hm, really? I see, I never fully understood it. I think the only other source which I've read that used it is Carroll, who has gone down significantly in my estimation
@qwerty That would imply the other camp has a consistent definition :P
I really think it's just they take the equivalence from the case of vector spaces/$\mathbb{R}^n$ and never think all too hard about it when they claim the same kind of equivalence applies to "coordinates changes" and "diffeomorphisms" on arbitrary manifolds
and, as I said, as long as you only work in a single chart you can get away with it - the computational results are correct!
it's just that when you try to formulate the argument in any proper mathematical fashion you realise there's a lot of confusion hiding under the rug
would there be a situation in GR say, where you'd need to work with more than one chart? not exactly something i'm familiar with, but I recall people knit together solutions to the EFEs, would it come up it situations like that?
>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 is it really about manifolds vs vector spaces, though? I would argue that it's the same category error to conflate the two things for vector spaces
and since we often even deliberately "conflate" two things if they are isomorphic, I would not have that much of a problem with the terminology if it was just used in that context
but he introduced Einstein's algebraic focused "form of the equation stays the same" general covariance vs minkowski's geometric methods /invariance without commenting on rigour or lack thereof?
@TobiasFünke the only time I engage with physics is in this chat or when I write an answer (but sometimes writing an answer might send me on an hour of skimming over references)
Another thing it's important to understand is that even if it's been a lot of time and ACM is undoubtedly pretty darn good at it, there are some thing he's been discussing here on a weekly to daily basis for 10 years now
> The authors of the original edition spent about a year locked in a cabin in Siberia just doing integrals. Supposedly, they measured an integral's difficulty by counting how many vodkas they needed to drink before finding a solution.
Like, do you have an idea of how many generations of users have been here in the chat? I have checked the transcript of many years ago and basically Slereah and ACM are the only regular users that stayed by
@qwerty I have that ambition too, but as you know far better than me, research itself is made using references. Everything is about references. Anyone needs to consult books and papers, you, I, ACM, Qmechanic, Slereah and everyone else that has ever been and will ever be
@TobiasFünke well, surely there are levels. I think most competent physicists would be able to do some derivations and write down correct fundamental equations. but I doubt I have those in my brain independently
the scam is 1) become a famous physicist, 2) force prospective students to study and pass exams based on a volume of physics books you wrote, 3) profit.
I think a lot of the books I hated in undergrad was just that they assumed familiarity with terminology that I didn't have, and I'd get stuck parsing them. at that stage I didn't really know how to look for what I didn't know or diagnose it
@SillyGoose that's true, but I find mathematicians don't like to waste words on hashing out explanations when they've already given you a definition. so I also never really liked pure maths books!
@SillyGoose the current generation have no appreciation lol
@qwerty Not really, but I bought them a long time ago. Also, I would care if it was originally in English, but since it's Russian it's a translation in any case
seemingly the transformation is mixing the dirac fields. so you maybe are considering a vector each component is a dirac field and you do matrix multiplication accordingly.
Now, consider the multiplication of terms $\bar{\psi}$ $1$ in the first bracket (don't forget the indices) $(i\not\partial-m)$ $1$ with jk indices and $\psi_k$
let's call $U := \exp(i\epsilon_a T^a)$. This unitary transformation mixes the dirac fields. without using index notation we could write something like $U \vec{\Psi}$ where $\vec{\Psi} = \{\psi_1, ..., \psi_N\}$. In index notation this is then $U_{ij} \psi_{j}$.
