The h Bar

naturallyInconsistent

01:28

@Allie the superscript or subscript that tells us which polarisation the particle is exhibiting

Silly Goose

You can think of Classically what the polarization for a transverse wave is

Tobias Fünke

07:11

morning

skullpatrol

07:22

hi pal

qwerty

how are you all?

Tobias Fünke

07:43

exhausted from the week :( was a bad week

but now I am enjoying the weekend and I will rest a bit.

How about you? :)

qwerty

08:00

:( sorry to hear you had a bad week. physics related or more personal?

I understand though as am finding myself constantly exhausted, I probably just need to exercise more

qwerty

08:17

btw @ACuriousMind I started listening to the Discworld audiobooks yesterday, do you have opinions on reading order? I did try one of them a long time ago and vaguely recall being told something about them being best read not in order

Tobias Fünke

@qwerty both

@qwerty do you do any sports? :)

qwerty

@TobiasFünke I used to go to a dance class 3 times a week for about 6 years but then stopped around covid due to lockdowns + phd stuff... I just started it again a few weeks ago :)

what about you?

Tobias Fünke

nice. what kind of dance? anything special or general class?

I used to do a lot of sports when I was in school/right after that, but then I had to move and somehow was never motivated enough to find something new :/ but I do keep telling myself that someday I will do sports again :d

qwerty

Pole dance haha

Tobias Fünke

hehe nice. it is very hard, no?

From time to time I play table tennis, in spring/summer quite a lot, actually!

qwerty

08:29

I'm not very good at dance in general (I am very uncoordinated and have no sense of rhythm) but I progressed okay actually :) it's very fun to learn crazy-looking tricks

Tobias Fünke

I see hehe sounds fun

qwerty

@TobiasFünke Cool! Is table tennis not played year round? or is it due to tournaments and stuff?

Tobias Fünke

no, I do it as an amateur, with friends, but mostly outside ^^

I am not thaaat bad I would say, though hehe. But yeah, I thought about doing it more "professionally", i.e. joining a club or so

but playing outside with good weather is sooo nice :) Is this a common thing in Australia, too? In Germany at least we have a lot of these spots where you can play table tennis outside

qwerty

08:49

hmm, maybe not as much/as popular as Germany? I have seen a few I think. the ones I definitely remember seeing are on university campuses maybe due to demographics

Ryder Rude

@PM2Ring yes. Chalmers tries to identify neural correlates of consciousness with superposition-resistant observables

Tobias Fünke

pingpongmap.net

that seems funny (although I don't know how they count these; probably these are not "real" numbers but only the ones users have added or so?)

since it says "buy me a späti beer", I'd suppose the website is from someone living in Germany :p

yes indeed

the ones in Iceland look...funny hehe

qwerty

@TobiasFünke yeah definitely haha i feel like china should have a lot more than the map suggests

Tobias Fünke

yes

PM 2Ring

09:43

@qwerty There's a bit of Early-Installment Weirdness as they say on TVTropes. I quite like the first couple of books, but they have a different tone to the later books, and some of the world elements & personalities of recurring characters aren't very consistent with later books. OTOH, Pratchett was never too concerned about keeping the Discworld universe totally consistent.

Here's a graphical reading guide that might help:

Many Discworld fans recommend starting on the Watch novels, or the Witches novels.

Another option is to try one of the more "standalone" works like Pyramids or Small Gods. Or Moving Pictures.

The Death novels are also excellent. :) They still have plenty of humour, but they also have death...

FWIW, I've read most of the Discworld novels multiple times, especially the earlier ones. Even most of the later ones I've read at least 2 or 3 times. Except for the last one...

qwerty

10:02

thanks for the suggestions :)

ACuriousMind

12:26

@qwerty As PM2Ring said, starting with the Watch (Guards! Guards!) or the Witches (Equal Rites or Wyrd Sisters) is a common recommendation. Personally I love Small Gods as a standalone (you do not need to read Pyramids before it) and the Death novels don't really suffer from being read out of order (Hogfather is my favourite but Reaper Man is also very popular)

Signor Feynman

@TobiasFünke complex numbers? :P

qwerty

@ACuriousMind thanks :)

Silly Goose

14:27

quentum

qwerty

que?

Silly Goose

A fun way to say it hehe

Signor Feynman

Quackntum

Ryder Rude

15:02

@PM2Ring this idea is good too if it works

but I have recently come around to the anthropic principle

like, ofc we find constants that r compatible with life. how could we not find them

i think the real problem is to condense the initial conditions of the universe into something compactly describable, like how the laws of physics r compactly describable

if we do that, we would have come a long way in "explaining" the universe. cuz all of existence would be compactly describable

Silly Goose

17:42

in thermodynamics, are the thermodynamic potentials genuine differential forms?

for instance, for fixed $N$ we could write the "differential" of $E(S, V, N)$ as $dE = TdS - pdV$, but $S$ and $V$ and $N$ are dependent on one another, so we implicitly are not just taking partial derivatives of $E$, we are taking partial derivatives with respect to one variable at a time while holding all others fixed.

hm maybe this is all fine actually

Tobias Fünke

@SillyGoose what should partial derivative mean if not what you wrote?

but yes, you have coordinates, e.g. $E,V,N$ and then a (assumed to be) well-defined function $S$ such that the one-form $\mathrm dS$ exists

sorry, I meant variables S,V,N and $E$ and $\mathrm dE$

Silly Goose

partial derivative is not the same as partial derivative with the other variables fixed when the variables are interdependent

Tobias Fünke

I cannot follow

Silly Goose

but it works out if one thinks of energy as the functional $E[S(V), V(S), N]$, which I think is the proper thing to do

Tobias Fünke

I really don't know what you mean. Check e.g. F. Schecks book "Statistical Theory of Heat", or the classic "The Geometry of Physics" by Frankel

Silly Goose

17:54

Tobias Fünke

What should $E[S(V), V(S)]$ denote?

$V$ and $S$ are independent here

Silly Goose

$V$ and $S$ are certainly not independent? e.g. the classical ideal gas entropy

Tobias Fünke

Sure, but then you express $S=S(E,V,N)$

it is the same old story in thermodynamics: abuse of notation

Silly Goose

yes but i think this artificial treatment of them as independent variables is misleading and also totally unnecessary

it all literally follows from taking the usual differential of $E$ considered as a function of $S(V)$ and $V(S)$

i mean it is plain wrong to say that $S$ and $V$ are independent variables. they are not even treated as independent variables. it is just that they appear in such forms after taking a differential.

Tobias Fünke

as I said. I cannot follow, but that's ok. you don't have to convince me.

but of course, e.g. also $p,V,T$ are usually not independent. Still, one can choose two independent variables, and the third is determined. I don't see where the problem is. Practically, it depends on what experiment you want/can realize

I gave you two sources. See also Baez' point of view

johncarlosbaez.wordpress.com/2021/09/23/…

so there is a totally fine differential geometric POV on thermodynamics, as far as I can see.

The Lagrange submanifold stuff is also explained in Scheck's book

I guess this formalizes what you mean (?) but I can just guess.

Silly Goose

18:12

i think i am just saying i have not seen an actually valid proof of why the maxwell relations hold true. only assertations or flawed reasoning. and it is actually quite simple to prove them using basic notions of differential forms.

but baez probably has something more in line with what i have in mind as justification

Tobias Fünke

@SillyGoose sorry, I think this really is not true

the usual textbook treatment of the Maxwell relations are totally fine

Silly Goose

to my understanding the usual treatment is to assert that the justification is that partial derivatives commute

is this a false impression?

Tobias Fünke

do you know Schwarz's theorem?

Let me clarify: I don't say that it is not useful to formalize thermodynamics with manifolds and diffgeo stuff; but from the little I understand from your messages, I think your "confusion" (?) is not related to that

Silly Goose

hm does schwarz's theorem have a stipulation on with respect to what variables the partials are taken?

Tobias Fünke

for the usual textbook treatment, it suffices (I'd say) to consider functions defined on $\mathbb R^n$, and then consider different functions depending on which independent variables we use. Much of confusion comes from the abuse of notation, but I think conceptually every thing can be clarified on this level of math. As I said, and mentioned two books, you can formalize even more, though.

@SillyGoose Can you please define what partial derivative means?

say, for a function $f:\mathbb R^2\to \mathbb R$.

there is a clear definition of $\partial_1 f(x,y)$ and $\partial_2 f(x,y)$. The theorem guarantees or shows that for sufficiently nice functions we have that $\partial_1 \partial_2 f(x,y)=\partial_2 \partial_1 f(x,y)$

maybe I am too stupid to see what your problem is. Anyway, I hope I could help at least a bit.

User198

18:32

@SillyGoose Here: physics.stackexchange.com/questions/96074/…

Tobias Fünke

and yes, you can get MR just by using the wedge product. E.g. it should hold that $\mathrm d^2E=0= \left(\partial_V T+ \partial_S p \right) \mathrm dS\wedge \mathrm dV$

Signor Feynman

18:57

@SillyGoose the thermodynamic potentials are functions. Their differentials are differential forms. As you know, not all differential forms in thermodynamics are differentials of a function (e.g. work and heat)

Tobias Fünke

@SignorFeynman yes, but they seem to think otherwise (?) at least this is what I've understood from their messages. but I probably did not do a good job in explaining

btw: Did you by now had your SC exam? :d sorry, I am curious :)

perhaps you in the end enjoyed it too much and now you stay in CMT ;)

Signor Feynman

In the language of differential geometry, the "independent variables" of thermodynamics are merely the coordinates of the manifold. An equation between differential forms can be cast in a coordinate independent way. $dE$ is a geometric object existing regardless of the coordinate representation you choose and you may write it in the usual way according to the first principle of thermodynamics, which you should read as a coordinate independent statement

@TobiasFünke Oh, I didn't read your replies, I just went straight for the kill :P

@TobiasFünke Monday! I chose a tie :P

I mean I'll do it monday.

Tobias Fünke

@SignorFeynman what do you mean?

ah ok

nice! good luck, you will do great, I am sure

Signor Feynman

Whatever happens, I'll leave it all to chance

Another heartache, another failed romance

Tobias Fünke

@SignorFeynman I'd overall agree with you, but then things are a bit more complicated, I think, because you have the equation of state and so on. See my message above with the two references. I think there the concept of "Lagrange submanifold" comes into play, but my knowledge there is really just superficial

lol

Signor Feynman

19:04

Does Frankel discuss thermodynamics specifically? I'll check that chapter, then

Tobias Fünke

yes, 1-2 chapters I think

the intro by Scheck is short but nice. (and overall it is a great book, I think)

Signor Feynman

In any case, as you said abuses of notations are a thing too

Before understanding them, I thought thermodynamics was a little too random. Good old freshman years

Tobias Fünke

but again: I agree with you, and I think this is exactly what I've said in my messages. For a simple start, just consider everything on $\mathbb R^n$ and it should be clear. But the abuse of notation in thermodynamics really is hell, so one has to take care if one considers a physical quantity as an independent variable or a (state) function

yes, indeed

Signor Feynman

At least there are no magnetic fields here

Tobias Fünke

not yet

User198

19:18

The canonical symplectic form {\displaystyle \omega =-d\theta =\sum _{i}dq^{i}\wedge dp_{i}} , in CM. So in CM we can talk about differential forms.

Look at this "weird" Wiki page: https://en.wikipedia.org/wiki/Helmholtz_theorem_(classical_mechanics)

it compares thermodynamic variables with CM variables.

And it "derives" the fundamental thermodynamic relation from classical mechanics.

Idk it seems like an odd Wiki page

Have you heard about this Helmholtz theorem before?

Tobias Fünke

19:48

nope, not in this context

Silly Goose

Let's talk about a single example, internal energy, for simplicity. Sure you can talk about the unconstrained energy function $E(x_1,x_2,...,x_N)$ with however many variables you want. This is just the statement that you have a defined energy function with domain charted by the coordinates $\{x_i\}$. But what is of physical interest is the constrained energy (thermodynamic equivalent of "on-shell energies") that the system is constrained to take on given known properties of the system.

This constrained energy I think is naturally a functional because it is a function of the constrained variables $S, V, N$ (for instance), which are all functions of each other for a real physical system.

I guess I mean to focus on the constrained energy, not just the energy function

I guess perhaps the usual notion is in agreement with your notions @TobiasFünke @SignorFeynman in that "thermodynamic potential" specifically refers to the raw (unconstrained) function (energy, free energy, etc.)

Tobias Fünke

a functional is, in essence, nothing different from a function

Signor Feynman

I always hate when people say that functionals are not functions

Tobias Fünke

who says that?

Signor Feynman

20:03

Sometimes people abuse language and say that, I have no references for soemthing so minor :P

Tobias Fünke

Goose: Sorry, I cannot follow (but as I said already, that's fine). If you think you've resolved your issues that's nice :).

@SignorFeynman minor???? hehehe

Signor Feynman

I mean, terminology. Of course who says that means functions of real/complex variables

Tobias Fünke

Well, in e.g. DFT, which is taught to many students with different background, it can get messy to introduces functionals and functional derivatives... and lecturers sometimes make a fuzz as if "functional" would be sooooo complicated or exotic or so.

and that is a bit annoying ^^

Signor Feynman

@TobiasFünke Don't even mention those obnoxious $\delta$ used for functional derivates in place of $\delta$

And then people fork up the EL equations :P

Tobias Fünke

$\Delta$ is for the real OGs

Signor Feynman

20:08

I would have used a less nice definition :P

Silly Goose

i believe functionals are also functions but one says functional to emphasize that they are functions of functions

Signor Feynman

I have also seen the reverse; functionals of (infinite) discrete variables $c_i, i\in\mathbb{N}$ (such as the coordinates with respect to a countable basis, if not finite), and then stuff like $\delta E/\delta c_i=0$ WHYYYYYY

Tobias Fünke

oO

some people just want to see the world burn

it seems

Signor Feynman

But at least I'm learning to sing a E1 subharmonics

It's getting more consistent

Tobias Fünke

haha

Signor Feynman

20:14

That's the 8th key of a piano

Tobias Fünke

wow. Amazon is such a s******

now even with prime they force you to watch ads

Signor Feynman

I don't know what happened but I agree :P

Tobias Fünke

unbelievable

Signor Feynman

Oh, I expected something bigger, come on! Ads are not that bad :P

Tobias Fünke

whut???

it is the worst!

Signor Feynman

20:16

I mean, just 30 seconds :P

Tobias Fünke

it is a matter of principles

Signor Feynman

I don't know, for me movies and series require probably more concentration than physics, so I need to let off steam

And those 30 seconds are perfect

Tobias Fünke

hmhm

Signor Feynman

Disney plus ads are torture, though

Tobias Fünke

ads are everywhere and I want to avoid as much as possible

I never had ads there

Signor Feynman

20:17

ACM is the ad of a secret AI company

Tobias Fünke

mhmh sounds reasonable

Signor Feynman

Disney plus give you about 1m30s of ads every 30 minutes (or maybe a 1m, not sure). Usually a bunch of 20 seconds ads. Sometimes there is a bug and you get the same ad like 4 times in a row. That's exhausting

But at least that happens in the least expensive plan which includes ads

Tobias Fünke

ah OK. I have access via the family account :d

so idk which level we have hehe

Signor Feynman

Do you get ads?

Tobias Fünke

nope

Signor Feynman

20:20

Now we now what level you don't have

Tobias Fünke

haha yeah

Signor Feynman

But honestly I only used disney plus to watch Shōgun

Tobias Fünke

never heard of. is it good? :)

I actually do not watch too many series or so. Mostly movies. but also not too often, I prefer to go to the cinema. but for older movies it is nice

imbAF

I read somewhere that according to Weyl the quantization of a classical system crucially involves understanding the Lie groups that act on the classical phase space.

The quantization of a system as a concept, is it the same as that of a field (for which I have an understanding)?
What it means for a Lie group to act on phase space?

Signor Feynman

@TobiasFünke I like it and I think it's pretty enjoyable (sometimes a little too gory for my own good hahah)

Silly Goose

21:10

@imbAF The poisson Bracket is a lie bracket

imbAF

And?

Silly Goose

Same story as quantum mechanics. Observances are functions on phase space. They live in some representation of some Lie algebra where the lie bracket is the poisson bracket

arxiv.org/pdf/hep-th/9411172 Maybe this is helpful

imbAF

"Observances are functions on phase space" what do you mean by this? The notion that something is a function in a specific space. From what I know, the phase space in CM is just the space of position and momenta and every point in it, it is a representation of possible state of the system

Silly Goose

Sorry i mean observables

imbAF

Still my question remains

Silly Goose

21:19

phase space can indeed be considered as the space of all pairs $(q,p)$ (though it can be more complicated than this). An observable is a function of such pairs: each $(q,p)$ (a state of the system) is an input into a Hamiltonian (an observable) and spits out a real number (the value of the observable given a state of the system).

in this sense an observable is (at the least) a function with phase space as its domain

imbAF

Ah so that's what you mean

with ...on phase space

as in phase space is the definition domain

Silly Goose

yes

imbAF

For the 2nd statement I need to read about Lie groups representations etc, which i am already doing

Silly Goose

generically, $\partial_\alpha x(\alpha) = \{O, x(\alpha)\}$ should imply something like $x(\alpha) = \exp({-\alpha O})x(0)$ where $O$ belongs to some (representation of a) Lie algebra and $\exp(-\alpha O)$ is necessarily part of (the associated representation of) the Lie group.

here $\{-,-\}$ are poisson brackets. but by generically i mean this is a fact about lie theory in general (not specific to the particular lie bracket in use)

imbAF

This is out of my depth currently

very abstract

But I am reading about Lie groups as of right now and I have a question

Can you explain to me in this: "The elements of a Lie group make
up a geometrical space of some dimension, and choosing local coordinates on the
space, the group operations are given by differentiable functions."

I understand that a geometrical space is a set of points that follow certain mathematical rules defining their structure, shape, and properties.

Silly Goose

21:28

i think what they mean is that a Lie group is an $n$-dimensional smooth manifold with group structure that is smooth.

imbAF

I don'\t understand the rest

Silly Goose

that would be the precise statement.

imbAF

smooth manifold is from what I read:

it means that the manifold has a continuous and differentiable structure that allows for calculus to be applied.

would you agree?

Silly Goose

sure that is fine

imbAF

then

what does it mean for a manifold, a geometrical space in other words

to have a continuous and differentiable structure?

Silly Goose

21:31

heuristically that you can do calculus on it

specifically, the manifold needs to be a topological space and also have the necessary structure to take derivatives of functions defined on it

imbAF

I find it hard to understand how geometrical characteristics (I assume that what they are) enable or are tied to calculus

Silly Goose

the initial notion of a continuity, a precondition for differentiability, relies on working in a metric space

a metric is a geometrical object as it immediately gives rise to notions of length and angles

Tobias Fünke

?

continuity does not need a metric space

imbAF

One sec

Silly Goose

that is why i said the initial notion of continuity

Tobias Fünke

21:34

it is a topological concept. or do I misunderstand what you mean?

Silly Goose

as it is introduced in say rudin's analysis

imbAF

how exactly is the derivative, which is tied primarily to a function of some variable, tied with the geometry

I understand that

Tobias Fünke

"tied with the geometry"--which geometry?

imbAF

the derivative is the rate of change of a function's value for a change of input

Silly Goose

@TobiasFünke indeed i consider continuity as real a topological idea

Tobias Fünke

21:36

@SillyGoose yeah, so a function is continuous if pre-images of open sets are open

imbAF

@SillyGoose This here. Continuity for a function and continuity in space, metric as you are calling it, is the same thing?

Silly Goose

@TobiasFünke yes i like this definition

Tobias Fünke

@imbAF what is continuity in space?

imbAF

@TobiasFünke If I have to give a butchered answer, that for every element of this space, one can map a value

of another space

with a function doing the mapping

Silly Goose

@imbAF Let $(X, \tau_X)$ and $(Y, \tau_Y)$ be topological spaces. A function $f: X \to Y$ is continuous if and only if for every$U \in \tau_Y$ we have $f^{-1}(U) \in \tau_X$.

every metric induces a topology, so every metric space has a natural topology.

Tobias Fünke

21:38

Finite topological space

In mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space which has only finitely many elements. Finite topological spaces are often used to provide examples of interesting phenomena or counterexamples to plausible sounding conjectures. William Thurston has called the study of finite topologies in this sense "an oddball topic that can lend good insight to a variety of questions". == Topologies on a finite set == Let X {\displaystyle X} be a finite set. A topology...

imbAF

I am sorry, but what are topological spaces?

Tobias Fünke

pick your favorite example from that

you can construct continuous function for these examples

does that help?

imbAF

I have to read what a topological space is first

to go back to your link

Tobias Fünke

@imbAF we cannot give you a whole math lecture here... very roughly speaking, it is a set and a choice of subsets which you call "open" (they have to obey some conditions)

Signor Feynman

::ad begins

Silly Goose

21:41

@imbAF you can always think of your good old friend $\mathbb{R}^3$

Tobias Fünke

@SignorFeynman hehe

Silly Goose

it is a metric space because it has a distance function (a metric), which is just $\lvert \vec{x} \lvert = \sqrt{\vec{x} \cdot \vec{x}}$

Tobias Fünke

yeah, sorry for interrupting your conversation, Goose and imbaf. I will be silent now

Silly Goose

twas no interruption

imbAF

So a metric is...a measurable quantity in the space

or...no?

Silly Goose

21:43

a metric tells you precisely how far apart points in your space are

imbAF

ahaaa

Silly Goose

that is it gives you some notion of angle and distance between points

imbAF

so distance essentially

Silly Goose

a topological space is a little bit of a weakening of this notion. in a topological space, you have a topology which tells you if two points are close, but not in as precise a way as a metric tells you

@imbAF yes a metric is a distance function on your space

imbAF

Ok

Silly Goose

21:44

these notions are all made axiomatically precise through their actual definitions

imbAF

And after reading your definition or statement about continuity of a function and how is tied to geometrical spaces, I would say that my butchered explanation, is not far from it

when asked by Tobias

@SillyGoose What is the notion of smoothness for a group? I assume different than for a geometrical space. Unless you can say that a group such as a Lie group = a manifold, which is a geom. space. And that way the notion/concept of smoothness can be "transfered" from the group to the geom. space?

is "transfered

@SillyGoose which notions?

Tobias Fünke

22:05

@imbAF all notions which have been discussed here.

imbAF

continuity, smoothness etc?

Tobias Fünke

sure

Allie

hiiiiiiii

Tobias Fünke

hey Allie

Allie

22:36

hi tobias

Silly Goose

22:47

@imbAF the group structure involves two functions: a binary group operation and an inverse. the smoothness conditions means both of these operations are smooth functions.

$\cdot: G \times G \to G$ and $^{-1}: G \to G$ must be smooth functions

Signor Feynman

So, also in real life, I "lag" during conversations and say later what had to be said before. Now it is the time. Today we finally got the ultimate version of Tobias. A swearing Tobias

3 hours ago, by Tobias Fünke

wow. Amazon is such a s******

imbAF

@SillyGoose so differentiable ?

Silly Goose

@imbAF smooth usually means infinitely differentiable

imbAF

ok

so you can have the binary operation

and have it be infinitely differentiable ?

Tobias Fünke

23:02

@SignorFeynman hehe

imbAF

23:51

Is this true: "If the set has a finite number of elements, the function space will be a finite
dimensional vector space." ?

Tobias Fünke

something is a vector space if it obeys the corresponding axioms

in particular, you have to define scalar multiplication and addition, and define a "0" element.

what have you tried so far? where is the problem in proving or disproving the assertion?

imbAF

In the way the statement is framed. I am not sure whether I understand what it actually means. So I thought of an example. The set is G={x}. It contains 1 element. the vector space considered is $S={f_1(x)=x+1,f_2(x)=x+2,...f_m(x)=x^2,...f_n=\sqrt{x}....}$

So, how does this example not violate the statement ?

Tobias Fünke

oO

I cannot follow

imbAF

A fundamental principle of modern mathematics is that the way to understand
a space M, given as some set of points, is to look at F(M), the set of
functions on this space. This “linearizes” the problem, since the function space
is a vector space, no matter what the geometrical structure of the original set
is. If the set has a finite number of elements, the function space will be a finite
dimensional vector space.

Tobias Fünke

what is $x$ here?

imbAF

23:58

a real number of your choice

as a concrete example

Tobias Fünke

ok

but now you should try to prove this statement

it should be a straightforward exercise (and in fact is a common homework problem, I think)

imbAF

which statement

Tobias Fünke

-.-

imbAF

I gave an example

as a counter to it

Tobias Fünke

@imbAF .

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