The h Bar

ACuriousMind

5:00 PM

I thought we were talking about all holonomies determining the connection

you can't just go and compute the holonomy along every possible smooth path in spacetime

Balarka Sen

lol

Slereah

@ACuriousMind Well, one can try!

Maybe the solution generalizes readily for simple cases!

You can certainly do it for empty space

ACuriousMind

as I said, the holonomy is exponentiated magnetic flux (that's what the A-B effect is all about)

Balarka Sen

holonomy of the empty bundle over empty set manifold, equipped with an empty connection?

Semiclassical

i do find it ironic how much i defend pilot wave, despite the fact that in practice I've found it pretty useless for any kind of research :P

Slereah

5:02 PM

Oh I guess that for the Coulomb field, it's basically gonna be that some curves will correspond to "accelerated observers"

Semiclassical

and i am still prepared to defend it as a useful/interesting perspective in cases where you don't have to worry about relativity

Slereah

and those guys will feel a magnetic field

ACuriousMind

as soon as you have a non-zero field (either electric or magnetic), you have some frames in which the magnetic field is non-zero, and you get non-zero fluxes. If you think about how often we usually exploit symmetry properties to make fluxes computable, I don't think "I'm gonna compute all fluxes" is a good idea :P

Slereah

@ACuriousMind It may take a while!

Semiclassical

(and i'd love if, say, pilot-wave thinking proved useful for people working on quantum-classical transitions)

ACuriousMind

5:03 PM

@Slereah lemme know if you finish before heat death

Balarka Sen

ok but now i want to know what is a thing which has holonomy on worldsheets not just loops

Semiclassical

but as an interpretation of what QM should "mean" it just seems hamstrung

Slereah

@BalarkaSen that's the higher gauge theory stuff

ACuriousMind

@BalarkaSen given the typical terminology for that kind of generalization, it's probably a 2-connection and that's not a world-sheet, that's a 2-loop :P

Slereah

It is called a 2-connection, yes :p

Balarka Sen

5:04 PM

lol

Slereah

Although as far as notation goes, it seems to be alphabetical

1-connection is $A$, 2-connection is the $B$-field, 3-connection is the $C$-field

Semiclassical

also, i'm very glad that i can talk about pilot wave theory here without having some people insist on the inevitability of a fluid paradigm :P @bolbteppa

Balarka Sen

is there a natural 2-connection occurring anywhere

ACuriousMind

does string theory count as "natural" :P

Balarka Sen

it might

Slereah

5:07 PM

The B-field is the "natural" connection usually used

It's the $U(1)$ 2-connection

ACuriousMind

then the Kalb-Ramond field is the example you want

Slereah

I mean there aren't really a lot of objects described by strings in physics

bolbteppa

We all know it's turtle-beables instead of fluid-beables all the way down

Slereah

It's either string theory or cosmic strings

Although

What about a guitar string in a magnetic field

Balarka Sen

lol

Slereah

5:09 PM

Is it under the influence of a 2-connection

put it on a guitar so that it has Dirichlet boundary conditions

Semiclassical

seems like mr fluid paradigm stopped coming by here

Slereah

Maybe I should ask this on PSE

For the stupidest question I've ever asked

6

Q: Solving free particles with Fourier series

Here's a silly idea : take the action of a free particle, $$S = \int_{t_1}^{t_2} \dot{x}^2 dt$$ Our configuration space is the space of $C^1$ functions over $[t_1, t_2]$, which is spanned by the Fourier series $$x^a(t) = \sum c^a_n e^{i\frac{2\pi n t}{T}}$$ So that every path is represented ...

So far this may have been my silliest question

bolbteppa

That thesis even sets up EM and the Dirac equation so it supposedly generalizes to these cases at least

Balarka Sen

take a principal bundle on $\Omega M$. is choice of a connection here a 2-connection

Semiclassical

yeah, there's definitely been work in that direction

Samyak Marathe

5:11 PM

greetings

Semiclassical

but my lack of comfort with QFT has meant i've never really come to grips with that stuff

Samyak Marathe

I have a question to ask

if anyone could help

I m very much interested in doing physics experiments, about electricity, magnetism, optics, and literally everything. But I don't know where to find the right material for it. Is there a place where I could buy custom physics experiment kit

Slereah

Sure, there's plenty of companies selling such things

In France Jeulin is the big company selling lab equipment for schools and such

jeulin.fr/produits/physique-chimie.html

it is very French, though

Samyak Marathe

ahh, I m indian

if there is something for India

I have the list of every experiment that I want to perform

Slereah

also these days you can just look up amazon

they will sell most anything

Samyak Marathe

5:17 PM

but every experiment requires different tools and instruments. In amazon there are mixed

not the one I m looking for

I don't want to buy lose parts separately, I want something as a bulk

something like custom designed kit

where I can choose items for my experiment kit

and obv available in India

ACuriousMind

I don't think you'll find something like that except those science kits targeted at children

Slereah

Well you can find "kits"

But they tend to be expensive

I have an example somewhere

lemme see

Samyak Marathe

I need to find the tools exactly I want

but the amazon sells premade kits with items which are mostly of not my concern

Slereah

lasershop.de/en

^the laser store, for instance

it is very expensive, though

Samyak Marathe

well I do have a laser already

and infact line laser

but I think I can't find what I m looking for

why is there not such place where we can buy custom science tools to make a whole science kit

Slereah

5:25 PM

there are plenty of such stores, it's just that usually only labs care to know about them

Samyak Marathe

Could help me to find one

Slereah

For instance here's a store if you need to buy cocaine : cerilliant.com/shoponline/…

Balarka Sen

Lol

Samyak Marathe

I saw earlier one the veritesium videos sponsoring such sotre

hahahaha

oh wait I remember I guess, there's a store something like kiviko

Slereah

although there is some paperwork to do if you want to buy cocaine there

Samyak Marathe

5:28 PM

well I will look for it, thanks

lol

kiwiko is of no use of mine

I guess I have to make a list of every item I want to buy

and then order every of them

well nevermind

I will make my own kit when it arrives, lol

Balarka Sen

@ACuriousMind @Slereah what is a conformal field theory?

Slereah

@BalarkaSen A field theory that transforms appropriately under the conformal group

Balarka Sen

what is a field theory

ACuriousMind

a field theory that has a conformal algebra as its symmetry algebra

@Slereah not really correct in the case of 2d CFT

there's no good "conformal group" corresponding to the Virasoro algebra

Slereah

@BalarkaSen Functional on the jet bundle of a vector bundle

Balarka Sen

5:30 PM

perfect thanks

Slereah

Schreiber has a nice dictionary about it somewhere

Balarka Sen

im not reading anything by Urs man

your answer is pretty good

Slereah

Balarka Sen

makes sense actually

Slereah

Mathematical physicists love to make little dictionaries like that

I have a bunch

Balarka Sen

5:36 PM

what is $U_1(1)$

and why is a circle bundle a "monopole"... i guess makes sense if i look at the Hopf bundle long enough

Slereah

Magnetic monopoles

Magnetic fields usually stem from something with two poles

ACuriousMind

@BalarkaSen no, this is physics - if you want to represent a magnetic monopole in the usual EM gauge theory, it's represented by the principal bundle being non-trivial

I give a handwavy account of that here

Semiclassical

is U_1(1) just U(1)

Slereah

I'm not sure what the extra _1 is for

Sometimes physics will use a subscript letter on $U(1)$ to specify what kind of gauge field this is, but not $1$

Balarka Sen

i dont know what the principal bundle associated to a magnet is man

Slereah

5:42 PM

It's $U(1)$

bolbteppa

What's a bundle

Balarka Sen

i expect the magnetic lines of a monopole in 3-space to look like a hopf bundle

lol

Slereah

@BalarkaSen Yeah IIRC the principal bundle for a basic monopole is basically the Hopf bundle?

You take Minkowski space, you remove a line

Your spacetime is $\mathbb{R} \times \mathbb{R}_+ \times S^2$

Balarka Sen

wait are you telling me those $U(1)$ fibers are actually field lines

i was just joking

Slereah

You add a twisted $U(1)$ bundle to that

ACuriousMind

5:45 PM

they aren't

Slereah

Well no, the $U(1)$ is the gauge, not the field lines

the field lines are the flow lines of the connection I guess?

bolbteppa

What is the em potential 'connecting'

ACuriousMind

no, the field lines are the flow of the electric and magnetic field when you split the curvature into them

the connection does not manifest directly - that's why the A-B effect triggered a comparatively heated discussion about whether the potential/connection is "more real" than they used to think

Balarka Sen

physics is way more abstract than math

Slereah

Fair enough

Semiclassical

5:47 PM

this is where i'd want to say something clever about Berry connection and the Hopf fibration but

nope

i dont' know stuff

Balarka Sen

heres a simple question; can you construct magnetic fields where field lines link/knot in R^3

actual physics

Slereah

Not sure

Not big on knot theory

Can you have a rotational vector field that have such lines?

ACuriousMind

the field lines are the flow lines of a divergence-free vector field

Balarka Sen

hopf bundle @Slereah

Slereah

Well the Hopf bundle is the fiber bundle, though

Not the spacetime itself

Balarka Sen

5:49 PM

think of it as foliating R^3, then take unit tangent field along each fiber

Slereah

You need to project that field down

Semiclassical

Start with the magnetic field for a current-carrying knot

Slereah

If you project that knot onto the base space idk if it's still a knot

Semiclassical

Then make that the vector potential

Balarka Sen

there's no knot, Slereah, but the flowlines link

@Semiclassical seems promising

Slereah

5:51 PM

If by "link" you mean that they are circles then certainly

That's the magnetic field's big feature

Balarka Sen

i mean any pair of circles make a hopf link

they cannot be unlinked in space

Slereah

I don't know enough about Hopf fibrations I'm afraid

You can check Naber's book on EM though if you want

It's all about that stuff

Balarka Sen

ill take a look

Slereah

it's like two wholes books about EM as a bundle

Balarka Sen

seems useful. but massive

sounds worth reading

Semiclassical

5:55 PM

The only problem i see with my idea is that I don’t know what the current distribution will be like

Balarka Sen

the magnetic field seems to be perpendicular to the siefert surfaces in your case semiclassical

Semiclassical

Neat

Balarka Sen

ok maybe one should ask the even more basic: take a {z-axis} U {circle x^2 + y^2 = 1, z = 0}, make it carry some current uniformly. what's the magnetic field?

Slereah

The neat thing about magnetic monopoles is that they have integer charges corresponding to the... Chern number, I think?

Something like thaty

@BalarkaSen do you mean a cylinder?

Balarka Sen

fixed

i have a conjecture for what it should be but i dont know enough physics to compute

Semiclassical

5:59 PM

Mine is probably singular as stated

Gotta smear it out a bit

Balarka Sen

i'd be happy if you can give magnetic fields on R^3 with some points removed where flowlines link

or knot

Semiclassical

@BalarkaSen like, a vertical current?

Balarka Sen

yeah, vertical current union circular current, the pair linking once

Slereah

physics.princeton.edu/~mcdonald/examples/knot.pdf

possibly related

Balarka Sen

you just superimpose the two magnetic fields or something

Semiclassical

6:01 PM

Ah

Balarka Sen

coz EM is linear or something

Semiclassical

Too bad the fields for a loop sorta suck

Slereah

"Laser beams have been generated in which some field lines have the form of trefoil knot"

Semiclassical

I don’t see a way around that tho

Slereah

I guess you don't even need monopoles

Balarka Sen

6:02 PM

@Slereah cool!

this is what physicists should study

concrete problems

Slereah

Apparently knots in the EM fields were considered as early as 1867

Balarka Sen

back when physics was honest

Semiclassical

Source for terrible circular current loop : farside.ph.utexas.edu/teaching/jk1/lectures/node52.html

Slereah

@BalarkaSen that's what people say all the time, though

People bitched at Newton all the time because action at a distance was complete nonsense

how can things happen if they don't touch

Balarka Sen

how did they explain magnets

dont tell me magnets were before 1600s

Slereah

6:09 PM

Magnets are pretty old

people used magnets in ancient greece

There's even a big book on the topic in the renaissance

Balarka Sen

so people were just dumb to not believe evidence of the eye

Slereah

De Magnete

De Magnete, Magneticisque Corporibus, et de Magno Magnete Tellure (On the Magnet and Magnetic Bodies, and on That Great Magnet the Earth) is a scientific work published in 1600 by the English physician and scientist William Gilbert. A highly influential and successful book, it exerted an immediate influence on many contemporary writers, including Francis Godwin and Mark Ridley. == Contents == In De Magnete, Gilbert described many of his experiments with his model Earth called the terrella. Gilbert made the claim that gravity was due to the same force and he believed that this held the Moon in orbit...

Balarka Sen

now connections, that's something you'll never see

Slereah

Well the usual explanation for such things was fluids

there is a magnetic fluid

Balarka Sen

thats not so bad

action at a distance is obviously wrong right you have to have a field

Semiclassical

6:10 PM

@BalarkaSen physics.princeton.edu/~mcdonald/examples/knot.pdf

Slereah