "charges" on the "scattered fixed points", then add them up to get $L(f)$
@ACuriousMind How can you not find this super interesting?
 
user116211
3:11 PM
@JohnRennie okayish.
 
or...maybe that's not what we're doing
now they're talking about something I know as the Gauss map
 
Have fun.
 
@BalarkaSen I'm almost done with what I want to read GP for...
on to Milnor next
 
@ACuriousMind I then had to point out your responses thus far exclude all those amazing MO expert answers, and that it should be troubling, I can easily find questions on this site where Lubos or Ron will give some amazing insight to a question like those
 
@MAFIA36790 Colloid science is absolutely fascinating, and underlies enormous parts of everyday life. It's a great area to work in. But I just don't see it working as an AMA.
 
3:13 PM
G'luck
 
@BalarkaSen do you know any Morse theory
 
Only an infinitisimal amount.
So, no, not really.
 
just like you don't know any topology?
 
@0celo7 Well...wouldn't that analogy work for any case where there is some invariant that can be computed from a variable number of points?
 
No, no. I genuinely don't know this :P
 
3:14 PM
@bolbteppa for Heaven's sake give it a rest
 
I know what's in G&P, and I know what it does. That's about it.
 
I.e., I'm not sure this actually tells me something about Lefschetz fixed points in particular.
 
@JohnRennie What's your problem with me discussing a comment he made on my question that I posted on the website last night?
 
@BalarkaSen Oh, damn
 
user116211
Well lemme see if I can contact Valter Moretti; he would be a very competent guest. Lemme see how to contact him. May be he would be busy but why not try ;)
 
3:15 PM
Luckily @ACuriousMind read Milnor and is an expert
 
Great, we won't see you in the math chat anymore then.
 
@0celo7 Ahahahahahaha...I hope that's a joke ;P
 
@bolbteppa my problem is you going on and on about it when no-one but you is interested
 
@BalarkaSen :(
what did I do
 
user116211
@Balarka, would you like to be a potential future AMA guest?
 
3:17 PM
Nah, I'm fine
I should get some work done before it's too late. I want to read/watch a movie today.
See y'all.
 
@ACuriousMind No, you're a super genius
I'm sure you can figure it out if you don't know it immediately
@ACuriousMind You can certainly help with algebraic topology I assume
 
@0celo7 Great, so you being mad at me for not actually knowing something because you think I'm just not telling you will become a regular occurence :P
@0celo7 Probably more than with Morse theory proper, at least
 
@ACuriousMind you read the book!??!
 
@JohnRennie I literally just responded to a comment directed at me, all referring back to a comment he first made on a question I posted on this site, I didn't start this, don't direct your problems through passive-aggressive comments at me please
 
@skillpatrol \o
 
3:20 PM
@yuggib gg breh
 
user116211
Welcome @yuggib ;)
 
@0celo7 I think we already talked about my reading being rather more cursory than what you consider "reading a book"
 
@ACuriousMind you skimmed?
 
@0celo7 I don't know what that means
hi all
 
@bolbteppa whether you started it is immaterial, you are perpetuating it. ACM stopped responding to you a while back. Take the hint.
 
3:21 PM
@yuggib "Good game my brother. You failed and I laughed heartily. Hahaha."
 
@0celo7 good game?
 
@0celo7 That's probably a better characterization for the purposes of what details I remember, yes
 
@ACuriousMind Suppose I have a linear isomorphism $A-I$ on $\Bbb R^k$. Then the image of the unit ball under this map contains some closed ball of radius $c>0$. I'm fairly confident I understand this point.
I'm not sure if the "unit ball" here is open though.
 
So @yuggib
Are you pumped for your AMA
 
Then, it is claimed that $|(A-I)z|>c|z|$ $\forall z\in\Bbb R^k$ due to linearity.
 
3:25 PM
@0celo7 Doesn't matter - it's true for the open version, and the closed is just a bit more.
 
Oh, I understand that too.
Er...do I?
 
@Slereah I am suffering sleep deprivation, and I had four very heavy courses in the summer school I am attending, and around 4 liters of beer yesterday night
 
@ACuriousMind Clearly the image of the open ball is open, so it contains an open ball, and any open ball contains a closed ball.
 
Well take 4 liters of coffee now
 
so, yes I am pumped ;-P
 
3:26 PM
@JohnRennie You are behaving very inappropriately now to be honest, I am only "perpetuating" getting an answer as to what his comment on my question that I posted on this site was really about for god sake, wow... If I did this through comments on my question it would direct me to the chat anyway. I mean are you kidding me, I just responded to his last comment to me, that's called a conversation. If you could answer the question I posted on the site you'd understand why I am following up.
 
@ACuriousMind If it's the closed unit ball, simply consider the restriction to the open one and repeat the last sentence.
Good?
 
@yuggib Perfect preparation!
@0celo7 Good
 
;-)
 
@ACuriousMind Now, for the other part...
 
One can easily heuristically motivate why the Hamiltonian is the way it is, why the wave function is the way it is, etc... but the creation/annihilation operators in terms of the wave functions is left un-answered in every book, it's basically just an inverse to those questions, obviously it demands a similarly heuristic explanation
 
3:28 PM
@ACuriousMind Hmm
I think I have it, one moment...
 
Yesterday I found an awesome one-line direct derivation of it $a(p) = \frac{1}{2}(2a(p)) = ...$ which makes now 3 separate derivations, but still a nice intuition is lacking
 
Let $z\in B_1(0)$. Then $|(A-I)z|$ is ...somewhere
crap
 
@0celo7 Hint: The boundary of the unit ball must be mapped to the boundary of its image.
 
maybe you need that previous result to bound $A-I$ wrt. the operator norm
 
@bolbteppa I am behaving as a room owner, specifically one of the owners of this room. I think your continued attempts to get ACM to respond when he is uninterested in doing so are not constructive. For the record I haven't, and won't, vote to close your question because I think it's a reasonable question. However it's beyond my ability to answer.
 
3:31 PM
@0celo7 What about this: Take $z$ with unit norm. Since the boundary is mapped to the boundary, $\lvert (A-I)z\rvert > c$, as the ball of radius $c$ is wholly contained.
Now, linearity gives the statement for all $z$.
 
@ACuriousMind yes, I think that's the idea
I had that thought too, but I could not put "as the ball of radius $c$ is wholly contained" into words
Reference: Lefschetz fixed point ball linearity GP
I am now attaching references to our chat conversations to make them easier to find in the chat log
 
lol
Actually not that bad an idea
 
@ACuriousMind Yup
I have said GP 61/81 times it has been said in this chat.
@ACuriousMind Oh lord, I need induction for an exercise
 
@JohnRennie He was interested enough to post his comment on my question and to justify it earlier in the chat until I illustrated his justifications betrayed something that should be worrying then he ran away and you're saying my attempts to give him a final moment to save face by answering are unconstructive, I guess I'll leave it at that so
 
I still do not understand induction, how do I fix that
 
3:39 PM
@0celo7 you would be a very bad constructivist
 
@yuggib I'm not sure I buy induction as a principle tho
I don't remember the proof of it we did in algebra class
 
It is not a principle
 
we reduced it to "every subset of $\Bbb N$ has a least element"
 
it comes from usual axioms and inference rules
and there is essentially nothing to prove
 
@0celo7 Well, you understand it in a first case, and then you prove that once you've understood the nth induction, you will understand the n+1th induction.
 
3:45 PM
@ACuriousMind Troll.
> Proof. Let S be the set of all natural numbers for which P(n) is false. Let us see what happens if we assert that S is nonempty. Well-ordering tells us that S has a least element, say t. Moreover, since P(0) is true, t is not 0. Since every natural number is either zero or some n + 1, there is some natural number n such that n + 1 = t. Now n is less than t, and t is the least element of S.
> It follows that n is not in S, and so P(n) is true. This means that P(n + 1) is true, and so P(t) is true. This is a contradiction, since t was in S. Therefore, S is empty.
Let's see...
 
@0celo7 that's not nice
Anyway, we have a chat session with AMA coming up
 
@ACuriousMind Oh, I solved one of the extremely hard problems from Milnor using winding numbers and intersection theory. No clue how one was supposed to do it with what Milnor presented.
 
Who's going to manage the AMA portion of the chat session? @yuggib should I just turn it over to you, or is someone going to introduce you?
I think @vzn said he wouldn't be around
 
@DavidZ Are you kidding me??
 
@DavidZ Well, I was trolling :P
 
3:49 PM
ACM, who has called me a troll many times, when I was NOT trolling, gets your protection?
Give me a break
Ridiculous moderation
 
Don't get me wrong, I know you were joking around (but that doesn't make it nice)
 
His profile pic is literally a troll
 
@0celo7 You didn't do it like the author wanted you to?!?! How terrible!
@0celo7 No, it's an orc.
Get your imaginary species right.
 
@ACuriousMind I know...
I might post an MSE thing about it
It's really bugging me
My prof said it can be done, but it's "too hard" and "much easier" using intersection theory
 
user147690
@0celo7 What intersection theory do you know?
 
3:51 PM
@AlexClark the stuff in GP
I needed Borsuk-Ulam
 
user147690
@0celo7 GP?
 
user147690
Gulpol?
 
No one calls it that, but yes.
Actually, I only even proved Milnor's result mod 2, I just realized.
Crazy
 
user147690
Couldn't remember how to spell their names, but remembered that it sounded approximately like that^.
 
user116211
8 mins to go
 
3:52 PM
@AlexClark Guillemin & Pollack.
 
user116211
Welcome @ArnoldNeumaier.
 
Apparently any linear isomorphism is homotopic to the identity or a reflection
@ACuriousMind do you know a quick proof
 
0
Q: Privacy error for SE link

Arnold NeumaierWhen trying to go to ther highlighted discussion Ask Me Anything 12th July: guest's introduction and questions my chromium browser gives an error message ''Your connection is not private Attackers might be trying to steal your information from meta.physics.stackexchange.com (for example, password...

 
18
Q: Ask Me Anything 12th July: guest's introduction and questions

yuggibI will be the next AMA guest. So here it is a brief introduction of myself, and you can suggest some topic/questions that can be discussed during the session. Academic Background. I obtained a master degree in theoretical physics almost ten years ago, working on the regularization of path integr...

 
@DavidZ I don't know
 
3:55 PM
@0celo7 That amounts to proving that $\mathrm{GL}(n,\mathbb{R})$ has two connected components
 
I guess someone can do moderator/intro
other than me I mean
 
@ACuriousMind !!! Is it equivalent!?!
 
@vzn is not here, so I guess we're looking for someone else
 
user116211
@yuggib @JohnRennie?
 
And now I recall that's exactly how Milnor did it
@ACuriousMind Please tell me it's equivalent
 
3:57 PM
@MAFIA36790 why not
or yourself
 
user116211
3 mins to go
 
;-)
 
I volunteer
 
you can, without trolling too much
 
@MAFIA36790 the problem is that I don't understand a word of yuggib's summary :-) I may struggle to ask any sensible questions!
 
3:57 PM
;-P
 
@yuggib NOT NICE
@DavidZ please reprimand him
 
user116211
@yuggib ;))
 
@0celo7 I think it is.
 
@0celo7 see the emoticon
@JohnRennie I am sure you understand the words...maybe not the context in which they're used ;-P
I think that as we discussed, the questions should be put in boldface
 
user116211
1 min to go
 
3:59 PM
the countdown seems exaggerated...
 
user116211
@yuggib ;}
 
We have lift-off
 
haha OK then
 
user116211
okay, the time comes....
 
Welcome to our chat session everyone!
Please keep discussion on unrelated topics to a minimum while the chat session is in progress
Here is our agenda for the day:

1. Intro, welcome newcomers, take policy questions (<5m)
2. AMA event with @yuggib
 
4:01 PM
hi you all
 
So with that in mind, is anyone here new to the site or new to chat or chat sessions?
Any questions or stuff to take care of before we move on to the AMA?
 
howto type here something in boldface?
 
** **
 
user116211
@ArnoldNeumaier __text__
 
@ArnoldNeumaier Enclose text in double asterisks: **text**
 
4:02 PM
In general, you can click on "help" at the lower right corner to see the formatting commands
 
*** *** gives the glorious italic bold
 
so @DavidZ, I am ready when you are
 
Well then. If there's nothing else, let's move on to the AMA with @yuggib!
@0celo7 are you doing introductions?
or at least one introduction?
 
Oh, I was joking
I'm at work
 
@sammygerbil There is a session at the moment. We should to go to other room. But I have a question here. So, after that we will go. Do you agree?
 
4:05 PM
Ah, whatever. OK @yuggib it's all yours
18
Q: Ask Me Anything 12th July: guest's introduction and questions

yuggibI will be the next AMA guest. So here it is a brief introduction of myself, and you can suggest some topic/questions that can be discussed during the session. Academic Background. I obtained a master degree in theoretical physics almost ten years ago, working on the regularization of path integr...

 
So, as I summarized in the post above ^
 
user218912
I have a question!!!
 
@lucas : Yes, I agree. Where do I go from here?
 
:30980439 (not yet)
 
I am a mathematical physicist
 
user218912
4:06 PM
lol
 
physicist by training, mathematician now
 
@yuggib What is the best introductory text for analysis in multiple variables?
 
@yuggib What is your opinion on Weinberg's exposition of the foundations of QFT via cluster decomposition and lorentz invariance?
 
user116211
@bolbteppa bold
 
OK what was that flagged question?
Is there an AMA going on?
 
4:07 PM
Why? Tell us something about your former visions and what came out of them.
 
@TIPS yeah
 
user116211
What let you switch from theoretical to Mathematical inclination?
 
@0celo7 this is subjected much to personal opinions
 
@DavidZ Why my question is deleted?
 
and it depends at which level you are interested in
 
4:08 PM
@lucas that question isn't appropriate for the chat event
 
@lucas It was flagged.
 
@yuggib Introductory.
And I'm asking for your personal opinion.
 
@DavidZ So, what does mean the "any"?
 
user218912
@yuggib Is it a bad idea to do a bachelors in math and ph.d in physics (reverse what you did)?
 
Alright, have fun with your Ask Us Anything event guys. Have a nice time. \o
 
4:08 PM
@3750 that's what I'm doing ;)
 
user116211
@TIPS o/
 
@bolbteppa I am not completely convinced about the cluster decomposition principle as it is commonly stated. I would say that I somewhat understand it in the algebraic formulation with nets of algebras of observables and their causal connection
but it seems still at times vague to me
 
cluster decomposition is invalid for quarks, although there are legitimate quantum fields for quarks. Why that?
 
@0celo7 I like the style by Rudin
 
user218912
@bolbteppa I guess we can discuss it later after the AMA.
 
4:10 PM
@3750 coolios
 
@ArnoldNeumaier TIL!
 
@sammygerbil Come to the next one chatroom. chat.stackexchange.com/rooms/13775/physics-meta
 
@ArnoldNeumaier I stopped doing theoretical physics because I don't like the lack of rigor that is usually done in physical arguments
and I think that mathematical subtleties often are the expression of some relevant physical problem
 
Ah, but this means you leave the work to make physics rigorous (which must be a mathematician's work) to tohers....
 
@ArnoldNeumaier not at all, I actually work in making the physical arguments rigorous
but often they have to be redone from scratch
 
4:12 PM
You work on the emergence of classical systems from quantum systems - how well does this emergence work as the "inverse" of canonical quantization? Does the same classical system come out that we quantized?
 
I think intuition is important, but also a doubled edged blade
@3750 I don't know if it is bad idea, but I personally don't know anybody that did that way
 
Actually, very interesting question, Bajoran.
 
and I think it may be somewhat difficult to do it
 
user218912
I see.
 
@ACuriousMind yes, it works extremely well in all QM cases
(almost all)
and in bosonic quantum field theories as well
for the very few results that we have
 
user218912
4:14 PM
brb
 
@Qmechanic Yeah. Also things like this comment. say what?
 
in the Nelson model for example, the classical limit even survives the renormalization procedure
and gives the right theory in the limit
 
Do you think QED can be made to exist rigorously?
 
@ArnoldNeumaier that I do not know, as I said I am not completely convinced of the absolute validity of the cluster decomposition principle
@ArnoldNeumaier That is our hope, but I may say that we are still quite far from that
 
Well the question is whether you believe triviality is due to approximating by a lattice, or something fundamental.
 
4:16 PM
@ArnoldNeumaier I believe the first actually
 
Do you have good reasons for it?
 
I have some intuition: the first is that the proof for high dimensions seems very procedure dependent
 
good = communicatable
 
and "thermodynamical" in a sense
it was essentially done for a $\varphi^4$ ferromagnetic spin system
 
@yuggib How far into your PhD were you before you started making progress?
 
4:19 PM
and even if there are many analogies with the "true" QFT, I fail to see wether the lattice approximation captures well the behavior of quantum fields
 
I noticed that in causal perturbation theory, a Landau pole (even when present nonperturbatively) is no longer anything lethal, as it only implies the lack of a conformal limit. Do you agree?
 
I am not so expert about perturbative QFT if I have to be honest
 
Well, any construction will have to recover what is known perturbatively, so it is better to learn the traditional stuff as well!
 
@ArnoldNeumaier mathematically there are many problems in doing that
mostly related to Haag's theorem and the alike
 
@yuggib What happens to fermionic objects in your classical limit? Is there an indication that they are "more quantum" than other quantum objects?
 
4:22 PM
to recover what is done perturbatively, you have to change rather deeply the scattering theory
@bolbteppa I started making the first relevant progresses around the half of my PhD more or less
@ACuriousMind fermions behave completely differently from bosons
and what happens in the limit is rather unclear; the intuition is that a classical fermion state would be a probability measure on the whole exterior algebra built on the infinite-dimensional one-particle fermion space
but, as you already see from this vague explanation, it seem to be quite a mess
I am planning to work on that sooner or later, but a lot of mathematical tools have to be developed in order to properly understand that
at the moment, we are not even able to prove that the free quantum Dirac field has the free classical Dirac field as classical counterpart (roughly speaking)
 
Wouldn't a probability measure on 2-forms suffice?
 
@ArnoldNeumaier no, because with interacting theories you have every $n$-form involved
the dynamics would "couple all the particles"
a clearer way to see it is thinking of many fermions, and of the Hartree-Fock approximation. It is thought to be an effective "classical" equation, but it is still a hierarchy of infinitely many mutually coupled equations
of course this is due to the exclusion principle, and to the fact that contrarily to bosons, fermions do not tend to behave all in the same way
 
I must leave now; thanks for answering!
 
for the free theory however, in principle you should be able to describe everything with just the one-particle evolution
@ArnoldNeumaier thanks to you for the interesting questions!
 
How do I bold
 
4:31 PM
is there someone who is feeling I have not answered adequately to something up to now?
 
user116211
Have you considered writing any book?
 
@MAFIA36790 It would be nice, but also very time consuming. I think I have a good expertise on some topic where a nice overview is missing (the classical limit of quantum theories of course)
but it would take a huge amount of time and effort to collect everything in a book
maybe when I am older... ;-P
 
@yuggib What exactly are your concerns with cluster-decomposition?
It says “It is one of the fundamental principles of physics (indeed, of all science) that experiments that are sufficiently separated in space have unrelated results…” These guys here physicsforums.com/threads/… say Weinberg is a bit loose and re-state it more rigorously (but to me it's basically the same thing)
 
@bolbteppa my concerns are that quantum theories are fundamentally non-local (at least in my opinion)
so if you state the principle as a requirement of commutativity of quantum fields that are spacelike separated I can accept it
(and as far as I know it is a requirement in both Wightman axioms and AQFT)
but I can surely imagine quantum thought experiments for which the cluster decomposition does not hold
(take your usual Bell states or the alike)
I would say that in the weak form of commutativity of spacelike separated fields the principle is reasonable
the stronger versions that I have heard do not sound completely convincing to me
 
What are your preferred axiomatics for a rigorous quantum field theory? Or do you think we have not yet found the correct axiomatization/formulation?
 
4:40 PM
@ACuriousMind I find the C* algebraic formulation of (any) quantum theory rather satisfactory
from a general point of view, and also Wightman's axioms are undoubtedly the correct axioms for a dynamical theory
nevertheless, I think that the only hope that we have of improving our knowledge on interacting QFTs is to better understand quantization
I know that it should not be the way it works (since quantum is more fundamental than classical)
but it is the best way we know according to our intuition, and it is a fact that classical theories emerge in the commutative limit from quantum systems
 
Interestingly, Weinberg mentions locality as an issue: "First of all, the argument I have presented is obviously based on perturbation theory. Second, even in perturbation theory, I haven’t stated a clear theorem, much less proved one. As I mentioned there are complications when you have things like mass zero, spin one particles for example; in this case you don’t really have a fully Lorentz invariant Hamiltonian density, or even one that is completely local."
 
@ACuriousMind To sum up, I would say that we have already the right frameworks for rigorous quantum field theories (in my opinion), but a poor understanding of infinite-dimensional quantization (as opposed to finite dimensional one)
@Slereah ** .... **
 
@yuggib Are there reasons to expect that emergence to be invertible? Naively one could expect different quantum systems to have the same classical limit, making it not immediately useful to understand the inverse map if the theory realized in nature may not actually be the quantization of the classical limit.
 
@yuggib Have you studied integrability of non-linear PDE's like KdV which links their integrability to evolution equations in the Lax representation? If so can you derive all those crazy matrices they use from first principles?
 
@ACuriousMind Indeed we can not expect full invertibility. That is reflected by the fact that a simple change of coordinates at the classical level may give, once quantized, the non-perturbative renormalization of the system (we have observed that on Nelson)
@bolbteppa No I haven't...I do not know almost nothing on integrability. This is because mathematicians tend to not care about explicit solutions, but only about existence of the solutions ;-P
@ACuriousMind nevertheless, this also suggests that renormalization/definition of the dynamics may be achieved by a suitable quantization of the right classical object (energy functional)
 
4:50 PM
Well, knowing that all the systems that give the same classical limit are related by renormalization would also be a rather "nice" result...
 
@ACuriousMind that is probably more or less already contained in what we can prove
 
haha true
 
because usually the problem is not defining a quadratic form, but a self-adjoint operator for the dynamicss
 
Which leads me to: Is a fully rigorous formulation of renormalization known?
 
and the classical limit may already give you informations on the convergence of the quadratic form to the energy functional
@ACuriousMind not that I am aware of
 
4:52 PM
Finally back home
Bloody bus
 
the closest thing may be the Wilson renormalization group formulation, but I don't know much about that and to what extent it is rigorous
 
@yuggib Do you think the correct theory of quantum fields will do away with renormalization and instead have some physical cuts of some sort
 
@ACuriousMind even Glimm in his paper about $(\varphi^4)_3$ defines renormalization as: subtracting divergent constants and operators, and dividing scalar products by a diverging quantity (the so-called wavefunction renormalization) :-D
@Slereah in which sense "physical cuts"?
 
@yuggib What is your opinion of the Reed/Simon functional analysis mathematical physics book? Is that level of physicist sloppiness too sloppy for you or a perfect balance? :p
 
4:54 PM
Well a lot of quantum gravity theories imply a quantization of distances, for instance
Which would be a pretty hard cut for short range phenomenon
 
@bolbteppa reed and simon books are usually considered the standard reference in the community of mathematical physics, and I think rightfully so
 
Yeah I am stunned at how nice it is, giving intuitive summaries of proofs before doing them
 
to my knowledge there is no non-rigorous result, and the physical intuition is restricted to the minimum
@Slereah Ah ok, I see your point. I think that we should try to get away with renormalization probably
 
@yuggib What's your favorite quantum gravity, by the way
 
the quantization of spacetime is a rather tricky subject, and I don't think that there is somebody able to say something convincing on the rigorous standpoint
 
4:58 PM
@yuggib Have you looked into LQG in studying the foundations?
 
@Slereah I have no preference...I would like to understand loop quantum gravity papers because I think I may help (dis)prove it studying its classical limit
unluckily, those papers are essentially inaccessible
 
Did you try Rovelli maybe
 
there are some by Thiemann that claim to be completely rigorous, but I am not able to follow them
@bolbteppa As I was discussing, I tried with not much success
@Slereah The fact is that I would like to see something mathematically convincing
 

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