Regarding the question on Recommended books for undergraduate electrodynamics, I've read some very positive reviews for Andrew Zangwill's Modern Electrodynamics. Has anyone read it?

The Feynman lectures are great for electrodynamics.

@Pulsar I think Griffiths' book is excellent, though basic.

I really can't make much sense of the comments on this answer... it feels like talking to a malfunctioning AI bot or something :-/

He's a real guy though

@DavidZ I think there's a language barrier there

There's some oddness about him too:

He got a downvote on an answer & thinks it was me

@KyleKanos Well, we all know that you're the big meanie giving out the downvotes left and right here :P

I don't downvote indiscriminately.

I kid, I kid (you've cast about as many downvotes as I have, and you've been here a year longer than me)

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

I'm trying to wrap my head around the idea that science and self-governed are even remotely incompatible

Yeah, it happens.

@ACuriousMind Hi, I've seen you've answered several questions about the Aharonov-Bohm effect.

@jinawee Yep, I have (and hopefully not said anything wrong).

@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.

@KyleKanos huh, that sort of thing is probably flaggable :-/

@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 ;)

@DavidZ Probably. So I did so now

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)

@JamalS wait.. what?

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...

@Danu You mean why we replace $\partial_\mu$ by $\partial_\mu + A_\mu$ (modulo some prefactors)?

yeah

Feynman has some motivation near the end of his volume on QM but it was wayyy too heuristic

@DavidZ I hope my comment cleared matters up?

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 $\mathrm{d} + A \wedge$

No, that's not what I mean

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

Ahhh, you want a reason why in the Hamiltonian picture that happens without saying "it comes from the Lagrangian"?

(If that's it, I'm not sure there is one, at least I know of none)

Yes, that's what I want

And I feel it has to exist

or, perhaps, the QFT reason could really be the fundamental one? Although taking classical limits etc etc is probably impossible to even properly define

@Danu: See Josh's most recent addition to his 'hard problems.'

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 $\partial_\mu \phi$.

@ACuriousMind Do we have any gauges in classical mechanics?

...of course we do once we use $A$, stupid question

@ACuriousMind Thanks for the comment. Would you recommend any basic reference (I'm reading Ryder's introduction)?

@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

@jinawee DONT DONT DONT READ RYDER (I think it's crap)

@ACuriousMind I'm only taking symplectic geometry next year... :\

@jinawee Hmm, Eichtheorie by Kugo is a brilliant work on gauge theories, but it exists only in Japanese and German (oddly enough) as far as I know

phermi.com/space-station-catch

^ That one

I've learned much of this from reading rather terse papers about 2D gauge theories, which I wouldn't really recommend didactically

I'm disappointed no one approves of my $0.02 on the E&M Books question

@Danu I've not heard such a thing at all, I hunted that part of gauge theory down for one of my answers here^^

@KyleKanos Is it really hard? It seems quite... simple?

@ACuriousMind Have not heard? Typical German English! :D I recognize it from my fellow students haha

@Danu Argh, dammit :D

Yes, in German we hear (or, better analogue, listen) to lectures/courses

It'd be more normal to just say taken

I know :)

@KyleKanos Yeah, well we all know that Gaussian units are a tool of the devil.

The word Vorlesung indeed captures the notion of a german lecture very well

...something that I wish wasn't so

@Danu I've not really looked at it

@dmckee What? Gaussian are the bestest units to use

@KyleKanos Begone, you demon!

F=kqq/r^2, k=1 is the obvious choice to anyone

@KyleKanos $k=\frac{1}{4\pi \epsilon_0$}$, and nothing else :)

In truth I don't care either way as long as I don't have to translate, but I've used SI more so I remember things in the SI form.

@Danu This $\epsilon_0$ thing. I know not what it means

@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?

@ACuriousMind I find a number of my lecturers quite... unwelcoming. They are here to say what they planned to say, and nothing else, it seems.

That's not my notion of a German lecture!

So please don't think we're all like this ;)

@Danu hopefully so, but I can only speak for myself

@ACuriousMind There's one very cool guy - but he used to work in Canada ;)

@ACuriousMind It may partially just be due to mathematicians hating to lecture for physicists

also... WTH Mr. Kugo?! Why would a physics textbook be translated to German over English?

@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)

@ACuriousMind They don't like people that like teaching? What an oddly paradoxical situation :)

Yeah, it's weird

But the entire system to train teachers is a bit...well, fucked up

@ACuriousMind Oh well, I "obtained" the book anyways, I think my German is good enough.

The teachers are basically taking the same courses as other people studying their subjects, except for the hard courses.

Which means that many people who do not want to do the hard stuff switch to the teaching track

@ACuriousMind Oh, you mean people that are specifically in a teaching track?

...which means that the teachers-to-be have a reputation for being dumb and/or lazy

those would be the high school teachers, then?

@Danu If you want to be a teacher in Germany, you have to be in a seperate teaching track

@ACuriousMind Someone with a PhD in math. won't be allowed to teach high school, then?

@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)

It's a weirdly archaic system

...as everything German

Haha :D

Yeah, our reputation for loving rules and regulations is perhaps really not exaggerated

What's this guy's problem in the comments on David Z's answer

what does he mean when he says elements of $L^p$ spaces are not functions

...oh, just the BS about equivalence classes maybe?

@Danu perhaps he's confusing them with the space of sequences summable in the p-norm?

this pedantism is really getting me angry

Meh, he's got to be thinking of the equivalence classes

(which are also sometimes called $L^p$m though $\mathcal{l}^p$ would be more common

but Reed&Simon in fact use $\mathscr L$ for the space before taking the equivalence classes, which live in $L^p$ in their notation

Hm, he (is that a male name?) seems insistent that the $L^p$ spaces do not contain functions as elements

Like I said, I think he's referring to the fact that you should consider equivalence classes if you want to talk about functions

f=g a.e.

that stuff

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

btw, mathcal doesn't work on non-capitalized letters, I think you were trying to use \ell $\ell$.

@Danu Yes, I was, thanks :)

@ACuriousMind Yeah, it's pedantism to the point that it comes seriously close to being wrong, if you know what I mean

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

And the "elements are not functions" should really rather read "not all functions are elements"

It's a language problem, I still think that.

I think it's a tendency to nitpick, combined with a condescending attitude, combined with a language barrier.

As you can tell, my jimmies are rustled.

....

That f wasn't supposed to be there

I just looked back in the conversation

That you're discussing, I mean

hmm?

Ah, another condescending comment. I'll just leave that guy to his ivory tower of L^p-nonfunctions then, I guess

I guess I'm just not precise enough (for him/her) in my comments, and don't really feel like spending more time writing them

@Danu The stuff about $\mathbb{Z}_n$ is nonsense, everybody thinks of them as integers :D

@ACuriousMind RIGHT?!

Ugh...

Also, I'm unable to fill out the blanks... What is he/she trying to say? Something about me getting angry?

I think he's trying to mock you

Somehow

I think the (blanks) are meant to be insults/swearwords, but I'm not sure

He's fairly confrontational

Oh, wait, I get it!

He's trying to say that I'm being condescending

and that he may stop responding etc etc

Ahhh

He just took your sentence and inserted blanks

lol

I dont even...

I have a very hard time not thinking about $\mathbb Z_n$ as integers, actually...

That's because they are...they're just a part of the line of integers $\mathbb{Z}$ that has been cut off and glued into a ring

That's, incidentally, the motivation for the term ring

ah man, I'll just drop it :)

Please do!

@KyleKanos It's quite rare for me to get into things like this, but some people... ;-)

The guy has some sort of "sandbox" issues

He doesn't like ours and appears to want to make trouble

It's a weird idea. "I don't like what they do over there. I'll go over there and participate"

This goes back to that highly-starred comment from a few months ago. It's just the human condition, I'm afraid.

You could group it under 'hypocrisy', to some extent. I like doing this because, in my view, hypocrisy is truly the mark of humanity ;D

@Danu chat.stackexchange.com/transcript/message/17295125#17295125 Found it ;)

@Danu Coming to think about it, that's...unsettlingly accurate ;)

@ACuriousMind and I came up with it all on my own! Maybe I should write a philosophical treatise.

Please do! And then advertise it in spammy questions on philosophy.SE ;)

Haha, I can imagine that this happens a lot over there...

If we get crackpots advertising their pet physics theory, I sure hope that the philosophers are plagued by similar stuff

Hey, should it be possible to physically 'derive' the correct Lagrangian for an electromagnetic system? (classical)

@Danu Hm...what's a 'derivation' for you? ;)

idk, a friend just asked me if I could justify the last term in fbcdn-sphotos-h-a.akamaihd.net/hphotos-ak-xpa1/v/t35.0-12/…

nvm, haha its obviously just the potential energy

but this does make me think though...

My first sentence would have been "That's just the conserved current coupled to the gauge field", but yes, it's also the potential energy for sure.

is it possible?

I know that, but he's not into gauge theories (or relativistic notation, for that matter), more like ED for qunatum optics and such

so I didn't want to bother him with fancy schmancy stuff

Understandable, I didn't say my answer was better

what does $\vec j \cdot \vec A$ represent, physically?

Uhhhh...

...hard, right? I have no intuition for magnetic fields, lol

...how much current flows in the direction of the vector potential?

(If you ask for the meaning of the vector potential, I've got nothing :P )

...lmao

I wasn't asking about the meaning of the dot product ;-)

I thought so :D

but really, I've got no intuition for ED either

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 ;)

@Danu There are no graduate level ED courses on the "normal" German curriculum that I know of

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"

hehe

@Danu Nay, Griffiths all the way ;)

I really dunno why, but there's also often no graduate level mechanics

So either you see phase spaces and Hamiltonian flows in your first two semesters, or never

@ACuriousMind Yikes!!! I really feel 'naked' as a physicist, if I'm not able to say I've worked from Jackson. Yes, also this. WTF

@ACuriousMind ...because it's a course for the graduate level!

I completely agree, but somehow...it's grown that way, and change is veeeeeery slow

@ACuriousMind It really bugs me :\ And I have no free time to study it on my own, because I'm so far behind on the math! LOL

btw

do you have a good book for basic diff geo?

not really liking any of the books my professor recommended

The one I like best so far is Lee's Intro to Smooth Manifolds

I would have said Lee

It's okay, but this *@%()%!^&% teacher decided to take a non-standard approach

so everything is backwards for me now

Ha...well, how non-standard can you do geometry? I'm having trouble imagining the basics being very different

Define differentials without having defined anything related to geometry

as just a linear map

no, no tangent spaces, nothing

Okay, that sounds weird :D

someone is TeXing notes to the course

copy.com/s/3LVn3jhPKgvMwFiL/scripts/diffgeo.pdf

Ah, I see

I've seen that somewhere before...and I didn't like it either

...I think it's ridiculous.

And highly inconvenient for total n00bz like me

Also, anyone using the term genericity assumptions more than once sounds exceedingly boring :D

@ACuriousMind because now I'm really quite lost. Hah, that was the talk of the week around here

WHAT DOES HE MEAN?!

Any particular reason why we're having gaming users over?

@Danu Hm, I think the genericity assumption is just "$dF$ is invertible", because the inverse function theorem holds for them

Yeah, I think so too.

It's throwing about big words with no real semantics.

Basically $dF\neq 0$ in practice

Man, that looks almost as mind-numbing as my DiffGeo course

...and I don't know any math!! :'(

I find this hard to believe. Less rigorous?

I feel like GR doesn't just most of the things DiffGeo has to offer...which I why I want to take a course on it

I can't recommend learning differential geometry the "mathy" way. It's horrible, and you take far too long to do anything else than bookkeepking

I highly recommend learning group theory (and all algebra, really) the mathy way, though ;)

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.

Haha, a wise decision (you skipping that, not moving the homework :D )