Well, honestly, I would never compute cohomology by the deRham way. The beauty of deRham's theorem is precisely that I can compute the cohomology however I want and still know stuff about differential forms.
But, guided by the fact that we know we should get $\mathbb{R}\oplus\mathbb{R}$, you might take a look at the forms $\mathrm{d}\theta$ and $\mathrm{d}\phi$. They are closed. Are they exact?
If you integrate over a point, the integral is zero. But if you integrate along the circle, it also has to be zero because the antiderivative should be single valued.
Well, "$\mathrm{d}\theta$" is, strictly speaking, not an expression for a form on the whole circle because the angular coordinate is not a coordiante for the entire circle (due to that non-continuity when it wraps around).
So writing $\int_{S^1}\mathrm{d}\theta$ is intuitively clear, but one has to say what one means by it.
@0celo7 Well, you have to cover the circle with two coordinate patches in which $\mathrm{d}\theta$ is well-defined, integrate locally, and sum up the result, substracting the overlap.
Well, there can still be arbitrary closed forms $\omega = t(\theta,\phi)\mathrm{d}\theta + p(\theta,\phi)\mathrm{d}\phi$ with $\mathrm{d}\omega = 0$, we've said nothing about those.
Okay, since we want to show that $\mathrm{d}\theta$ and $\mathrm{d}\phi$ span the cohomology, we need to show that every other closed form $\omega$ has constants $a(\omega),b(\omega)$ such that $\omega - a(\omega)\mathrm{d}\theta - b(\omega)\mathrm{d}\phi$ is exact.
No, I'm pretty certain my last two statements are entirely correct. I think the problem is that I have no clue what the picture in your head of the situation looks like. Let me try a bit more verbose:
When we say a cohomology group is $\mathbb{R}^n$, this means there are $n$ closed but not exact forms $\omega_i$ such that for every other closed form $\xi$ there are constants $a_i$ such that $\xi - \sum_i a_i\omega_i$ is exact.
@0celo7 deRham cohomology is really not suited to understand what one is computing, don't worry.
The thing about the constants follows directly from the definition as the quotient of the closed by the exact forms.
@0celo7 Okay: Do you see that, in general, given a $n$-dim. vector space $V$ and a $k$-dim. subspace $W$, the quotient $V/W$ is generated by $n-k$-vectors that lie in $V$ but not in $W$?
And this means that "$n-k$ vectors span $V/W$" is the same as saying "There are $n-k$ vectors $u_i$ in $V$ but not $W$ such that for every vector $v\in V$ there are constants $a_i$ such that $v - \sum_i a_i u_i$ is in $W$".
Okay, since we want to show that $\mathrm{d}\theta$ and $\mathrm{d}\phi$ span the cohomology, we need to show that every other closed form $\omega$ has constants $a(\omega),b(\omega)$ such that $\omega - a(\omega)\mathrm{d}\theta - b(\omega)\mathrm{d}\phi$ is exact.
I think you're confused because you did never stop to think about what the definition of the cohomology group meant (and Arnold evidently didn't care to clearly explain). What, exactly, do you understand by the "quotient of the closed by the exact forms"?
@0celo7 Okay: Let's say the line $L$ I divide out is generated by $v\in\mathbb{R}^2$, that is, elements of $L$ are multiples $\lambda v,\lambda\in\mathbb{R}$. Now, for two $w,w'\in\mathbb{R}^2$ to lie in the same equivalence class, $w-w'\in L$, so $w-w' = \lambda v$. But that means we can just pick any $w_0$ in a class and all other vectors in the class are $w_0 + \lambda v$ for some $\lambda$.
This shows the equivalence classes are lines parallel to $L$.
Okay, since we want to show that $\mathrm{d}\theta$ and $\mathrm{d}\phi$ span the cohomology, we need to show that every other closed form $\omega$ has constants $a(\omega),b(\omega)$ such that $\omega - a(\omega)\mathrm{d}\theta - b(\omega)\mathrm{d}\phi$ is exact.
$\partial_\phi \int t\mathrm{d}\theta = \int (\partial_\phi t)\mathrm{d}\theta = \int (\partial_\theta p)\mathrm{d}\theta = 0$ since the integration is over a circle without boundary.
So $a(\omega)$ is constant in $\phi$ and so is $b(\omega)$ in $\theta$.
Thus $a,b$ are constants, and $\omega - \frac{a}{2\pi}\mathrm{d}\theta - \frac{b}{2\pi}\mathrm{d}\phi$ is exact. (use the integration condition from the beginning)
@0celo7 It's really just plugging the form I claimed is exact into your result from the start that $\int t\mathrm{d}\theta = 0 = \int p\mathrm{d}\phi$ means exactness.
@ChrisWhite Ah, that explains the questions we've been getting on Astronomy about that star. I knew that it had made the news somehow, but I was unaware of some of the other claims being made.
Ok so (at least for spinless free propagating tachyon field) either it is already nonlocal to start with, or that it is local but subluminal or it exponentially grows (unstable)
But how to formulate the interacting case with ordinary fields/particles, or is the free propagating case sufficient to illustrate everything?
dirac equation solution for tachyons? or some EOM that can handle spin of any number?
So far I only seen the non interacting case (because they rule the interaction case out by runaway generation of cherenkov radiation), but is there really no go for the interacting case also?
yeah, the only relevant ones I found are in response to the OPERA neutrino anormally, which I am not sure if they are good reference given the experiment itself is a cable error
Currently trying to solve dirac equation with tachyon mass term
"Finally, covariant wave equations are given for each unitary irreducible representation of the Poincar´e group with non-negative mass-squared. Tachyonic representations are also examined."
Oh little groups
You never let me down
"As shown in Wigner’s classic work [1] in four dimensions, there are four types of irreducible representations of the Poincar´e group. Two describe massless and massive elementary particles with definite helicity and spin, respectively. However, Nature does not seem to use the other two: one that describes particles with space-like momenta [2], tachyons which move faster than the speed of light, and the others that describe massless states with an infinite number of integer or half-odd integer unit-spaced helicities, dubbed by Wigner Continuous Spin
The horror
"There is no analogous cure for the CSR, for which the obstacles are indeed formidable: negative norm states, non-locality, and acausality [3], and according to Wigner himself [4], infinite heat capacity of the vacuum."
@0celo7 Well...technically, it's not the quotient as vector spaces, but the quotient as abelian groups (which coincide in this case). The article is sloppy to not explicitly state the category the quotient is taken in, though.
I was reading the quite nice answer of QMechanic on the topic of compact support tachyon fields not propagating faster than light, but this case is a rather simple one, free scalar field in flat space.
Are there any more general theorems on this? Let's say, for a field transforming under the ta...
I was reading the quite nice answer of QMechanic on the topic of compact support tachyon fields not propagating faster than light, but this case is a rather simple one, free scalar field in flat space.
Are there any more general theorems on this? Let's say, for a field transforming under the ta...
