The h Bar

John Rennie

05:19

Sunrise in Chester. It is almost worth starting work at 5 a.m. when the view from my desk includes sunrises like this.

Almost ...

Loong

06:20

@EmilioPisanty next round: meta.stackexchange.com/q/332974/271002

Emilio Pisanty

07:00

@Loong thanks for the heads up

Slereah

@ACuriousMind Achtung

Slereah

07:28

0

Q: Translation invariance of point particles as a field theory

The case of point particles, relativistic or not, can be treated as a field theory in general, ie for the $(1+1)$-dimensional case this is the theory of a field theory on the vector bundle $$\pi : \mathbb{R}^2 \to \mathbb{R} $$ with $x(t)$ a section of that bundle. I thought therefore to look a...

plz halp

Actually thinking about it, why don't we talk about symmetries of field theories for scalar fields?

$\phi \to \phi + \alpha$ is also an entirely fine field symmetry for massless fields

Although I guess that may simply be the conformal symmetry

Although... not quite, no

same problem as before

Slereah

07:45

Are we supposed to use an affine bundle

Slereah

08:05

"An affine space or affine linear space is a vector space that has forgotten its origin."

sad

peterh

@KyleKanos Typically well-educated people can be more motivated by secrets. The urge to know and to learn things, is more strong in them.

bolbteppa

Well there's $A_{\mu} \to A_{\mu} + \partial_{\mu} \alpha$

Slereah

08:56

@bolbteppa True

Is the tangent space of the $U(1)$ bundle an affine bundle

bolbteppa

What's a bundle

Slereah

I'm sure if the answer is somewhere it's in Naber

Naber is literally two whole books on EM from the bundle perspective

bolbteppa

The tangent space to a $U(1)$ principal bundle is going to be (part of) the $u(1)$ Lie algebra right

Slereah

Hopefully yes

books.google.fr/…

It does seem to be affine

Does this derive from some fundamental idea or is it just decided by fiat

We need the bundle to do this therefore it is an affine bundle

I guess for generality it is best to treat the fields as affine bundles

bolbteppa

09:18

Skimming it they seem to be saying the connection bundle is affine (like in GR) but not the $U(1)$ starting point

Slereah

I don't think there's any general issues with that

Well $U(1)$ isn't even a vector space

it's not gonna be affine

Though the $U(1)$ action on it is basically the equivalent of an affine action I s'ppose

Intellex

11:17

@JohnRennie, Hi. Could you please answer my query - Do you feel Halliday, Resnick and Jearl Walker is a good book for physics. I am interested in physics and know till High School level.

John Rennie

@Intellex I can't help I'm afraid. I'm not familiar with the Halliday book.

Intellex

@JohnRennie, Or could you please suggest a good book for Physics?

bolbteppa

@Intellex yes it is, the calculus-based one, and GRE's use it

John Rennie

@Intellex I can't help with books I'm afraid as I'm out of touch with what books are good these days.

bolbteppa

@JohnRennie most good books are before all of our time :p

Intellex

11:39

@bolbteppa Thank you for your reply. I am not preparing for GRE but I wish to read since I am interested in Physics.

@JohnRennie No problem. Thank you for spending your time helping me.

@bolbteppa Yes. I agree.

What are your views on Feynman Lectures?

bolbteppa

You will learn how to do physics better from Halliday

Intellex

What do you mean by "do physics" ? Do you mean numericals?

bolbteppa

Sure

Intellex

Thanks @bolbteppa !

Slereah

12:43

@bolbteppa I always read your name as Bob Teppa

bolbteppa

You're supposed to read it as

Vito Volterra

Professor Vito Volterra (, Italian: [ˈviːto volˈtɛrra]; 3 May 1860 – 11 October 1940) was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations, being one of the founders of functional analysis. == Biography == Born in Ancona, then part of the Papal States, into a very poor Jewish family: his father was Abramo Volterra and mother, Angelica Almagia.Volterra showed early promise in mathematics before attending the University of Pisa, where he fell under the influence of Enrico Betti, and where he became professor of rational mechanics...

Ryan Unger

adjusting my sleep schedule to normal hours is proving difficult

bolbteppa

@RyanUnger it's impossible to do

I really want to know the whole integral equation stuff properly

Naveen Balaji

13:32

0

Q: Weak Solutions to the Einstein Equation across a Junction

Consider the principle part, i.e., the part which contains the highest derivatives of the metric (which is the $2^{nd}$ derivative) is $$\mathcal{P}\{R_{ab}\}=\frac{1}{2}g^{cd}\left(\partial_{a}\partial_{c}g_{db}+\partial_{b}\partial_{c}g_{da}-\partial_{a}\partial_{b}g_{cd}-\partial_{c}\partial_{...

general-relativity metric-tensor curvature boundary-conditions branes

For those interested in GR

Slereah

13:47

@NaveenBalaji There's the treatment of stress energy tensor at junctions in Visser if you want

Naveen Balaji

14:25

@Slereah Yeah, in fact, that was what I referred when I learned the subject (and Poisson too). This question addresses the shell problem in specific with only the principle parts of the Einstein tensor considered.

bolbteppa

17:02

-2

Q: If the string is inelastic (conserves Hamiltonian engergy) then by Gauss's theorema egregium there is only one equation of motion solution, right?

The idea the string equation of motion has many solutions results because the differential wave equation cannot define a Riemann integral on mass-and-spring elements. As the atoms get smaller under large numbers they do not converge on a limit while the spring constant is inexplicably conserved. ...

string-theory hamiltonian-formalism

"Since the tangent-cotangent product space of the Hamiltonian is fiber-co-fiber space, the manifold is time and space averaged."

ACuriousMind

17:21

@Slereah I'm not sure what you're really asking - since you haven't given any reference that would claim that you need to restrict yourself to vector bundles, nothing is stopping you from considering the tangent bundle as an affine bundle. So what's your problem?

Slereah

@ACuriousMind yeah I suppose that is the solution

Although now I'm wondering

Do I even have to consider affine bundles at all?

Are there any physical reasons to not consider non-linear transformations?

ACuriousMind

Not really

But generic fiber bundles are a rather weak structure, so there's a mathematical reason to not want to relax the notions that far if one doesn't need to :P

Slereah

Are there any reasonable system that has nonlinear symmetries?

Sine Gordon has a bunch of symmetries IIRC, mb that one

let's see

ACuriousMind

@Slereah Gauge symmetries are realized non-linearly in spontaneously broken phases with Goldstone bosons.

Slereah

yeah so I guess it's best to consider them

Although I guess it's probably useful to sometimes restrict them to affine or linear transformations

Are all Lie symmetries linear btw?

I guess all the ones generated by a principal bundle are

ACuriousMind

17:29

@Slereah Gauge symmetry groups are usually Lie groups, so the SSB statement above applies to them.

Slereah

Dang it :V

Why is nothing ever simple

I wonder if it's doable to show that the point particle has no nonlinear symmetries

IF IT DOESN'T

dun dun

ACuriousMind

@Slereah However, you can get away with only considering the linear representation of a subalgebra and inducing the non-linear realization of the full group as in en.wikipedia.org/wiki/Nonlinear_realization

Slereah

thx

I should check Henneaux too

i'm sure there's some weird ass symmetries in there

ACuriousMind

Alas, in full generality, gauge algebras are not even required to close, i.e. there are "gauge theories" in the broad sense of constrained Hamiltonian systems which do not really have a gauge group and cannot be described in the principal bundle language at all

Henneaux/Teitelboim also has some examples for the failure of the Dirac conjecture, which provides another class of theories we can't really treat with standard methods.

Slereah

yeah that is what I suspected

I mean

just consider this dumbass action :

$$S[\phi] = \int_\Omega 1 d^nx$$

It has ALL THE SYMMETRIES

enumaris

17:38

I like that name

ACuriousMind

@Slereah Well, you likely should restrict your action functionals to those that actually have equations of motions :P

Slereah

$0 = 0$ is a fine equation of motion

ACuriousMind

Of course the theory that predicts nothing is symmetric under everything!

Slereah

Although $\phi$ has some (all) gauge freedom

ACuriousMind

@Slereah Let me say it like this: You should require your action functionals to have isolated extremal points.

Slereah

17:39

I'm trying to come up with really dumb actions with weird symmetries but it's hard to think of ones that are continuous

@ACuriousMind Could be worse

Henneaux had $$S[q] = \int q(t) dt$$

With EOM $1 = 0$

why is basic physics so complicated :V

ACuriousMind

@Slereah Because it needs to support a world as complicated as ours :P

Slereah

Even zero dimensional QFT is bad

Why is it complicated

it's just a point

Although fortunately zero dimensional classical mechanics is alright

It's just the stationary point $x = 0$

Slereah

18:23

Heyyy

I've got a dumb idea

$$S = \int \ddot{x}^2 dt$$

hm

Semiclassical

isn't zero-dimensional QFT = stat mech?

Slereah

no I'm not sure there's any stranger symmetries here

nvm

Semiclassical

(up to wick rotation etc)

Slereah

@Semiclassical Zero dimensional QFT is just the Lebesgue integral

Semiclassical

mmkay

ACuriousMind

18:25

@Slereah No, bad Slereah! (physics.stackexchange.com/a/44183/50583)

Slereah

Since functions from $\{0\}$ to $\mathbb{R}$ are $\approx \mathbb{R}$

@ACuriousMind I know it's bad!

I just wonder about its symmetries

Just call it symmetries of a functional, if you prefer :p

Ryan Unger

what are you talking about

@ACuriousMind this answer is wrong, there are many natural higher order actions

say the Willmore energy

Slereah

@RyanUnger math

ACuriousMind

@RyanUnger Maybe you missed that Qmechanic is talking about quantum theories?

Ryan Unger

why would he be doing that

ACuriousMind

18:39

...because he's not Cmechanic? :P

4

Emilio Pisanty

Yo @Semiclassical

Semiclassical

hey

Emilio Pisanty

I'd like to have a crack at the 2D quantum harmonic oscillator, but with funky stuff in the middle

Semiclassical

what sorta funky stuff?

Emilio Pisanty

i.e. I don't care about irregular behaviour at the origin, I'll fix that later

Semiclassical

18:44

gotcha

Emilio Pisanty

I just want the solutions to the radial equation, for arbitrary energy, that are regular at infinity (i.e. $\mathrm{poly}(r) \exp(-r^2/2)$)

for zero angular momentum I've pinned them down to the parabolic cylinder functions

Semiclassical

hmm. remind me what the radial part of the SE equation looks like?

Emilio Pisanty

do you know off the top of your head what class of special functions I should be looking at?

Semiclassical

nope

Emilio Pisanty

the relevant Schrödinger equation has the form $$-\frac12 \frac{\partial^2\psi}{\partial r^2} -\frac{1}{2r} \frac{\partial \psi}{\partial r} + \frac{ \ell^2}{ 2r^2}\psi(r) + \frac12 r^2 \psi(r) = E \psi(r)$$

Semiclassical

18:47

hmm, yeah.

Ryan Unger

what are the special functions if you do assume it's bounded at the origin

Emilio Pisanty

if I feed it in that form to Mathematica it returns a confluent hypergeometric and a Laguerre polynomial, times a bunch of prefactors and an exploding exponential, and neither is regular at infinity except for the usual energy eigenvalues

@RyanUnger laguerres times gaussians

Semiclassical

the change of variables I'd be inclined to consider is $r=e^u$, so that $\partial_u = r \partial_r$

and $\partial_u(\partial_u-1)=r\partial_r( r\partial_r-1)=r^2\partial_r^2$

Emilio Pisanty

@Semiclassical wooot

that looks insane

Semiclassical

meh. it's the usual change of variables to turn a Cauchy-Euler ode into one with constant coefficients.

it turns the ode into $$-\frac12 \partial_u^2 \psi +\frac{\ell^2}{2}\psi+\frac12 e^{4u}\psi=E\psi$$

the fact that you're only interested in good behavior at $r=\infty$ translates into behavior at $u\gg 0

Emilio Pisanty

18:59

I mean, sure

but this has to have been done before, right?

Semiclassical

yeah

i'm just trying to find a change of variables which makes it look familiar

Emilio Pisanty

I'm hoping I just need to identify the ODE and that'll lead me to some suitable Authoritative Resource

Semiclassical

oh. should've been $e^{2u} E \psi$ after the change of variables

Emilio Pisanty

I tried setting $\psi(r) = r^\ell e^{-r^2/2} \phi(r)$, which then has one of its solutions as a bare confluent hypergeometric

but it's not regular, and the other solution is $r^{-2\ell}$ times a different confluent hypergeometric

ODE comes to $$ -r\phi''(r) + (1-2r^2-2\ell)\phi'(r) + 2r(1-E+\ell) \phi(r) = 0 $$

in case that tells you anything

Semiclassical

looking at the solutions I get when $\ell=0$ in Wolfram, I see why you resist my change of variables

it doesn't make things any nicer

i think your ode is in the form of the generalized hypergeometric ODE, but pFq gives me headaches

c.f. dlmf.nist.gov/16.8

oh, wait. ignore that. i was thinking of the wrong thing

Emilio Pisanty

19:19

Bam, got it

hypergeometric $U$

thanks, guys =)

Semiclassical

oh, the tricomi one? (had to look it up) neat

Novalium Company

20:27

My heart, your heart, tide up like two ship.

Drifting, weightless, waves try to break it.

Why is it so hard to say it.

I don't wanna wait till you're gone. Cuz you make me strong...

I'm sorry if I say I need ya,

but I don't care, I'm not scared of love.

Slereah

Hm

How to prove that the action is only invariant under diffeomorphism of the field if it's a Galilean transformation

$x(t) \to f(x(t))$, with the time derivative being $$\frac{d}{dt}f(x(t)) = f'(x(t)) \dot{x}(t)$$

enumaris

anybody here have visual studio 2019?

ACuriousMind

@enumaris Yes, but I've never used it :P

Slereah

Can I say that $$\int f(x) g(x) dx = \int f(x) dx$$ for all $f$ iff $g(x) = 1$

Is that a theorem that is a thing

enumaris

Do you know if this functionality is in it: blogs.msdn.microsoft.com/onecode/p/devassistant

Slereah

20:36

Oh wait

Novalium Company

@enumaris I'm pro

Slereah

Shuffling it it's the fundamental theorem of calculus of variation

enumaris

Specifically the part where you can search a "how do I" question and it pops up example code snippets

Slereah

So indeed all transformations are $f'(x)^2 = 1$

So $f'(x) = \pm 1$ or $f(x) = \pm x(t) + a$

Heyyyy

That works

now to prove all the symmetries of the MSSM

I think $n$-dimensional point particles should be about as complicated

enumaris

I don't have access to visual studio to see if the functionality is already fully integrated or if they just discontinued that functionality

Slereah

20:39

just $(\vec{f}')^2 = 1$ instead

ACuriousMind

@enumaris Dunno. I could check tomorrow

enumaris

All the links to that plug in give me 404 error now

I know they integrated that functionality into Bing itself and you can try it

but

I wonder what happened to it in the IDE lol

@ACuriousMind that would be very helpful :D

enumaris

20:51

@NovaliumCompany if you could check if this functionality exists in VS2019 that'd be helpful :D

Slereah

github.com/nvbn/thefuck

enumaris

o.O

Emilio Pisanty

@Slereah great find

bolbteppa

22:02

@Slereah "fundamental lemma" not "theorem" ;)

Fundamental lemma of calculus of variations

In mathematics, specifically in the calculus of variations, a variation δf of a function f can be concentrated on an arbitrarily small interval, but not a single point. Accordingly, the necessary condition of extremum (functional derivative equal zero) appears in a weak formulation (variational form) integrated with an arbitrary function δf. The fundamental lemma of the calculus of variations is typically used to transform this weak formulation into the strong formulation (differential equation), free of the integration with arbitrary function. The proof usually exploits the possibility to choose...

enumaris

what the difference is

bolbteppa

Going by the etymology 'lemma' means 'something assumed' which means it's basically something not important enough to be a theorem but something you absolutely need to assume to do more important things

Slereah

Lemmas are theorems

"Lemma" just means you think it's not important enough!

But I think it's plenty important

enumaris

In that case a "fundamental lemma" might be an oxymoron...

but lemmas should be proved...unlike axioms

Slereah

22:20

Sure

Not a whole lot of axioms in analysis tho

They tend to be lower down on the totem pole

enumaris

I just make every statement I make an axiom

this way I can never be wrong

Slereah

It certainly makes proofs easier

Although you may end up with an inconsistent axiom set

bolbteppa

It is an oxymoron until you look at the lemma, it's like 'if $\int f(x) g(x) dx = 0$ for all $g$ then $f$ must be zero', it's completely obvious, but it is used all the time to invoke the EL equations, and it can fail for some monster examples no sane person would think of, so it's trivial and the most fundamental thing ever in CoV at the same time

I would call it an oxygenius

Slereah

I don't trust any obvious statement in mathematics

enumaris

See Jordan's Theroem: en.wikipedia.org/wiki/Jordan_curve_theorem

bolbteppa

22:29

Physics takes all the good from math and thankfully leaves all these boring counterexamples that fail (usually for bizarre real functions it took 200 years to think up)

enumaris

intuitive but very hard to prove

Slereah

@bolbteppa oh sure

Until they pop back up!

All the paths in path integrals are continuous nowhere differentiable!

enumaris

is that all paths or almost all paths

Slereah

The other paths are of measure 0

enumaris

so almost all paths

Slereah

22:33

My favorite convoluted physics example is probably the spacetime constructed to be geodesically complete but with a singularity

It's the worst spacetime ever concocted, and I've seen a few

enumaris

what definition of singularity is that one using

cus isn't geodesic incompleteness the usual definition

in which case you'd have a contradiction

like it's geodesically complete but some curvature scalar is not bound?

Slereah

There are non geodesic curves of bounded acceleration which are of finite proper time

It is an extremely convoluted example

enumaris

oh

so it's...geodesically complete but some "attainable" curves end in singularities

lol

yeah good luck with that...

like some singularity you'd have to accelerate into to hit and can't hit if you just moved naturally

Slereah

Yeah

enumaris

who had enough free time to come up with that?

Slereah

22:38

It has to be of bounded acceleration because of course, if unbounded, even Minkowski space is singular

It was Robert Geroch

enumaris

right

Slereah

Basically it's an infinite stack of spacetimes designed to trap geodesics

So that only an accelerated curve can emerge

enumaris

I wonder how much free time he had to concoct it lol

Slereah

Hell he's not the only one to have done it

There are two such examples

enumaris

fun times...

does it violate the energy conditions?

Slereah

22:44

Well it's based on AdS and also has a lot of conformally minkoswki stuff

So in places yeah

enumaris

hmmm

Slereah

Although I don't know if it's mandatory for such a spacetime

Nobody cares that much about the topic

I don't think there's a proof for it

enumaris

I mean there's a bunch of space times with CTCs...but we can sort of just exclude those...

like the Godel spacetime

Slereah

GR is just full of pathological cases we just sort of hope real hard aren't physical

Earman wrote an entire book on the topic

It's entirely possible that our universe could at some point just naturally evolve with a naked singularity for no reason

Just take the natural spacetime development of our universe, remove an open set from it

This is still an entirely acceptable spacetime

bolbteppa

"Thus, the lesson to learn is that Energy is only conserved if there's translational time symmetry in the problem.

Which brings us to General Relativity: in several interesting cases in GR, it's simply impossible to properly define a "time" direction! Technically speaking, this would imply a certain global property (called "global hyperbolicity") which not all 4-dimensional spacetimes have. So, in general, Energy is not conserved in GR."

Slereah

22:50

It's kind of assumes any spacetime will be inextendible but really there's nothing against it

bolbteppa

26

Q: Is energy really conserved?

In high school I was taught energy was conserved. Then I learned that nuclear reactions allow energy to be converted into mass. Then I also heard that apparently energy can spontaneously appear in quantum mechanics. So, are there any other caveats with the conservation of energy?

energy-conservation

What's the deal with energy conservation in GR

Slereah

Define energy

Although really

Things are even worse than you'd assume

It's not just the lack of time translation

bolbteppa

Time translation charge

Slereah

Take Minkowski space with the origin removed

That's a singularity at every time t

From that singularity, an arbitrary EM radiation can emerge or get absorbed

You could start with zero energy in flat soace and end up with arbitrary energy

I think this is one of those "we don't define the action properly at the boundary" sort of issue

bolbteppa

math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html is interesting

Slereah

22:56

We just always assume that there's onlynboundary conditions on the hypersurfaces and everything works out wrt the action

bolbteppa

Basically this is a mess

Slereah

Oh yes

It is

Although you can of course do the split $g = \eta + h$

And then it's just an awful field theory that is time invariant

Once you start fucking with the topology and geometry of spacetime you have to throw out a lot of notions

bolbteppa

"We will not delve into definitions of energy in general relativity such as the hamiltonian (amusingly, the energy of a closed universe always works out to be zero according to this definition),..."

enumaris

from my understanding you can define energy conservation in different ways in GR. Locally, you can describe it as the covariant derivative of the stress-energy tensor has to vanish. But globally you would require some time-like Killing field to exist. But not all space-times have a time-like Killing field. Even non-pathological ones. For example...the FLRW universe...

Slereah

That's why a lot of times people just assume a very mild spacetime

enumaris

23:01

this is due to a problem in GR that global quantities are often ill defined

but my understanding could very well be flawed :)

At the end of the day, when you remove a gravitational force from the equation and equate geometry with gravitation, you can no longer easily define gravitational potential energy (except when you have a time-like Killing field).

And so...things like gravitational red-shift become a purely geometric effect...but now you have no good way to describe where the lost energy is going

Slereah

Well it's not that bad

enumaris

but IF you have a time-like Killing field like in the Schwarzschild solution then it's all restored :)

Slereah

You can define a gravitational potential energy

Einstein whatever pseudotensor

enumaris

in spacetimes without time-like Killing fields?

Slereah

Sure

enumaris

23:04

I think there's other ways you can try, but it's always not quite satisfactory...

but when you do have a time-like Killing field it's pretty satisfactory

Slereah

Some people don't like it because it's not a tensor

But I'm not prejudiced

enumaris

right pseudotensors are eh

Slereah

en.m.wikipedia.org/wiki/Stress–energy–momentum_pseudotensor

enumaris

I remember something in Wald about it

but it seemed like you need to have some particular set of coordinates to make it work or something?

Slereah

Well it's a pseudotensor

Ergo not covariant

enumaris

23:08

right...so...

eh...

Slereah

But that is fine

enumaris

like in the sense that the dog trapped in a burning building is fine?

bolbteppa

"One way to get round this is to pick one coordinate system, and transport vectors so their components stay constant. Partial derivatives replace covariant derivatives, and Gauss's theorem is restored. The energy pseudo-tensors take this approach (at least some of them do)"

"Indeed, the issue of energy in general relativity has a lot to do with the notorious "problem of time" in quantum gravity... but that's another can of worms." oh good

enumaris

seems legit

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