@Danu: I updated the instanton answer with Donaldson invariants, but after staring at the paper for some time trying to figure out how to say any of it without just retyping the thing, I have no clue whether what I've produced makes any sense to someone just reading the answer.
@MikeMiller Uh...the gauge group is actually not fixed here, and the grading has nothing to do with the gauge group - it comes from the fields' transformation behaviour under the rotation algebra $\mathfrak{so}(4)\cong\mathfrak{su}(2)\oplus\mathfrak{su}(2)$, but I didn't want to explain the twisting procedure how that relates to their grading because it's not important at all for the result.
@MikeMiller Yes, exactly! But writing that index-free would have required me to chase prefactors all across what follows, and I would have probably gotten at least one sign wrong
(To be honest, I think the indices are one of the biggest obstructions to my ability to engage with the physics literature...)
@ACuriousMind I'm still a little surprised I haven't heard about it. Whether or not we need it, we still tend to like extra data. I also should note that, as far as I know, the only place it is possible to define Donaldson polynomials is $SO(3)$, $SU(2)$, $U(2)$. (You can write them down for say $S^1$ but they're all zero.) And for both $SO(3)$ and $U(2)$ you need to modify the procedure; it does not work the same as $SU(2)$.
@MikeMiller Only the $A$ is a connection - and the $F$ is still the curvature. The other things are some $\mathfrak{g}$-valued fields. Some of them are meant to be something like a basis of one-forms on the space of connections (they are in the three-dimensional Floer theory on which Witten builds this, and you see a trace of this in the zero-modes being as many as the allowed instanton perturbation, i.e. they span the instanton tangent space).
but a part of them is also just there to make the Lagrangian gauge invariant and supersymmetric, for which I don't know a mathemati
@MikeMiller Witten never restricts the gauge group. However, there are a few ifs there - such as the non-anomalous transformation of the path integral measure, or the independence of the assignment of the signs of the path between the instantons - for which I cannot say whether they might just fail for other groups.
@ACuriousMind It is not obvious to me where they are in the Floer theory. This is a hard dictionary (for me) to build!
(I hope, BTW, this isn't sounding like an interrogation. Your answer is helpful but due to my lack of physics knowledge it's hard to engage with a lot of the, well, words.)
@MikeMiller I only know the few things I gleaned from Witten's short expo, I haven't read Atiyah's work where this is apparently done, so I don't know if you mathematical approach even has something equivalent to these physical fields
I don't think I've ever actually read anything by Atiyah other than some of his 60s stuff on index theory and the Atiyah-Bott paper; wouldn't know, to be honest.
@ACuriousMind: Your section 2.2 makes a lot of sense to me, very clear, thanks.
If your goal is to build an actual dictionary for Donaldson theory, is seems it might be appropriate to start with the dictionary for Floer theory - The 4D SYM model here is motivated as the Lorentz-covariant generalization of the 3D Hamiltonian gauge theory Atiyah seems to have used for that.
To me, the (historical) motivation for Floer theory is as a way to calculate Donaldson invariants! (Of course, it now has a billion applications of its own.)
This, in turn, is essentially based on Witten's work in relating supersymmetry and Morse theory, which seems like a nice co-evolution: Floer theory is to Morse theory as quantum mechanics is to quantum field theory. (the step from finite to infinite dimension)
@ skill patrol My latest try suggest if I want to model time travel back to the future style that is compatible with general relativity, I will end up with highly nonlocal dynamics where the dynamics of a certain event X=(x0,y0,z0,t0) will depend on a finite number of other (x,y,z,t) that is in the past light cone of the event X
However, more calculations needed to see whether my naive guess is self consistent
If I don't want to ping someone but still refer to it, I tend to do this I only ping people if I have something important to ask or discuss
I know which of my messages are important, which of mine are just space fillers of the chat because of getting lonely and bored easily when trying to scratch my head deriving the stuff needed to solve the problems or investigate something
having said that, some users don't take my space fillers well especially when I started drawing pictures without elaboration (which is a sign that the picture is to be treated with a grain of salt)
If a picture is important, I will often elaborate it (even if it might be just a futile attempt at visualising something)
Referring to the pull ups performed by Oliver Queen in the TV series The Arrow.
[]
My hunch is that such pull ups are impossible because it violates physics principles: as Oliver tries to pull himself up, his hands applies forces on the bar, and therefore his hands can no longer lift the bar up...
@0celo7: take the points $A,B,C,D = (\pm 1, \pm 1, 0)$ and $E = (0,0,12)$. I'm trying to rotate and translate them to different integer points such that they aren't as nice anymore (in a coordinate plane). I tried rotating by x and y axis by $\pi/4$ and then translating and scaling by (multiples of) $\sqrt{2}$ but can't get rid of all squareroots. maybe you can? ._.
@Huy Since no further information is given on the translation used, let m1, m2, m3, n be integers
For ease of writing down, create an augmented matrix $M$ using the position vectors A,B,C,D,E
Now rotation will be carried out by the matrix $R_{\frac{\pi}{4}}$ Translation by the matrix $T$ and scaling (assuming uniform scaling) $S$ (which is just a scalar
After that the transformed coordinates can be extracted from the column of the 3x5 matrix
The result is that the transformed position vector E' will always contains $\sqrt{2}$ unless there is no translation along the z direction
What's wrong with this community? Only here, I am experiencing such blunt closing.
If spacetime is discrete: what would space expansion mean? -- This question has been corrected. Nevertheless, no one seems to bother to reopen it
Implications of observable Universe growing indefinitely -- what i...
If the units of action are $[E][T] = [MV^2][T] = [ML^2/T^2][T] = [ML^2/T]$ so that the string action is $S = - [M/T] \int dA = ...$ how do you (easily) get from $[M/T]$ to string tension $T_0$, $[M/T] = T_0/c$?
A local symmetry can be written $\delta f(x) =f(x)+\partial f (x)$. Would it be appropriate to consider the function to be $f(x)=q(t)$ so that $\partial f(x)=\dot q(t)$? Within the context of Noether's theorem for local symmetries.
@Qmechanic Thank you very much for the update in the 2-cocycle answer! It was extremely interesting! :D
@ACuriousMind Have you ever heard of this definition before? Let $\{z_1,\dotsc,z_{n-1}\}$ be an orthonormal basis on the subspace of $T_pM$ orthogonal to some $x\in T_pM$. Then the Ricci curvature in the direction of $x$ is $$\mathrm{Ric}_p(x)=\frac{1}{n-1}\sum_{i=1}^{n-1} \langle R(x,z_i)x,z_i\rangle$$
and the scalar curvature is $$K(p)=\frac{1}{n}\sum_{j=1}^n\mathrm{Ric}_p(z_j)$$
The homework tag doesn't help the fact that your question is explicitly what we don't want posted here. We want questions that deal with physics concepts, asking "how do I solve this" isn't a physics concept. — Kyle Kanos4 mins ago
The homework tag doesn't help the fact that your question is explicitly what we don't want posted here. We want questions that deal with physics concepts, asking "how do I solve this" isn't a physics concept. — Kyle Kanos8 mins ago
While most questions/answers seem to be pretty good about including the link to articles (etc) where they take images from, it seems like it might be good to consistently encourage that the original URL be provided along with the included images.
@0celo7 About your Maupertuis question, you are asking whether it is possible to treat the potential energy as though it is of kinematical origin, this was a question asked by Hertz and the result is Hertz forceless mechanics, something that motivates Kaluza-Klein and extra dimensions, c.f. Lanczos Variational Principles for starters, no more freebies
Referring to the pull ups performed by Oliver Queen in the TV series The Arrow.
[]
My hunch is that such pull ups are impossible because it violates physics principles: as Oliver tries to pull himself up, his hands applies forces on the bar, and therefore his hands can no longer lift the bar up...
What's wrong with this community? Only here, I am experiencing such blunt closing.
If spacetime is discrete: what would space expansion mean? -- This question has been corrected. Nevertheless, no one seems to bother to reopen it
Implications of observable Universe growing indefinitely -- what i...
While most questions/answers seem to be pretty good about including the link to articles (etc) where they take images from, it seems like it might be good to consistently encourage that the original URL be provided along with the included images.
