@Relativisticcucumber You should be able to see this rigorously directly from Maxwell's equations. The relevant equations are $\nabla\cdot E=\rho$ and $\nabla\times E=-\partial_t B$ and if you assume magnetostatics, then it also kills the $\partial_t E$ term in Ampère-Maxwell law, and then it is extremely clear that nothing moving is contributing to the E field. @VincentThacker @qwerty @Mr.Feynman @SirCumference
@SirCumference This is actually sad. If you really look at the theory as a whole, there was never any good reason to define the units in such a haphazard way; Heaviside's suggestion is absolutely sensible, and that is why HEP basically all uses Heaviside-Lorentz units. In such units, E field and B field have the same units, and for light waves, they have the same magnitude. Instead, what happens is that the Lorentz force law becomes $F=q(E+\frac v{\boldsymbol c}\times B)$ and that totally
explains why magnetic forces act on moving charges with a magnitude smaller by a factor of $v/c$
@qwerty JD Jackson is better, but beginners should not be starting with a graduate book. Especially for those who are not used to the maths of vector calculus and all, Jackson is just wayyyy too difficult. Griffiths was very happy when miao miao emailed him to thank him for his nice little book. @SirCumference his QM book is nice too, a simplification of others, just like his EM was a simplification of Jackson. Sadly he didn't use bra-ket.
@SirCumference it is quite a problem. sometimes miao miao finds myowself teaching English (as She is broken) at day job because they really didn't catch the meaning of the words in the code they are trying to work with.
@naturallyInconsistent interesting, I was told "Americans use Jackson, we use Griffiths" by someone at some point, so I didn't realise Jackson was graduate level. isn't Silly Goose TAing an undergraduate(?) course which uses Jackson for instance?
@naturallyInconsistent yeah the seasoned editor knows how to classify/tag and the correct things to say. part of why I asked here if anyone had experience.
@qwerty i am TAing an undergrad course that uses Purcell’s text on electromagnetism
In my (small) experience, jackson is quite a nice book to reference electromagnetism concepts. However, quite a needless time sink as a pedagogical resource. But that is my opinion.
The EM course i am currently taking takes some problems from Jackson which have been pretty good. The course content itself is taught out of prof’s lecture notes, though.
The only thing i think i learned out of jackson is solving laplace’s eqn in cylindrical coordinates as that problem is not treated in general in griffiths or zangwill (?).
its content is very long-winded and relies heavily on “physical intuition”. The problems in the book also heavily rely on “physical intuition”.
But the “physical intuition” employed is not very convincing to me.
Although to be fair this is the first of three required EM courses that physics students at this institution have to take. I assume they get a more systematic view of EM in the latter two courses.
It can go either way. Later courses can be extremely technically challenging, and that might require absurd levels of physical intuition to even attempt solution. Or it can go very mathematical and redo the absolute basics in diff geo. I don't think there is any course that combines the tools of maths phys and solving actually horrible problems.
> In the context of GR, we may in practice forget the conceptual distinction between a point in a manifold and its coordinates, and so we may use 'diffeomorphism' and 'coordinate transformation' interchangeably, despite the fact that the former is strictly speaking more general
this gives me heeby jeebies, this idea of blurring distinctions. is it just standard though?
> what physicists call GCTs (General Coordinate Transformations) is what mathematicians call diffeomorphisms, perhaps modulo some insignificant nuances like whether only the part of the diffeomorphism group that is connected to identity constitutes as a GCT or whatever else. https://physics.stackexchange.com/a/638191/92181
This error of acting as if coordinates are conceptually equivalent to the abstract objects they describe isn't only a problem in GR, by the way - the question about Noether's theorem being a trivial consequence of how integrals behave under coordinate change I answered here would never have come up if the physicists were more careful about that point
@naturallyInconsistent could it be because you can't add images to comments? i've not tried...
@ACuriousMind i've had working out the veracity of “The fact that the coordinate values do not change, while the tensor fields do, distinguishes the diffeomorphism from a simple coordinate transformation. An important implication is that, in integrals over spacetime volume, the volume element $d^4x$ does not change under a diffeomorphism, while it does change under a coordinate transformation...
... By contrast, the volume element $\sqrt{|g|}d^4x$ is invariant under a coordinate transformation but not under a diffeomorphism.”
on my to do list for a while now because it was from that pdf you said was absolutely trash months and months ago (before I was a regular here)
>what about gauge transformations? In GR, this notion is extremely specific and application-oriented, because it only concerns perturbation theory.
this is from the person I respect, but, did not quite ring true. but perhaps it's just (another) overloaded term. iirc there isn't even a definition of gauge that pleases mathematicians; it's the next thing I need to nut out after the diffeo stuff.
@ACuriousMind I know you hate the whole passive v.s. active thingy, but reading your answer on the Noether's theorem, it seems like you are very much pushing that it is only interesting in the active sense; does it not also work in the passive sense? That is, does there not exist a way in the rigorous mathematical formalism to express the passive kind of isometry?
And again, I don't like the active/passive terminology to begin with - it's silly and I've never seen any actual mathematicians talk that way about transformations or coordinate changes. I cannot explain its details and will not defend it.
If it has bad rep, it would not be in such common use. ACM has a rigorous objection that he can properly state and explain. Yours may be consistent, but it is much more likely to be yet another example of your many mistakes.
um, @ACuriousMind is it okay if you please remove the starring of the "in bad practice" comment as I quoted someone I know. no worries if you don't think it's uncalled for, but I'm just slightly uncomfy with it being starred
In this image, the author is trying to derive a general expression for diffracted planar waves on a sample. He gives a formula for the wave inside the sample by: $\Psi_p(t)=\Psi_o exp(ik(L+r)-i\omega t) $ (so far so good!)
He then says, we will allow diffraction. Each point p inside the sample sends out a spherical wave, which its amplitude and relative phase are modeled by complex scatter density. (i understand this part) So we multiply $ \rho$ with original term.
What i do not understand is the last part. Where he writes $exp(ik'(L'-r)$ divided by the norm. What is this exactly? The pl…
If i am standing at B and only watching P. Then the exponential term would be just the equation for the wave emitted from P. and psi and rho are its amplitude. but why divide by the norm of the vectors? i am trying to find an explanation to the reason why we multiply all these quantiites
@Madder your problem lies with the first statement. These are not plane waves. They are spherical waves, which is much more physically sensible than plane waves in this context.
@imbAF The usual definition of the Heaviside function is the opposite, yes, but no one forbids you to define some arbitrary step function the other way
@naturallyInconsistent See, the reason I hate the active/passive distinction is because it doesn't map neatly onto the (much more useful, in my view!) coordinate/abstract object distinction I'm making here. I don't even really know how a "passive" transformation (or, really, the "coordinate change" = "diffeomorphism" logic) on a manifold with more than one coordinate chart is supposed to work.
People define this notion in the context of vector spaces and then act as if it still makes sense on manifolds without elaboration :P
@Relativisticcucumber I was saying it makes perfect sense from the viewpoint of a non-native speaker in whose native language the referent of the pronoun is grammatically masculine
@Relativisticcucumber in the same vein, if I don't think about it I have the grammatical instinct to use 'she' for you not because of any belief I have about your natural gender but simply because the word for cucumber is feminine in German :P
> no eye, wherever in matter it might be placed, has a sure criterion for telling from the phenomena where there is motion, how much motion there is and of what sort it is, or even whether God moves everything around it, or whether he moves that very eye itself
@Slereah when it comes to jargon there should just be one jargon word per technical meaning. i find even mathematicians are like that, just synonyms everywhere.
Also, I am having trouble proving the claim that $I \otimes A \otimes I = B \otimes I \otimes I$ iff $A, B$ are simultaneously diagonal.
IT is conceivable that this statement is true, but I am wondering if there a way of proving it using a tensor factor permutation operator and showing that only diagonal matrices commute with it
I haven't really made a secret out of my identity, you can find youtube videos of me giving talks for work stuff :P (I'm the first guy here, for instance)
Australia in major cities is very multicultural, imo more so or differently than Europe (at least the parts I visited when I last travelled there; i guess things may have changed). when i was living in southern europe briefly I used to have people (to be fair, mostly middle-aged/older people and immigrants) refuse to believe where I was from.
If I have some curves described by $\varphi=\text{constant}$ where $\varphi$ is a complex scalar field, the tangent vector is proportional to $\nabla \varphi \times \nabla \varphi^*$ where $*$ is the complex conjugate. Why?
@ACuriousMind But that just means that you cannot mathematically implement the equivalent of a passive transformation; I'm not sure whether that means you don't understand it... I mean, it is clear and obvious that the concept of an active transformation could be implemented as a global vector flow that transforms every curve; surely something similar exists for the passive case too?
@naturallyInconsistent 1. The group of diffeomorphisms is, in fact, not the same as the set of transformations that are flows of vector fields, cf. MO. (But this is a nitpick)
2. There is no "group of coordinate transformations" I know of. If you have two coordinate charts $(U_1,\phi_1),(U_2,\phi_2)$, then the coordinate transformation between them is merely a diffeomorphism $\phi_2\phi_1^{-1}$ on the coordinate spaces of the overlap $\phi_1^{-1}(U_1\cap U_2) \to \phi_2(U_1\cap U_2)$, not on the manifold, and you cannot stitch this together to get a global diffeomorphism
the reason physicists always get away with this identification on a technical level is that they never take manifolds seriously and work only in a single coordinate chart that covers the entire manifold (e.g. spherical coordinates which cover the entire sphere) so that these overlaps are the entire manifold
we might have a different understanding of what "could be implemented" means in what you said
but what I'm saying is that if you use "active transformation" to mean "diffeomorphism", then it is not true that every active transformation is the flow of a vector field
The only kinds of active transformations that physicists talk about are invariably rigid global translations, global rotations, global Lorentz boosts, and so forth. i.e. the standard Noether's theorem set.
@ACuriousMind just to be painfully clear, are you saying that if the overlaps cannot cover the entire manifold, then the passive viewpoint fails to exist?
@naturallyInconsistent I'm saying it's at least not as simple as saying that a coordinate transformation "is" a diffeomorphism when we have more than one chart.
and I've never seen anyone who bothered to work this out properly - mathematicians don't use this confusing terminology, physicists don't care about charts
@naturallyInconsistent but now that I read your original question again: Yes, I guess I am saying there isn't really such a thing as a rigorous notion of a "passive" isometry (or diffeomorphism).
diffeomorphism, isomorphism, isometry, automorphism, really can melt faces off
@ACuriousMind phew, ok. Does this mean you know it cannot exist, or that it might exist, that just nobody had ever found it (and published widely enough to make it known)?
in each individual instance where physicists talk about such things I can figure out what they really mean, but it doesn't map onto a single equivalent mathematical notion
They do, I'm sorry I don't have any example, but I'm 100% sure they are different to the point that knowing a sound in one language doesn't help you recognize it in other laguage (in writing, that is)
For the following lagrange density of N fermions with mass m:
$\mathcal{L}=\sum_{i=1}^N\bar{\psi_i}(i\not\partial-m)\psi_i$ I was trying to calculate the equations of motion. I simply want to know if what I got is accurate, that way I know that my understanding is also accurate:
@qwerty I just looked at it and yes, sure, he defines the terms exactly as I would have above - the active transformation is a diffeomorphism on the manifold, the passive transformation is a switch on a single chart/on the overlap of two charts. But that's a silly definition because it just means those are two different things, while everyone else who uses active/passive terminology acts as if these are two different viewpoints of the same thing
