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9:13 AM
@ACuriousMind Is the function space of group inclusions $\phi : H \hookrightarrow G$ basically indexed by the quotient space $G / H$
and for every $x \in G/H$ there is a corresponding inclusion $\phi_x$
 
@Slereah unless $H$ is normal in $G$, there is no $G/H$
 
Well let us say there is
For bundle reduction purpose
 
you mean "let us assume $H$ is normal"?
because you can't just assume the existence of a quotient that you can't actually form :P
 
Yes
 
hm, wait
 
9:22 AM
Checking if I understand the reduction $\sim$ section of the associated homogeneous space bit
 
this doesn't even work
 
Although I think the issue is slightly more complicated with bundles because I don't think $G/H$ has to be a group quotient
 
when you write $G/H$, you have fixed one particular embedding of the "abstract $H$"
but normal and non-normal subgroups can be isomorphic, so writing $G/H$ doesn't make a lot of sense
@Slereah you'll have to be more specific what kind of quotient it is, then
 
A very good question
 
I mean, if you define $G/H$ as the space of distinct inclusions $H\to G$, then there's no problem, but that's also not what we usually mean by a quotient
take $H=G$, then this just looking for group automorphisms
but we would expect $G/G$ to be trivial in any "nice" notion of quotient, so I think you're on the wrong track here
 
9:37 AM
Ah it's not a quotient, but a coset space
Except they use the same notation because why not
 
well, if $H$ is normal then the left and right cosets are exactly the quotient :P
so it's acceptable notation
but what I said above about $G=H$ is still true, I don't think this classifies inclusions without further constraints
 
Hm
How does the reduction correspond to a section on the coset space, then?
ie
If I have let's say the frame bundle $GL(n)$ and I pick a section of $\mathbb{Z}_2 = GL(n) / GL^+(n)$, does that not pick an inclusion of $GL^+(n)$ in either the right or left handed set of frames?
 
@Slereah the cosets are not subgroups (except for the coset associated with the identity), so while this picks an inclusion of sets it does not pick an inclusion of groups
 
So does it span the inclusions as a function of sets?
 
no, of course not
as sets there are much more inclusions
 
9:50 AM
Dagnabbit
 
and of course the inclusions of groups that there are are also inclusions of sets
to see that this doesn't classify inclusions of sets, just consider that a set with n elements has $n!$ possible bijections to itself, but again we would arrive at $G/G = 1$
 
Does the coset space classify anything related to the inclusions in some sense?
Or am I on the wrong track
It seems somewhat important since it's the space that generates the actual structures in GR
 
I think you're entirely on the wrong track :P
 
What does the cross section of that bundle imply?
The existence implies that the reduction exists, but what do specific cross sections imply as far as that reduction goes?
I would guess that different metrics of $GL(n)/O(n)$ select different equivalence classes of frames
 
10:07 AM
I think what confuses you is the following statement: The possible reductions of a $G$-principal bundle to a $H$-principle bundle are given by global sections of a $G/H$-bundle.
 
It certainly does
 
This has nothing to do with inclusions $H\to G$ directly, but simply with how the reduction works on the level of the transition functions
 
Doesn't it imply anything on a local level?
 
sure, locally it tells you the transition functions
 
I mean point-wise, not on a neighbourhood
 
10:09 AM
the requirement that what you do locally to reduce the bundle must be globally consistent gives you in the end the requirement that there be a global section
@Slereah I don't really know what you mean by "pointwise"
 
for $p \in M$ there is some value in the fiber $G / H$
 
sure, but as I said that value comes essentially from thinking about transition functions
the value of a transition function at a single point means nothing
 
I see
 
10:47 AM
One thing I wonder sometimes
Is picking a left or right action actually important
Or do you just pick one and keep it consistent
 
11:01 AM
Sometimes a definition will just say "It has a left/right action", and I am like sure
 
11:28 AM
whats the deal with so many users having names like "leftOverMonica" or "reinstateMonica"
..?
 
@satan29 it's a reference to the events surrounding judaism.meta.stackexchange.com/q/5193, just search Meta Stack Exchange or the wider internet for more info
 
 
5 hours later…
glS
4:10 PM
so, ncatlab [writes](http://nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/Liouville-Poincar%C3%A9+1-form) that the canonical one-form $\theta\in\Omega(T^* X)$ is defined as the unique such form such that $\sigma^*\theta=j(\sigma)$, for any smooth section $\sigma\in\Gamma(T^* X)$, and with $j$ the isomorphism $j:\Gamma(T^* X)\to \Omega^1(X)$.

Anyone understand why this makes sense? For one, I thought differential one-forms were *defined* as smooth sections of the cotangent bundle, so wouldn't $\Gamma(T^* X)=\Omega^1(X)$? And then, $\sigma:X\to T^* X$ and $\theta:T^*X\to T^* T^* X$, so $\s
I feel like I'm missing something obvious here
ah, wait, I got it. They mean the isomorphism connecting the two ways to understand 1-forms: as smooth sections $X\to T^* X$, or as smooth maps $TX\to\mathbb R$. So $\theta:TT^*X\to\mathbb R$, $\sigma:X\to T^* X$, $\sigma^* \theta=\theta\circ d\sigma:TQ\to \mathbb R$, and $j(\sigma):TQ\to\mathbb R$, so it checks out
 
yes, exactly
 
It's the secret behind why Lagrangian mechanics is awful
the jet space v. the regular space
 
that's where the name "tautological" for this one-form comes from - if you choose to define the 1-forms as sections, then you can omit $j$ and you just get $\sigma^\ast \theta = \sigma$
 
glS
is it standard to understand $\Omega^1(X)$ this way though? I don't see them using that definition when introducing differential forms. I've seen this switch done "silently" often, but I always assumed it was just some abuse of notation used to simplify expressions
 
wikipedia does a decent demonstration of it
In differential topology, the jet bundle is a certain construction that makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form. Jets may also be seen as the coordinate free versions of Taylor expansions. Historically, jet bundles are attributed to Charles Ehresmann, and were an advance on the method (prolongation) of Élie Cartan, of dealing geometrically with higher derivatives, by imposing differential form conditions on newly introduced formal variables. Jet bundles are sometimes...
 
glS
4:24 PM
@Slereah wait, is the $j$ in jets the same one used here?
 
@glS well, because $j$ is an isomorphism and because there is no consensus on which of the two sides really is "the" definition, whether or not you're abusing notation, "omitting $j$" or just doing the correct thing is a matter of viewpoint in any given situation
@Slereah what does any of this have to do with jets?
the tautological 1-form is just a nice thing that appears when you think too hard about Hamiltonian mechanics :P
@glS the nLab article is a bit weird in that it seems to assume it's clear that we define differential forms via their action on the tangent bundle, but you're just equally weird in assuming that it's clear we define differential forms as sections of the cotangent bundle ;)
 
glS
@ACuriousMind in my defense, that's how I've mostly seen them defined? Even though they then often end up using the $TX\to\mathbb R$ definition when actually calculating things
I'm actually happy to have finally seen the mapping between the two notations explicitly written down (or at least, its existence acknowledged)
@ACuriousMind the tautological one-form is something I've been trying to get a good grip on for some time. Though every time I end up getting lost in some tangentially needed definition or some corner of ncatlab. I'll get there eventually
the fact that it seems like every source writes it differently doesn't help lol
 
@ACuriousMind I may be thinking of a different canonical form
Too many things called the canonical X
 
glS
is there an easy way to see why the definition $\theta(\alpha)=\alpha\circ d\pi\big|_{T_\alpha T^* M}$, where $\pi:T^*M\to M$ is the canonical bundle projection and $\alpha:TM\to\mathbb R$, is consistent with the local expression one often sees as $\theta=\sum_i p_i dq^i$?
 
4:58 PM
@glS where have you seen that definition?
 
glS
@ACuriousMind you mean the coordinate-independent one? It's on wikipedia for example en.wikipedia.org/wiki/Tautological_one-form
In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle T ∗ Q {\displaystyle T^{*}Q} of a manifold Q . {\displaystyle Q.} In physics, it is used to create a correspondence between the velocity of a point in a mechanical system and its momentum, thus providing a bridge between Lagrangian mechanics with Hamiltonian mechanics (on the manifold Q {\displaystyle Q} )....
 
I don't see what the definition there has to do with what you wrote down
what's $T_\alpha T^\ast M$?
why is $\theta$ eating a function $TM\to \mathbb{R}$ and not a vector field on $T^\ast M$?
 
glS
it's in the linked section of the wiki page. They write it as $\theta_m=m\circ d\pi_m$, where $m:T_q Q\to\mathbb R$
 
I'm saying I have no idea what your notation means, but I wholly agree with the definition on Wiki :P
 
That's a lot of work just to take a derivative
 
glS
5:04 PM
@ACuriousMind I'm probably mixing the two ways to define differential forms, cue previous discussion. In my mind, a differential form on the cotangent bundle is a map $\theta: T^*M\to T^* T^* M$, and $\alpha\in T^* M$, so that $\theta(\alpha)\in T^* T^* M$, and thus $\theta(\alpha): T(T^* M)\to \mathbb R$
 
this is already getting extremely confusing because you've been using $M$ for the base space but Wiki uses $Q$ for the base and $M=T^\ast Q$
 
glS
@ACuriousMind sorry, that's my fault. Yes the wiki is using $M=T^* Q$. I'm just using $M$ as base space (I kinda find it confusing to use $M$ as they do)
 
here's how Wiki's definition relates to the local expression: At $m=(q,p)$, $\mathrm{d}\pi$ projects out any $\partial_p$ and keeps only $\partial_q$. And the map that "$m$ represents" is just applying the co-tangent vector $p$ to the resulting vector on $Q$. If you take a vector field $\alpha = \sum_i \alpha_q^i\partial_{q^i} + \sum_i \alpha_p^i \partial_{p_i}$ and apply the local expression for $\theta$ to it, the result is $\sum_i p_i \alpha_q^i$
which is the same you get from applying $p$ to the "$q$-part" of $\alpha$ that's kept after applying $\mathrm{d}\pi$ to it.
 
glS
@ACuriousMind $\pi:T^* M\to M$, so $d\pi:TT^* M\to TM$, and $d\pi\big|_{T_\alpha T^* M}$ is the restriction of the map to the fiber in $TT^*M$ over $\alpha\in T^* M$. Is this a weird notation?
 
oh, I see, the part of your notation that's weird is writing $\theta(\alpha)$ for "the value of $\theta$ at the point $\alpha$"
when I see $\theta(\alpha)$ for $\theta$ some 1-form, I always assume first that $\alpha$ is a vector field being fed to the form
 
glS
5:09 PM
@ACuriousMind but isn't that what a section of the bundle $T^* (T^*M)\to T^* M$ should do?
 
feeding a point to it is $\theta_\alpha$, like Wiki does with $\theta_m$
@glS yes, but that's not how people usually write it :P
we usually don't put this emphasis on them being sections
anyway, just notational differences then, and my reply above about the local expression applies, regardless of which notation you use :P
 
glS
@ACuriousMind so, in that notation, $d\pi_m (\alpha)=\alpha^i_q \partial_{q^i}$ and $m=p^i \partial_{q^i}$ (which is shorthand for $m_q=p^i \, \partial_{q^i}\big|_q$ I suppose), and thus $\theta_m (\alpha) = p^i \alpha^i_q(q)$, where $q=\pi(m)$
 
why is your $m$ expanded in $\partial_q$s?
 
glS
@ACuriousMind right. Should be $m_q=p^i \, \mathrm dq_i\big|_q$?
 
it's a point in the cotangent bundle, and so $m = (q,p)$, with $p = p_i\mathrm{d}q^i\in T^\ast_q Q$, and this $p$ is the map they mean
 
glS
5:21 PM
ok, yes, that's what I was trying to write lol
so, the idea is that $\theta$ takes curves in $T^* Q$ passing through $m$, and looks at the increments of the curves when projected on the base space, and multiplies that with the "height" (i.e. the output values) of $m$ iself?
so kinda, just uses $m$, which normally eats curves in $Q$ (modeling tangent space with curves), only making it work on curves on the larger space $T^* Q$, but "forgetting" about its "vertical" structure somehow?
 
I would just say "it takes velocities $\partial_{q^i}$ and replaces them by momenta $p_i$"
that's the idea, all the rest is technicality
 
glS
@ACuriousMind but isn't that just what $m$ does?
 
$m$ is just a point in the cotangent bundle, it does that just at a single base point $q$
$\theta$ is the thing that does this "globally"
 
glS
@ACuriousMind is it that simple? When you say "replace velocities with momenta", you are assuming to replace velocities with some values of the momenta. But $\theta$ takes cotangent vectors and sends them to the map replacing velocities with the corresponding momenta?
I mean, $\theta$ can be fed with any covector $m$, so it can give us any "replacing velocities with momenta" map
 
5:39 PM
Look again what it does to our vector field $\alpha$ up there: It forgets about the $\partial_p$ part, and then it takes the $\alpha_q^i\partial_{q^i}$ and just replaces the $\partial_{q^i}$ by $p_i$
it doesn't replace the velocities with any specific momentum - the result is a scalar function on $T^\ast M$
it's action at any specific point $m=(q,p)$ is to replace the velocities at $q$ by the momenta $p_i$
 
glS
6:04 PM
@ACuriousMind hang on. So now we are using a third way to understand what "differential form" means no? A "differential one-form on $Q$" could mean (1) a section $Q\to T^* Q$, (2) a map $TQ\to\mathbb R$, or (3) a map $\Gamma(TQ)\to C(Q,\mathbb R)$, which is what you now mean? Takes a vector field and gives a functional
except we are doing it on $M\equiv T^* Q$, so we have a map $\theta:\Gamma(TT^* Q)\to C(T^* Q,\mathbb R)$,
 
I don't really think of (2) and (3) as distinct, but you're technically right
what's happening here is very similar to currying
you can think about a 1-form as having "2 slots" - one that accepts points and one that accepts vectors
depending on which slots you're filling or not, your view on what a 1-form is changes
note that the operation that goes from (2) to (3) is just concatenation - a vector field is a section $Q\to TQ$, and so we can just concatenate it with our map $TQ\to \mathbb{R}$ to get a scalar $Q\to\mathbb{R}$
 
glS
@ACuriousMind that makes sense. Definition (1) fits in there too I'd say no? With (1) we isolate base points, and get functionals on tangent vectors. With (3) we (effectively) isolate tangent vectors, and get functionals on base points. With (2) we directly assign numbers to each pair of basepoint+tangent vector
 
glS
so, going for example by the second definition, as a map $TQ\to\mathbb R$, which in our case is $TM\equiv TT^* Q\to\mathbb R$, we understand $\theta$ as a map sending $(m,\alpha)\in T_m T^* Q$ to $m_2\cdot \alpha_1\in\mathbb R$ (where with $m_2$ I mean something like "second coordinate of $m$ etc), i.e. $p_i\alpha^i_q$
nice
or going with (3) above, we get a map sending each "velocity" $\alpha$ to a map $T^*Q\to\mathbb R:m\mapsto \theta_m(\alpha)$, which we like to think of as "momentum"
 
glS
6:31 PM
so.... to connect these definitions above, given $\alpha:Q\to T^*Q$ 1form as a section, $j(\alpha):TQ\to\mathbb R$, and $\ell(\alpha):\Gamma(TQ)\to C^\infty(Q,\mathbb R)$, we have $$\color{red}{(}\alpha(q)\color{red}{)}(X(q))=\color{red}{(}j(\alpha)\color{red}{)}(X(q))=\underbrace{\color{red}{(}(\ell(\alpha))(X) \color{red}{)}}_{\equiv \alpha(X)}(q)\in\mathbb R,$$ where $q\in Q$, $X\in\Gamma(TQ)$, $X(q)\in T_q Q$. This is evil lol
 
6:44 PM
You know what I hate
The electroweak interaction
We have unified the $SU(2)$ gauge and the $U(1)$ gauge into an $SU(2) \times U(1)$ gauge!
 
the deceptive thing about that is that the unbroken EM $\mathrm{U}(1)$ isn't actually the right-hand factor in $\mathrm{SU}(2)\times\mathrm{U}(1)$
it sits a bit strangely in there, and that's why you can't usefully think about a separate SU(2) and U(1) above the unification scale
 
I know :p
 
7:10 PM
"To see explicit examples of the abstract formalism that follows, one may want to glance from time to time at the examples of Sect. 3."
Can't they put the examples during the explanation
 
7:55 PM
Ummm ... So basically we need to displace the Hamiltonian density by say $c$ and not the Hamiltonian by $c$ to make the connection to the cosmological constant?
How does one go to the cosmological constant from this? One cannot. $c$ will only displace Hamiltonian by a constant factor. But we can't say that $c$ will also displace Hamiltonian density by a constant factor. Cosmological constant should be viewed as a space time density. — KP99 1 hour ago
 
8:24 PM
Cosmological constant is non-trivial in curved space where the factor is $\sqrt{-g} \Lambda$ not just $\Lambda$
 
Doesnt the $\sqrt{-g} $ come from making it a volume integral?
 
If the metric is constant, $\sqrt{-g} \Lambda$ is constant so the integral in the Lagrangian can be ignored, otherwise it can't be ignored it's a non-trivial contribution because of the dynamical $\sqrt{-g}$ part, which is needed for making a coordinate invariant volume integral
 
So I cant add a term to this stress energy tensor and the spacetime around it?

$ T^{\mu\nu} = \sum_{n} \frac{p_n^\mu p_n^\nu}{p_n^0} \delta^3(\vec{x} - \vec{x}_n(t)) $
 
9:00 PM
Hmm ... Okay ignore me I think it makes sense
 
9:14 PM
Hey folks, quick question.
Can I ask a question about optics here?
 
sure
 
If a beam of light comes back to the observer, will the observer see his/her own reflection?
 
I mean, how else would seeing one's reflection work?
 
Something like this:
@ACuriousMind Ah, ok.
I was thinking of a funhouse-mirror kind of scenario
Or a room of mirrors
In a room of mirrors, then, why does every line of sight land on your own reflection? Shouldn't only certain light paths come back to themselves?
Not sure if my question makes sense, but I'm open to clarifying.
 
every different reflection you see in such a scenario corresponds to a distinct path the light can take from you to the mirrors and then back to you
if "every direction" has your reflection, then someone simply engineered the room such that there is such a path in every direction
 
9:19 PM
So, to clarify, every reflection you might see in a hall of mirrors (for instance) is a beam of light "coming back to where it originated from"
 
yes
or, well
 
Wow
That's powerful
That's really powerful
 
you don't really radiate light, so the light does not "originate" from you
 
I see
 
you're just reflecting light from some other light source
but the idea is the same
 
9:20 PM
@ACuriousMind Thank you! You answered my question!
 
for a qm. harmonic oschillation the energy is $E= \hbar \omega (n + frac 1 2)$. Now, if $\omega$ is constant, what do we observe when $n$ gets bigger values?
 
uh, we observe higher energies? :P
 
yes
which physically translates to what?
more oschillations over time?
with the given constant frequency ?
 
not really
 
if $\omega$
is constant
 
9:33 PM
what is even "more oscillations over time with constant frequency" supposed to mean?
 
how, what changes physically that implies higher energy?
 
it's a quantum system
you don't get some ontological description of "what" it "is"
 
look
 
if you draw the wavefunction, you'll see it gets more nodes, i.e. looks like higher "overtones" the higher the energy gets
 
but more nodes doesn't imply a different value of $k$ ?
higher value
 
9:35 PM
what's k
 
wavenumber
 
wavenumber of what
 
in a line with N atomes
a solid
 
that's not a single harmonic oscillator
 
thats multiples
but the idea behind what I need is the same
For example in a solid with N particles in 3D, you have 3N modes. In a mode you have the energy $e_j=\hbar \omega_j(n_j + frac 1 2)$ where $j$ implies a mode. Now $n_j$ is the number of phonons. You can consider an oschillation through the solid, as a phonon with a specific $\vec k$ and polarisation
therefore $n_j$ is the nr. of phonons, in other words the number of oschillations
that propagate in the solid
 
9:45 PM
yes, but there the $n_j$ means something completely different from the $n$ for a single oscillator!
you're saying you have $n_j$ times the first excited state
 
I am considering multiple non interacting oschillators
 
while for a single oscillator, you just have the $n$-th excited state once
 
yes
but if we agree to have $omega$ constant, even for one oschillator
while $n$ changes values
first of all is that possible ?
 
I don't understand what you're saying, sorry
 
ok
if you have two identical oscillators, in the same exact mode, but with different $n$ values, how can you tell that one has bigger $n$ value then the other
?
 
fqq
10:00 PM
you measure the energy?
 

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