In this textbook $k = \Delta m \omega ^2$

as the spring constant

is that derived from $F = ma = -kx$?

@Obliv I would guess this is a spring with a mass on one end being rotated around the other end, so the centripetal force mrω² is equal to the force due to the extension of the spring kx.

Learning GR is an adventure because once in a while villains I thought I would never see again just pop back up

It's the return of contact manifolds

Apparently Coleman's so-called direction field is just a contact element

Hard to find much on the topic that doesn't try to shove Weil algebras down your throat as well

Not another generalization :(

There is such a thing as being too general

Even worse I think that's how nlab defines the jet bundle

@Slereah some would consider that statement heresy :P

Maybe I should have seen the contact geometry coming when I saw that jets were about submanifolds being in contact

Also I am seeing that all contact structures are locally the same, which may explain why there is a description of thermodynamics similar to GR

that's just the contact version of the symplectic Darboux' theorem, right?

Yeah

According to Coleman, every geodesic equation only depends essentially on the projective structure and only really needs direction fields

Wondering what's the statement of that idea in more modern terms with contact geometry

I mean when I say "modern", he wrote all that stuff in the 80's, I think he just didn't know about it

Mr. Cartan had already written on that topic in... 1924

@ACuriousMind How does one show that the equivalence class of equivalent maps $X : M \to N$ up to diffeomorphisms on $M$ is equivalent to the image of $M$ in $N$ as a submanifold?

I'm not really sure what that means

what exactly is the equivalence relation you're quotienting out of $\mathrm{Hom}(M,N)$ here?

Two elements are equivalent if $\exists \phi \in \mathrm{Diff}(m), X_1 = X_2 \circ \phi$

Ah: $\mathrm{im}(X_1) = \mathrm{im}(X_2\circ \phi)$, and since $\phi$ is a bijection, $\mathrm{im}(X_2\circ \phi) = \mathrm{im}(X_2)$

I don't think there's anything more to it

Short and sweet

very good

Not 100% sure how much of all this relates specifically to contact structures since they are only on odd dimensional manifolds

I guess the projectivized tangent bundle is, but not entirely sure this is because of that

Hey! Is the relativistic particle mass-shell constraint $p_\mu p^\mu - m^2$ admissible? I think I might be misunderstanding the admissibility conditions in "quantization of gauge systems", page 7: the variations of the constraints must be of order $\epsilon$ for arbitrary variations of the momenta $p_\mu$ and coordinates $x_\mu$. Seems like $p_\mu p^\mu$ is order $\epsilon^2$.

maybe its admissible to use non-admissible constraints? :)

wouldn't the variation of $p^2$ wrt $p$ just be $2p$

Oh wait I guess it would lead to a $\delta p^2$

hmm yep an infinitesimal variation would behave that way: $delta p^2 = 2p \delta p$ and maybe at this point we say $\delta p$ is order $\epsilon$. But then any differentiable function of the $p$ and $x$ would be admissible

For mysterious reasons Ehresmann gives different names for contact elements on the tangent and cotangent bundle

Oh a contact structure is just a specific field of contact element

No need to worry too much on them

slightly annoying topic to research because it seems that most people talking about contact elements do so in the context of contact structures

@ACuriousMind Some places seem to say that the jet space $J^k(M,N)$ is a fiber bundle over $M/N/M\times N$, while others seem to say that no, it is only a fibered manifold

Do you know what this is about

is there a nuance somewhere I am not seeing

are they not locally trivial

I mean, the difference between fibered manifolds and fiber bundles is local triviality

I don't know if this is something trivial or specific about second quantization in condmat, but why single particle creation operators can be said to be always of this form ψ†(x) = [ρ(x)]1/2e−iθ(x), where pho(x) is the density operator? (it is said in this notes ggi.infn.it/sft/SFT_2019/LectureNotes/Giamarchi_book_Chap_3.pdf eq (3.10))

I think you usually construct explicitly the stalks $J_p^k(M,N)$ for $p\in M$: This construction gives you a fibered manifold (disjoint union of the stalks), but in order for this to be a fiber bundle you'd need to show that there are charts on $M$ where locally you have that $J^k(U,N)$ is $U\times F$ for some generic fiber $F$

the problem is - what is $F$? The stalks $J^k_p(M,N)$ do not carry a lot of structure (e.g. no vector space structure!), so you can't do what you do for vector bundles and just say $U\times V$ for $V$ fixed dimension and there's just some natural isomorphism between the stalks and this $V$

I've seen books claim it is a bundle

Like Michor's big book

I mean...you can say this is a bundle

Is it just that it's hard to prove and they don't bother proving it most of the time?

I don't know why the distinction matters for a structure-less fiber, really

as long as the fibers all have the same cardinality you can just pick a random bijection between them

@Ratman are you asking why there is a polar decomposition, or why $\rho$ appears there?

A lot of the time the total space is important though

Since that is what the tangent bundle or frame bundle is

but the fibers in the tangent or frame bundle carry structure!

a generic $J^k_p(M,N)$ does not

the situation is much better when you choose $N=\mathbb{R}^n$, then you get vector bundles

@fqq why $\rho$ appears there

The fiber doesn't depend on the source and target manifold though

In all cases it's the space of polynomials from Rn to Rm

@Slereah wrong; the fiber inherits any algebraic structure the target has

that's because it's a quotient of $C^\infty(M,N)$, and functions valued in $N$ of course inherit any structure $N$ has (like addition, multiplication, etc.)

Michor claims that if M is modelled on a vector space V and N on a vector space W, then the typical fiber is the space of polynomials from V to W

Although maybe he has a different definition of jets

"is" in what sense

In the subway rn so I will get back on you for that later :p

I previously posted it actually:

I guess if the jets aren't jets with fixed points, they may have properties that depend on the global structure of the manifold?

Since that leaves their "translation" part

Still that wouldn't explain why Michor and Kolar claim that they are bundles

I guess I'll have to look into Ehresmann's original big book

@Ratman well once you admit a polar decomposition $\psi = A e^{i \theta}$, then $\psi^\dagger\psi=A^\dagger A = A^2$ so $A$ is a square root of the number operator (which is $\rho$ or $N\rho$ depending on conventions I guess?)

yes in Giamarchi's notes $\rho$ is not normalised so it's correct