::sigh:: I should have spend the time I spent on BBS reading topology instead. Now I know how bosonic string theory works (maybe) and just got confused after that.
@0celo7 The math of QFT works on all linear fields. That may not prove that there are gravitons, but it makes it easy enough to start writing equations that have them in. Up until the point you try to renormalize, I mean.
Neutrino mixing was described as a possibility before Ray Davis discovered the solar neutrino problem, but the paper was ignored languishing in obscurity.
in a college where the EP program is one which offers you a lot of good pure physics courses, in which like half the people want to go for pure physics
I did want to do physics for a long time, but last year I decided to go into software instead. That always existed as a backup plan and was one of the reasons I chose an engineering college (where I can be in touch with CS as well)
@ACuriousMind Hmm, maybe because the typical vector is $v$. The next typical vector is $w$. Since having $\nu$ and $v$ in the same calculation might be confusing (thinking typewriter days here), they went with $w\to\omega$ for the typical form.
@ACuriousMind What exactly is the difference between $\mathbb{R}^n\times\mathbb{R}^m$ and $\mathbb{R}^n\oplus\mathbb{R}^m$? Please dear god no category theory...
@ACuriousMind I ask because Sharpe, in his section on forms, has: Let $M=\mathbb{R}^m$, then $$TM\oplus\cdots\oplus TM\cong \mathbb{R}^m\times(\mathbb{R}^m\oplus\cdots\oplus\mathbb{R}^m)$$
@0celo7 Right. So it is different from the sum of vector spaces, since $TM\oplus_\text{Whitney} TM = \mathbb{R}^{3m}$, but $TM\oplus_\text{vector space} TM = \mathbb{R}^{4m}$.
@0celo7 Yes and no. There is no $\oplus$ on manifolds. What the r.h.s. is supposed to be is of the form $B\times F$ for the base $B$ and the fiber $F$ of the bundle. Writing $B\oplus F$ doesn't make sense.
@0celo7 $\mathrm{d}\vec A$ is the normal vector. So express the normal vector to the sphere as a function of $\phi,\theta$, compute its dot product with $\vec v(\phi,\theta)$, integrate the resulting number over $\phi,\theta$.
@0celo7 The pullback is actually a bit hidden here. You have that ${\star}v\wedge n = \langle v,n\rangle\omega$ for $n$ the 1-form belonging to the normal vector.and $\omega$ the volume form. So what you pull back is actually the volume form (and it becomes the induced volume form on $S^2$), the $\langle v,n\rangle$ just gets expressed as functions of $\theta,\phi$.
Somehow I don't think my prof who is a physicist if you close one eye will be too impressed if I write $\star v\wedge n=\langle v,n\rangle\omega$ on a quiz/exam ;)
There are "nice" pictures like a 1-form in 3 space being a bunch of planes, and integrating it along a line is counting how many planes the line intersects. A 2-form in three-space is a bunch of lines, and integrating it over a surface is counting how many lines the surface intersects
@0celo7 And a vector field is not a finite set of vectors stapled to points on a grid, yet we draw it as one
It's a picture, it's not meant to tell you how to actually compute the numerical value
Note that there is an arbitrary normalization in this picture of forms - which density of points, for instance, corresponds to the canonical volume three-form?
