The h Bar

12:11 AM

Is the reason the gauge potential $A_\mu$ in electromagnetism has only one index while the Christoffel symbols have 3 because the Lie algebra of $U(1)\sim i\Bbb R$?

They're both Lie algebra valued connections unless I've missed something

naturallyInconsistent

12:59 AM

@Obliv It is only an approximation. A real capacitor is finite and will have leakage of the fields.

Maxwell's equations forbid fields to be large and non-zero one place and zero right beside it.

(without charges causing such a situation right there)

ACuriousMind

@Charlie yes, the two other indices on the Christoffels are from thinking about $\mathfrak{gl}(n)$ as matrices with two indices, see footnote 2 in this answer of mine

Slereah

3:04 AM

You can give two to the EM potential but then it's a 1x1 matrix

Sanjana

5:59 AM

What is the proper mathematical equation which defines Index which equals the determinant of a Cartan matrix. Also, where can I find the proof of the statement "Index=determinant of Cartan matrix"?

Got some references here but not any exact definition/explicit calculation in the links therein.

On this other wiki page the term is used in a different context because clearly it doesn't match the determinant as shown on the adjacent table.

nickbros123

@naturallyInconsistent if i remember it correctly the fringe field can have a contribution to the force when we insert a dielectric in between capacitor plates, though i dont remembr that calculation exactly

Ryder Rude

6:49 AM

hi

what are the merits of Bohm's implicate order-type hypotheses?

Mr. Feynman

7:11 AM

@lucabtz the gang

And now back to sleep as I only got 9h50m of sleep

naturallyInconsistent

@nickbros123 Yes, you remembered the general gist correctly.

Mr. Feynman

7:34 AM

I failed

naturallyInconsistent

@Mr.Feynman ???

123

8:03 AM

Hello Everyone...

Mr. Feynman

@naturallyInconsistent read my message above :P

nickbros123

8:26 AM

@Mr.Feynman there was a running joke that one of my computer instructors (who sleeps a lot, even in the lab xD) sleeps 18 hours a day, and my friend quipped: "he's 6 hours away from being dead" 😂

2

Mr. Feynman

8:48 AM

Good one

Cats sleep that much btw

ACuriousMind

wish I was a cat

9

Mr. Feynman

Cats don't have to deal with renormalization, yeah

Ryder Rude

cats have their own difficulties with hunting and stuff

naturallyInconsistent

@Mr.Feynman It could have been something else, even though that was the most likely possibility. Let's fight insomnia together. But in meow case it is because of too much sexi time. miehehehe.

@nickbros123 I just retrofitted the lyrics of a love song to talk about how shockingly much I sneepppuuu

@ACuriousMind sameeee

Mr. Feynman

@naturallyInconsistent I thought you were a cat

naturallyInconsistent

9:03 AM

@Mr.Feynman have you considered feline... activity...?

And feline insomnia happens when the sun is up

Ryder Rude

smilodons were the early relatives of cats

animals probably dream too

sometimes they wake up terrified

bolbteppa

9:19 AM

@Sanjana Maybe p.55 of Huang 'Lectures on Representation Theory', or P323 of Procesi 'Lie Groups: An Approach through Invariants and Representations', or P.68 of Humphreys, is what you're looking for

ACuriousMind

@Sanjana it's just the standard notion of index of a subgroup - the lattices in question are groups (under the addition of lattice vectors), so a sublattice is in particular a subgroup

Mr. Feynman

9:45 AM

@naturallyInconsistent cats are not so durable :P

I guess their activity lasts ~ 1 minute

DIRAC1930

10:03 AM

@bolbteppa This Fermi liquid theory is incredibly interesting. He is essentially saying that close to the Fermi surface (which is essentially low energy states close to the ground state), the excitations can be treated as essentially non-interacting. These are "an ideal gas of quasiparticles". However it looks like this approximation fails for higher excited states which he details exactly how in the early part of the book however I need to understand this better

naturallyInconsistent

10:36 AM

@Mr.Feynman but they can do it frequently...

Ryder Rude

10:54 AM

why do u think physics cannot explain consciousness?

Ryder Rude

11:55 AM

Sean Carroll - Is Time Real?

►

123

Hello @ryder rude

We know from N2L force is proportional to the acceleration. What if object can be deformed/deshaped by some amount by the application of single force.

If two objects having same mass but object1 is rigid and object2 is not rigid. If same amount of force applied to both objects in free empty space, does both objects accelerate by same amount?

Where object2 has ability to change its shape or deform permanently by the application of force.

Ryder Rude

12:21 PM

@123 the center of masses of both objects accelerate by the same amount, yes. u can prove this

123

@RyderRude Why nonrigid object accelerates by same amount? What i think some of the force being waste in changing the shape of the body.

Ryder Rude

suppose it was possible to "waste force" by deforming an object. u hit a squishy object and the entire force gets wasted and the object gets deformed

this would violate conservation of momentum as the total force was wasted as the resulting body is at rest

the correct idea is that kinetic energy can get wasted in deformation, impulses dont get wasted

123

@RyderRude Ooh i see

Ryder Rude

u provide an impulse of $F\cdot t$ to a body, the body must gain exact that amount of momentum, doesnt matter how squishy the object is

123

But rigidity and elasticity are different things.

Ryder Rude

12:31 PM

note that momentum is conserved in both elastic and inelastic collisions

123

@RyderRude Yes,

Ryder Rude

only kinetic energy can get wasted

123

It means $F = ma$ is true for all real objects?

Ryder Rude

@123 hav u seen the proof of the center of mass theorem?

123

@RyderRude Yes i have seen.

Ryder Rude

12:32 PM

so yeah. the proof is true for any system

energy conservation has two components : kinetic and potential, which is why one can get wasted by getting converted into the other

momentum doesnt have anything like that

123

How same acceleration is produced when object deformed on the application of force? Pls explain particle by particle

@RyderRude Yes i understand LOCM for elastic and inelastic collision. But here i am not considering collision between two objects. Here i am considering single force applied on object which is norigid.

Ryder Rude

in the initial phases, some of the portions of the body are getting accelerated while some are not. this results in an accleration of the center of mas

becuz center of mass depends on the positions of all the particles in the body

and in the final phases, the force has propagated to every part of the body using internal forces. so the entire body has acquired the momentum and is moving as a unit

the key point is just that center of mass depends on the position of all particles

123

@RyderRude Oookay. It means force is always proportional to the acceleration irrespective of the nature of the body whether elastic/inelastic and rigid/real.

Ryder Rude

@123 yes, but state it precisely using the center of mass

it's the accelerations of "individual particles" that are different for rigid and non rigid bodies

for rigid bodies, the entire body must have a common acceleration at all times

123

@RyderRude Oookay. Thanks for your good explanation. There are two more confusions about N2L.

Ryder Rude

12:39 PM

what r they

123

@RyderRude Let say i have 3 noninteracting objects (these 3 objects i considered my system) and the 3 external forces applied on each objects individually. Can we apply N2L in this situation?

Ryder Rude

yes. u can apply N2L to the whole system by doing $F1+F2+F3=(m_1+m_2+m_3)a_{cm}$

123

Or it is always necessary to apply N2L system should be interacting and net external force apply to objects of system.

Ryder Rude

and u can apply N2L to each particle individually too

@123 no

123

@RyderRude But here particles are not interacting. How COM moves?

Ryder Rude

12:42 PM

@123 u shud look at the center of mass theorem and the assumptions that go into it about the nature of internal forces

@123 Again, CoM depends on the position of all objects in the 3 body system

it need not accelerate if $F1+F2+F3=0$ btw

but in general, it does accelerate

123

@RyderRude What i am thinking interaction between objects ensures that system as whole move. and we can track COM by interacting particles.

Ryder Rude

no, the bonds are not necessary in general. just look at the definition of CoM

123

@RyderRude Oooooh i seeeeeee....

Ryder Rude

u can consider any unrelated objects far from each other as a system

the theorem allows u to apply it anywhere

u just need to understand the assumptions of the theorem

as a trivial example, take two particles of equal masses, such that CoM is in the middle and no interaction forces

if one of them moves, does the CoM move?@123

123

@RyderRude It means to apply N2L , it is not necessary particles interact each other. also it doesn't matter external forces act on one particle or two particles or all the particles?

Ryder Rude

12:47 PM

note that the CoM is $\frac{x_1(t)+x_2(t)}{2}$

@123 yes

123

@RyderRude Yes CoM moves for non interacting particles. I was confused because knK takes system of interacting particles. When i see law of conservation of linear momentum where the two particles are noninteracting.

@RyderRude Yes

@RyderRude Thanks a lot to clear my confusion.

If we have three masses along the x-axis horizontally, $m_1$ is at origin then $m_2$ is at $r_1$ from the origin and $m_3$ is at $r_2$ from the origin, where $r_2>r_1$.

How does $m_1$ can apply same amount of gravitational force on $m_2$ and $m_3$. Suppose $m_1 = m_2 = m_3$

What i am thinking. If i have a maximum force of $10N$ and i apply this force on one box which will be $10N$. But if i apply this force on two separate boxes by each hand i can not apply $10N$ to each box which will be less than $10N$ on each box but the total force will be $10N$ for both boxes.

@RyderRude Why gravity does not act that way?

Also effect of gravity does not change on $m_3$ by $m_1$ because there is hinderance of $m_2$. Where $m_2$ is sit between $m_1$ and $m_3$

Ryder Rude

1:05 PM

i dont understand the question. the maximum force of gravity that an object can apply does not have any limit

123

@RyderRude I comparing force of gravity by my muscles force. As i explained in my above box example. Sorry I don't have simple words to explain my question.

Ryder Rude

@123 note that u r studying a simplified model of gravity. gravity is non linear in general relativity, for instance.

in newtonian gravity, "the hinderance" does not really hinder anything

123

@RyderRude You are saying effect for hinderance of $m_2$ does effect . But in GR it encounter this problem?

Ryder Rude

@123 in GR, the "force" on the third mass is not a simple sum of the forces from the first and the second mass

123

Did you understand my box example? What about this in GR about gravity.

Ryder Rude

1:10 PM

in the newtonian model, every mass creates its own independent field which obeys the superposition principle

so no hinderance

123

@RyderRude Yes

Ryder Rude

so it's just a simplified model of gravity

123

If i have $10N$ force and apply this force on two separate boxes by each hand so the force on each will be less $10N$ but total force on both boxes will be $5N + 5N = 10N$

@RyderRude But gravity act differently. Gravity act same amount of force on each object. It will not distribute by fix total amount.

The same is true for electrostatic force.

Ryder Rude

@123 see this answer physics.stackexchange.com/a/77446 for why big bodies have a "force limit"

gravity works nothing like muscle force :P

the "force limit" is more like the maximum force the big body can withstand without breaking

123

@RyderRude I read the thread Thanks.

Ryder Rude

1:17 PM

@123 also see the maximum acceleration that humans can withstand in aircrafts

this stuff has to do with big bodies breaking apart

it doesnt have to do with limitations on fundamental forces

123

@RyderRude Thanks . It means gravity/electrostatic does not work like muscles force

I have searched a lot about pressure vs stress. But there still there are a lot of confusion in both topics.

Ryder Rude

i do not remember this distinction myself ...

123

@RyderRude Oooh.. Oookay. No problem. You helped me a lot BTW. Thanks

Ryder Rude

this answer looks good physics.stackexchange.com/a/107826 @123

123

@RyderRude What i am comparing how single force/multiple forces act on a body contribute to deform object and produce acceleration of the object

Ryder Rude

1:26 PM

it seems like stress is a tensor..while pressure is the trace of this

but i have never learned this formalism.. sorry @123

123

@RyderRude Can gravity produce stress within the object. Like sun produce stress or deformation of Earth. I am not asking for tidal effect. Tidal effect is due gravity difference due to size of Earth.

@RyderRude Yes i have also learned this type of explanation. But wiki says pressure is scalar not tensor. Stress is tensor. There is no clear distinction i found about pressure and stress.

Ryder Rude

@123 u shud ask what problems u have with the definition in ur book. then maybe someone else can help :)

123

Which book helped me in understanding tensor by geometric interpretation?

Ryder Rude

is the book using tensors to describe this? if yes, did they not introduce tensors?

Ryder Rude

1:51 PM

@123 this post may help physics.stackexchange.com/questions/672094/…

123

@RyderRude Thanks

Monty

2:13 PM

what kind of conclusions can we draw from the first result derived here web.stanford.edu/~peastman/statmech/interpretation.html

the author claims to derive the equipartition theorem, I follow all the steps but I feel like some steps are skipped to claim "If a system is in equilibrium with a heat bath at temperature T
, then its average kinetic energy is kT/2
per degree of freedom."

If $E(x_1,x_2,\ldots,x_n)=E_1(x_1)+E_2(x_2)+\ldots E_n(x_n)$ where $E_i(x_i)=C_i x_i^2/2$ are all quadratic then I can see that $\langle E \rangle = n k T/2$ since the result derived in that link is independent of $C$.

DIRAC1930

youtube.com/watch?v=lAkuJXGldrM lol

Monty

But i have some questions from this, firstly isnt it a very big assumption to assume that the energy can be separated in this quadratic way, moreover how should I interpret the meaning of the phrase "degree of freedom" here

ACuriousMind

@Monty "degrees of freedom" is the usual meaning, i.e. essentially an independent state variable of the system, e.g. position, angular position, momentum, etc. And sure, it's a "big" assumption, but it's a valid assumption for at least three very common cases: Energy of free particles (quadratic in linear momentum), energy of rotating objects (quadratic in angular velocity), energy of oscillators (quadratic in displacement).

Monty

2:30 PM

@ACuriousMind hi !

But I dont think "degree of freedom" is in that usual sense,

I was trying to understand this physics.stackexchange.com/questions/206303/…

ACuriousMind

@Monty I wouldn't worry about this phrase so much, "degree of freedom" can have slightly different meanings in slightly different contexts - as the answer you linked says, in this case we essentially mean the (generalized) coordinates that occur in the Hamiltonian/energy

Monty

Also a follow up question : one of the reasons the FPUT problem (en.wikipedia.org/wiki/…) was so famous is that they added non quadratic terms to the energy and then observed that the system did not exhibit equipartition. For some reason they said this went against their expectations, why would they expect equipartition if they had terms that wernt quadratic...

@ACuriousMind right so in this context it really just means the coordinates appearing in the hamiltonian.

bolbteppa

2:52 PM

@DIRAC1930 It vaguely hurts me not to know that stuff

DIRAC1930

Me too

It is difficult to find a source that does it completely but also fleshes out all the details

bolbteppa

If I was going down this road I'd do L&L5, 9, and that Abrikosov book

DIRAC1930

I am not a big fan of that Abrikosov book. L&L 9 is partially easy to atleast partially
follow the QFT stuff but understanding physically the ideas behind Fermi Liquid theory are mostly lost on me. I need to sharpen up my basic statistical mechanics

bolbteppa

That first chapter of 5 is incredibly deep and its difficult to make sense of everything e.g. why can't you just do multi-particle qm what is the distinction

DIRAC1930

3:10 PM

Thanks this stuff is amazing

bolbteppa

Scroll down around here for some puzzles on this

DIRAC1930

Man I wish I had a job where I could just learn this stuff all day

Instead I just get upset that my current PhD work doesn't involve any of this stuff

Sanjana

3:52 PM

@bolbteppa Thank you so much

@ACuriousMind Hmm

Given a sequence there is a (highly non-unique) way to write down "its next term". We simply make a polynomial go through the given numbers i.e. $f(x)=\sum_n a_n x^n$ such that $f(x_i)=s_i \forall x_i \in \mathbb{N}$ where $\{s_i\}$ is the sequence. Then we find out the function via matrix inversion or Lagrange interpolation whatever and "act" like we have a recipe to cook up the next term in the sequence!

I was thinking of generalizing this to matrices i.e. I want to construct a function $f(x,y)$ where $x$ and $y$ are row and col. no. of a matrix respectively, such $f(1,1)=a_{11},f(1,2)=a_{12},...$ where $a_{ij}$ is my matrix.

ACuriousMind

@Sanjana where's the generalization? that your sequence is a sequence of entries of a matrix does not change the procedure one bit

it's just a special case of what you said before

Sanjana

@ACuriousMind Its bivariate now? But ok...I am facing a problem. What should I take my $f(x,y)$ to be? Whatever I am taking is giving me a system of linear equations with the determinant of coeff matrix being zero, so I am not able to invert it

I could do it for just an easy $2 \times 2$ symmetric matrix case with $f(x,y)=a(x+y)+b(x^2+y^2+xy)$

ACuriousMind

@Sanjana What do you mean "bivariate"? A bunch of matrix entries is just a sequence: Choose any bijection $d : \mathbb{N}\times\mathbb{N}\to \mathbb{N}$ (preferably one that map the tuples in (1,1)...(n,n) to 1..n^2 for n the size of the matrix) and a bunch of matrix entries $a_{ij}$ is just a sequence $s_i := a_{d^{-1}(i)}$

you don't need to come up with a new function with two parameters or whatever, your previous case can already cover this one

Sanjana

@ACuriousMind That will indeed be trivial. In programming jargon, I can just "flatten" the matrix and use the single variable methodology. Yeah :)

But I insisttttttttttt...on using two parameters and complicate stuff :)

ACuriousMind

given that you're aware this "solution" is highly non-unique, I don't know why you'd want to make it any more complicated than it needs to be :P

Sanjana

4:07 PM

But that's something good, I guess? because now I can use the row number as $x$ and column number as $y$ which are more natural. For example $13th$ element of a $4 \times 4$ matrix or $4$th row $1$st col?

ACuriousMind

and in any case that problems that are easy in one variable become intractable due to coupled equations in two variables is also not uncommon

Silly Goose

@DIRAC1930 it’s time to sell out and then retire early ;)

Sanjana

@ACuriousMind Hm... Actually I saw one Mathematica tutorial where they printed 10i+j and made i run from 1 to 3 and j run from 1 to 3. So I thought "ok, they got one cute matrix...I wanna do this for all!"

Ryder Rude

@Sanjana hi. is this an infinite sequence?

nickbros123

@Sanjana wont this lead to redundancies / inconsistency since u need only n terms?

Sanjana

4:11 PM

@RyderRude You mean my $f(x)$ in the single variable case? No.

Silly Goose

@bolbteppa schlosshauer’s text on decoherence seems to cover density matrices better than is described in the alluded to conversation

i learned that you need density matrices to describe subsystems

Sanjana

@nickbros123 Yes you are right. It is a finite polynomial only. I didn't include the upper and lower limits. Sorry, bad notation.

Silly Goose

and it is just the mathematically correct way to do so

I.e. local expectation values of the full density matrix is just the expectation value of the reduced density matrix

nickbros123

@Sanjana you are letting your f(x_i)=s_i take the first n values of s_i right?

Sanjana

@nickbros123 Some finite number, yes (not $n$ cz I used it as a running index in the sum to define $f(x)$ :p). The point is to take a finite number of terms from a sequence and "guess" the next term...

My JEE teacher taught this as an "application" of Lagrange interpolation btw

nickbros123

4:16 PM

right, but ur life would be easier if u take an nth order polynomial and feed in n points right?

Ryder Rude

@Sanjana for a 3x3 matrix, u can assume f(x,y)=a+bx+cy+dx2+ey2+fxy+gx3+hy3+ix2y$. this wud give u 9 equations in 9 variables after u feed in the integer co ordinates

nickbros123

@Sanjana lagrange interpolation in JEE? thats some next level shit

DIRAC1930

@SillyGoose Or just get a job and have 2 tabs open ;)

Silly Goose

I think also i have a better appreciation of semi classical statistical mechanics after being told that semi classical approximations are what is actually used to match experimental results for quantum matter :P

At least in some cases

@DIRAC1930 hehe

But idk it still seems like you can just introduce quantum stat mech proper then do semi classical approximations on it

nickbros123

@Sanjana the maximum amount of linear algebra I did in jee was cramers rule

Silly Goose

4:22 PM

I feel like i got to go back and learn about canonical matrix forms 😭

Sanjana

@RyderRude For some reason the determinant is coming out to be zero in many similar cases. Didn't try for $3 x 3$ matrices yet...

Ryder Rude

@Sanjana but i think we need to make sure that this gives an invertible matrix. so $ f(x,y) \neq f(y,x)$ shud hold

Sanjana

@nickbros123 Many of us took statistics instead of biology, and this math teacher took some good advantage of that.

Ryder Rude

@Sanjana yes. we need to make this invertible..

Silly Goose

What is something interesting that u guys have recently learned

nickbros123

4:24 PM

@SillyGoose my reaction to everything in math

Ryder Rude

@Sanjana if all the terms are symmetric wrt exchange of x and y, then the determinant wud b zero

Silly Goose

XD

Ryder Rude

but my formula.had this term $x^2y$

so maybe it's still not invertible in some non trivial way

nickbros123

@Sanjana i envy u really. the calculus and matrix theory we learnt in JEE was pure crap. it was all tricks and hardly any rigor. simple stuff like RREF procedure on the identnty gives us the inverse, it was taught as a "trick" when they couldve presented the theory

Sanjana

David Gross, Witten's Ph.D. supervisor wrote a one line recommendation letter to Weinberg about Witten which is something along the lines of
"He is smarter than me and is probably smarter than you, so you better accept him."

Silly Goose

4:28 PM

What does one do RREF calculations for in a LA class? You can read off the rank and nullity but what else

Oh solve linear systems

nickbros123

use lin alg in python

but writing code for rref from scratch is...some use of your time

Silly Goose

Lol witten worked with weinberg (steven?)?

Sanjana

@RyderRude I tried this for a symmetric matrix, though. It works out nicely. But now it appears it works out for the general case too if we include a polynomial which has a functionally homogeneous form...need to find out why...

nickbros123

@SillyGoose you can actually prove some cool stuff from just rref calculations

@SillyGoose that steinitz exchange lemma holds for arbitrary sized subsets of any vector space whatsoever

Ryder Rude

@Sanjana oh

Sanjana

4:32 PM

@SillyGoose I dunno if he worked with him, but yeah he went to Harvard for Postdoc during a time when Weinberg was there

What I do know is: Burgess was Weinberg's PhD student. Yet he wrote no paper with him!

Silly Goose

Oh i didnt know weinberg was at harvard at some point; i associate him with UT Austin

Sanjana

He was there for 10 years only afaik

I just remembered Weinberg Witten theorem...Yes, he did work with him

Silly Goose

4:48 PM

Oh i see

Is there a reason people always refer to homology spaces as groups? If I have a chain complex of R-modules, the homology spaces will then be R-modules. We can additionally forget about the R structure and deal with the underlying group.

nickbros123

ive been having second thoughts about my method of learning--> basically i look at the axioms of the chapter and use the remainder of the chapter as a problem book, ie all theorems are problems. ive noticed though, in 3 months time, ive just completed 3 chapters of Hoffman Kunze, and 3 of rudin. how do u guys go through books that make it efficient, cuz there will be semesters where my study method could indeed get me killed

Silly Goose

For core maths i read through the text thoroughly and tried to see if I could outline the proof or intuit the results of theorems i thought were important. For less core maths I read through what seemed relevant and skipped things that seemed not. Usually if i skipped something relevant it would come up again in another result later on, so I would just go back to read what i skipped

I only really did problems for the maths i took classes for though, which is probably not the best way to do things :P

but, e.g., i wasn’t learning Lie theory to do lie theoretic computations just to obtain the vocab and conceptual understanding to know of its existence and scope in physics, so in this case it was appropriate to not really do any problems imo.

But idk if u do what u are doing and it has been working then you probably only need to do that method for the first few chapters of core maths: abstract algebra, linesr algebra, analysis. Then u can pick up whatever (not extremely exotic) math book u want and probably read through it pretty straightforwardly

nickbros123

@SillyGoose i suppose its ok? for physicists that is..

@SillyGoose i hope thats the case, cuz there are semesters like this one: complex analysis, fields modules algebras, general topology, Theory of ODE, probability and stochastic processes, Statistical mechanics, Classical field theory

ACuriousMind

@SillyGoose it seems to me you've already said the reason: They are groups! :P

bolbteppa

@nickbros123 what is your goal

nickbros123

5:03 PM

@bolbteppa I dont know really :(

I do want to become something physicist adjacent though

ACuriousMind

and in particular $\mathbb{Z}$-modules are just groups (every group is a $\mathbb{Z}$-module) and by the universal coefficient theorem the homology groups over $\mathbb{Z}$ determine all the other homologies with abelian coefficients

Silly Goose

@ACuriousMind hm i guess I’m confused say we’re doing vector space homology, so taking scalars from the rationals to be concrete. Aren’t we forgetting structure by just talking about homology groups

bolbteppa

You could spend forever reinventing the wheel, most people do not do this, there's a name for this approach and most people don't follow it, if you've even done this for the first chapter you know you could do this for the rest of the book but you see what a massive time sink this is, and this will basically be useless in physics

Silly Goose

@nickbros123 this can go one of two ways depending on how much you like math XD

ACuriousMind

@SillyGoose I mean people aren't just "forgetting" that

if it's relevant, people will talk about "homology with real coefficients" etc.

Silly Goose

5:06 PM

Oh okay

ACuriousMind

but due to the universal coefficient theorem, the "default" case is just $\mathbb{Z}$-modules, i.e. groups

Silly Goose

Groups in general? Or finitely generated abelian groups? I guess wiki says abelian groups

ACuriousMind

yes, abelian

Silly Goose

I suppose i shall see if such a distinction ends up mattering at some point i have been staying agnostic and calling them homology spaces :P

ACuriousMind

homology with non-abelian coefficients does not exist in general

Silly Goose

5:08 PM

oh

right befause you need normality

or some condition to allow always well defined quotients?

nickbros123

@bolbteppa idk if i want to become a physicist in the conventional sense--> one who builds theory or something. i realise i may not actually like that. mathematical physics, as it appears to me from afar, seems like it would be up my alley--> solving math problems related to physics / building math theory for physics etc.

Silly Goose

Which i guess so far all i have seen is that it ultimately comes from considering structures built on top of underlying abelian group structure

nickbros123

@SillyGoose i definitely like math, but even for the strongest of hearts that semester shall be "character building" to say the least

bolbteppa

I'd suggest a 'standing on the shoulders of giants' approach i.e. learning what came before so you can at least try to be in a position to advance things further in the future if you continue, e.g. Rudin is basically just doing calculus which you already know, yes its good to be able to trace everything back to some axioms but you've seen how that goes, that time spent reinventing a more rigorous form of what you already know could be better spent on something new

Silly Goose

@nickbros123 to my impression there is a good amount of opportunity to do this in actual physics: quantum information, condensed matter theory; but maybe it is not enough mathematics for what you have in mind

bolbteppa

5:13 PM

and 6 months out there is a chance you'll probably only remember the calculus version and forget a lot of these pedantic details

Silly Goose

Well rudin also covers basic topology which is conceptually new—compactness, the inklings of a generalized definition of continuity, and so on

nickbros123

@bolbteppa right, this makes a lot of sense.

@bolbteppa this hits way close to home ngl

bolbteppa

It took basically a century to iron out the nitty gritty on limits in calculus, you are trying to recreate all that in a month or so hoping that the hint of the structure of the book will be enough to do that, in reality most people just don't do this its too hard even for the people who discovered this stuff

nickbros123

I had about 4 methods to prove that the 2 harmonic converges. someone asked me that question 4-5 months later, and I was actually blank. i forgot that niche ad hoc trick, and i remembered one of the methods

which was straight forward

DIRAC1930

americanscientist.org/article/an-interlude-with-dirac This is a wholesome article

Silly Goose

5:17 PM

Some theorems will also appear again and again if at the very least as technical justifications if you care about that sort of stuff. Like Heine-Borel theorem

bolbteppa

Lets say you've only got a year left, do you want to waste months of it trying to prove the Weierstrass Approximation Theorem without ever having seen things like Bernstein polynomials and trying to rediscover these insane tricks for yourself, or do you want to do something interesting like QFT

nickbros123

youre right

bolbteppa

@SillyGoose You don't need density matrices to describe a closed system, a wave function exists for the closed system, a discrete set of stationary states even exists, yes we can artificially restrict to subsystems and ignore information outside the subsystems to use density matrices for the subsystem, but this is just artificial, my guess is that's what your source is doing which is not what my comments were doing, from this perspective density matrices appear artificial

naturallyInconsistent

@nickbros123 This is why it is important to recognise when a trick is niche and ad hoc and ought to be admired but put aside. Only learn something if it is so short and beautiful that it would be easy to remember

Silly Goose

Hm i always thought of the condition of computing local expectation values (e.g. of $O \otimes I$ in a bipartite system) to be natural and not artificial

I don’t know how it works in real life though :P

ACuriousMind

5:22 PM

That you need density matrices to describe subsystems is just a fact, I wouldn't try to classify facts as "natural" or "artificial"

Silly Goose

@naturallyInconsistent you can massage many complicated things into something short and nice ;)

like the maxwells

Or classical mechanics

nickbros123

@naturallyInconsistent yeah, i really dont know why i used to do this. some insecurity of some sort i guess

naturallyInconsistent

@SillyGoose Sadly not the spin statistics theorem

@nickbros123 That's just the usual critique of mathematicians: They are cowards who would not dare take anything they cannot prove.

Silly Goose

Hehe yes qft perhaps would evade such a nice framing

nickbros123

@naturallyInconsistent I am a totally different guy when im doing physics vs when im doing maths hehe

Silly Goose

5:32 PM

@nickbros123 the difference for me is happy vs trying to just get it over with xD. Hehe only slightly joking.

nickbros123

@SillyGoose in physics the physics takes precedence, like calculating stuff at the end of the day, i ergo take the maths for granted there. cant take the math for granted in math :P

naturallyInconsistent

@nickbros123 then maths nickbros is courting death and needs to learn to let go

Silly Goose

Even mathematicians will work off of conjectures tho

naturallyInconsistent

I mean, the least you can do is to just read the presented proofs. Or at least make a judgement and see if you would be wasting a lot of time doing useless proofs of stuff that are only needed that once, or if the theorem will be important enough to actually be worthwhile to learn the proof thereof.

Silly Goose

If the Riemann hypothesis were true, then…

Or are willing to speculate and work off of conjectures*

nickbros123

5:38 PM

@naturallyInconsistent as far as ive seen, what im doing is definitely a "me" problem, not a math-wide problem :p

even math students just read proofs and move on in life

a remodel of how i learn is in order, for sure. perhaps less focus on doing this masochistic stuff and more on problems / applications would serve me better

DIRAC1930

@nickbros123 Do you have an example of a papers in physics that you could see yourself doing research in (and then the same question for maths)?

Silly Goose

If u have applications of interest in mind then it is a nice guiding light to efficiently learn

nickbros123

@DIRAC1930 oh, im not that advanced. I do have long term plans for the "topics" ill want to eventually learn in both math and physics

Silly Goose

it is easy to feel that you need to know everything at the same time, but it is impossible to know everything :P

DIRAC1930

@nickbros123 What is your current level of physics? And also your current level of math?

nickbros123

5:48 PM

@DIRAC1930 newtonian mech (like, kleppner/ morin stuff if u know that book), Classical EM (not field theory), Classical thermodynamics (currently doing- callen & zemansky), waves stuff optics etc. in math its a hodgepodge of stuff- 3 chapters in hoffman linear algebra, 3 (some parts of 4) in rudin, ive read analysis on R fully from bartle, Lang multivariate calc (the whole thing), probability and statistics, some ODE from math methods class, currently also doing "metric spaces" by Searcoid

More Anonymous

any fans of biology here?

0

Q: Probability in the logistics equation?

Assumptions Conjecture $1$: Assumption for a species genetically/mimetically/etc there "life" prefers the state found in the maximum population for a species. Conjecture $2$: All members of a species influence each other and each influence and member is unique Conjecture $3$: There is only one do...

Discussion is welcome :)

DIRAC1930

@nickbros123 I would strongly recommend you learn QM. It will last you for the rest of your life and inform your decisions about what topics to learn

Silly Goose

Quantum is the best :D

nickbros123

@DIRAC1930 I definitely plan on that next semester. First, though, Goldstein in the summer :)

DIRAC1930

Yes that's probably a good idea. It will help the QM stuff

bolbteppa

5:57 PM

I've either bought, used or partially used those math books, even Bartle... apart from Lang they were all basically useless for physics, even HK is largely useless, and Goldstein is like 600+ pages

Course of Theoretical Physics

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

Just give in

DIRAC1930

There's a book called "Mechanics and Electrodynamics: A Shorter Course Of Theoretical Physics" which I recommend or classical mecahnics by L&L. It is way shorter and only covers essential material

bolbteppa

Yeah that would be a lot better, there are 2 of them, one on mechanics and EM, another on QM and elementary QED

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