The h Bar

Obliv

12:01 AM

You sound like sheldon from the BBT to me right now :D

and I'm penny

Is this undergraduate QM? lol

ACuriousMind

don't make me rant about TBBT again :P

Silly Goose

@Obliv i think it depends on where you go for UG

Obliv

haha, but it's the only popular example i can think of.

Silly Goose

it is not UG QM at my institution because my specific school only has one semester of QM. but at the school 15 mins walking distance they have two semesters of QM. This probably would be covered in the second semester

ACuriousMind

May 7, 2020 at 16:41, by ACuriousMind

I've never wanted to punch anyone more than the people who installed Sheldon as the face of theoretical physicists in the public mind

Silly Goose

12:02 AM

LOL

this is content from the 4th chapter of Sakurai, a canonical graduate quantum mechanics textbook. but to my understanding you probably would use sakurai or something similar in a second semester QM course. you maybe just wouldn't cover the entire text

Obliv

I've honestly barely watched it so idk if it's any good. I don't really like shows with laugh tracks

Silly Goose

but i don't think you would try to "mathematicize" it as i have tried to :P so it would be more like a pithy result: "adiabatic evolution is blah" and then you move on

Obliv

Is.. that a typical approach to understanding QM? I do that to get by but I don't feel good about it lol

Silly Goose

i feel like people can do what works for them

Obliv

Well they say in UG you should just "shut up and calculate" and save your questions for graduate school

Silly Goose

12:07 AM

i personal find that formulating the physics i learn into more mathematical statements makes them easier to understand and more interesting

well not "easier to understand". I think I just cannot understand certain physics things to a personally sufficient degree until I can understand it as a mathematical-esque statement, or at least that it can be turned into a mathematical statement.

Obliv

it's the opposite for me, I use the math to "map" the physics to my mind like so I can visualize what's going on.

I see math and physics as completely distinct for some reason.

Silly Goose

@Obliv my UG advisor did their training in string theory so i get to ask all sorts of welcome questions lol

Obliv

you plan on doing string theory as well?

Silly Goose

hell no

i currently hope to explore condensed matter theory/quantum information/open quantum systems. it is looking like open quantum systems right now

Obliv

But you like math :D

Silly Goose

12:12 AM

there's lots of math in the above mentioned areas too :D

i think i'd like to do something a little more grounded, and the above three fields seem to be a pretty good compromise between interesting math and being empirically related to reality

Obliv

Yeah but you could learn about BRANES

then you'd really be sheldon

Silly Goose

@ACuriousMind hm wait so is the A-B effect not contingent on using an eigenstate as the initial state?

DIRAC1930

Iwould highly recommend Allan Adam's lectures on QM

They are on youtube

Silly Goose

i also find the zwiebach (also a string theorist lol) lectures on QM personally enjoyable. the content is the same but the way it is done is slightly different than is done in sakurai. often better.

ACuriousMind

@SillyGoose no

Silly Goose

12:20 AM

oh no. does no mean it is indeed not contingent?

ACuriousMind

yes

Silly Goose

ah okay

so AB as you previously stated has nothing to do with adibatic evolution

okay i think this was a good conceptual thing to clear up :P i am satisfied

so it is just something special about the form of the A-B hamiltonian that leads to this type of evolution which roughly takes on a similar form to adiabatic evolution

ACuriousMind

I wouldn't say it's the form of an "adiabatic evolution", it's just that you can frame both cases in terms of a connection

Silly Goose

okay i see

ACuriousMind

and then transporting values from one point in the parameter space to another will be the phase/parallel transport/holonomy of that connection

Silly Goose

12:25 AM

this QM I lecture "Experimental Facts of Life" by Allan Adams (again, another string theorist lol) was also quite good in my opinion. this is content not covered in quantum textbooks I have read and it gives you an idea of quantum mechanics in real life

DIRAC1930

For anyone interested Allan Adams left String Theory and went into ocean sciences

Silly Goose

okay now i have to finish topology hw :P i will see u all later

DIRAC1930

@SillyGoose If you are interested in these you are at a significant advantage

Just make sure you choose the right topic

and supervisor

naturallyInconsistent

1:44 AM

@Mr.Feynman I'm also the same; that's why I can tell you the relevant constraints that you actually have to satisfy.

@Mr.Feynman But it is not a mess?

@SillyGoose One particular example of a flat space is a cone. If you cut a straight line from the tip of a cone to the base, you can unroll the cone surface into a sector of a flat circular disc, so there is not curvature. This is to be contrasted with a sphere, which you cannot do any such thing.

@ACuriousMind And in the end, even after all your replies, he continues to assert the same. Why don't you see it as a moderation event?

@SillyGoose Why do you not take the extremely canonical example that is impossible to miss in a textbook on Hamiltonian mechanics, which is the bead on a hoola hoop, where the hoola hoop is being rigidly, mechanically, and uniformly, rotated about the vertical axis?

@SillyGoose Part of the problem is that the term "adiabatic evolution" is just so badly coined. It is really quasistatic evolution. And for simplicity of exposition it is always sensible to restrict our attention and exposition to first cover quasistatic evolution when discussing geometric phase issues. It is the equivalent of considering equilibrium thermostatistics before doing any actual dynamics. Just the prudent thing to do.

@Obliv I see that the wacky fowl also explained it later on, but if you didn't get his explanation, my reply comment immediately preceding this very one, is probably relevant / helpful.

@SillyGoose See, I knew you must have been conflating something along this line.

nickbros123

4:26 AM

@naturallyInconsistent JD Jackson's uses the term quasistatic for his derivation of work done against induced E fields (the derivation for $\frac{1}{2\mu}B^2$) which assumes quasistatic motion of the charges in loops. The derivation for poyntings theorem doesn't assume such things, though, and that's the canonical energy conservation in Classical EnM

naturallyInconsistent

5:49 AM

@nickbros123 Correct. Poynting theorem should be general, or else we would have additional terms. There is a lot of nuance involved in getting all of these things correct.

nickbros123

6:17 AM

Btw is which book should I buy: dirac lectures in QM, which is quite cheap, or dirac "principles of QM" which is costlier.

What is the difference: to those who've seen both

naturallyInconsistent

The two books are very different

It is nice to read them both but it is probably not worth the cost. They are mathematically rather neat to read, but they are both mostly of "scratches curiosity itches" than exactly illuminating.

bolbteppa

The little one is an advanced one on the constrained Hamiltonian formalism and trying to quantize theories with constraints, the big one is on QM from first principles, it's a hard book with old notation which you risk getting lost in and giving up but obviously the most important stuff is there and really reading it would give deep insight

@Mr.Feynman The CFT approach is usually an afterthought in string books, it provides another way to set up the basic stuff in an advanced manner and it has strengths and weaknesses, this is not a good reason to ignore the most canonical book in the subject but lose out on the insight it gives if you want

bolbteppa

6:44 AM

@DIRAC1930 Did you read Blok on 2nd quantization? This (eq. 3 to 9) is a summary of what he does, however the position of the indices should cause serious stress

Silly Goose

I am looking at this classical lagrangian density $D_\mu \phi D^\mu \phi^* - m^2 \phi^*\phi$ where $\phi$ is a scalar field and $D_\mu = \partial_\mu + iqA_\mu$. A gauge symmetry here is $\phi \to e^{-iq\epsilon(x)}$ with $A^\mu \to \partial^\mu \epsilon(x)$

bolbteppa

(Unless I'm wrong, (8)-(9) do not follow from (5) because of the position of the indices, despite what Blok and this claims)

Silly Goose

so this is the classical klein-gordan equation with the partials replaced by a "covariant derivative"

Ryder Rude

en.m.wikipedia.org/wiki/Tree_of_life

it's about the tree of life

Silly Goose

now, I get that the Noether current due to the "global" phase symmetry is dependent on $A^\mu$ itself. this seems a little bit nonsensical? Is it not?

bolbteppa

6:52 AM

Why is it nonsensical

Silly Goose

well I guess I am not sure physically what that would mean. to me it seems to imply either 1) this noether current is not what we should associate with charge density, or 2) that the charge density depends on the applied electromagnetic field/potential, but the charge density associated with a particle subject to an electromagnetic field probably should not change due to interactions with the electromagnetic field

or these thoughts could be totally off

or i should say electric charge/current density, not charge density

also a separate question: the EM lagrangian just happens to be linear in $j^\mu$, right? the interaction coupling coefficient for a physical theory could be quadratic and so on in general?

bolbteppa

7:08 AM

You would exactly expect $J^a \sim \phi^* \partial^a \phi - ...$ to get replaced by $J^a \sim \phi^* D^a \phi - ...$ which is what happens, in terms of physically interpreting it read below (17) in there maybe that will help

Mr. Feynman

@naturallyInconsistent yeah it's not :P

@bolbteppa ok, thanks. I needed some feedback to be sure just it wasn't obsolete, I understand

And since I don't plan to work on ST, I guess that's fine

naturallyInconsistent

@SillyGoose It is good showmanship. But what I am more worried about is that you are saying that you didnt get the content from quantum textbooks, and that just means that the textbooks you are using isn't that good. All of the covered stuff are pretty standard fare for quantum textbooks.

There are some small mistakes though. Like, he said that Einstein came up with the energy packet of photons; which is silly because he himself also said that Planck resorted to curve fitting: Einstein called it Planck's constant because Einstein was referring to Planck's desperation and proposed that it is true for all photon behaviour rather than just the blackbody radiation curve fitting.

bolbteppa

@SillyGoose Fermi's theory (see 2.1) for example was a $J^a J_a$ interaction, this was really motivated by the $A^a J_a$ form of Maxwell, but it is quadratic in currents

naturallyInconsistent

The video is also really insufficient. There is a whole other set of motivating experiments that cover the 2nd half of quantum phenomena.

bolbteppa

7:29 AM

Coincidentally, this $J^a J_a$ Fermi stuff is another motivation to consider string theory, e.g. this Fermi stuff was non-renormalizable, the fix was basically inserting a propagator (see the discussion around (9)-(10)), for gravity strings offer a similar-ish.... fix

Silly Goose

@naturallyInconsistent i sort of brushed over the experimental stuff when i first learned quantum since i was not interested. but nowadays i think it would be nice to have a good understanding of the "real" side of things :P

bolbteppa

> While the unification of the non-gravitational interactions within the framework of quantum gauge theories does not meet fundamental problems, this changes once one attempts to include gravity. One way to see this is due to the perturbative non-renormalizability of the gravitational interaction.1

> In this context, the main differences between the gravitational and the renormalizable Yang–Mills gauge interactions are (1) the graviton has spin two while Yang–Mills gauge bosons have spin one; (2) the gravitational coupling constant has negative mass-dimension while the gauge couplin…

Trying to mimic how the non-renormalizable Fermi theory was patched up into a renormalizable theory as in the standard model for gravity, and finding strings as the closest natural analog of this process, it's just undeniable that it's a very good idea and simply crazy to ignore

naturallyInconsistent

@SillyGoose At some point in a theorist's or a mathematician's life, the complexity of the contradictory stuff being considered would be so great as to cause a fundamental reevaluation of one's life, and it is then that having experimental motivation for all the postulates will be a life affirmer.

Silly Goose

did u do ur training in exp or theory?

or perhaps both

@bolbteppa hm okay thanks for the pointer. i will try to digest your further comments at a later time :P

Ryder Rude

7:47 AM

@bolbteppa hi. do u not think it is somewhat ad-hoc to smoothe out lines into tubes? especially since this ignores qft from which the lines are derived in the first place

bolbteppa

It does not ignore QFT it will contains QFT as a special case if you just restrict to the lines, it's as 'ad-hoc' as inserting a propagator in the Fermi case, nobody has had any better ideas

Ryder Rude

oh

yeah..that sense, it does not ignore qft. still, i think it is wrong to derive qft from lines. the lines are derived from qft, as qft is a more complete theory

basically, the route we're taking is : qft ---> lines --->tubes ---> hope that tubes come from a more complete theory

bolbteppa

The lines are not derived from qft, we begin by assuming a point particle theory i.e. lines which implies lines are all we'll ever find, if you instead begin by assuming tubes i.e. worldsheets you are doing string theory and your qft will be a qft of worldsheets instead of a qft of particles, both of which can be described by fields, either fields made up of particles or of strings

Ryder Rude

8:05 AM

yeah... maybe it works out that the particle view is just as good as a starting point as the field view

lucabtz

8:32 AM

@bolbteppa the string fields aren't do well understood yet though right

bolbteppa

Right it's a massive area of research

lucabtz

There was this conference by Sen last year who works on those

But I missed it because I got the starting time wrong

naturallyInconsistent

@SillyGoose I'm a theorist. But I can do expt too

@lucabtz it is known that I abhor strings but to be honest there is a lot of things in particle QFT that we do not understand well either. Of course, the comparison itself is facetious since strings are known so little that we cannot even begin to compare with experiment, but then that is a far better critique of the state of the art.

Mr. Feynman

9:24 AM

yesterday, by lucabtz

Graduated today

49 mins ago, by lucabtz

But I missed it because I got the starting time wrong

Now the meme is complete

nickbros123

@lucabtz this is a cardinal sin

Ashoke sen was to come to my university, I was super excited, but it was cancelled cuz he got COVID

That week was just sadness

ACuriousMind

9:45 AM

@SillyGoose You need to be careful - something can "depend on $A$" and still be gauge-invariant. See Qmechanic on the scalar QED Noether current.

Mr. Feynman

I want to see qmechanic's grocery list. Imagine "comment to groceries of DD/MM/YYYY (v4): I updated the list"

3

Ryder Rude

11:31 AM

anyone want to play this? lichess.org/fWd0OZTu

Monty

11:48 AM

scholarpedia.org/article/… can anyone give me some idea what the normal mode coordinates are and why they are introduced? Been a long long time since I did Fourier transform stuff

Ryder Rude

12:15 PM

what is a weird philosophical position u align urself to?

ACuriousMind

@Monty It's just the usual decomposition of vibrations of a string: The sines there are the fundamental and the overtones, and any shape of the string can be described as the linear combination of these (countably many) overtones. The $Q_k$ tell you "how much" of the $k$-th overtone is needed for the current shape $q(j)$ of the string.

Ryder Rude

ncatlab.org/nlab/show/worldline+formalism the diagram here summarises the relationships between worldline/worldsheet formalism and qft/string field theory

i think theyre taking this approach : 1. start with a classical theory of a relativistic point particle/string. 2. first quantise it : this gives the Feynman propagator. ( but i dont think this is a consistent quantum theory becuz evolution isnt unitary. it is first quantisation only in name)

3. "second quantise it": i think they're calling the imposition of sum over topologies and Feynman rules on the above as "second quantisation"

the second quantised theory is a valid quantum theory

@bolbteppa do u think this is correct

i dont think this is the definitive approach tho. i once saw an approach where they started with a point particle classical theory where particle trajectories could split apart

so the feynman diagrams were there since the beginning

and finally, the map from this theory to qft/string field theory isnt well defined becuz qft contains non perturbative information

bolbteppa

12:41 PM

I don't know about your unitary comment, but the diagram looks fine, what it's basically saying is because we know the 2nd quantized field result, we know what we need to get from a worldline perspective where things are ugly and need to be summed over by hand, this becomes a serious issue because string scattering is all stuck in the worldline perspective and it's too difficult to take a string field perspective which does everything blindly

Ryder Rude

@bolbteppa i mean that the Feynman propagator involves an integral over proper time in order to get $G(x't', x,t)$ so this is not unitary in a relativistic theory

in the momentum space, there r two terms for forward and backward in time, and there is the $\frac{1}{2\omega _p}$ factor @bolbteppa

bolbteppa

1:05 PM

Is $\Delta_F(x-y) = <x|\frac{1}{\Box-m^2}|y>$ not unitary

Ryder Rude

@bolbteppa yes

they do the relativistic particle path integral and dont even care that it's not unitary and hence not a quantum theory

but unitarity is restored when do the sum over toplogies

bolbteppa

1:27 PM

Well now you've confused me, how do you see it becomes unitary when you sum over all topologies

Ryder Rude

1:41 PM

@bolbteppa it's becuz the sum of topologies is sort of equal to $e^{-iHt}$ after some touch-ups, where $H$ is the field Hamiltonian

so this satisfies the Schrodinger equation

but this only after some additional touch-ups, which is the LSZ formula @bolbteppa

DIRAC1930

@bolbteppa Hi, I haven't looked in Blokhintsev just yet

I wonder if this Fermi Liquid Theory applies to low energy electrons in QED

However first I need to understand FLT properly

Relativisticcucumber

2:01 PM

@SillyGoose "working" being used loosely here i assume? XD

bolbteppa

@RyderRude If you have a simple reference discussing all this it would be good

Ryder Rude

2:19 PM

@bolbteppa this is just generic qft. e.g. Schwartz. im just saying that : correlation function + LSZ = sum over topologies + LSZ= $\lim _{t_1\to -\infty, t_2 \to \infty} e^{-iH (t_2-t_1)}$ and the last expression is manifestly unitary

Obliv

Do people write $\vec{\nabla}$ or $\nabla$

what's more standard?

bolbteppa

@RyderRude It's not generic this is all very non-trivial, this worldline stuff is very complicated

Ryder Rude

@bolbteppa wait, what part of my statements u sort of disagree with? is it : point particle path integral is not unitary, or sum over topologies is unitary?

@bolbteppa yes. i agree unitarity is non trivial to achieve in interacting theories

bolbteppa

All the stuff I have on this doesn't even mention unitarity so idk what to say

DIRAC1930

2:44 PM

@RyderRude I'll play you in a blitz 5 min game

@RyderRude lichess.org/UtQOOn6G

Ryder Rude

@DIRAC1930 great :)

DIRAC1930

I am doing terrible in this game lol

Mr. Feynman

Does the free particle einbein action really have the "same" form as the Polyakov action?

Ryder Rude

@DIRAC1930 same :)

Mr. Feynman

It looks like the mass disrupts the analogy (?)

Ryder Rude

2:59 PM

@DIRAC1930 i havent joined. r u plying someone else

Mr. Feynman

$$S=\frac{1}{2}\int\sqrt{-g_{\tau\tau}}(g_{\tau\tau}\dot{X}^2+m^2)$$

While the Polyakov action is $$S_P=-\frac{T}{2}\int d^2\sigma\sqrt{-g}g^{\alpha\beta}\partial_\alpha X\cdot\partial_\beta X$$

The term $g^{\tau\tau}\dot{X}^2$ should be "like" $g^{\alpha\beta}\partial_\alpha X\cdot\partial_\beta X$

DIRAC1930

@RyderRude I thinks so lol

They are winning

Ryder Rude

oh. i thought u were playing white

Obliv

LOL

Sorry I joined thinking I could spectate but it made me play

Ryder Rude

lol

Obliv

3:04 PM

Also I resigned because I didn't want to further embarrass myself with that ending

Ryder Rude

but u wer winning

Obliv

I was.. but Dirac had too much fight left

Honestly not sure what I was supposed to do

Ryder Rude

yeah it's not easy to win that

maybe a draw

DIRAC1930

Here's another link

lichess.org/s53P2Idu

Obliv

it's only a draw if you have a bishop/knight left. rook or knight+bishop should win

Ryder Rude

3:08 PM

oh

@DIRAC1930 ok

Mr. Feynman

@bolbteppa do you have any insight? :P

DIRAC1930

Lol

Thanks

Ryder Rude

i didnt see that :(

lol

DIRAC1930

Lol technically you won though since you let me take back my queen lol

Ryder Rude

yeah let's call it a draw then :P

DIRAC1930

3:14 PM

Your pawn on the LHS was getting me nervous lol

Ryder Rude

ye my plan was to get it to ur king

it is used to open kings

do u want to play one rematch

DIRAC1930

Maybe later

Ryder Rude

okay :)

Obliv

How do I get to $\vec{\nabla}\cdot(\vec{\nabla}\times\vec{A})=0$ from this

grr. It's compiling fine in overleaf

$$\left(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right)\cdot\left[\hat{i}\left(\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z}\right)+\hat{j}\left(\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x}\right)+\hat{k}\left(\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y}\right)\right]$$

was missing a backslash woops

Ohh I get it

all the terms will cancel out

bolbteppa

$g^{\tau \tau} \dot{X}^2 = g^{\tau \tau} \partial_{\tau} X^{\mu} \partial_{\tau} X^{\nu} \eta_{\mu \nu} = g^{\alpha \beta} \partial_{\alpha} X^{\mu} \partial_{\beta} X^{\nu} \eta_{\mu \nu}$ with $\alpha,\beta=\tau$

Obliv

3:26 PM

is there an implicit assumption of equality of mixed partial derivatives

Obliv

3:46 PM

What's more standard, $$B(\textbf{r}) = \frac{\mu_0}{4\pi}\int_C\frac{Id\textbf{\ell}\times\hat{\textbf{r}}'}{|\textbf{r}'|^2}$$ or $$B(\textbf{r}) = \frac{\mu_0}{4\pi}\int_C\frac{Id\textbf{\ell}\times{r}'}{|r'|^3}$$

DIRAC1930

@bolbteppa I'm not sure I fully understand L&L 5's arguments regarding why a macroscopic body cannot be in a stationary state

Obliv

for Biot-Savart law and $\hat{r}'$ denotes the unit vector of $r'$

ACuriousMind

@Obliv what do you mean "assumption"? For smooth functions, the order of derivatives doesn't matter, and you should be used to physics treating every function as if it's smooth

Obliv

crap, ran out of time to edit that. tried to denote the vectors with boldface

Okay, but at discontinuities then it's not guaranteed?

ACuriousMind

smooth functions don't have discontinuities :P

also at discontinuities derivatives don't exist :P

Obliv

3:52 PM

Well in my thermo class we were doing phase changes of pure substances and at phase boundaries we say the gibbs energies are the same for the substances, but I guess we can't take the derivative then

ACuriousMind

what does that have to do with $\nabla\cdot(\nabla\times A) = 0$?

Obliv

Nothing, I think. Was just mentioning that because we did take mixed partials to get maxwell relations for the thermodynamic identities but they're not guaranteed to exist at phase change boundaries I guess

like it's not always the case we're dealing with smooth functions

Relativisticcucumber

@naturallyInconsistent flex

bolbteppa

@DIRAC1930 What page

Relativisticcucumber

@SillyGoose if i go the exp amo route im sure u will hear all ab it in the next 5 years XD

Sanjana

3:57 PM

Is there a way to find the coefficients of highest root in Dynkin basis or its dual basis formed by the simple roots directly without constructing the root system from simple roots?

bolbteppa

@Mr.Feynman see above

Sanjana

One thing for sure all the coefficients of the highest root must be greater than $1$

DIRAC1930

Section 5

Page 14

He gives 2 arguments

Sanjana

I found a cute formula (coefficients of the highest root+2)*rank of algebra=dimension of the algebra!

Relativisticcucumber

@SillyGoose what is better pillow talk than BECs

Sanjana

3:59 PM

Another thing: does anyone know some resource (it might be even a textbook exercise) which discusses scalar-fermion derivative interactions in QFT i.e. the interaction term is $\overline{\psi} \gamma^\mu \partial_\mu \varphi \psi$ along with the usual kinetic terms for the scalar and Dirac fermions.

DIRAC1930

@Relativisticcucumber en.wikipedia.org/wiki/Dicke_model this without the "e" lol

Mr. Feynman

@bolbteppa yeah, that's what I mean. My point is that the $m^2$ term has no analogue

Silly Goose

4:25 PM

@Mr.Feynman i actually saw a qmechanic answer recently that was in an absolutely different style than their answers these days. It was jarring

it was all like “but why and how is it this way…” and proceeded to motivate and answer and etc. in several paragraphs.

@Relativisticcucumber teach me about BECs >:D

@Relativisticcucumber hehe actually it works just fine i just have no idea what these “problems” are :-)

bolbteppa

@Mr.Feynman You can do this conversion for any $p$ using the method in this answer:

2

A: Derivation of the Polyakov Action

The solution can be found in the article in here. The idea is, assume you have a starting action for an object whose extent is (spatially) $p$-dimensional. It can be described by introducing an auxiliary "abstract" worldvolume $\Sigma$ which will be embedded into spacetima via $X:\Sigma\rightarro...

The $p=0$ case is the particle action, you can re-scale the answer there so that you end up with this $m^2$ term as in your point particle action, you see that the $p=1$ case is very special

@Sanjana No idea but I'm sure the answer is buried somewhere

Mr. Feynman

@bolbteppa oh, looks like strings are the weird ones, not particles

Silly Goose

4:45 PM

Does anyone have a resource showing just how “unique” our choice of electromagnetic lagrangian is given that we want X, Y, ….

naturallyInconsistent

@SillyGoose You have to first state what your X and Y are...

@Obliv These are two representations of the exact same integral, and so both are about equally commonly appearing in textbooks.

Silly Goose

Well maybe a better question is for what physical constraints is the EM lagrangian unique and the proof for such a claim

DIRAC1930

@Sanjana This is not an answer to your question but this might be a useful resource for you just in general billcookmath.com/sage/Lie_algebra/root_demo_prelim.html

In regards to the representation stuff you were doing earlier

naturallyInconsistent

@SillyGoose I doubt there can even ever be such a proof. Back in the day when people are desperately trying to overcome the infinities of QED by some other method than renormalisation, they came up with many alternative electrodynamics, some classical and some quantum, that look completely different from the usual EM Lagrangian, and yet give rise to either equivalent physics, or slightly different ones.

Obliv

@naturallyInconsistent Does this make sense for a vector that's origin is at a point on a circle in the xy-plane (concentric around the z-axis) pointing to some point on the z-axis: $\mathbf{r}=(R\cos\theta,R\sin\theta,z)$ or in spherical coordinates, $\mathbf{r}=(\rho,\theta,\varphi)$ where $\rho = \sqrt{R^2+z^2}$, $\theta = \tan^{-1}(\frac{R\sin\theta}{R\cos\theta})$ and $\varphi = \tan^{-1}(\frac{z}{R})$

bolbteppa

4:51 PM

@DIRAC1930 They are considering macroscopic bodies as closed systems. In QM a closed system always has a discrete set of stationary states, so it always has a wave function, see section 10 of QM. End of story, or so it seems. In stat-mech, they want to say there are so many particles in a finite volume that the discrete energy spectrum is now practically continuous. It never actually is, if it was then it would imply QM itself breaks down for large enough N which it obviously doesn’t.

ACuriousMind

@SillyGoose It's the unique (up to scaling and total derivatives) Lagrangian that yields the Maxwell equations, what more do you want?

bolbteppa

Instead, really they should invoke the thermodynamic limit i.e. a theoretical infinite number of particles in an infinite volume with a finite density, maybe they say they do this in the beginning I forget. They do not use the words thermodynamic limit, instead they basically say a continuous spectrum arises because any value in it can be approximated to any error by an energy eigenvalue from our spectrum of particles i.e. the energy spectrum is now continuous for all intents and purposes,

but really you need to take the thermodynamic limit to transition from QM to stat mech. Thus they are now trying to talk about a macroscopic body as a closed system with a continuous spectrum. But a closed system only has a discrete spectrum. Thus there is no way to describe the system as ever living in a stationary state which would have to live in the discrete spectrum, thus a thermodynamic system technically has no wave function description.

The resolution is to approximate the continuous spectrum by a discrete spectrum and to use this discrete spectrum as the basis for a density matrix description of the system, encoding the information that is being missed by the use of a stationary state basis in the complicated evolution of the coefficients in the density matrix.

Thus even if you assume at an instant that a wave function describes everything, it will then evolve into a general density matrix with $w_{mn}$ coefficients. This is the extra notion of statistical averaging they discuss beyond the usual QM missing incompleteness that they reference.

naturallyInconsistent

@Obliv This question is not making sense. Those two integrals are the same thing regardless of where you picked the origin to be.

Silly Goose

@ACuriousMind i am interested in seeing the proof because i would like a flavor for what it would look like to prove the uniqueness of an action for a physical theory

Obliv

Silly Goose

4:53 PM

@naturallyInconsistent oh is there a name for this sort of stuff?

Obliv

@naturallyInconsistent Wasn't asking about the Biot-Savart integrals but whether that vector I described is correct

ACuriousMind

@SillyGoose modulo topological obstructions it's a general fact that Lagrangians that yield the same equations of motion differ only by total derivatives or scaling, see physics.stackexchange.com/q/131925/50583

naturallyInconsistent

@SillyGoose I dont know, but there is a short list of them in Feynman lectures. Things like Mie's theory. Some other electrodynamics textbooks sometimes cover them.

ACuriousMind

there's nothing specific to EM about this

naturallyInconsistent

@Obliv I cannot make a judgement until I see a precisely enough stated question that can be answered at all.

Silly Goose

4:56 PM

@ACuriousMind What are topological obstructions? In the domain of the action ?

Obliv

What part needs clarification?

Silly Goose

Hm so would it be correct to interpret differential form theory as the appropriate tool set to prove such claims that an action is the unique one doing this and that

ACuriousMind

@SillyGoose it means when the domain of the action is not $\mathbb{R}^n$ but something that's not contractible, you might get two Lagrangians with the same e.o.m. that are not related by total derivatives - it's the variational analogue of the Poincaré lemma only holding on simply-connected domains

Obliv

@NaturallyInconsistent Imagine a circle on the xy-plane and the z-axis at the center. The radius of the circle is $R$ and the current running through it is $I$. I want to find the magnetic field at a point $(0,0,z)$ on the z-axis

Silly Goose

Oh okay

Obliv

4:59 PM

So I need to construct a vector $\mathbf{r}$ which points from a point on the loop to the point $(0,0,z)$. So $(0,0,z)-(x,y,0)$ or something equivalent for spherical coordinates

Silly Goose

So for instance the action for a particle outside a solenoid is not simply connected on its domain

naturallyInconsistent

@Obliv That is usually solved by Biot-Savart integral, yes, but it seems like you are using it wrongly. This is a very common homework problem and have known answers, so I don't think I want to debug it.

@Obliv this statement is correct

Obliv

I wish I was a cat

naturallyInconsistent

meow

ACuriousMind

@SillyGoose that's the wrong phrasing - it's the domain that's not simply-connected, not the action - but yes

Silly Goose

5:01 PM

I see

Obliv

I want to define and use the term "domain expansion" in some capacity in math/physics some day.

ACuriousMind

it's like how there are vector fields with vanishing curl that are not the gradient of a scalar field when you exclude the origin

Silly Goose

oh i don’t recall that example but i believe you

ACuriousMind

but for the argument I'm making this is really irrelevant: The EM Lagrangian is the correct one because it yields the Maxwell equations on $\mathbb{R}^4$ and any contractible neighbourhoods, that this hypothetically might not be unique anymore on some weird domains doesn't really matter

Silly Goose

Simply connected spaces are awfully nice

naturallyInconsistent

5:07 PM

@Obliv The original definitions of sines and cosines is only for the first quadrant. We also initially only prove the angle addition and subtraction identities for just the first quadrant. It is perfectly valid to wish to insist that we want the angle addition and subtraction identities to hold for a wider domain; i.e. by postulating the extended validity of those identities, we can extend the definition of sines and cosines. Thereby enact domain expansion on those functions.

@SillyGoose They will put you into the comfy chair

Silly Goose

@naturallyInconsistent i become the chair

naturallyInconsistent

@SillyGoose if it fits, i sits

Silly Goose

Also abt the discussion on the A-B effect yesterday. I am still trying to connect all the facts. So the restriction to physical space outside the solenoid interior breaks the bijection between equivalence classes of potentials and physical EM fields. Is this a separate thing to deal with in addition to the fact that the picked up phase depends on the electromagnetic potential?

ACuriousMind

@SillyGoose who said it breaks that bijection?

again, the A-B effect depends only on the flux through the solenoid, which is gauge invariant

Silly Goose

Wald and Tong have statements that the bijection breaks over not simply connected spaces

hm okay i see

Obliv

5:15 PM

@naturallyInconsistent A function is guaranteed to be defined in that domain expansion.. unless there is a simple domain within the expansion that doesn't allow that.

cats are liquid

ACuriousMind

@SillyGoose what they're (hopefully) talking about is that technically there isn't really a single $A$ covering all of the non-contractible space

Silly Goose

@Obliv i would like to use or see a use of the sylow theorems in physics :) maybe they show up in p-adic stuff

Obliv

not sure what p-adic stuff is but yeah I'm banking on abstract algebra to eventually be relevant in my life.

Would be kool

oh wait i recall my professor mentioning what they were but I forgot

ACuriousMind

the equivalence between gauge potentials and the field strength tensor never breaks down, it's just that actually the gauge potentials are only defined locally, while the field strength is defined globally

Silly Goose

p-adic is doing stuff by using prime numbers as the space of scalars to my understanding. The sylow theorems have to do with finding primes or something—it hs been a while :P

Obliv

5:19 PM

I thought sylow theorems were about finding the number of subgroups of a finite order

Silly Goose

Oh then maybe ive just completely forgotten XD

@ACuriousMind oh ._. hm maybe i misinterpreted their statement then. It seems misleadingly stated if not…

ACuriousMind

well, to be honest the discussion of gauge theory in most standard resources is lacking :P

Silly Goose

Can never trust these physicists am i right :)

DIRAC1930

@bolbteppa Thanks

Silly Goose

well i think i will learn the actually maths behind these results (hopefully) at the end of algtop 0.o

ACuriousMind

5:25 PM

you won't :P

at least it would be unusual for a standard algebraic topology course to do anything about principal bundles

DIRAC1930

Bolbteppa should start a school of theoretical physics just like Schrodinger did

Silly Goose

Oh

Yeah we are not doing any bundle stuff other than we introduced the definition of a bundle…

@Obliv according to my notes the sylow theorems are statements about Sylow p-subgroups of a group, which are subgroups with order prime number $p$ to a power $\alpha$

Obliv

I think if the order is prime, then there can only be $1$ unique subgroup or something

bolbteppa

That will very quickly expose everything I don't know, absolutely not

DIRAC1930

For anyone interested, Schrodinger released some lectures on Statistical Mechanics that he lectured on in Dublin

You can get them cheaply store.doverpublications.com/products/…

Silly Goose

5:35 PM

Are gauge fields distinguished from other non-gauge fields by being described as principle bundles?

DIRAC1930

However it is not for the faint hearted

naturallyInconsistent

One would have to point out that a library consisting solely of L&L is not a library, merely one single series, and loudly exhorting that students who do not think of L&L as the end-all and be-all of physics education are wrong will not make a university education. By that notion, the Bolbteppa school of theoretical physics will be rather difficult to get accredited.

Obliv

what about if the library also has an introduction to thermal physics by dan schroeder?

naturallyInconsistent

lol

ACuriousMind

@SillyGoose I mean...yes?

or rather, the difference is that they're connections, not "normal" fields

naturallyInconsistent

5:39 PM

L&L and Schroeder, I think the students would be shredded away enough to rather do some chemistry than physics.

Silly Goose

@naturallyInconsistent some of the materials science and CMT i was looking into sounds just like chemistry XD

naturallyInconsistent

@SillyGoose When I sent my first paper to my ex-boss prof, he was like, hmm, your paper looks like chemistry...

@Relativisticcucumber yes, this kitten goes to the gym now...

Obliv

@NaturallyInconsistent I'm such a bad student :( I didn't study until the last minute for my thermal exam today and ofc the one problem I didn't study was on the exam. Involved finding relations about thermodynamic identities rip. yeah the 2nd third of this semester has a lot of overlap with chemistry (which is normal i think)

last 3rd will be more stat mech so hopefully we cover bb radiation to some degree.

naturallyInconsistent

@Obliv there is nothing normal about that...

Obliv

well it's not a stat mech course. It's thermodynamics so it's only natural to cover a lot of topics used in chemistry/applied physics.

Silly Goose

5:44 PM

When i was briefly studying chemistry i was working on a project that was looking at the structure of some molecule. I go to oak ridge to find them doing the same thing :P just with a massive neutron scattering set up XD

naturallyInconsistent

@Obliv I suppose I have to congratulate you for surviving that...

@SillyGoose at day job miao miao was calculating how long the neutrons would survive. Because if we do not stop the stupid experiments, we would be contaminating nVidia and Foxconn next buildings over with them.

It is very annoying when basically everybody with us now understand that neutrons are very scary things, but the one main guy wanting to do the experiments wont stop doing the ones that make neutrons.

Silly Goose

XD oh my

Obliv

I can tank it.

Silly Goose

pls don’t hit my nuclei !

Do you know how the neutron analogue of “fiber optics” works?

naturallyInconsistent

@Obliv Again, it is extremely uncommon for thermodynamics in a physics course to cover chemistry. It is not useful, since once the stat therm is properly introduced, the results are trivial to introduce to students.

Obliv

5:49 PM

My friend works at a bio-pharm research company and has f*ked up many-a-times spilling chlorofoam on his hands, picking up dry ice without gloves, accidentally mixing ammonia with a surfactant, etc lol

Silly Goose

You can “touch” dry ice momentarily :) however the other stuff sounds a bit worse lol

naturallyInconsistent

@SillyGoose oh, they make waveguides for neutrons too? What materials?

Silly Goose

i have no idea. But on the tour they said you can wave guide small wave length neutrons

or it might be large wavelength lol

Obliv

aren't neutrons neutral? How do you guide them?

naturallyInconsistent

I dont think spilling chloroform on hands is that immediately toxic as to be life threatening, but yes, if he is making a routine of having these mishaps, guide him towards a different career

Obliv

5:51 PM

I guess law of reflection like in optical cables

@naturallyInconsistent It melted through his gloves and burned his skin lol.

ACuriousMind

that wasn't chloroform then :P

naturallyInconsistent

@Obliv still not the one drop and death. And if he can burn his skin many times this way, it only just means that he never did learn from it; which means it is not deadly enough

Silly Goose

@Obliv they have a nonzero magnetic moment and so i think can be directed very weakly by an appropriate application of magnetic field

They also might have nonzero electric dipole moment but that is speculatory i think and if it was nonzero it would be extremely small

Classically a dipole would try to align with the external field and so in my head I imagine you can change the direction of a moving neutron via application of an external field. It is probably pretty hard to do :P idk

ACuriousMind

@naturallyInconsistent @Obliv chloroform doesn't "burn your skin"! It's a mild irritant and pretty toxic, but it doesn't attack your skin from a brief exposure

Silly Goose

These slides on neutron optics are from UTK (the uni ~40 mins away from oak ridge) they have some slides on neutron-material scattering and also on wave guides: phys.utk.edu/neutron-summer-school/lectures/soldner-01.pdf

ACuriousMind

5:58 PM

and it would be very strange for a chemist to wear gloves that can quickly be dissolved by chloroform when they're handling it

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