The h Bar

Ryan Unger

01:51

@Slereah arxiv.org/abs/2101.08324

user 85795

02:36

skullpatrol says, hi pal @RyanUnger

123

03:19

Hello World.

user 85795

yo

Sir Cumference

04:01

@DanielSank Huh? They publish astro papers

though tbh most of the astro papers i've read from nature have been very old

RewCie

din din

boooooow

user 85795

🙏🏼🙏🏼🙏🏼

RewCie

Namaste

Nihar Karve

05:47

@bolbteppa do you know any good resources on 2D CFTs?

Manas Dogra

So what I am confused about here is that we need to add the proper spin projections and get the unknown right?
Now Addition of spin implies these spin projections can take a range of values...Are we supposed to try all...or is there some other trick?
Because if we tried all there would be so many possibilities and so many answers( some like 7/2 which is not possible...but some even like 1/2 which is not mentioned in the correct answer)..

For example $2-1/2=x+1$ gives x=1/2 so it can be a spin 1/2 particle right?
But it cannot be according to the answer!!!
So what's wrong with the above equation?

I think I am missing some concept.(I read addition of spin angular momentum from Zettili's book of QM and Kamal and Griffith's book on particle physics...)

John Rennie

06:06

← 3 messages moved from Problem Solving Strategies

Can anyone help with Manas' question. It was asked in the JEE room but none of us there can figure out the answer.

Slereah

07:18

@RyanUnger Thx

DanielSank

07:34

@SirCumference I meant that I heard that astrophysicists largely avoid Nature, not that Nature disallows astrophysics.

Slereah

08:11

@NiharKarve why not Di Francisco

Nihar Karve

@Slereah last I checked there didn't seem to be an online version

Slereah

There are always online versions

if you know where to look

wink wink

nudge nudge

say no more

Nihar Karve

ok I'll see

but I'll probably have to settle for some lecture notes

thanks

John Rennie

@NiharKarve This book ?

Nihar Karve

08:49

@JohnRennie that's the one, yes

John Rennie

@NiharKarve cough ... Z-Library ... cough

Nihar Karve

@JohnRennie thanks, never heard of that one

Jasmine

@ACuriousMind I need your guidance if you could please

I created a room by mistake and want it to be removed please :(

Nihar Karve

these online PDF troves are a great use case for 10minutemail

ACuriousMind

@Jasmine just don't use it and it'll get deleted for inactivity automatically in a while

Jack Rod

09:05

@ACuriousMind Is there exist a procedure too delete old rooms?

ACuriousMind

Rooms that have more than 15 messages by at least 2 different users will be frozen if they are inactive, but not deleted. SE doesn't seem worried about the space the rooms take up on their servers, so there's no real deletion - if needed, all rooms can be unfrozen or undeleted after an indefinite amount of time.

Slereah

09:31

It's not like SE is overrun with chat

There aren't a lot of active chats here

ACuriousMind

09:42

@ManasDogra I don't really see a reason why either 1/2 or 7/2 should be impossible

Shing

09:58

I was calculating the multiplicity of N oscillators, get the same result as Kettle, $g(N,n)=\frac{(N+n-1)!}{n!(N-1)!}$; however, when I plug in N=2 (2 oscillators), n=2 (total energy is 2hw), I get a $g=3$!!

user586228

@JohnRennie Since yu said that you will look into the standard state problem when you have time,when is the earliest that I ping you?

Shing

shouldn't $g=1$?

John Rennie

@user586228 I probably won't have time today.

user586228

No probelm tomorrow?

problem*

fqq

10:52

@NiharKarve DI Francesco is considered the standard reference so it's probably worth getting a copy. I think most string theory books/notes have at least a chapter on CFT, eg Weigand's and Tong's notes

these notes are also often cited, but I confess I never read them arxiv.org/abs/hep-th/9108028

for a slightly different perspective (mostly different applications) there's the statphys literature, Cardy's book has some of the basics, but I think he also has some notes specifically on CFT

yes arxiv.org/abs/0807.3472

Mussardo's book also has CFT chapters. Then there are more maths heavy/rigorous expositions.

bolbteppa

Yeah, the yellow book is worth getting if you are going to devote any serious time to cft, but there's four sets of notes cited here that are often recommended, and try some of BPZ to get a sense of what cft is really about which is comparatively scattered through Di F, a random thesis on ads-cft will also do the bare minimum cft without going into the weeds on minimal models

Nihar Karve

@fqq haha, my main motivation was to find something more comprehensive than Ch4 Tong String Theory. But yeah, that's where I'm coming from

Thanks so much for the refs guys, really appreciate it

bolbteppa

The perimeter videos follow Di F pretty closely and it's useful to get some extra input on this stuff

More Anonymous

@NiharKarveup for a game of chess again sometime?

Slereah

12:10

@bolbteppa I'm afraid I can never look at your name and think it's "Bob"

Bob Teppa

bolbteppa

To make matters worse that b actually stands for a V (and the second one is silent, and the p's are actually r's)

Slereah

Bob will do

John Rennie

@bolbteppa Cyrillic?

bolbteppa

Yeah

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...

Nihar Karve

12:49

@MoreAnonymous sure :)

More Anonymous

@NiharKarve One moment busy reading an interesting physics answer

for a question I asked

Nihar Karve

@bolbteppa how did л -> L

bolbteppa

13:18

El (Cyrillic)

El (Л л; italics: Л л) is a letter of the Cyrillic script. El commonly represents the alveolar lateral approximant /l/. In Slavic languages it may be either palatalized or slightly velarized; see below. == Allography == In some typefaces the Cyrillic letter El has a grapheme which may be confused with the Cyrillic letter Pe (Пп). Note that Pe has a straight left leg, without the hook. An alternative form of El (Ʌ ʌ) is more common in Russian, Ukrainian, Belarusian, Bulgarian, Macedonian, and Serbian. == History == The Cyrillic letter El was derived from the Greek letter lambda (Λ λ). I...

eL google translate agrees with me :p

ACuriousMind

13:31

@bolbteppa but if you go by letter-relations, then you have to transcribe the cyrillic r also with r, i.e. bolbterra

although the modern P looks like the ancient $\rho$, it is not the case that P developed from $\rho$ - $\rho$ developed into R, the similarity between Latin P and Greek $\rho$ is accidental

bolbteppa

Not according to google translate :p

ACuriousMind

13:47

I'm sorry to hear Google translate doesn't understand the history of letters :P

but I'm not sure how you're using it, when I enter Вольтерра into Translate, it outputs Volterra

in contrast, bolbteppa becomes something different:

bolbteppa

What does (English) Volterra output as in Russian

Nihar Karve

keep iterating

see if you find a fixed point

@bolbteppa Volterra -> Вольтерра, like in your profile

bolbteppa

There we go

user 85795

14:03

Aha! We finally know who you are!

~~are~~ were :P

Slereah

14:18

I have to use the mathjax method to do diagrams on my site

Using the way that nlab does it

It works but boy it's not pretty

\begin{equation}
\array{
&T_xX&\\
\overset{f_1}{\swarrow}&\downarrow_{f_1 \oplus f_2}&\overset{f_2}{\searrow}\\
T_{q_1} M_1 & T_{q_1} M_1 \oplus T_{q_2} M_2 & T_{q_2} M_2 \\
\underset{g_1}{\searrow} & \downarrow_{ g_1 \oplus g_2} & \underset{g_2}{\swarrow} \\
&T_yY&}
\end{equation}

Maybe I shall put them as png instead

Slereah

14:47

Also I'm trying to define the exterior tensor product properly but that involves too many projectors

I'm running out of letters

they are getting unwieldy

Nihar Karve

Start a new trend

Use Klingon letters

ACuriousMind

@Slereah what's the "exterior tensor product"?

Slereah

@ACuriousMind the structure defined for bitensors

Ryan Unger

@ACuriousMind you know K theory don't you?

ACuriousMind

@Slereah I'm not sure what that is either :P

@RyanUnger not really

Ryan Unger

14:58

Given X1,X2∈H the external tensor product over these is the functor

⊠:Mod(X1)×Mod(X2)⟶Mod(X1×X2)
given on A1∈Mod(X1) with A2∈Mod(X2) by

A1⊠A2≔(p∗1A1)⊗X1×X2(p∗2A2)∈Mod(X1×X2),
where p1,p2 denote the projection maps out of the Cartesian product X1×X2∈H.

Slereah

@ACuriousMind A tensor but bi

Roughly speaking a tensor defined at two points of the same manifold

Your bundle is something like $$\pi : V_1 \boxtimes V_2 \to M \times M$$

Ryan Unger

I feel like this is one of those things where if you just write it in coordinates there's no issue

Slereah

@RyanUnger pretty much

Ryan Unger

it's an object $v_i(x)w_\mu(y)$ etc

Slereah

That's what most papers do certainly

Ryan Unger

15:00

doesn't need to be $M\times M$ right

just some product

Slereah

Well it could be any two manifolds, yeah

Ryan Unger

yeah

Slereah

Although in the case I need it for it's the same one

the most famous bitensor being the worldfunction

Also the parallel propagator, I guess

Manas Dogra

15:23

@ACuriousMind Yes you are right...By the way this question is from an entrance exam to TIFR,India.

Jake Rose

15:46

For an equation like $(\mathbf{\nabla u })\cdot \mathbf{S}$

eek

what's dot product in mathjax

Is there an ambiguity in which index is dotted

oh actually never mind

Shing

16:33

3I was calculating the multiplicity of N oscillators, get the same result as the Thermal Physics wirtting by K&K, $g(N,n)=\frac{(N+n-1)!}{n!(N-1)!}$; however, when I plug in N=2 (2 oscillators), n=2 (total energy is 2hw), I get a $g=3$!! I think $g$ should be $1$?
There should be only one state for min energy for a system of two oscillators. No idea what I have gone wrong.

123

Hi All..

1drv.ms/f/s!AozWlUoG8z4tiD3A5_aG6NLda7Vt

Pls see the link the what is sentence of underline.

In mechanics what is the meaning of "To avoid reaction at point we take moments about it."

ACuriousMind

@Shing why are you using $n=2$ and think that's the minimal energy?

If I understand correctly, in your formula $n$ represents the sum of the excitation numbers of all the harmonic oscillators involved. The lowest possible excitation number of a QHO is 0, so $n=0$ is the non-degenerate ground state of a system of oscillators, too

for $n=2$, you have three possible distributions of the excitations among 2 oscillators, namely $(2,0), (1,1), (0,2)$, so $g=3$ is the expected result

s.harp

17:03

Hey I was a bit curious - I have some questions that are too ill formed for main sites.
In particular I am kind of interested what "algebraic quantum field theory" refers to today and what its "status" is. I've been told that the program has kind of "failed" (and I think the person talking to me was referring to the stuff in Haag's book "Local quantum fields") is that accurate?

I also found a book by Baez Segal and another guy that is literally called "Introduction Algebraic and Constructive Quantum Field Theory", but its content / perspective is so different from Haag's book.

Slereah

17:34

Depends what you mean by "failed"

s.harp

I don't know - maybe something like "people no longer think its a good approach"?

Slereah

It's used in some fields, most QFT people do not use it because the "classic" methods work well enough, but also it's not enough to solve standard model calculations by itself

Shing

@ACuriousMind Yes, you are correct, thank you so much! I was having each's oscillator's energy in mind, and I just "shifted" the 1/2 to 1. this is something very wrong, I think. the total energy of two oscillators (if I use n+1/2) in the states (0,0), (1,1) are $1\hbar\omega$ and $3\hbar\omega$ respectively. However, if I use n+1, then (0,0), (1,1) are $2\hbar\omega$ and $4\hbar\omega$.

Slereah

But modern methods for it will also use methods from AQFT

It's not lost to the wind

s.harp

do you understand AQFT to be the net of local operator algebras approach?

Shing

17:36

okay, now I remember why the "1/2", I think it is for symmetry of the ladder?

Slereah

That's usually the one, yes

s.harp

If I pressed you what language would you use to describe the mathematical foundation of QFT? (or what such a foundation would/should look like)

bolbteppa

@s.harp is there a single result related to that field

Slereah

@s.harp Bit of a complicated question!

There's quite a few

and how they relate to each other, and also to QFT as it is practiced, isn't easy

s.harp

@bolbteppa I know next to nothing about the theory, but here is one result: A state gives a time evolution which makes the state "thermal", ie an explicit version of the connecction between statistical physics and time usually (which is usually vaguely suggested by everybody)

Slereah

17:42

You can start from scratch and build the QFT yourself or you can take a classical theory and quantize it, you can define it from an operator theory, an integrator, functors, graphs, algebras, and also amidst all that you have to do some kind of regularization procedure because I think we gave up doing QFT without renormalization these days

bolbteppa

Not sure what that means, I would bet a lot that algebraic qft doesn't show anything like that

Slereah

also there's the lattice version

I don't know if there's a theory that's considered more modern QFT than the others

The true QFT is the friends we made along the way

4

s.harp

I need to eat, but thanks for your comments I will come back later

But to say something about how I would answer that question: I always used to think that the mathematical foundation of QFT should be represenation theory - a QFT should be a projective representaiton of an appropriate symmetry group (ie wightman axioms)

(but I know very little about this)

Slereah

I mean that's certainly part of it

but not all of it I'm afraid

According to nlab, this is how you define QFT :

$$\mathrm{PolyObs}(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] \otimes \mathrm{PolyObs}(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] \overset{\star_{\tfrac{i}{2}\Delta}}{\longrightarrow} \mathrm{PolyObs}(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]$$

Hope that helps

bolbteppa

It's simply not possible to understand any qft without the BV-BRST polyobservables :p

Slereah

17:53

Well, you don't need it for a free theory without gauge, I suppose!

bolbteppa

Luckily Bhabha had eaten his broccoli and learned higher topos theory before getting to scattering amplitudes!

@s.harp Folland and Ticciati are two books which are most likely to strike a balance between rigor and a normal qft book, but they are probably way too much if one hasn't learned it the normal way first

Bohemian relativist

is perturbation theory the protagonist of quantum field theory?

Slereah

@Bohemianrelativist I haven't studied QFT literary analysis

ACuriousMind

@s.harp ...would you say that an ordinary QM system with its Hamiltonian is "a projective representation of an appropriate symmetry group"?

Certainly the space of states is always a representation of the algebras and groups of operators acting on it, but the core of the physical theory, the time evolution generated by the Hamiltonian, doesn't really have all that much to do with representation theory

Slereah

I mean, I think when people say "QFT", they mean "the standard model" specifically

Not a random QFT

Bohemian relativist

18:07

@Slereah It's like Coleman's preface says perturbation theory is the main contents of QFT.

Slereah

Perturbation theory is pretty important, and on a practical level, that's mostly what people do

ACuriousMind

There is no reason the mathematical tools needed to explain a physical theory should correspond neatly to one of the subfields of mathematics, and indeed they often don't

Slereah

I don't know if that makes it the protagonist

Or the love interest

ACuriousMind

@Bohemianrelativist perturbation theory is what many people do because its computationally tractable and gives excellent results for a large variety of situations

but e.g. strongly-coupled QCD isn't amenable to perturbation theory and certainly also an important part of applied QFT

Slereah

also you could technically solve a QFT without ever using perturbation

ACuriousMind

18:09

and I'm not sure how perturbative various condensed matter applications are

Slereah

But that would be challenging

ACuriousMind

the "QFT is perturbation theory" viewpoint is usually something from people who mostly look at high-energy colliders :P

Slereah

Those damn S-matrix fetishists

ACuriousMind

@Slereah I think the problem is that we actually don't have the techniques to do that :P

Slereah

@ACuriousMind I didn't say it would be easy!

Ryan Unger

18:11

just do the sum

Slereah

The perturbative sum is divergent

So it's pretty easy

$$D(x, y) = \infty$$

Ryan Unger

incredible

Slereah

18:23

There is some notion that the perturbative expansion converges to the actual solution up to some specific order IIRC

And diverges after that

I'm not sure if there's an actual theorem for that or if it's just vague

bolbteppa

18:44

Not sure that a non-perturbative perspective makes sense tbh

Slereah

It may or may not, depending on the theory

Nihar Karve

@Slereah en.m.wikipedia.org/wiki/Asymptotic_expansion

Slereah

@NiharKarve I know

I don't know if there's a theorem as it applies to QFT, though

IIRC it's tied to the coupling constant

Slereah

19:02

arxiv.org/pdf/1201.2714.pdf

of interest

s.harp

@ACuriousMind I mean thats because global time is special for QM (vs QFT). Unlike (classical) field theories the way you talk about QM has kinematics and dynamics - here the representation of the observables (not the symmetry group, right?) is the kinematics.

I guess my second question is trying to get an answer to "what do you think a QFT is", regardless whether or not its possible to DO something with this "what"

Slereah

You can have a QFT without any symmetries

bolbteppa

QFT is quantum mechanics with the Galilean group replaced by the Poincare group

s.harp

@Slereah You mean like you can have a completely inhomogeneous solution to Einstein equations for GR? Where the isometry group is trivial? Or do you mean something else?

Slereah

yes

Although the Poincaré group still plays a role, 'course

since it's always a local symmetry

bolbteppa

19:06

I should say relativistic qft is...

Slereah

I mean rly I think QFT is good enough to define what QFT is

it's a quantum theory of fields

Defining what that means is a bit tricky but that about encapsulate it

s.harp

Describing an "inhomogenous" configuration really is something that a hypothetical grounding should be able to do - so I agree that moving away from the "representation of the Poincare group" viewpoint is good
unfortunately everything is so muddled in my head

ACuriousMind

@s.harp 1. QFT is not inherently relativistic, you can do QFT in Euclidean/Galilean space. 2. How do you think the dynamics of a QFT - its Lagrangian - are encoded in a representation?

Slereah

there's a lot of things you could include or exclude under "QFT"

bolbteppa

It's pretty much a shock to actually see how to go directly from the usual QM to non-relativistic qft

ACuriousMind

19:10

@s.harp "A QFT" is what we do to get our usual predictions, really

i.e. some very non-rigorous muddling about with Feynman diagrams or path integrals that in the end results in some scattering amplitudes or expectation values

s.harp

The problem with the viewpoint "QFT is that what works to describe it" is that its probably philosophically sound (and probably philosophically superior to Platonism which is for mental adolescents like me)
but I didnt get interested in physics for reasons like that - it doesn't satisfy the craving for order or truth that I had as a "kid"

ACuriousMind

If you phrase everything in terms of path integrals there aren't any "states" or "representations", there's just this strange procedure where you put in some polynomial in the fields and get out expectation values

Slereah

if you want order and truth I'm not sure you're in the right field :p

s.harp

well I'm a mathematician

Slereah

it's a bit of a mess

If you want some fancy QFT you can try Reed & Simon

for free fields

s.harp

19:14

wha? Simon & Reed didn't write a book on QFT did they? just operator theory I thought

Slereah

If you read it you'd notice that there's a few chapters dedicated to QFT!

Book 2 has the theory for Klein Gordon QFT

ACuriousMind

@s.harp physics isn't about platonic truth, it's about providing working approximate models that can usefully describe reality

Slereah

except GR, 'course

bolbteppa

You wont get a platonic perspective from any math book on qft, you will from L&L :p

Slereah

GR is perfect and beautiful

bolbteppa

19:16

Course of Theoretical Physics

The Course of Theoretical Physics is a ten-volume series of books covering theoretical physics that was initiated by Lev Landau and written in collaboration with his student Evgeny Lifshitz starting in the late 1930s. It is said that Landau composed much of the series in his head while in an NKVD prison in 1938-1939. However, almost all of the actual writing of the early volumes was done by Lifshitz, giving rise to the witticism, "not a word of Landau and not a thought of Lifshitz". The first eight volumes were finished in the 1950s, written in Russian and translated into English in the late 1950s...

Slereah

L&L is a bit old timey

They don't even talk about Hilbert spaces

bolbteppa

Hilbert spaces are a lie

Slereah

tell that to Hilbert

bolbteppa

First off they need to be rigged, and few people care to be careful about that

Slereah

"It is thus clear that such a theory cannot lead to healthy correlation functions,in other words for any negativeλthe power series (12) will diverge. From this w ecan conclude the radius of convergence being R= 0!"

noooo

bolbteppa

19:18

It's old timey in the sense that Yang-Mills and SSB were being worked out when those books came out, but up to QED it's all solid

Slereah

The Game Was Rigged From The Start

►

s.harp

there are dynamical formulations of (classical) electromagnetism but there are also non-dynamical ones that involve more machinery. Its much harder (and stupider) to use sections of fibre bundles to design a telegraph wire.
But the perspective that (classical) field theories are static and not dynamic is more common in theoretical and mathemtiacl circles - right?
so my distinction QM - QFT is not non-relativistic - relativistc | but finite dimensional phase space - infinite dimensional phase space

Slereah

You still need a way to find the dynamics even in very fancy formulations

You need to define a Hamiltonian vector field or what have you on that bundle

ACuriousMind

@s.harp yes, finite vs. infinite d.o.f. is the correct distinction, but you said something about "global time", which is a relativistic/non-relativistic distinction usually

Slereah

Simply because given the same representations or whatever else, you can define multiple theories

ACuriousMind

19:23

there's nothing "static" about a field theory - the Euler-Lagrange equations are usually still thought about as answering the question "given an initial field configuration at an instant, what will it evolve to?"

Slereah

you need a form on your

JET BUNDLE

dun dun

s.harp

I need to think about the things you said before I reply further. But I also want to ask one more thing about the "practical" perspective ("QFT is anything that can usefully describe reality").
Because this perspective is fundamentally at odds with the very popular (among physicists) string theory! I don't want to be polemic, but that seems like a counter-example to any claim that people aren't interested in what QFT "is" and that the utilitarian perspective (if it works then thats QFT) is king

Slereah

Well not everything that works is QFT

Otherwise Newtonian mechanics would be QFT

s.harp

everything that works to describe a certain regime of observation

Slereah

QFT is a specific field in physics

It's a quantum theory of fields, that's about the best I can describe it without getting more specific!

You have a classical theory of fields, you quantize it

Or you have a quantum theory from the wavefunctions on the space of fields

Or you have a C^* algebra for field operators

bolbteppa

19:34

Quantum mechanics/qft says classical mechanics doesn't exist, but it still needs to reduce to this non-existent theory in a limit that can only be defined by the existence of the theory that doesn't exist

Slereah

I mean it does exist, but I'm still not 100% sure if there's a generic process for it?

The dequantization

We talked about it a few times here

How do you go back from the Hilbert space to the configuration space

s.harp

somebody told me to do a Rieffel deformation for that direction, but I'll be quiet for a while now

bolbteppa

The first thing QM says is that the variables upon which all of classical mechanics is founded on do not even exist. If they did we could theoretically repeat experiments exactly which is exactly what experiments do not show

Slereah

I mean yeah, but [something something limit process]

s.harp

(sorry I looked it up again, deformation is [classical] -> [quantum] not the otherway around)

Slereah

19:39

Although I'm not 100% sure this works for everything

bolbteppa

So instantly we have absolutely nothing, no theory. The only way to define QM from then on is by the fact that classical mechanics has to exist in some vague sense [yeah limiting, which also has to be defined]

Slereah

Like if there are correlations involved, is there a classical limit?

bolbteppa

I'm not sure about that stuff yet but if there's a serious problem with it it would mean Bohr etc were wrong about QM and that's very unlikely

Slereah

Classical mechanics cannot be contextual

ACuriousMind

@s.harp your perception of popularity of string theory among physicists is probably exaggerated - many physicists don't even know all that much about QFT!

Slereah

19:42

I know a physicist IRL

One time I showed him the stuff I was writing about and he burst into flames

ACuriousMind

that's a vampire, not a physicist

it's an easy mistake to make

Slereah

He is a rheologist so he is not used to math beyond "a function"

mostly he inflicts punishment on fluids

Slereah

20:23

that paper actually explains quite nicely why perturbation doesn't work and also works

Although as usual, the warning

QFT is divergent in several different ways

Taming all these beasts is the main issue

bolbteppa

21:04

Classical field theory is also divergent and it's directly due to special relativity and point particles

QFT just exacerbates those problems

Charlie

where does classical field theory become divergent?

bolbteppa

Renormalization

Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering values of these quantities to compensate for effects of their self-interactions. But even if no infinities arose in loop diagrams in quantum field theory, it could be shown that it would be necessary to renormalize the mass and fields appearing in the original Lagrangian.For example, an electron theory may begin by postulating an electron with an initial mass...

This, the self-reaction and I think some fluid example are basically what motivated Kramers etc... note the 'classical electron radius' violates special relativity so it's an incompleteness in classical physics

Charlie

ah i see

Charlie

22:53

regarding the induced metric on the worldsheet in string theory, along which map do we pull-back the background spacetime metric?

ACuriousMind

@Charlie along the coordinates of the string, usually denoted $X$

Charlie

hmm ok

ACuriousMind

you don't sound convinced :P

Charlie

23:10

I guess it's odd to me that we pull back a flat metric onto flat spacetime

ACuriousMind

I'm not sure what that means - you're pulling back a flat spacetime metric onto a worldsheet that a priori has no metric

Charlie

but is the worldsheet not just like, a submanifold of spacetime?

ACuriousMind

The $X$ makes it one

Charlie

If we want to measure distances/areas on the worldsheet, why not just do it in a chart? it seems like the string is being treated as something that isn't just a sheet sitting in spacetime

ACuriousMind

The worldsheet is a 2d manifold $\Sigma$ and the coordinates $X : \Sigma \to\mathbb{R}^{26}$ are an embedding that identifies it as a submanifold

@Charlie I think you're trying to imbue the mathematics with a bit too much ontology here

Maybe an analogy helps: When we consider a curve in a manifold, the curve is not just "a submanifold of dimension 1", it is an embedding of the abstract manifold "the unit interval" into the manifold, i.e. an embedding $\gamma : I \to M$. Likewise, a worldsheet is an embedding of a 2d manifold into another manifold

you wouldn't ask "why are we treating the curve as an interval that's not sitting in spacetime", would you?

Charlie

23:20

oh, is the metric begin pulled all the way back to the 2d manifold itself?

that actually would make more sense

my problem was that it seemed trivial to give an submanifold a metric obtained by pulling back the metric of the surrounding space to it

ACuriousMind

since the 2d manifold is isomorphic to its image under the embedding, I'm not sure that's a meaningful question :P

The reason why "pulling back" a metric is the natural thing to do is precisely because the pulled back metric gives the same results as "restricting" the metric to the submanifold would

Charlie

ohh

If it's the same as just restricting the metric that makes sense, both Tong's notes and my lecturer made a point of saying it's the pullback of the metric, which made me think there was something special about it

a bit like taking a circle, then considering a small part of the circle and measuring distances on it by just using the metric on the surrounding circle :p

that seemed like a strange point to labour

ACuriousMind

it's just standard differential geometry, but for some reason some physicists feel they suddenly need to pretend to do actual math when doing string theory (and then still not do it :P)

Charlie

oh god wait the "coordinates" $X$ we're talking about here go from the 2d manifold into spacetime, right? We're not talking about manifold coordinates in the sense of charts

or hm

ACuriousMind

23:36

@Charlie yes, they're maps $X: \Sigma \to M$ where $M$ is the spacetime.

Charlie

you've written the coordinates as $X:\Sigma\rightarrow \Bbb R^{26}$, is $\Bbb R^{26}$ the Euclidean space that is the target space of charts on spacetime or is it spacetime itself?

ah

now it seems like a slightly less trivial thing to do, because $\Sigma$ is now a separate manifold

ACuriousMind

but when the physicists write $X^\mu(\sigma, \tau)$, that of course is $X$ "expressed in charts" - the $(\sigma,\tau)$ live in some $\mathbb{R}^2$ chart of $\Sigma$ and the $X^\mu$ values live in some $\mathbb{R}^{26}$ chart of $M$

it's common not just in physics to not care much about the distinction between expressions "in local coordinates" and the abstract map, don't worry about that too much

Charlie

I seee

ok ty, that is clearer

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