The h Bar

yuggib

10:00

but you need to enlarge all the structure, not only the norm

you enlarge the basis, etc.

Slereah

I know

The norm was just the annoying point in the usual theory

yuggib

the resulting space is separable as a hyperreal/hypercomplex vector space but as a real/complex space has an uncountable basis

well, the theory is developed a bit in robinson's book

anyways, building theories from scratch is usually rather complicated

Slereah

I'll bet

Especially qft

yuggib

also, one sure problematic point is that maybe you can sum to unbounded operators when transferred to the hyperreals, but the result may be hyperreal everywhere, i.e. you can't interpret it as a standard operator

Slereah

I'll have to do non archimedean poincaré group

The horror

Well yes

That is part of the point

yuggib

10:04

yes, but then the result is moot

Slereah

Manipulating formally divergent quantities

yuggib

since we do not observe hyperreal-valued quantities

Slereah

With the axiom that all observables must be finite

yuggib

then what you obtain is not observable

Slereah

Yes

yuggib

10:05

the point is that in renormalization two ill-defined / unbounded quantities may conspire to get a finite result

this is extremely hard to see on hyperreals

Slereah

Is it tho

yuggib

let $N_1,N_2$ be unbounded. Then $N_1-N_2$ can be unbounded, bounded or infinitesimal

Slereah

By analogy with the Colombeau formulation, H will be infinite, but H - lambda will be finite, hopefully

By a judicious choice of the vacuum energy

yuggib

example: $N_1=(n+1)_{n\in\mathbb{N}}$, $N_2=(n)_{n\in\mathbb{N}}$

or $N_1=(n+\frac{1}{n})_{n\in\mathbb{N}}$, $N_2$ as above

Slereah

Yes

Basically the hamiltonian normalization, actually :p

yuggib

10:08

or $N_1=(n^2)_{n\in\mathbb{N}}$ and $N_2$ as above

@Slereah that is what I am saying

you do not solve the real problem

Slereah

I know

yuggib

because it is as difficult to predict the outcome with hyperreals as it is in standard analysis

Slereah

It seems an interesting topic tho

yuggib

it is interesting perhaps

but not so promising in my opinion

Slereah

Promising topics are all too hard or boring

Or already done

I'll do the chaff

Danu

10:12

The resignation is (hyper?)real

yuggib

@Danu from what?

;-P

Slereah

Eh

I'd rather do this than real physics

Danu

^ heh

Slereah

Meson cross sections or whatever

Danu

How are you doing @yuggib

yuggib

10:13

@Danu not bad

Danu

Great

yuggib

you?

Danu

Pretty good---math is still new and exciting :)

Slereah

Just an endless stream of BLIPIDIBLOOP MESON CROSSSECTION AT S = 9000 MeV

yuggib

@Danu math is much better than physics ;-D

Danu

10:14

I'm learning about complex geometry. Just got through the section on Hodge decomposition on Kahler manifolds

yuggib

at least, I find it much more satisfactory

the problem is that being a working mathematician is very complicated, especially when you're young

many papers to write, conference to try to be invited to, positions to get, ....

in physics is probably more or less the same though

Slereah

Just invent a new number

Like threeven

Half three, half seven

Danu

@yuggib I assume so.

Getting a PhD spot will already be difficult enough for me.

Slereah

I tried and didn't get one

With a real topic!

yuggib

@Danu what about moving, france should be a safe choice for geometry and the alike

or maybe Cattaneo in Zurich if you want to make your physics background count

Danu

10:20

@yuggib Is it? I don't know... I am not yet actually searching for places. Right now, I don't even know anything yet.

@yuggib But he's not interested in things that I find interesting. I'm interested in smooth manifolds, mostly. EDIT: Wait, I think I misread what he was working on.

It's funny, by the way, I've recently found some reasons to perhaps return to some analysis soon, in geometry. This Hodge decomposition stuff requires some machinery on Sobolev spaces etc. to really prove.

yuggib

@Danu Mathematics in France is very strong, and I think it is not so hard to enter some PhD program (especially outside of Paris/écoles normales)

Danu

@yuggib Right, I guess with a "I'll take anything" attitude it should be possible. We'll see...

yuggib

@Danu Well, you have plenty of time to shape/change your interests

and of course you can't hope to do mathematics without any sight of analysis ;-P

Danu

I guess not :P

But you seem to manage to avoid any interesting geometry ;)

I guess that manifolds etc are inherently "higher structures" in the sense of relying on/incorporating many different aspects of mathematics.

yuggib

@Danu Well, I don't avoid geometry competely; for example I often used optimal transport techniques

i.e. analysis with a lot of geometry in it

Slereah

10:27

Also sometimes he has to cut pizzas

Lots of geometry involved

Danu

lol

@yuggib Hmmm. Okay. But are there any manifolds involved?

yuggib

@Danu in some case yes; anyways manifolds don't behave so well with respect to quantization

and I am mostly a quantum guy (i.e. I do functional analysis)

Danu

Yeah, I always like classical field theory most, in physics :)

Slereah

Do you know yuggib

There's a bundle theory of QM

Hilbert bundles

I looked into it

It is entierly written by one guy

yuggib

yeah I know, but nothing more than a bunch of formal statements

Slereah

10:32

Another guy who needed ideas for papers

yuggib

probably more than one guy worked on it

btw who is the guy?

Slereah

I forget

Just google hilbert bundle

Should be him

His papers have the BOZO logo

Probably a clown college

arxiv.org/abs/quant-ph/9902068

That guy

yuggib

let's say that I tend to be cautious about almost self-referenced work that pretends to be mathematical but is published in J.Phys.A or the alike

Slereah

Heh

No love for clown colleges?

yuggib

it's bozho not bozo

the "h" is probably important

Danu

10:39

haha

Slereah

If I do a nonarchimedean Lorentz group I'm guessing it's best to take the finite subalgebra

yuggib

today is one of those days when you're lazier than a sloth

user228700

@Danu: Hi :-) Can u help me with a math question? I've asked at the MSE chat...

Slereah

Don't want infinite boosts

@yuggib wot

yuggib

@Slereah not you as you slereah, you as "on" in french

I was referring to myself

@Kaumudi ask and, if not Danu, someone else may answer

Danu

10:46

@Kaumudi Nah, sorry. Please don't follow people around with your questions :P

If I don't reply by myself, I am not interested in answering.

user228700

OK, sorry.

Danu

No problem :)

Slereah

Welp

Let's try to transfer Hilbert spaces, I guess

Emilio Pisanty

Yo folks

↑ This could have been handled better

@MAFIA36790 @rob

It was a pretty bad post but from a well-meaning user

physics.stackexchange.com/a/284132/8563

John Rennie

Does this make any sense?

0

Q: Can anyone please explain how the energy of vacuum is related to Zeta function?

Please, explain in the most simple words, how the energy of vacuum is related to Zeta function, preferably if the formula would show relationship to Hurwitz Zeta function, and well as show where the series undergoing regularization arose and whether it is related to infinities, infinite energy le...

Isn't the zeta function one of the methods used in renormalisation? Why would it be related to the vacuum energy?

Danu

11:00

@EmilioPisanty I think rob's part in it is fine. But @MAFIA36790 that was a pretty unnecessary reply :P

@JohnRennie Regularization.

So maybe you know of the Casimir effect---you can derive it using zeta function regularization.

(explained in Mukhanov's book)

Slereah

you can use it with any regularization!

It's the magic of regularization

Danu

There is definitely no deeper claim than "this is just one way to derive some effect" in the sense of "this is some way to make sense out of infinite sums".

Emilio Pisanty

@Danu It could've used a bit more compassion, is all I'm saying

I'm pretty sure @DanielSank would agree

@JohnRennie It comes up in calculations of the Casimir effect

Danu

@EmilioPisanty rob's contribution has nothing to do with compassion. It's an automated result of a review...

I agree that MAFIA's response is unconstructive.

Emilio Pisanty

adding up the vacuum energy inside of two parallel metal plates

@Danu That's sort of beside the point. Automated or not, anything that produces comments needs to be seen from the perspective of the person receiving the comments.

if the automated review comment is too harsh, it's a good idea to pop in with a personal comment explaining what's going on

Danu

11:05

@EmilioPisanty If you're suggesting that someone completing a review should go back to each post and remove the automated comment, then I think that that's unreasonable.

Emilio Pisanty

I'm not saying anything is terrible

It's just that it could have been handled better

Danu

If you're saying there should not be an automated comment---maybe. Take it up on meta.SE.

Emilio Pisanty

@Danu for every comment, sure

just taking some thought about whether it's appropriate or not.

Danu

@EmilioPisanty I don't think the automated comment is very harsh, certainly not too harsh, in this case. It's pretty brief and to-the-point, but I don't think that should be confused for hostile.

Slereah

Hm, what's the proof that in vector spaces, $0v = 0$

It doesn't seem to be an axiom, nor is $-1 v = -v$ one apparently

user116211

11:12

And now I get Kelley! Yess!!

user116211

@Danu ._.

Slereah

Oh apparently the proof is (0+0)v - 0v

user116211

Are you doing linear algebra?

user116211

It seems so.

Slereah

Which is $0v - 0v = 0v + 0v - 0v \to 0 = 0v $

fortunately transfering vector spaces is rather simple

Banach space will be a bit more complex

user116211

11:15

@Slereah Is vector space a field?

Slereah

No

user116211

I need to investigate.

user116211

@Slereah okay.

user116211

A field is a non-trivial division ring with additional structure. So, ring product with zero trivially applies to it too.

Danu

@MAFIA36790 Do you disagree?

user116211

11:18

@Danu Hmm. Well maybe that was unnecessary; but I am kinda vexed seeing such lol attitude.

Danu

I agree with that too.

If you think it's inappropriate you can also flag it.

user116211

Anyways, back in Kelley...

user116211

@Danu sure, noted.

Slereah

A hyperreal banana space

You know I'm glad that there are only so many useful non-archimedean fields

Otherwise it would have been quite an escalation

Hyperreal, surreal, megareal, gigareal, ultrareal

user228700

12:05

@MAFIA36790: Hi again :-) When u were studying the syllabus of 11th, did u understand a concept called the motion of a body on banked and unbanked curves?

user228700

If u did, from where did u learn it properly?

user116211

@Kaumudi learnt it from French.

user116211

Verma was good too.

user228700

@MAFIA36790 French? O_o

user116211

12:21

Apr 6 at 13:36, by ACuriousMind

@yuggib A. P. French.

user228700

He's written a book with these concepts in them and u read that, I assume..?

Slereah

Ah yes, the inventor of the french language

0celo7

@JohnRennie I think I have a pretty good post on the issue

back from when I knew/cared about physics.

user228700

@MAFIA36790 U mean H.C.V?

user116211

yes.

user228700

12:27

@MAFIA36790 Ohk. It isn't that good :/

user228700

Thanks :-)

user116211

@Kaumudi The first volume is quite good; but I didn't like the 2nd volume. I'm not talking about the exercise though.

user228700

@MAFIA36790 Hm, OK. I haven't read all of it so I'm unfit to comment but the section about banked & unbanked circular motion is not very good.

user228700

Is this covered by RHW?

user116211

Googling RHW

user228700

12:30

@MAFIA36790: Resnick, Halliday & Walker.

0celo7

I'm fascinated by this culture

Slereah

Important choice

user116211

@Kaumudi The electromagnetism part in his book is wholly based on Maxwell's treatise which lacked Vectorial treatment; I vehemently hate this approach. That's why I didn't like the second volume. Modern Physics content is okay, albeit.

user116211

@Kaumudi There is an example concerning the concept; but that's it.

user228700

@MAFIA36790 Oh, shucks :/ OK, thanks.

0celo7

12:38

What did JR have to say about your spring problem @Kaumudi

Slereah

How do I note complex conjugates

Should I use a bar or a star

0celo7

Star

user116211

@Slereah star star

user228700

@0celo7 DS helped me out and then we talked about another problem for like 3 hours.

0celo7

What did he have to say about it?

user116211

12:43

Balarka changed back the profile pic ;/

user228700

8 hours ago, by DanielSank

You pick up the spring by its end and start whirling it around you like a lasso.

user228700

Follow that thread.

Slereah

12:58

@yuggib does this look okay for a transfer

0celo7

What is that supposed to be.

Also typo in (18)

Slereah

what typo

0celo7

You need a square root.

Slereah

Oh right

yuggib

seems enough

user116211

13:06

Welcome @ValterMoretti.

Valter Moretti

Hi

user116211

Nice to see you here in the h bar ;)

yuggib

however you should star the V and the norm/bracket as well to remind you that those are the enlarged versions

Valter Moretti

Just few seconds for an issue regarding a short discussion with @Ocelo7 and @Slereah

we have had yesterday

Slereah

With the complex conjugates and the $C^*$ algebra that's gonna be a lot of stars :p

Valter Moretti

13:07

A relevant reference is arxiv.org/pdf/0806.1036v1.pdf

Slereah

Thanks

user116211

WTH! Greek yogurt isn't a yogurt made in Greece

Slereah

Welp I guess now I just have to prove all things ever in a Hilbert space

Except larget

Shouldnt be too hard for the most part I guess

Most of them I can just transfer

Valter Moretti

on p. 34 you see the general operators used in this work. Section 3.2 is devoted to study the global and causal properties of the solutions

@Ocelo7 and @Slereah I am referring to the fact that solutions of hyperbolic equations in curved spacetime, like KG with the wrong sign of m^2 are however causal

our discussion of yesterday night

This reference is much more advanced of the works of Frieadlander and the content of Wald's textbook, Ok, I have some work to do now. see you.

Slereah

bye

which one is the big book on hilbert spaces for quantum theory again

I forget

Reed or something

yuggib

13:14

reed simon

actually, there are four books

Slereah

four even, no?

Hm, what should the Hilbert space of states be

yuggib

that is a badly formulated question in qft

Slereah

$\| x \| = 1$ or $\|x\| \approx 1$

not the question you think :p

yuggib

well, what does it mean for the total probability to be in the shadow of one

to me, it is unclear

Slereah

well both will give the same probability

just not sure if they'll have the same structure if I do that

Oh yeah I also need to restrain the distributions I guess

Should drop off to $\approx 0$ outside of finite values

Shing

13:26

I have just found A.Zee wrote a group theory book!!

not sure if I should get one. just won some $$ from a reading group competition :D

yuggib

there are better ways to waste money...

Shing

why? lol you don't like him?

yuggib

drugs, sex and gambling are pretty popular nowadays

user116211

How do Lubos know such facts about Greek Yogurt? hmm.

Shing

well I surely will go with sex... so I am practicing picking up girls.

yuggib

13:28

@Shing I think that physicists should stick to physics

user116211

@yuggib There is no Bourbaki option ;(

Shing

@yuggib but that's a book about group theory in a nullshell for physicists!

yuggib

@MAFIA36790 I said wasting money, not putting them to a good use ;-D

Slereah

@yuggib Said the mathematician working in QFT

user116211

@yuggib Aha!!

Shing

13:30

@yuggib ohoh I see, ha! sweet

user116211

@Shing You want to learn Group Theory?

Shing

@MAFIA36790 yeah

yuggib

@Slereah well, I don't write a book on experiments for mathematicians

Shing

always think my math is not good enough

rob

@EmilioPisanty You may be right right. The answer was a correct statement, but not really relevant to the question at hand. I picked the automated delete comment rather than writing my own because it has helpful links in it. What would you have done differently?

yuggib

13:31

I do mathematics that can be (sometimes vaguely) interpreted as a physical theory of quantum fields

user116211

@Shing Then take Modern Algebra by Werner Seth.

Shing

@MAFIA36790 a good book for self-teaching?

Bass

@yuggib Do you have any papers that are publicly accessible?

yuggib

@Bass all my papers are publicly accessible

but they are rather technical

Slereah

How much diagram chasing?

yuggib

13:40

@Slereah "diagram chasing"??

Slereah

that kind

they tend to invade technical paper

aka abstract nonsense

yuggib

@Slereah no I don't use diagrams

I have only one (commutative) diagram in all my papers

user116211

@Shing yes.

yuggib

and it is almost only for the sake of it ;-P

Slereah

But AQFT loves diagram chasing!

yuggib

13:43

@Slereah I am not an AQFTist

Slereah

what is your religion then

yuggib

I am QFT-agnostic

Slereah

Which QFT formalism is that

Do you at least believe in the Hilbert space

Shing

what exactly is the difference between QFT and Relativistic Quantum Mechanics? Are they just the same thing?

Slereah

Relativistic quantum mechanics is an approximation of QFT in the low energy limit

It's basically the one particle approximation

It's QFT without variable number of particles

Shing

13:48

@Slereah I see, thanks!

Slereah

The difference is that in QFT, $\phi$ is an operator valued distribution, but it's a wavefunction in RQM

Shing

so their physics interpretations are not quite same either

Bass

@yuggib Where can I find them? (I don't expect to understand them..)

0celo7

14:05

@yuggib halp

Frechet derivatives are too hard

ok there is a proof in the book of the thing I don't understand

it's just a statement about limits along lines, ok

@DanielSank Arnold.

Seriously? $\hbar\ne 1$?

user218912

14:51

@Slereah I need help please.

Slereah

Perhaps Jesus can help you

yuggib

@Bass you can find them here:

user218912

@Slereah this I don't get what it's asking.

user218912

can you help

Slereah

Isn't Falconi a Batman crime lord

yuggib

14:54

no

it's with an "e" in the end

Falcone

Slereah

that's exactly what a crime lord would claim

yuggib

accidentally (?) the same name of an attorney general that was killed with a car bomb by mafia in the 90s

@0celo7 they're not so hard

(definitions are not hard by nature)

Slereah

@yuggib Don't lie

Here's a definition

⊢ DProd = (𝑔 ∈ Grp, 𝑠 ∈ {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp ‘𝑔) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ ‘𝑥) ⊆ ((Cntz ‘𝑔) ‘(ℎ ‘𝑦)) ∧ ((ℎ ‘𝑥) ∩ ((mrCls ‘(SubGrp ‘𝑔)) ‘∪(ℎ “ (dom ℎ ∖ {𝑥})))) = {(0g ‘𝑔)}))} ↦ ran ( 𝑓 ∈ {ℎ ∈ X𝑥 ∈ dom 𝑠(𝑠 ‘𝑥) ∣ (◡ℎ “ (V ∖ {(0g ‘𝑔)})) ∈ Fin} ↦ (𝑔 Σg 𝑓)))

yuggib

I'm not saying things cannot be complicated

I'm saying that a good definition is:

1) simple

2) the hypothesis of an interesting theorem

0celo7

15:10

@yuggib I was just confused by a limit

$f:E\to F$ (Banach spaces) is cont. at $x\in E$ iff $f$ is cont. along each line through $x$

maybe not iff

but definitely $\Rightarrow$ is true.

Danu

@Slereah That's hardly diagram chasing :P

Slereah

Show us a true diagram

Danu

I don't really know

yuggib

@0celo7 "each line" is a bit strange in a general (infinite dimensional) Banach space, but yes the idea is more or less that

0celo7

@yuggib well we defined directional derivatives on banach spaces

by going along lines

yuggib

15:14

ok, but you may have a huge lot of them

0celo7

yes

if $\lim_{x\to y}f(x)=f(y)$, then $\lim_{t\to 0}f(x+tv)=f(y)$ for each $v\in E$

something like that.

yuggib

that is of course true

0celo7

"of course"

yuggib

for linear maps, also the converse should be true

@0celo7 by definition of limit in a metric/normed space ;-P

0celo7

that's not what we mean by limit...

yuggib

15:18

?

does not continuity mean that $\lVert f(x) -f(y)\rVert_F\to 0$ whenever $\lVert x -y\rVert_E\to 0$?

0celo7

What does that even mean?

What are those arrows

yuggib

"tends to"

0celo7

I know that

But what does "tends to" mean if not "limit"

yuggib

so?

0celo7

The proof of what I said is not hard, but not immediate. I'll write it formally when I get home

yuggib

15:23

of course it is immediate, don't be silly

0celo7

I will write the proof later, then we will see.

yuggib

define $\lim_{x\to y}f(x)=f(y)$ and you're done, noting that $\forall v\in E$, $\lim_{t\to 0}\lVert x+tv - x\rVert=0$

0celo7

Yeah that's the proof.

yuggib

I can't see almost nothing more immediate than that

0celo7

I disagree

You are a PhD analyst, of course it's trivial to you.

yuggib

15:31

maybe

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