The h Bar

vzn

17:04

@0celo7 what do you mean by that?

0celo7

@JohnRennie hello

John Rennie

Evening

It's quiet in here these days - at least apart from discussions about maths :-)

0celo7

have you started on Wald yet

John Rennie

No. Realistically the chances of me getting time to read a big textbook properly are non-existant. There is just too much life to cram into too few hours. Maybe in a few years when I get fed up of the IT work ...

0celo7

"big textbook"

John Rennie

17:16

But when I do get the time my priority is to learn QFT not improve my GR.

0celo7

Wald isn't large

John Rennie

Compared to an elephant Wald's GR book isn't large.

ACuriousMind

@JohnRennie I approve :)

0celo7

@ACuriousMind shoo

ACuriousMind

::hisses::

0celo7

17:18

Analysis homework done.

On to more fruitful endeavors...NOT

John Rennie

@ACuriousMind I learned GR by getting the basic idea first then filling in the maths. This is a slow and inefficient way to learn, but it does work and puts me in an ideal position to answers questions for other learners.

0celo7

Probability homework ;_;

John Rennie

But this approach is failing completely for QFT.

It seems as though there isn't anything except the maths.

0celo7

QFT isn't even real

like what is a field

John Rennie

I do wonder if we've missed a trick with QFT somewhere.

0celo7

17:20

you think there's a trick to GR?

the trick is sweeping differential topology under the rug

John Rennie

QFT is so profoundly unintutive that I wonder is somewhere around the 1940s we took a wrong turn and have been doing it the hard way ever since.

0celo7

what's unintuitive about it?

John Rennie

@0celo7 I think it's possible to grasp what a metric is and how you use it to describe a spacetime with relatively little maths beyond simple calculus.

0celo7

@JohnRennie It takes a nontrivial amount of topology to define what a manifold is.

And god-forbid you want to prove four-dimensional is well-defined!

Off into the land of (co)homology!

John Rennie

@0celo7 Well yes, I remember you and Barry Carter rambling on about that for hours. But you assume I care exactly how to define a manifold.

0celo7

17:23

@JohnRennie The biggest thing really is this:

We postulate spacetime is 4-D, but we also postulate diffeomorphisms leave the geometry invariant

So how do we know our spacetime is not diffeomorphic to a 5D spacetime

or a 3D one

There are local arguments using tangent spaces, but the global proof is very tricky

John Rennie

We don't know and we don't care. GR - with all our assumptions - works just fine.

0celo7

@JohnRennie You do care.

If you use the technology of manifolds, then you must care whether the object "4-dimensional manifold" makes sense

John Rennie

You know when I said: you assume I care exactly how to define a manifold ...

ACuriousMind

@JohnRennie That's possible, but somewhat unlikely - QFT is the straightforward continuation of the quantization programme that has served us so well in the standard QM setting

John Rennie

@ACuriousMind I'm veering into crackpot territory here aren't I. This is the good old I don't understand it so it must be wrong approach so beloved of some of our site members.

0celo7

17:27

Oh, I take it back.

Diffeomorphism invariance is immediate from some tangent space stuff.

It's homeomorphism invariance that's very hard.

But we don't care about those in GR, so all is well

John Rennie

@0celo7 See, I knew that subconsciously and that's why I didn't care :-)

Balarka Sen

@0celo7 Err, what?

0celo7

@BalarkaSen If $M^m$ is homeomorphic to $N^n$, then $n=m$.

Balarka Sen

Ok, nevermind, I do see what you mean. It takes a nontrivial amount of work for topological manifolds.

ACuriousMind

@JohnRennie I think many people are somewhat dissatisfied with QFT, for a variety of reasons. But its success and the lack of a viable alternative make betting on an actual alternative existing rather unreasonable

0celo7

17:29

The usual proofs involve (co)homology

Balarka Sen

For smooth manifolds it is obvious.

Yes, this is a special case of Brouwer invariance of domain.

0celo7

@BalarkaSen speaking of invariance of domain

is the invariance of domain theorem in Hatcher equivalent to the one on wiki?

John Rennie

@ACuriousMind Nima Arkani-Hamed went through a phase of thinking QFT could be better formulated in a non-local way, but things seem to have gone quiet on that front.

Balarka Sen

I don't know what the version on wiki is.

ACuriousMind

Even the more fundamental theories supposed to "replace" QFT, like string theory or LQG, do not abandon the essence of quantization - the main difference is in the starting point of what we are quantizing.

0celo7

17:30

@BalarkaSen Homeomorphism invariance is obvious for smooth manifolds?

Or do you mean diffeomorphism invariance

Balarka Sen

Invariance of domain, to me, is that if a manifold embeds into a manifold of same dimension then it's a connected component.

0celo7

how do you prove that

Balarka Sen

@0celo7 I meant diffeomorphism. But you can prove for homeomorphisms too, I believe: Whitney approximation.

Approximate by smooth maps better and better.

0celo7

Invariance of domain

Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space Rn. It states: If U is an open subset of Rn and f : U → Rn is an injective continuous map, then V = f(U) is open and f is a homeomorphism between U and V. The theorem and its proof are due to L. E. J. Brouwer, published in 1912. The proof uses tools of algebraic topology, notably the Brouwer fixed point theorem. == Notes == The conclusion of the theorem can equivalently be formulated as: "f is an open map". Normally, to check that f is a homeomorphism, one would have to verify that both f and its inverse...

John Rennie

@ACuriousMind I don't think anyone doubts the principle of quantising a field, but the question is whether we've chosen the best mathematical techniques for doing it.

0celo7

17:31

@BalarkaSen You can smooth, but do you get a smooth inverse?

Balarka Sen

If you approximate in the uniform norm sufficiently better, yes.

Scrap that. I think it's still true, but I'd have to think a bit.

0celo7

That seems like a long shot. Much easier is to use de Rham cohomology

Balarka Sen

Should be a stability argument.

0celo7

the proof using de Rham is in Lee, you look at punctured open balls or something

and use the cohomology of $S^n$

Balarka Sen

That's not even a proof-sketch.

ACuriousMind

17:34

@JohnRennie We clearly haven't because the mathematic rigor is still lacking in many areas, yet the one thing which almost certainly doesn't really exist - the Lorentzian path integral - is thought by many to be the fundamentally right way to think about QFT. And I do think that even without the math, the "sum over all possible histories" is a very nice intuition. The trouble is that without the (pseudo-)math this intuition doesn't let you deduce anything.

0celo7

@BalarkaSen Because I do not know the argument any more than that.

I can think hard about it

(or look it up)

@BalarkaSen so how do you prove your connected component theorem

Balarka Sen

Local homology. That's the go-to analogue of tangent spaces for C^0 manifolds.

0celo7

ah, homoogy

@BalarkaSen is it in Hatcher?

Balarka Sen

Sure

Somewhere in section 2.1

After excision I believe

@0celo7 It's not true that every homeomorphism between smooth manifolds can be smoothly approximated to be a diffeomorphism. Oh well. I already knew this, didn't come to mind immediately.

0celo7

@BalarkaSen I didn't think so, but why not?

Balarka Sen

17:45

If you could do that, every homeomorphism would be homotopic to a diffeomorphism (approximating in the uniform norm better and better makes it homotopic to the limit: this is a standard fact).

0celo7

Ok, so?

Balarka Sen

Let me remember what the counterexample to that was. It should be an exotic sphere with a self-homeomorphism.

0celo7

Ah, of course

Or an exotic $\Bbb R^4$

or is compactness needed

Balarka Sen

Ah, I remember after googling. The group of exotic 7-spheres under connected sum is $\Bbb Z/28$ - pick an element which does not have order $2$. That does not admit a orientation reversing self-diffeo: if it did, then connected summing it with itself would be diffeomorphic to the standard $S^7$ - contrary to assumption. But every sphere admits an orientation reversing homeomorphism: antipodal map!

0celo7

is that the kind of topology Mike does

Balarka Sen

17:50

Da.

Or so is my impression.

0celo7

too topological for me

Balarka Sen

it's the cool topology

0celo7

meh

Sanya

18:06

@2physics hi :o

John Duffield

@JohnRennie : no. You're veering out of it.

@ACuriousMind : don't bet on that.

vzn

18:40

@ACuriousMind agreed (w JD). "absence of evidence is not evidence of absence"

Space Otter

19:18

Is it true CuriousOne doesn't believe in photons?

ACuriousMind

What kind of question is that?

Why don't you ask him what he believes in?

Space Otter

I'm not sure where i would reach him

I'm just trying to work out how that works

It's what ive been told

yuggib

I think that matters very few what any person believes, in general

Unless you're a theologist of course

0celo7

or a head of state

(or really anything to do with the real world)

Space Otter

At one point we believed that the earth was the centre of the universe and that everything orbited us on a series of spheres, so that kind off mattered. I'm only trying to see if he refutes the idea of wave-particle duality altogether. And so I could ask to see how he might explain the quantum like nature of light.

@yuggib So what actually matters?

0celo7

19:29

@ACuriousMind Does the limit definition of continuity $\lim_n f(x_n)=f(x)$ carry over to topological limits?

Or is it just analysis

yuggib

If you're a scientist, it matters that you are able to draw conclusions from your hypotheses that agree with experience

ACuriousMind

@0celo7 Sequential continuity is not always the same as continuity, but it holds at least in all metric spaces

yuggib

Or that you are able to quantitatively investigate nature (e.g. through experiments)

0celo7

@ACuriousMind That it holds in metric spaces is clear.

What about in Hausdorff spaces?

(the target is Hausdorff)

yuggib

@0celo7 forget about it

You need nets/filters

0celo7

19:32

Forget about it?

ACuriousMind

@0celo7 It's not a separation axiom that is the problem, but a countability axiom; Wiki says first-countability is what you need for sequential continuity = continuity.

0celo7

I don't even know what first countability is

ACuriousMind

Then look it up!

0celo7

I don't have a topology book with me

countable neighborhood basis?

seems legit

ACuriousMind

@0celo7 You evidently do have internet with you; why would you look up a basic definition in a book?

0celo7

19:36

I'm trying to show that if a continuous function $f:A\to Y$ ($Y$ Hausdorff) can be extended continuously to $\bar A$, then this extension is unique. If sequential continuity = continuity, this is pretty easy.

@ACuriousMind I'm running the chat on my ~~calculator~~ potato, not enough power to use the internet

ACuriousMind

You're not @BernardMeurer, you can't run anything on a potato

0celo7

I have the same potato he does

This problem is harder than the boundary one

I didn't think that would be possible

Noah P

Are you guys proud of me?

0celo7

no

Noah P

:'(

ACuriousMind

19:46

meow

6

0celo7

@ACuriousMind wtf is with this PhD level general topology

ACuriousMind

@0celo7 ?

Dog

@NoahP No

var firstName

Dog is here

djsmiley2k

meowwww

var firstName

19:56

I love cats.

Dog

Meaow!

Noah P

@DavidZ

djsmiley2k

So, according to Brian Cox (I asked him myself) gravity isn't a force

\o/

Why do we insist on calling it one :(

var firstName

gravity is a force

ACuriousMind

81

Q: If gravity isn't a force, then why do we learn in school that it is?

I have studied some of Einstein's Theory of General Relativity, and I understand that it states that gravity isn't a force but rather the effects of objects curving space-time. If this is true, then why are we instructed in middle school that it is a force?

djsmiley2k

19:57

Wow \o/

var firstName

Newtonian gravity is close enough

djsmiley2k

cept it wasn't for me, when i was learning

/me sighs

Took me until last year of sec school i think for a teacher to actually explain things properly to me

var firstName

SU outings are fun

0celo7

@ACuriousMind I can't do these topology problems

also I just wasted 30 minutes on parenting.SE

var firstName

nice job, 0celo7

WAIT

YALL MOFOS LIED

This guy's a cat ^

yuggib

19:59

@0celo7 in your proof for metric spaces, substitute the word "sequence" with "net" and it should work for hausdorff spaces

ACuriousMind

@0celo7 oO

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