I think the reason why QM is so often presented in an axiomatic way is twofold: First, it has historically grown that way, and second, it is not satisfactory to start from classical mechanics which we know isn't valid at small scales to derive something that suddenly is valid at small scales
It's backward, like deriving the convergent sequence from the limit
Well it's using specific reasons (Heisenberg + experiment) to select specific convergent sequences given that we know the limit in specific cases, and we can do this in every case to get the right answers. Taking axioms is philosophical and no more justifiable than starting from CM, Heisenberg and experiment, the thing is that those axioms are derived from the CM, H, E approach anyway. Either choice is philosophical I guess :)
That post is great, thanks!
I'm reading about quantum groups and trying to find out how they relate to geometric quantization
@bobby Hm. I know a categorical formulation where the path integral over the exponentiated action forms a functor between the category of $n$-manifolds and cobordisms and the category of Hilbert spaces
@bobby Mhh, you cannot do that. But the category of paths is a nice category to get comfortable with categorical notions in a phyiscal context, I think
Okay, I just like it cuz I have a geometric picture of a category now. I see the objects as points, the morphisms as paths between those points, and that composition rule as composition of paths!
The graphical representation of a category is supposed to be analogous to Feynman diagrams. Is the graphical representation of this category literally the image of the paths?
Fact I just thought of: The category of paths a category consisting of $0$-manifolds (points) with certain $1$-manifolds (lines) as morphisms, and hence is just a cobordism category
@ACuriousMind hey, just had a glance at your discussion with The Quantum. you guys seemed pretty at it :d nice though. btw how did your bachelor thesis finally come about? were you personally satisfied with it?
And it is a nice post indeed. That is one of the few descriptions of gauge theory I've read that acknowledges that gauge degrees of freedom are "not real" without calling them "redundant".
I don't understand what "A general thing -- when people say "gauge theory" they often mean a much more restricted version of what this whole discussion has been about. For the most part, they mean a theory where the configuration variable includes a connection on some manifold. These are a vastly restricted version" is intended to mean, though
@bobby Do you know what a homotopy is? The notion that natural transformations are homotopies between functors holds in general.
Well that's the way I think of them, the book says we want to define "structure-preserving maps $from one functor to another$" (I hope that is italicized), so perhaps this is how you slip in the notion of homotopic deformations?
Look at it like this: The categories are the structured objects, and the functors are structure-preserving maps between them, just like spaces and continuous maps are
Now, a homotopy is essentially something providing a notion of smoothly deforming one continuous map into another, the deformation parameter "sliding" along the closed interval
So take C = P([0,1]x[0,1]) & D = P([2,3]x[2,3]) are our categories of paths in subsets of the R^2 plane S : C ---> D as a functor mapping points and paths in C to points and paths in D, how do I define this idea of a homotopic deformation of paths? Man it seems so cool but the def of a natural transformation makes no sense :(
A natural trafo t between functors F and G assigns to every object c in C a morphism between F(c) and G(c)
...such that this assignment commutes with taking any morphism from c to c' and then applying the transformation
The homotopical definition of a natural transformation would be: Let 2 be the arrow category 0 -> 1. Then a natural transformation is a functor t: C x 2 -> D such that t(-,0) = F(-) and t(-,1) = G(-).
@bobby I don't know if you should, but they much behave the same way, and you even often identify functors "up to natural transformations", which is the same as identifying maps "up to homotopy" in topology.
It may give my math prof a heart attack, but that's fine, I always say things that are correct but are a different way of thinking about topic X lolz this will be an extravagent one hehe
What you've given me tonight is real mathematics, thanks dude!
@ACuriousMind I know it would be pretty time consuming, but I think you write really well in general, which usually seems effortless for you, when in fact you're reasoning about physical problems etc. so I think someone like you could actually be also a good blogger
or certain interest at that time, that leads you to writing about x or y
e.g. all the posts that have inspired you in SE, surely there are a couple of topics among them that you'd like to write more about etc. Anyway, with your studies considered, you should also be careful with your time-priorities :D, I just brought it up^^
@bobby The Hom-functor in the category of paths is the functor that takes a point x to all paths starting at that point and a path from x to y to the morphism that just glues it to the end of another path. Now, for any other Hom-functor at a point x', we can take the Hom-functor at x' and turn it into a Hom-functor at x by taking any path x -> x' and gluing it to the beginning of all paths in Hom(x',-).
(That's the Yoneda embedding statement. For the general Yoneda lemma, I got nuthin)
@Phonon I think, "Erwin Schrödinger and Werner Heisenberg devise a quantum theory" could be part of almost every timeline to every major physics topic in the last half century or so ;)
@bobby: The snake lemma is actually trivially true in a category of paths because all morphisms are isomorphisms. (To see that - just run the path in the opposite direction!)
@Phonon: I don't think it would interest him, considering the question is very broad, and stems from a basic confusion by the OP about Lie groups and algebras.
@Phonon Note I VTC that one. Though I like Lie theory, the question is essentially "Please explain Lie theory to me without me having to actually learn it".
@ACuriousMind I believe that if you start an answer before the close, lose contact with the server and then submit without first re-copntacting the server it will accept the post. Or something like that.
This behavior has been around for a while and is even documented somewhere, but I can't recall where.
@Danu On the exact solutions of two-dimensional (Yang-Mills) gauge theories. Nothing new, but it is a bit more detailed in the steps than what you'll find in most of the papers on that.
It is a "pseudo"-topological theory, so if you know the partition functions for arbitrary 2D surfaces with boundary, you basically know everything about the theory, since the propagators/"scattering" amplitudes are given by such surfaces having the spatial "initial" and "final" slices as boundary
I had a H&R in High School and had to teach lab for students using it in grad school and have now encountered it again as a junior professor. My displeasure is growing with every contact.
@dmckee I think Feynman's lectures are pretty bad to learn from, tbh. I actually read most of the first volume on my own during my first year of uni, and it didn't teach me much.
It's one of those books that you can look into later and be like: Ah! That's a nice way to make it look really simple!
Although I've heard the 3rd volume is pretty good: I read the last chapter where he heuristically explains the Josephson effect (and superconductivity), that was pretty good
It's also the place I managed to correct Feynman (woo-hoo)
It's weird. Our university doesn't use books for courses at all. They always recommend you to look into some books if you don't understand the lecture, but the lectures are essentially supposed to be self-contained, and almost all lecturers write their own notes instead of teaching from a book.
@Danu Almost everyone uploads their hand-written notes here, and the good profs have TeXed notes available (if not at the start of the semester, at least at the end)
@Danu A Messe is also a kind of exhibition that some industries or artists or other group do, kinda like a convention (e.g. the biggest book convention is the Frankfurter Buchmesse)
reddit is so weird. There news besides funny stories besides horror stories besides serious discussion besides vile propagana...I don't "get" reddit at all
@Danu After I found that (and I don't remember how), I finally knew why BBT rubbed me the wrong way (aside from being a sitcom, which I generally don't like very much).
@ACuriousMind xD I see. Any similar one you'd suggest written by theoreticians?
@ACuriousMind in all honesty I have a good experience with books written by experimentalists, one example that comes to mind is: Quantum Chaos by H-J Stöckmann
Normally octane percentage mentioned on petrol (95% octane). How this is defined? Is it percentage of chemical or some other parameters?
