The h Bar

0celo7

00:00

yes

DanielSank

The $k$-volume of that parallelogram is equal to the square root of the sum of the squares of the $k$-volumes of the projections of that parallelogram onto the various $k$-planes in $\mathbb{R}^n$!

0celo7

Page?

DanielSank

I forget.

Anyway, $\sqrt{\text{det}(J^t J)}$ is that $k$-volume.

I think.

That's why that expression shows up all over the place.

0celo7

Hmm.

DanielSank

Yeah, so @ChrisWhite perhaps now you share my distress.

How, for example, would we do the problem if instead of a 1D ring in 2D space, we have an $n-1$ sphere in $n$ space?

I feel like we need a procedure to rewrite the probability distribution in a way that allows integrating over the variables.

Slereah

00:08

"These are turning points of the Lagrangian which, in the Euclidean approach to QFT, correspond to maxima of the Lagrangian instead of the minimum. At first sight, they do not correspond to a normalizable state of the system. On a second look, one finds that they do not make sense at all."

:D

DanielSank

Using the machinery I described above, we may be able to write $T_{\delta \circ f}: \phi \rightarrow (1/2) \int_{f^{-1}(0)} \phi(s) / \left \lvert \sqrt{x^2 + y^2} \right \rvert \, ds$.

How do we "integrate out $y$" here?

Perhaps as this is already a 1D integral, it doesn't make any sense to integrate out $y$.

If that's the case, then is there any way to express $\delta(x^2 + y^2 - r^2)$ such that we can integrate out $y$?

Finally, @ChrisWhite, yet another approach is to Taylor expand $f$ about the points where it's zero. That idea is explored here.

AngusTheMan

00:59

Anyone know how the third step of this proof multiplies out the bracket?

0celo7

@AngusTheMan You need the formula for evaluating a 2-form.

I.e. use the definition of $\wedge$

@AngusTheMan something like $(\alpha\wedge\beta)(X,Y)=\alpha(X)\beta(Y)-\alpha(Y)\beta(X)$ modulo an overall number

AngusTheMan

@0celo7 Gotcha! :)

0537

01:29

it's finally snowing.

FenderLesPaul

01:44

it's finally cold

yuss

Sir Cumference

04:50

@0celo7 Okay, I took your advice and looked into some calculus. I think I'm now pretty good with differentials

0537

05:00

@SirCumference Learn all of calculus.

It will open doors for you.

surenbharath

hellow

Stan Shunpike

@NeuroFuzzy What is a four-force? I am habung trouble imagining this.

Sir Cumference

I imagine, @0537. Everything cool I want to learn requires it

GR, QM, etc.

Stan Shunpike

@SirCumference Personally, I have found tom apostol's Calculus Volumes 1 and 2 most helpful in learning calc

That + wikipedia is a hard combo to beat

Sir Cumference

@StanShunpike Holy...it's $200+

Stan Shunpike

05:13

Uh.....library?

Sir Cumference

Ain't got none near me

I gotta get one online

Stan Shunpike

I also suggest reading books by David Griffiths for E&M and QM.

Even if you dont understand them initially, just having them is good imo

Sir Cumference

E&M?

Stan Shunpike

Electromagnetism

Sir Cumference

Oh

Well I'm pretty good with a lot of QM's main concepts

That and GR are incredibly interesting to me

Stan Shunpike

05:15

Me too!

Sir Cumference

I gotta learn the math behind it though

Ya in college or...?

Stan Shunpike

Yup

Econ major

Sir Cumference

I'm in high school...

Stan Shunpike

Second year

Nice!

Sir Cumference

Not nice

I don't get to learn the cool stuff

I gotta stick with Newtonian physics and Galilean relativity

Stan Shunpike

05:17

Well, it depends. My high school teacher did research in supergravity with higgs

Soooooo

All depends on where ya go i guess

Sir Cumference

Yay...

Stan Shunpike

But I have friends who hate hs

Cuz they are ready for better things

Sir Cumference

Exactly

Stan Shunpike

U sound in that boat

Yeah

Sir Cumference

Why are they teaching basic chemistry to people who know a good amount of QM?

sans the math...

Stan Shunpike

05:18

Because most people dont care

At all

Sir Cumference

Sigh...

Well yeah, my teachers hate me because I ask so many questions

Stan Shunpike

Ha! I have the same problem

Sir Cumference

Like "Why doesn't electron capture happen in protium?"

Every time I ask something, they're like "Sigh..what?"

Stan Shunpike

Many people find it annoying when i get curious. They just want to get by, not learn

Sir Cumference

Same!

The other students get annoyed when I ask big questions

This is why students should be separated by performance, not age

Stan Shunpike

05:21

I totally agree

Sir Cumference

On a side note, what branches of physics have ya had to study?

Stan Shunpike

Age is not irrelevant, but it is often used in a discriminating way.

Sir Cumference

^

Stan Shunpike

Me? Uh I like GR and QM

I want to learn QFT

Sir Cumference

How about special relativity?

Conflicts a bit with QM but still interesting

Stan Shunpike

05:22

I am just starting to really master that

Sir Cumference

And then there's relativistic QM

Stan Shunpike

I finally understand what an affine space is

But i need to learn like SO_+(1,3)

And the generator stuff in SR

But at least I get the basic ideas now

Sir Cumference

I padded myself on the back for understanding vector and scalar fields... ;-;

Stan Shunpike

needed for a deeper understanding

Sir Cumference

And spin...

Haven't even begun with tensor fields

Stan Shunpike

05:24

John Lee's books are really good imo to learn differential geometry

Sir Cumference

Sigh...I gotta learn more calc first

I also wanna as many math branches as I can...

Stan Shunpike

@0celo7 can probably rattle off a bunch of good titles for u. That guy knows this stuff cold.

Sir Cumference

Differential geometry, noneuclidean geometry, mathematical analysis

Well then there's the (somewhat) uninteresting parts like statistics

Stan Shunpike

Statistics is not the most interesting for physics, but for econ i have found it incresingly interesting in ways I had not anticipated

Sir Cumference

Well yeah, I took 2 years of a stats class

Stan Shunpike

05:27

What's been the best class u have had so far?

Sir Cumference

I learned some interesting things but it wasn't the kind of "OMG THIS MAKES SENSE" that I got with other maths

Hmm...I've never really learned much from a class besides stats

Since I mostly research on my own

That's where I learn all the cool things, like why $e^πi = -1$

wait

Stan Shunpike

Do u have access to a lab? Thats a great way to break out of the hs bubble

Sir Cumference

screw it

Stan Shunpike

I worked in a genetics lab in hs. It was nice to be taken seriously

Sir Cumference

Nope, not at all

We have one lab, but we're shielding from almost any interesting project

*shielded

So yeah, I have to learn the cool things on my own

And figure out the confusing stuff...like this

Stan Shunpike

05:30

Have u watched Susskind's lectures?

Sir Cumference

archive.ncsa.illinois.edu/Cyberia/NumRel/…

Haven't heard of it

Stan Shunpike

U should, he's really well known and is great for learning advanced physics

He's a stanford prof and leading string theorist

One of the most interesting lecturers i have ever watched.

Sir Cumference

Eh...string theory

Stan Shunpike

No not string theory

Sir Cumference

It runs on mathematics and assumptions

Stan Shunpike

05:32

He does EVERYTHING

Sir Cumference

Oh...

Stan Shunpike

like he has a course on every intro level physics topic

Sir Cumference

Huh, I'll check it out

Stan Shunpike

Its like 120 lectures or something

Each topic gets like 10

Sir Cumference

He do GR and QM?

Stan Shunpike

05:33

Yeah in depth

Its a lot of fun

Sir Cumference

Cool, I'm loading one

Internet sucks...

On the mean time, ya maybe know the answer to this question?

0

Q: A camera and time dilation?

If I travelled near a black hole, my time would progress slower relative to someone on Earth. This is clear enough. However, what if we sent a probe with a camera to a black hole? When we watch the screen, would we see time through the camera's perspective — that is, would the Universe appear to ...

Stan Shunpike

@ChrisWhite likely can answer that.

Anyways gotta run. Gl with ur learning

Sir Cumference

I actually wrote a 52 upvoted answer on that site

Thanks

0537

06:03

@SirCumference Where are you learning calculus?

Sir Cumference

On a bunch of different sites

in a random order

I'm heading for differentials and integrals first

@0537 Why ya ask?

0537

Good luck, finish before winter break ends.

Sir Cumference

Unlikely

Calculus is a big subject

0537

Depends.

Sir Cumference

?

0celo7

09:57

Great, jet lagged.

@Huy $d(f,g):=\int_0^1|f(x)-g(x)|\mathrm{d}x$ is not a metric if $f,g$ are only integrable functions because they can differ on a set of measure zero. How do I show that if $f,g$ are continuous, then they don't differ on a set of measure zero?

(so that this is a metric)

@Huy I watched Goldfinger on the flight home.

The scene where he says "Man talk" and slaps her on the ass is delicious.

I could taste the SJW tears

Huy

10:18

@0celo7: take $h = f-g$, then $h^{-1}(\mathbb{R} \setminus \{0\})$ is open

0celo7

@Huy Ah, and a set of measure zero is closed?

Huy

maybe

but an open set can't be of measure zero

because any open set contains an interval

and an interval has a length $> 0$

0celo7

Yes.

Well, that's good enough for me.

Huy

yes, that scene is hilarous

Craig should have done that in every movie

0celo7

I also watched Age of Ultron. That movie was basically Mass Effect.

Secret

10:25

I have a physics slash philosophy question:

Consider a system of two isolated spins $\lvert \uparrow\rangle$ and $\lvert \downarrow\rangle$

Now consider the possible states of this system as a whole:

$$\lvert \uparrow\rangle,\lvert \uparrow\rangle$$
$$\lvert \uparrow\rangle,\lvert \downarrow\rangle$$
$$\lvert \downarrow\rangle,\lvert \downarrow\rangle$$

We know that the quantum number $M_S=\sum_i s_i$ where $s_i$ is the spin quantum number of each spin

Therefore from the above, we have
$M_S=-1,0,1$ respectively that describes the possible macrostates of the system in question

Huy

I started watching Marvel movies

never had time for them

I mean the ones from Marvel studio themselves

pretty good stuff

ACuriousMind

12:07

@Secret Once again, your basic treatment is wrong. There are not three possible states of the combined system, but four: $\lvert \uparrow \rangle \otimes \lvert \uparrow \rangle, \lvert \uparrow \rangle\otimes\lvert \downarrow \rangle,\lvert \downarrow \rangle\otimes\lvert \uparrow \rangle$ and $\lvert \downarrow \rangle\otimes\lvert \downarrow \rangle$. Furthermore, $\lvert \uparrow \rangle\otimes\lvert \downarrow \rangle$ doesn't have spin 0.

However, it is well-known (as in, you should have seen this somewhere) that the combined space of states of two spin-1/2 objects decomposes as the direct sum of a one-dimensional spin-0 space of state ("singlet") and a three-dimensional spin-1 space of state ("triplet"), so there is an altogether two-dimensional subspace which has $M_S = 0$, i.e. this doesn't uniquely identify a state.

0celo7

@ACuriousMind Well what spin does it have then

ACuriousMind

@0celo7 It sits in a mixture of the spin-0 and spin-1 representation, so assigning $m_s = 0$ to it is nonsensical, since the $\lvert s,m_s\rangle$ scheme needs a fixed $s$.

(However, it is an eigenstate of the spin operator with eigenvalue 0)

0celo7

@ACuriousMind Yes...I knew that.

ACuriousMind

@0celo7 Why are you asking me, then?

0celo7

@ACuriousMind But this is what I was really asking :P

ACuriousMind

12:13

Ah

0celo7

Juuuuust checking

0celo7

13:01

@Huy did you ever complete that proof of the coordinate invariance of the divergence

Huy

no, that's PhD analysis

0celo7

damn...

Secret

13:29

...I do recall seeing something similar in the context of constructing molecular orbital diagrams by using character tables and symmetry representation, in particular, the two spin state that you mentioned seemed to be similar to the treatment in a hydrogen molecule electronic states (because it can be shown to have a singlet and triplet state). However, the fact that I still made the mistake (and not aware that
I made the mistake, and I am still pretty confused about singlets, doublets and triplets means I should redo my exercise on this to understand it better

Secret

However (as I later realise right after I finished typing the original question) the semi philosophical question can exist independently of the two spin treatment above (I was originally trying to choose a simple enough system to illustrate my philosophical question, but as Acuriousmind pointed out, I made some mistakes

Given some observed outcomes from the system (including energy that can be measured) and processes that connects them, what are the ingredients needed in order to uniquely specify the (time) evolution of a system (that is, what else is needed to tell us that the system evol…

---
^ *I should reread the relevant chapters on spin multiplets*

0celo7

@Huy what do affine maps do that's so special

Huy

preserve some geometry, but I have never seen the expression outside of geometry courses

0celo7

13:50

@SirCumference what

Slereah

14:10

We haven't had a starred comment in three days

Say more amusing things people

ACuriousMind

@Slereah There are two singly-starred ones from yesterday.

They're...not really amusing, though :P

0celo7

@ACuriousMind Anything JD says is amusing

Slereah

JD does not renew his repertoire, though

He always says the same things

ACuriousMind

@0celo7 Our tastes in humor obviously differ :P

0celo7

@ACuriousMind do you like Louis CK or George Carlin?

or Larry David?

ACuriousMind

14:13

@0celo7 Carlin yes, CK no, never heard of David

And now I gotta go again...holidays are exhausting

0celo7

@ACuriousMind surprised that you like Carlin

Larry David wrote Seinfeld

and starred in Curb Your Enthusiasm which is p. amazing

Slereah

0celo7

and wrote it, but as I understand it a large part of that show is imprompt

Qmechanic

14:36

@DavidZ : I recommend to first agreeing on the main principles and contents of the homework-and-exercise policy. Which semantic names to best describe this policy should then be decided later.

David Z

@Qmechanic I couldn't care less about names

The idea is that we stop trying to determine whether the intent of a question is educational, and instead close questions because they are asking us to do a calculation, rather than asking about a concept

2

This should cover a lot of the homework-like questions we close already - perhaps not every single one, but most, I think - but it also covers a lot of questions that are not anywhere related to homework, and not even educational, where somebody just wants us to do some math for them

We could even consider generalizing it to a wide-ranging "insufficient effort" close reason

So to be clear, this is not about a name change or wording change, this is about a major expansion of the class of questions we can address with a single close reason

John Duffield

15:22

@Slereah : I'm constantly extending my "repertoire".

ADG

hey anyone here?

Slereah

15:33

Nope

Aaryan Dewan

hi!

John Duffield

Lo!

0celo7

15:50

My cat is sitting on the floor crying pls help

He needs to learn English

I have no clue what he wants

Terry Bollinger

@0celo7 Check water, food, cat litter first of course, but also try gently petting his head.

Slereah

Maybe he just wants you to give him attention

He demands your attention but denies your affection

Terry Bollinger

If his food/water is OK, picking him up and talking softly may help.

Slereah

Do you actually have a cat Terry

Most cats are not fond of picking up

Terry Bollinger

Many cats, many years, all of mine like being picked up, but you have to do it right, with full support of the legs and freedom to leave.

My wife, Sponge Bob, and Homer.

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