I'm clearly not the first to voice concerns regarding homework questions. Most people here know this is not a check-my-work or give-me-the-answer site, and so don't ask poorly-phrased homework questions that aren't about a physics concept. But we still have a lot, especially from new users. Ther...
@Kyle Kanos: will your ever dare to criticize me (namely my hypothesis in this context) publicly? Or will restrain to Physicsoverflow.org-style −1s because of personal reasons, without explanations? — Incnis Mrsi2 days ago
He also comments in his profile This site is supposedly useful for “active researchers, academics and students of physics”. If you want science, then never rely on such self-governed Internet sites (Stack Exchange, Wikipedia, or whatever), that are not supervised by an academic organization. They are all flawed.
And the beauty of irony is that he posts answers here! As if, somehow, his answers are exempt from the "flawed" science
@ACuriousMind I would like to learn the mathematical aspects behind it. I've seen that it's related to (co)homology and homotopy, plus differential geometry and group theory. But I don't see why those topics are so useful. Can you calculate something you woudn't be able to calculate with Schrodinger's equation?
I would appreciate if you could give some motivation, because using homotopy just to say that the fundamental group of a non simply connected region is not trivial doesn't seem very useful.
@jinawee It's the simplest example of a holonomy, which is the integration of a gauge field along a path (also called a Wilson line). The holonomies along homotopic paths are the same, so the fundamental group (counting exactly the homotopy equivalence classes of loops) tells us how many different holonomies we should expect along closed paths. Also, since homotopy groups are a topological invariant, this means that the possible values of a Wilson line are also only dependent on topology.
The Schrödinger equation doesn't really tell you much about this sort of things, that the AB effect can also be explained with QM at that level is because we write the gauge field $A_\mu$ into the Hamiltonian without really knowing what we're doing ;)
In a pure gauge theory, these holonomies/Wilson lines are the natural observables to look at, and in 2D, they're almost the only interesting observables
Also, in QCD models (often on a lattice), they can provide an order parameter for confinement (they're also called Polyakov loops in that setting)
Basically, they're "useful" because they are almost completely determined by topology and thus non-perturbative, which is always an interesting thing to have.
@ACuriousMind Now that you mention this... it's been a question that I've been walking around with for a long time: Why do we introduce $A_\mu$ into the momentum term when considering a particle in a field? I find the answer 'it's just the canonical momentum from the Lagrangian' unsatisfactory
Do you know of any good discussion? Or do you know the answer yourself? Some professors I asked were unable to help...
Well, fields have to transform in a representation of the gauge group. If you have $\mathrm{d}\phi$, it won't transform in a representation since $\mathrm{d}U(x)\phi \neq U(x)\mathrm{d}\phi$ for a local gauge trafo $U(x)$
So, we seek to have a covariant derivative that transforms in the same rep as the original field, and thats given by $\mathrm{d} + A \wedge$
This is a much more basic thing: It appears already in classical mechanics. I am aware of the covariant derivative thing in QFT, and am not bothered by it
I just want an explanation why $p\mapsto p+ieA$
I'd prefer to have an intuitive reason that is also rigorous, or at least can be made rigorous
or, perhaps, the QFT reason could really be the fundamental one? Although taking classical limits etc etc is probably impossible to even properly define
Well, the Hamiltonian has to be gauge invariant just like the Lagrangian, and so it could be that, if you examine $p$, it would transform just as wrong as $\partial_\mu \phi$.
@Danu Yes, and I even know them there in a phase space, i.e. Hamiltonian formulation as symplectomorphic group actions, but I've never seen a Hamiltonian being constructed other than by Legrendre transform
@Danu I've found it really depends on the lecturer - some are very engaging, while others really only read/write their script. Is the Dutch style very different?
@Danu I think he just wrote it in Japanese and didn't care who else could read it :D
Which is really a shame because it is quite lucidly written and thorough
But since it's not very famous (due to being not in English) it seems no one ever bothered to translate it into English
@Danu Most math profs I know are quite happy to lecture physicists...it's the teachers in spe they despise (again, oddly enough, and it explains a lot about the kind of math teachers running around)
@Danu Well, it's possible - schools are allowed to pass individual judgement about candidate, and especially when "real" teachers are scarce, they'll take anyone that applies
But it's not encouraged to teach without having studied for it
It's also a bit difficult to get tenure as such a "Quereinsteiger" since you haven't passed the special exams to call yourself "Studienrat" (which is the title of a teacher that has completed the teaching track, but isn't used anymore to address them)
I could only understand that if one was confusing them with a sequence space, but if it's really about the equivalence classes, what's the point? We're taking the equivalence precisely because we don't care about zero sets
And reading the comments again, the point seems rather to be that David writes "all possible wave functions" without stating that the requirement for a wavefunction is to be in $L^2$.
I must admit I found some satisfaction in the fact that Reed & Simon define the spaces such that the standard notation is in fact precisely the correct one.
Yes, that's it. David could've simply writting all allowed wave functions, and it'd all be over with. That's why this nonsensical 'mathematical insight' he (she?) is displaying annoys me
This is an important thing I feel students at LMU (maybe all German unis?) are missing: Physical intuition, or a graduate level course in ED for that matter
...perhaps the trade-off for the math they have is a good one though ;)
That is completely insane to me. Have you guys never heard of "Jackson"?
...I told the guy that the two terms go nicely together in relativistic notation: "but its just not relevant to think about them as relativistic if you just put an emitter in a cavity"
I'm praying that LMU offers a course on Lie groups next semester... there's nothing on groups here in the Winter Semesters except "Finite groups and their representations" at the TU Munich, and the TU building is about an hour away :(
@ChrisWhite Ever sat in a particle physics lecture where essentially all they do is talking about representations, but they just muck about with the explicit matrices and call them $n$-plets?
It's really one of my less favourite experiences, especially because one can derive the possible $\mathrm{SU}(2)$ reps in such elegant ways.
user54412
@ACuriousMind Oh well, to be honest, I skipped those lectures ;)
user54412
That got awkward the time the prof moved the homework collection box inside the lecture hall, and homework was due at the beginning of lecture.