@JimdalftheGrey People have won arguments with Shog, but you have to be right, and be patient and articulate about it, and probably need a little luck, too. If you do win he's a gentleman about it.
I love using RSS-feed and I wanted to use it for this site too. The problem is that my RSS-Reader does not render Latex so I get the source code of Latex and trying to render Latex in my head is not so easy! Wouldn't it be better if the Latex formulas are sent as pictures so that the RSS-Reader c...
It's supposed to be the mapping degree. Def: If $f(P)=P'$ and $f$ preserves (resp. reverses) orientation in a sufficiently small neighborhood of the point $P$, then we set $\operatorname{deg}_Pf:=1$ (resp. $-1$)
The only application I know are non-trivial gauge transformations and such. The possible mapping degrees often classify the type of non-isomorphic bundles your theory can have underneath it, and each isomorphy class of bundles corresponds to some kind of topological charge/effect
E.g. different instantons essentially arise from different mapping degrees of a map from the three-sphere.
@0celo7 Yeah, it should definitely show up there, but it could also turn up earlier, I certainly don't know every occurence of the mapping degree in physics.
@ACuriousMind This book is pretty cool; I'm learning a lot now. The only problem is that he outsources a lot of the proofs to his earlier books. I guess that's understandable since it's already a 1000+ page book.
@0celo7 I have no idea what the square brackets are doing there. I'd have written either $(\psi_\alpha(p),\psi_\beta(p'))$ or $(\psi_\alpha \times \psi_\beta)(p,p')$ on the RHS.
@ACuriousMind Zeidler writes: "For the rotation group of the 3-dimensional Euclidean space, $SO(3):=\{(\mathbf{n},\varphi):\mathbf{n}\in\mathbb{P}^2,\varphi\in S^1\}$. Note that each rotation about the axis $\mathbb{n}$ can also be described by a rotation about the axis $-\mathbb{n}$. We briefly write $SO(3)=\mathbb{P}^2\times S^1$." Do you see a way to get to $SO(3)=S^3/\mathbb{Z}_2$ from here? (Remember, no quaternions!)
There has been a general sense among a number of users, myself included, that simplistic, almost throwaway answers can get very high scores, while thoroughly researched, high-level answers can languish with few to no votes. See the discussion here for example.
To what extent is this backed by da...
@dmckee I saw a question that was put on hold as unclear. Alfred Centauri tried to help, but he asked the user questions involving concepts from GR that the user didn't understand. From the user's comments I saw that what he asks is whether the curving of the space in presence of masses, which is an axiom in GR, may be a consequence of a more fundamental property* of our universe.
@dmckee I re-edited the question and I suggest it be re-opened. There is nothing wrong in asking a very fundamental question. Of course, the ultimate answer may eventually be "we don't know", but let the user get an answer.
@innisfree As far as I understand, QBism basically just constitutes a complete rejection of realism, and then following through the consequences of this with the QM formalism.
@innisfree I think it is rather similar to Copenhagen in spirit, but one does not distinguish between "quantum" and "classical" operations.
The only "real" operations/processes/entities are whatever goes on inside your own head.
@Mark thanks, but how can one eliminate objective probabilities? If I measure whether an electron is up or down, my uncertainty about the outcome does not stem from my ignorance about the system. I know everything there is to know about it. The probability is physical.
The probability is not a reflection of my mind or my ignorance - it's a property of the system.
@innisfree That is a philosophical position that you have chosen to take, which a quantum Bayesianist would not agree with.
There is no experimental evidence supporting your claim that "probabilities are objective/physical". It doesn't make any difference to your experimental predictions which stance you take. Physics is the same either way.
Some people like subjective probabilities, in which case QBism is nice for them.
@Sofia Why are you pinging me about it? You edited it so it went into the review queue. High rep user will look at it. If @AlfredCentauri also edited it then you can likely count on another vote.
@mark But surely it isn't? Bayesian probabilities stem from incomplete information. If I know all possible information about a system and I still can't predict its behaviour, that uncertainty is not Bayesian
Of course, it is possible that people are responding to identity of the poster. In principle they shouldn't and most people don't go looking to tag questions be user but I notice that really persistent bad posting will eventually draw people's attnetion none-the-less.
My attempts to guide Ray Kay towards a more mature and self-sufficient approach to toward working through the questions that have occurred to him don't seem to have made much of an impression.
@innisfree Yeah, that's correct. But you are assuming that the wave function represents complete knowledge about the system, which is obviously unproven/unprovable (although falsifiable, in principle).
@innisfree However, there do exist certain no-go theorems which have recently come to light, as initiated by Pusey, Rudolph and Barrett (PBR), which restrict the extent to which the wave function may really be epistemic.
However, these theorems ultimately work in a framework where an underlying reality is assumed to exist.
A Bayesian would not make that assumption. Really it is just the logical extension/conclusion of the anti-realist "shut up and calculate" stance taken by most physicists.
Ah, so QBism is assuming there is something else? Some information that I don't know that's determining the outcome of a QM measurement? Isn't that a sort of hidden variable theory?
@0celo7 I feel that what you say is unjust - it's a honest feeling, you please don't feel offended. If a question was put on hold, we don't have to find reasons to support that. Our top intention is to help. The history of this question is that Alfred Centauri asked the OP what the OP means, but in a terminology the OP does not understand.
@innisfree For example, normal common-or-garden quantum mechanics is a hidden variable theory: the wave function is unobservable yet determines the physics.
But some people get round this by taking an anti-realist viewpoint, where the wave function is "information". So then the question is "information about what?". Depending on your viewpoint, you are then either led to some variant of Copenhagen/QBism etc. (In my opinion Copenhagen is not even logically self-consistent in its usual form)
@mark honestly, I think QBism is incoherent from a philosophical point of view. That's ironic, because the authors have a very verbose, philosophical style in their presentation.
@Sofia Too young for what? If anything, it makes us young people mad when people ask uneducated answers and they claim "they are only 18" or whatever. I learned GR while sitting in Pre-Calc.
@0celo7 probably too young for understanding that a general question, can be answered in general terms, without profound knowledge of the specific domain.
@mark i don't mind Bayesian interpretations of probability, but unless you radically alter QM, it is clearly a source of objective, physical probabilities. To claim otherwise is IMHO advocating a hidden-variables theory.
@Sofia To put it in QM terms, he asked a question like "why is energy quantized" and Alfred asked him if he has seen the Schroedinger equation solved for a square well and Ray Kay said "what's the Schroedinger equation". (Run-on sentence, oh well.)
@0celo7 Copenhagen is basically a dualist description in terms of quantum and classical systems which are fundamentally different in behaviour. But classical systems are presumed to be fundamentally composed of quantum systems. The inconsistency lies somewhere in there, but I admit it is a gut feeling.
Basically, anti-realist descriptions describe the quantum world as simply outputs on classical dials resulting from twiddling classical knobs.
There is nothing more than this, according to copenhagen. So then how is it possible for classical knobs and dials to exist, if they are composed of quantum systems.
@MarkMitchison I've always thought that classical mechanics is explained by taking the stationary phase approximation of path integrals. Then we have a bunch of quantum systems. Law of large numbers and all that.
@0celo7 What Alfred asked was out of point. The OP asked if there is an explanation of the curving space outside the GR, not inside the GR. I.e. some more fundamental property.
@MarkMitchison what do you mean Copenhagen is inconsistent? Hidden variables seem to be a way to deny the Uncertainty principle (which says there's no such concept as the path of a particle) and say that classical mechanics fundamentally rules nature, I may be wrong
@0celo7 What you have described is basically the correspondence principle in the path integral language. I would say it's perhaps a bit more subtle than that. But in any case, the problem with the emergence of classical physics is related to the "measurement problem".
It is easy to show that only classical configurations survive in the $\hbar\to 0$ limit. What is not easy to explain is why only one classical configuration is observed, when in fact unitary quantum evolution generically predicts an incoherent mixture of many.
@bolbteppa Sorry, I already explained my thoughts above. If you don't agree then fine, it's philosophy after all! But your conception of hidden variables is not really right, or perhaps I should say it is not universally agreed upon. The term has a more general meaning. There is no way to rule them out from physics. It's just a philosophical preference!!
@0celo7 I am an engineer not a philosopher. And I ask why bodies attract one another. Is it a fundamental law, or there is one law more fundamental, that justifies the attraction. Please tell me a yes, or no.
@0celo7 I'm not personally a fan of the idea, but I suspect that the user base might go for it. Such a post should probably have a meta-consensus before you begin and would need on-going, active curation.
@innisfree I just mean that the wave function is an unobservable input to the theory, which is assumed to be physically real and also determines the measurement outcomes.
But probably we define hidden variable in a different way. Allow me to reiterate that I hate the term hidden variable.
@dmckee Obviously. But there are some things like Carroll's online GR lectures (don't know if you're familiar) that are very well written and accessible.
@Danu I like to think of the Yoneda lemma as essentially the statement "Things are defined by what they do". Things that do the same are the same, no matter how different they look at first.
@Danu Of course, the connection of category theory to logic is much deeper, because every category has an "internal logic". I don't understand enough of that yet to say whether it elucidates anything about epistemological reasoning.
@Mark I'm still unsure what you mean. Suppose I *know* a wavefunction. The indeterminism in my measurements is not Bayesian.
In real life, there may be extra Bayesian uncertainties because I don't know what wavefunction I'm looking at. But regardless of lots of extra Bayesian uncertainty, there is always the objective physical probability in QM.
@0celo7 Chasing down duplicates and posting the links is not actually rewarded by the SE system as things stands, and it is easiest if you spend a lot of time on the site.
has someone experienced the formatting problems with chrome that John Rennie brought up on meta some time ago? I never had those, but yesterday for no reason the QUESTIONS ecc. buttons became too big, exactly in the way Rennie described
@dmckee first the question was put on hold, and only after my vocalizing was indicated an address of a duplicate. W H Y? Why this hard hand with novices?
@innisfree " there is always the objective physical probability in QM" Yes, unless you assume that the probabilities are not objective, but arise from ignorance.
@Sofia Look, the site does not exist to be "nice" or "easy" or "friendly" to novices. I know you don't like or belive that, but it exists to accumulate good answers to good question.
Coddling people who don't know much and aren't willing to work gets in the way of the real purpose of the site.
@mark, i think one of us is confused. If we're talking about QM, you can't assume probabilities arise from ignorance, because there is nothing more to know. QBism is just a interpretation of QM, so how can it posit that there is something more to know? If it does, it's a different kind of theory, a hidden variables theory. Do you agree?
@innisfree Clearly you are not a fan of QBism, which is totally fine. But you can't just dismiss things because they are hidden variable theories. There is no experimental evidence that QM cannot be described by hidden variables. We just know various things about them (non-locality, contextuality, no preparation independence).
@Danu Uh, in principle, nothing. It's general nonsense. Algebra helps because it provides many examples of interesting categories, but the abstract theory relies on nothing else
And if you want better and faster duplicate identification (something we always need) then spend some time hunting down the previous version of fresh questions.
@Danu it's not easy to explain. Basically I believe the font looks different everywhere, on wikipedia it's just more apparent. Looking carefully even this chat is "broken" (I'll explain what that means in a minute). Your screenshot is fine.
@innisfree I think the point about it is that they do not assume any model for the hidden variables theory. They just say, "there are hidden variables, we probably won't ever know anything about them, let's not take the QM formalism too literally". Something like that
@Danu Hm. Yeah, it certainly helps if you are used to shoving things around in equation like in most algebra proofs, because that's the essence of diagram chasing, but then again, you only do diagram chasing in "algebraic" categories
@innisfree Ha, no, I would agree with you there. I don't think anything that is manifestly philosophy can be claimed to be a significant development really.
NB: I opened up the queues this morning to find 7 close votes, 1 reopen, 4 first posts, and 4 low quality. Haven't seen anything that big in a while (mostly on the latter two)
@dmckee , @ManishEarth , I looked at the supposed duplicate. Why is it a duplicate? It deals with the equivalence between space curvature and gravitational attraction. But the discussed question is about a possible more fundamental law implying the curvature (or gravitational attraction).
Gravity holds stuff together at large scales, E&M is responsible for your iphone, the strong force holds nuclei together and the weak force is responsible for certain radioactive decays.
@ManishEarth Aaaaa! Indeed, it is not a duplicate. Then, let me answer to the discussed question, as follows: for the moment, no researches are known to us in the questioned direction.
@0celo7 There are no particles getting "moved" by the weak force, it's just an energy differnce, so there is no proper notion of "attrack or repel". A proton, electron and antineutrino represent a lower energy state than a neutron.
@Sofia They both go exactly to the question of whether GR is fundamental. Same thing, only one of them is a good statement of the question and the other is not.
@Danu I find it rather strange. Especially because it all worked fine until a few days ago. I remember first seeing the John Rennie's meta post on that problem and I didn't have it then
@Danu by the way, the behaviour disabling the feature seems to be expected, see e.g. this post on chromium's blog. I guess I'll just have to find some other work around
@dmckee do you realize that the discussed question is a Nobel Prize question? Could the OP answer by seeking about gravity, curve and mass? He asked us if we know of some research in this direction, because he doesn't know - probably he was inspired by Higgs' discovery.
@dmckee we are physicists, aren't we? So, he simply relied on that maybe we know something. Isn't that clear? And what did we do? Showed him the door, because we don't know an answer.
@Danu looking around it seems that the DirectWrite feature is for high-DPI displays (and mine is), which explains why not everyone is experiencing the problem with the the fix. What's strange is that I didn't have the toolbar problem before, as I already had the DirectWrite enabled (I checked it when I saw the post). I'll find a way, thanks for the time anyway!
@Sofia It wasn't blocked because of that, though. It was blocked because 1) it is a duplicate and 2) OP didn't know basic material about the subject matter and 3) it was/is vague.
I seriously have to get off this chat and do homework/scholarship essays. Bye.
@Danu Wait a bit. It seems to me, but I am not absolutely sure, that the curvature of the space was introduced as a completely equivalent language to the Newtonian attraction + modifying it according to the relativity. I understand that this replacement was more amenable to mathematical treatment. Am I right?
quick question: does anyone know of any resource/reference about the Sommerfeld-Watson method for summing series?
funnily enough, a google search returns as a second result my very question on math.SE about this, and little more. Just advanced application which I'm not even sure are related to what I mean
@ACuriousMind yea I'm starting to think that our lecturer just made up that term. Even on math.SE the best I obtained was someone pointing me to literature about "Regge Pole theory", which I didn't even know existed, and a book which I was unable to find around
@ACuriousMind but did you ever see/applied the method of using residue's theorem to find explicit epressions for series?
@ACuriousMind well you can say that. One could also say that it's just a straightforward application of residue's theorem on meromorphic functions, but I thought there was some more to be learned on this particular application, given that they (or maybe just my lecturer) had given a name to the technique.
@ACuriousMind I can't find much more examples than those two I gave either. The best is the generalization using instead of $\cot(z)$ or $1/\sin(z)$ the logarithmic derivative of a function with finite-order zeros suitably positioned.
@ACuriousMind It's a pity :P. I was hoping for some cool examples of this method applied to find summations of nasty series
or even just some more rigorous mathematical results about it. I guess mathematicians don't bother much on this techniques to calculate specific series
@glance Ah, yeah, the mathematicians probably don't care for examples. This (Mittag-Leffler) kind of constructing meromorphic functions is interesting from an abstract point of view because it is part of showing that the meromorphic functions are a sheaf
@Danu I once translated Funktionentheorie as functional analysis during a discussion. It took us about five minutes to find out why we were both suddenly talking past each other.
@ACuriousMind and I thought it was just cool just because you obtained nice series and product expansions.. but now that you mention it, that's the most obvious implication! :)
about funny SE questions.... did you catch this one on the sidebar? anime.stackexchange.com/q/19540. I was on math when I randomly saw it, and my first thought was some strangely named series expansion... I don't know if this means something
@ACuriousMind were it you who voted for closing? (Please don't be upset by my question, I just know that you are quite a radical person.) The question, as I am parsing it, begins to make sense, and people begin to upvote it. So please, you and all the friends in the room, don't hurry, wait patiently for me to do my job. (And have my kind regards.)
@Sofia Yes, I voted for closure because the question, as it stands, is wholly unclear to me. If it is clarified, I will gladly retract my vote or vote to reopen, but it is universal SE policy to vote based on the current state of the question, and to vote immediately, as is discussed in this meta.SE post.
What is this and how does it work
