@SillyGoose because, unlike Galilean relativity, it has a lot of interesting consequences on spacetime, on the quantum fields, and so forth?

@Slereah this is just the front part. He would later restate it in slightly more modern form.

@ACuriousMind I envy your parents :P mine told me to suck up a 3 hour per day commute and also were unsupportive of me doing physics/STEM -- I had to double degree with something else (that I later dropped behind their back) that I was no good at anyway

@ACuriousMind what miao miao tries to do is to fit everybody. Notation is fitted to the occasion; precise enough to preempt the incoming confusions that will soon appear, yet still follow-able by the weaker students

@SillyGoose did you write this, or is this a text?

@qwerty enjoyment in progess

@naturallyInconsistent :3

@naturallyInconsistent this is actually kinda opposite to how we were handled in my undergrad - it was precisely the strong students who were able to follow overloading of notation and terminology and imprecisions

I was once told by someone senior to me it's the "German style" (ordered and precise) versus "Russian style" (handwaving ensues)

but who knows, I had sample sizes of 1

hi

I wonder how many times I have attempted to read the Big nlab thing

ncatlab.org/nlab/show/…

Every time I understand a little more

nlab just won't shut up about bundles

"bundle" appears 700 times in that link

> Heisenberg proposed that even when space and time are unreliable, the notion of momentum state, which is defined far away from the experimental chamber, still works. The physical quantity he proposed as fundamental is the quantum mechanical amplitude for a group of incoming particles to turn into a group of outgoing particles, and he did not admit that there were any steps in between

"and he did not admit that there were any steps in between" this is crazy

i strongly dislike this approach to fundamental physics. just guess a S-matrix and hope u will figure out the rest

no wonder physicists have ended up with a jigsaw puzzle

@naturallyInconsistent 1. I don't think you can fit everybody, no matter how hard you try (though that doesn't mean you shouldn't try!). 2. Notation wasn't really the problem in this context, more the rapid onslaught of new concepts while many students were still reeling in general because university was so different from school.

@qwerty I'm not disagreeing. Basically, because miao miao gets to teach the first years, every lecture miao miao had to slightly change the notation, because miao miao would be building up their tolerance for notation as things go along. Note that the notation is not overloaded in the computer science sense, but maybe in the visual æsthetics sense. Still, it is very much an artistic judgement of which class of students can accept what.

@ACuriousMind yes, fitting everybody at once is a fool's errand. Still, my students really like meow, measured in the fact that they would come to find meow outside of class for their other modules' work. So, there. It is mostly a matter of fitting to the mean of the cohort.

@naturallyInconsistent g'evening :). yeah. teaching is a fine art. I admire good teachers very much.

@naturallyInconsistent You also teach?is that a side job that u have?

Good teachers are why miao miao want to do this well

@Arjun used to. Currently just one on the side. That kid is too poor to have any other option.

@naturallyInconsistent I wish I had you as my teacher lol,most of the teachers that I got were terrible and the good ones haven't taught me a course yet

@naturallyInconsistent I thought maybe I would pick up tutoring again for my uni after finishing my PhD but they don't hire graduates cos they legally have to pay more :(

@qwerty That's terrible to hear,how are graduates supposed to feed themselves? :'(

@Arjun everybody knows bad teachers. But just one or two great teachers can make all the difference.

@Arjun EXACTLYYY

@qwerty here, PhDs command an increment in tutoring fees and are highly sought after.

@Arjun well I suppose we're supposed to get real jobs. PhDs also have it tough cos technically they're paid below min wage as they are on scholarships and aren't employees

I think I've long come to the realisation that I am a mediocre physicist at best and my special talent in making it as far as I did was being insanely, ridiculously stubborn

As each day passes..I'm reconsidering doing a phd and going down the academic pipeline

also teaching is a performance art. I think of it similar to stand up comedy

@qwerty Did you do a phd?

@Arjun yes

@qwerty Agreed and I think good teachers are also often quite patient

when I said they didn't hire grads I mean PhD grads

@Arjun yes - good at public speaking, thinking on the spot, ad libbing. patient, dose of humour or theatre.

@qwerty Looks like I'll never make a good teacher lol

@Arjun well if you really want to be, you can be. I think it's possible to gain those skills by practice.

despite what @naturallyInconsistent said about not needing to know the territory though I feel that is essential for myself

@qwerty The problem is, to teach I have to go down the acdemic pipeline,i.e get a phd=>postdocs=>apply to universities for which the competition is like 200 people for a single post in my country,the other way is to be a highschool teacher for jee exam,which I think I'll hate teaching

Jee is an undergrad entrance exam in my country,the coaching centres are billion dollar businesses..they will pay you a lot,but you'd have to trivialize the subject a lot and often times reduce it to a bunch of tricks to solve the problems

@Arjun private tutoring is popular for many kids here, it pays well. there's also private tutoring companies - the one I worked for back in the day exclusively hired PhD students and postdocs, and ex academics. they were committed to doing a good job and not just forcing rote learning which I liked

@qwerty sounds amazing,are you from Australia?

uh not to dox myself too hard but a famous physics YouTuber used to work there. I admired him a lot (before he blew up on YouTube) and he was a great tutor

yeah I'm from Australia

@Arjun if you get credentials and are a good businessperson you could start a coaching college with at least some intellectual integrity, if that's the sort of thing that would appeal to you.

@qwerty Is it veritasium by any chance lmao?

no comment lol

Dayumm

@qwerty True..but i highly doubt if I posses the required business skills

well it's just one path. many paths. only wrong paths are paths that hurt yourself or others

@qwerty I'm thinking of studying machine learning and coding as a side skill

I've heard from friends that ml is math heavy(which appeals to me) plus they get paid a lot

learn whatever skills interest you :) beware of fads and buzzwords. coding though is essential in pretty much any STEM job these days - I hear kids are learning coding in high school now

@qwerty Advice taken 🙏 ,yupp I've had html,c++,python intro courses in highschool : )

But i didn't like it back then lol,I'm thinking of re-studying them,this time from proper sources

@qwerty I had (elective) coding classes in school in 2010, it's not that new an idea :P

@ACuriousMind since you're in tech now,you think it's worthwhile to invest some of my time in learning ML,or you think it's just a fad that'll die out?

@ACuriousMind you were in school in 2010?!?! if so we might be about the same age... I thought you were older :o . cos I was told no one is really into string theory since the 2000s

@qwerty I'm 31 years old :P

@ACuriousMind see the schools AND universities are better in Germany :p

@ACuriousMind *high five *

@Arjun machine learning as such will not go away (it has valid use cases and is valid research), but the current AI hype will (hopefully) die down when people realize they can't keep most of the promises being made and it's hideously expensive (OpenAI is losing a lot of money)

I just posted a meta discussion question on our GenAI policy. Please consider submitting answers, or at least comments. ;)

I posted it in response to

AI Snake Oil: What Artificial Intelligence Can Do, What It Can’t, and How to Tell the Difference aisnakeoil.com/p/starting-reading-the-ai-snake-oil

@qwerty Me, too.

@qwerty Might be, but I went to at least above-average institutions in both cases; my high school was considered one of the best in our city

@ACuriousMind :eyes: same in the case of high school - but as the saying went for the kind of school I went to "they select the students not the teachers"

my high school physics teacher was a biologist T_T

no physics training and it showed.

that's...bad :P

anyone see The Thing?

@ACuriousMind WHO was saying this in the 2010s?? :p

@qwerty pop science and string theorists

I thought the string theorists all migrated to the maths departments at that point

qft does have deepest insights imo

@qwerty not at all, at least not here

I had at least four different lecturers who were string theorists

:O

I think your university must be a lot bigger than the ones I went to

but really don't focus on the part about string theory too much, I was also burnt out on QFT - I loved learning it, but except for the Standard Model most of the hep-th QFT models also have no relevance to reality :P

@qwerty Heidelberg has around 30000 students

when I began, we were around 450 physics students in my cohort, I think

@ACuriousMind I had no idea about this. could you please name a few of the popular ones you mean - I may have seen the names in the lit without knowing...

sorry, you lost me there with the grammar :P "the popular ones" - what "ones" do you mean?

@ACuriousMind okay so my first uni I went to we ended up with 10 physics students by the end. hundreds taking first year physics - but lots of engineers and the like. second one I went to... maybe 50? but I'm not as sure.

@ACuriousMind sorry thinking and typing too fast :p - um I meant a few of the popular hep-th QFT models that people look at that aren't SM. Is that stuff like SUSY?

@ACuriousMind miao miao's not that much older than ya but mine was a full decade earlier than yours lol

@qwerty Lots of stuff - 2d conformal field theories, GUTs (i.e. SM extensions that unify electroweak and strong forces), non-Abelian gauge theories beyond the specifics of the SM, e.g. with magnetic/'t Hooft monopoles

@naturallyInconsistent are you saying you finished high school as a 10 year old?

@qwerty no engineers here, most of the applied sciences go to nearby Karlsruhe, so those 450 were actual physics students + a few mathematicians with physics as their applied field

@Arjun miao miao taught from age 12 upwards all the way to uni and enjoyed teaching every age group. My students never complain that there is "teaching to the exam" and "rote learning" because even though it is clearly motivated by "teaching to the exam", miao miao always had more than enough excess time to have plenty of fun in class and explore beyond syllabus. Very little rote learning

@qwerty no. A combination of being taught computing way earlier in the school system and being older.

@qwerty what did miao miao say that you are referring to?

@ACuriousMind I honestly thought that GUTs were supposed to be well-motivated, just difficult...

the motivation is mostly aesthetics in my eyes (wouldn't it be cool if all forces were secretly the same?)

but arjun, miao miao is quite different. miao miao likes kids. so might not be so easy to translate

ahhh

@ACuriousMind interesting! I never heard of that opinion before actually

Well, if it is pre-university topics, you definitely should know the territory pretty much completely.

although I notice you didnt mention GR + QFT

It is deep lore that you can openly point out that nobody knows the territory

@qwerty I mean that's quantum gravity research, of which string theory is one branch; to me QG is not part of "QFT proper" but I don't mean any value judgement by that

@naturallyInconsistent OK, but our syllabuses were messed up at the time I was doing it -- we were teaching superconductors and cooper pairs BEFORE a simple harmonic oscillator

it was this stupid "we need to make physics engaging and involve history" thing that was going on at the time

also was teaching cosmology without GR... I mean... you can wave your hands

but that's all youre doing

yes this was high school >.<

WTF

all just waving hands

burn them down

I heard they improved it since my time

but they always say that

change the syllabus every five years

who knows

@qwerty well, of course it's usually not framed as flippantly as I said it here - they'll talk about naturalness problems and being able to predict some of the current fundamental constants from the GUT symmetry breaking, but I never really accepted this because you hid the input by not having a constant but instead having to specify the GUT symmetry group;

I don't see this as more elegant and I think it's not actually "fewer constants", we're just hiding the constants inside the choice of group/geometry.

it would be interesting if we had any experimental indication that there is unification of strong and weak forces at high energies, but there aren't any afaik

@qwerty just a few days back, my student was telling meow meow that the examining body decided to change the syllabus so that energy is no longer "converted", but is "transferred". That's perfectly fine and good. But then they also insisted that we now say that "Energy is transferred from the kinetic energy STORE to the gravitational potential energy STORE and heat energy STORE and sound energy STORE" omg kill mew plz

@naturallyInconsistent as I mentioned my anxiety is pretty bad (for deep seated reasons inappropriate to mention in a chatroom) and I regularly blank out when put under any kind of time pressure or IRL social circumstance. so I am comfy really only when I know things inside out and from many levels higher... I have a suspicion that is the level where you are at shrug but who can tell.

@qwerty how did you get interested in philosophy

@naturallyInconsistent >.<

i prefer to learn from books. but it depends on the teacher ofc

i was watching some Susskind lectures and there was too much vague-ness

i didn't feel like I learned too much, especially things like how charges emerge in string theory

@ACuriousMind that's super interesting to me! it reminds me of how sometimes people will say symmetries somehow "explain" conserved quantities, but there's just a correspondence, not an explanation.

@qwerty yes. And Noether's theorem only shows that this correspondence holds one way : symmetries ---> conservation

idk about results about the other way

@RyderRude ...i don't think that's true >.<

@qwerty I'm not sure about the other way correspondence...

@qwerty For school level work, yes, I'm definitely at that level. It is at uni level that we have no way to know the territory fully. But even at school level work, once they press us on the quantum underpinnings, it can sometimes exceed QFT knowledge. In which case miao miao just says miao miao dunno. But those are so rare as to be negligible.

but i haven't seen any conserved quantity that didn't come from a symmetry

@naturallyInconsistent i wrote it

@SillyGoose yeah, I don't think that is correct. Even though ACM did not challenge it.

maybe one could work out Noether's theorem backwards

the other way correspondence should hold to a great extent

@RyderRude have you read the original paper??

@qwerty no...

maybe you should start there

i have read derivations in Lagrangian and Hamiltonian mech

i think the Hamiltonian mech proof def works backwards

and the Lagrangian mech one does too

so the other correspondence holds, sorry

@SillyGoose I specifically gave you the ingredients so that it can be seen as everything being done on one time slice. And I tried to entice ACM to talk about the crazy shit that is done in condensed matter physics that goes even further. I do, however, give it to you that it is not clear what it means to use the free Hamiltonian in the interacting theory, nor if we still stay in one single Hilbert space if we do so.

@naturallyInconsistent goin' to work for my job at the energy warehouse

@naturallyInconsistent very impressive indeedy

@ACuriousMind ded

some people think of energy as currency

@qwerty I don't think of it so? As in, I find no lack of profs around that can easily outdo meow on this

@qwerty now we're getting into semantics :P I would say Noether's theorem is an "explanation" - conserved quantities don't appear randomly, but they occur almost always together with a corresponding symmetry, and the more symmetries a system has, the more independent conserved quantities you can find

that's a much better story than "sometimes stuff is just conserved, I dunno"

plato.stanford.edu/entries/scientific-explanation

@naturallyInconsistent I think if you're comparing yourself to full profs you're probably ahead of the curve though. although I will say not all profs are alike.

We could get into semantics over what an explanation is :p

hi slereah!! how slereah of you to drop in with the philosophy xD

Is it deductive-nomological, statistical relevance or causal mechanical

@qwerty thanks~~ I have a feeling, though, that you can be at the same level too, and is just too modest to admit so

@ACuriousMind ok, but are symmetries more fundamental than the conserved quantities, or are they just easier to spot from the Lagrangian? :p

@qwerty the whole point of spending so much time in QFT classes to teach symmetries is precisely that symmetries are extremely difficult to spot!

@Slereah i have a philosophy of "explanations". It distinguishes between two kinds of explanations : 1. A logical explanation 2. A causal explanation

Some people still find symmetries in modern times of well-known equations

a logical explanation is giving some logical premises which can imply the thing u r explaining

it is a good explanation if the premises are acceptable

a causal explanation is more limited

e.g. u see a dead body with blood and u explain it by "a person must have been killed". this is a causal explanation

it is using the physical past to explain a phenomenon

i also think a causal explanation is a subset of a logical explanation

Noether's thereon qualifies as a "logical explanation"

@naturallyInconsistent may I ask is this possibly Taiwan?

@qwerty "fundamental" is a meaningless word :P

I would say the interesting thing is that "having a symmetry" is a kinematic property of a system that you can tell without solving the equations of motion, while "having a conserved quantity" is a dynamical property you can only check by using the equations of motion

@ACuriousMind Obviously it is its position in the great chain of being

@ACuriousMind but also, Noether's theorem is not about kinematical symmetries but about dynamical symmetries

or maybe the Proclus one idk

A kinematical symmetry says that predictions are invariant for different states at a particular moment of time

A symmetry is clearly a cause of paradigm

a dynamical symmetry is about dynamical predictions

@ACuriousMind hmmmmm

@Slereah i also see reality like a chain. What's more fundamental and what's not is an identification made by humans

> a dynamical symmetries is a set of transformations of variables that appear in the equations of motion, in the field equations, that preserve the form of the equations themselves. For example, the Euler-Lagrange equations. \textit{Not every dynamical symmetry is a Noether symmetry.}

> For example a scaling is a common case -- you could scale all your parameters by a constant amount, e.g. if the equations of motion contains nothing but derivatives. In that case your equations are covariant under rescalings. \textit{However the action may not be invariant under a rescaling, so that is not a Noether symmetry and there is no corresponding conserved quantity, or even continuity equation.}

@qwerty yes. Im aware

for kinematic symmetries, Wigner's classification is relevant in QFT

@qwerty wait, u r sayiing not every dynamical symmetry is a Noether symmetry

there may be exceptions. but Noether's theorem still used dynamical symmetries

@RyderRude tis what this lecturer said in the lecture I was following last night

but also note that i made the converse statement. That Noether's theorem uses dynamical symmetries, not that every dynamical symmetry is a Noether symmetry

@ACuriousMind may I trouble you to define what you mean by a kinematic property and a dynamical property please?

@qwerty no. TW would have everything in chinese. miao miao would be less annoyed; the insanity would be confined to that language.

@Slereah is this chart serious :P

in many religions, they say that not having a causal explanation of the universe is incoherent

but causal explanation is not even defined for the universe as a whole

causality is an within-universe thing

so i would say the phrase "causal explanation of the universe" it itself an incoherent idea

maybe they should look for "logical explanations" instead

the anthropic principle could qualify as a "logical explanation", but its premise itself is more questionable than the existence of our universe. this is y anthropic arguments are not very good imo

@naturallyInconsistent ah, sorry. you mentioned SG earlier but that you had left there

@qwerty Ignore the words, I just used them to mean what I said explicitly after that - for one of these you have to use the equations of motion, for the other not

@qwerty yes?

@naturallyInconsistent sorry, was just curious. nevermind

np

Brian Green tells a heart-breaking story about a "grand-ideas" letter he received

"it is deeply sad that someone who is deeply passionate just as we are about figuring it all out has been misled by graphs and patterns that have no bearing on the answer"

this is a very good take on the topic

@naturallyInconsistent hm well i was following an answer from stack exchange. i am now consulting Simon & Reed

That's definitely way better

hm well S&R seem to not state what the maps are

I think maybe it is easiest (though heuristic) to just suppose that $\mathcal{H}_0 \subset \mathcal{H}$ or something

Heuristically, if there is an interaction Hamiltonian $H = H_0 + V$, then one can subtract the interacting terms from it to obtain $H_0$, which is still an operator over the interaction Hilbert space since $H$ originally was an operator over the interaction Hilbert space

@Arjun lol, that one evening sitting with the ex-pure math, ML / data science convert truly did a number on us xD

@SillyGoose Of course they do, look at the discussion after theorem XI.109, in particular the partial isometries $\Omega^\pm$

Simon and Reed is not a book one can skim :P

"dont waste time on analysis man"- ex cmi student

oh i see

also why are in states denoted with a $+$ even though they are from the negative limit?

who denotes in states with a +?

it seems like weinberg and also S&R and some people on stack, but i might be misreading

It seems to only be Valter Moretti and Gold who use the more intuitive notation ($-$ for in-states)

Or maybe it is that the Møller operator sign is opposite of the in-/out-state sign

@SillyGoose Where do S&R use "+" for an in-state? They consistently denote the in-states with $\mathscr{H}_\text{in}$, no?

They have this on page 17

oh there is perhaps an explanatory remark

> Sometimes you hear in GR, that for the action to be ``physical'' it must be a scalar invariant, so it is well defined and diffeomorphically invariant. However we have just discussed that the Lagrangian density is only defined up to a four divergence -- so you could take the Einstein Hilbert action and add a divergence term which gives the Einstein-Gamma-Gamma action, which is \textit{not} a scalar invariant.

I hate it when a lecturer casually tells you he thinks your textbooks are wrong

So it seems that this $J$ operator is swept under the rug in standard treatments of textbook QFT? Is that accurate? It seems that this $J$ operator is kind of crucially what relates the free to interacting Hilbert space

because $U_0: \mathcal{H}_0 \hookleftarrow$ and $U: \mathcal{H} \hookleftarrow$ and $J: \mathcal{H}_0 \to \mathcal{H}$

yes, the whole construction of $J$ and the subtlety that there are a priori different "scattering subspaces" $\mathscr{H}_\text{in/out}$ is what standard QFT lore ignores

Okay I see. now the domain/codomain issue at least makes sense

that is satisfactory for now hehehe

but it's not so simply that you can just "add" this to the standard treatment, since standard QFT isn't really done in the context of the Wightman axioms, but Haag-Ruelle theory is based on them

@nickbros123 Lmao the ex pure math guy did have an impact on me :' )

@JohnRennie Have some time to take a look at this? physics.stackexchange.com/questions/828960/…

@Arjun your still young. its for old folks like me whos should be worrying

@Arjun what does he say

what results in physics do u find to be the deepest secrets of the universe

I think QM is deep. Non realism was a secret of the universe

looking at classification of materials in this picture, i was wondering why semiconductors have different properties than insulators. the idea i have is that if bands are filled and the fermi energy is in the band gap, we cannot "excite" electrons to the next band, so this should be insulating. if the fermi energy is in the band itself, we can easily excite electrons, so the material should be conducting. however, this analysis must be somewhat naive since it [...]

[...] doesnt distinguish between insulators and semiconductors?

@nickbros123 bro ive been on the e&m STRUGGLE BUS and everytime i think of u

we have a whole pset dedicated to method of images and i just would like to absolutely DIE. the stupid problem of yesterday required like 3 assumptions that one would only make out of desperation to make the problem solveable

@Relativisticcucumber Haha, I presume you are working through Griffiths infinite conductive sheet problem? Note that some of Griffiths problems are like that, you need to make assumptions that are so bold only god could make them

I encountered a few like that in the materials chapter

@nickbros123 jackson im afraid

the problem was an infinite conducting cylinder with a line charge iirc

but indeed the assumptions scare me

its giving stat mech vibes

@Relativisticcucumber because in the strictest sense, semiconductors are insulators. Their difference lies only in the realisation that, when doped, their behaviour can be in-between metals and insulators in such a way that neither is a good approximation and something specifically made for them should be used. People didnt even imagine that there could be semiconductors before quantum theory!

@Relativisticcucumber one must be comfortable with making bold assumptions in physics. Feynman was very upfront with that.

@ACuriousMind that's part of why miao miao wanted to get your attention on the obscene stuff in cond mat...

I have $$ x(t) = \cos(\omega_0 t) + B\cos(\omega t), \dot{x} = ....$$. Does it make sense to define $$\alpha = [\frac{\omega_0 t}{2\pi}]_1, \beta = [\frac{\omega t}{2\pi}]_1, \text{where} [\cdot]_1\equiv \cdot mod 1 $$ and study the eom's in the $(\alpha,\beta)$-plane instead of the usual phase space?

I'm being asked to study the two cases $\omega/\omega_0 \in \mathbb{Q}, \omega/\omega_0 \notin \mathbb{Q}$ but I'm not sure where this is going tbh

I'd say yes

It provides some useful results for the latter case like $r \notin \mathbb{Q} \Rightarrow \{[kr]_1, k \in \mathbb{Z}\}$ is dense in $[0,1]$

if I write $x(t) = F(\alpha) + G(\beta)$ then I get $F(\alpha +1) = F(\alpha), G(\beta+1) = G(\beta)$. This periodicity tell me to restrict myself to the square $[0,1]\times[0,1]$ in the alpha,beta plane

no wait, I don't know where I'm going lol

@naturallyInconsistent you have any idea why the exercise is telling me to do this hahaha?

@Claudio yes, it is obvious if you know what is about to happen next

honestly, I have no idea, is it some kind of closed orbit in the usual phase space?

yes

no wait I meant to say bounded orbit

not closed

it is trying to push you to see something in it. It is not exactly closed like ellipses are, but it will give you something you can visualise

how do I visualize it in the $\alpha,\beta$-plane tho???

More precisely, what's the relation between the two spaces? Like can I quickly switch from one to another? I suppose the alpha,beta plane should make things easier to visualize but I dont know

@Claudio note that $\alpha$ and $\beta$ are 1 exactly when $t$ is the respective period

so their values tell you what fraction of each of the two individual oscillations has elapsed at each point in time $t$

I would be surprised if your exercise says this is instead "of the phase space", though

because time $t$ is not a coordinate of phase space to begin with, so $\alpha(t)$ and $\beta(t)$ are not functions of the phase space, either

they're definitely interesting variables (for some value of interesting :P) but not phase space variables

Let $\nu_i = \omega_i / 2\pi$, Suppose I pick a generic time $t$ such that $[t]_1 = \frac{\delta}{\nu_0}$

where this last quantity lies obv in [0,1]

now $\beta = \frac{\nu}{\nu_0}\delta$

but the coefficient is dense in [0,1] according to the previous result

No wait

I guess for each value of $\delta \in [0,1]$ I have infinite values of $\beta$ so I can span the whole vertical line $\alpha = \delta$??

No wait, this doesn't make any sense anyways

what you're supposed to find is that as you let $t$ run from $0$ to $\infty$ and you draw $(\alpha(t),\beta(t))$ in the $\alpha$-$\beta$ space, the result looks very different depending on whether the ratio of the frequencies is rational or irrational

oh ok, I can see what may be happening in the irrational case, maybe

like I have infinite straight lines intersecting the vertical line $\alpha = \alpha^{\ast}$

and this for every value of $\alpha \in [0,1]$

the slope is given by the ratio of frequencies obv

ncatlab.org/nlab/show/biology

can category theory be used for everyday thinking? does it change ur view of everything

Can someone tell me where does equation 1.14 on page 37 gravitatoin come from?

@RyderRude he the kind of guy who studied some pure math and has burnt out a bit,now he's like what's the point of studying all this bs if it can't even land me a job,he got into more applied side of the game,stuff that could potentially land him a high paying job

@nickbros123 Brah I'm gon turn 20 in 6-7 months,I'm no longer gonna be a teen!Should also prolly get on the grindset soon..😭

@ACuriousMind Whoever he was (I don't know him, my bad), I would have loved to have such a guy as Prof

People like him make theoretical physics fun

@Mr.Feynman I believe so as well

if I have a sum of two electrostatic potentials $\phi_1(\vec{r}) + \phi_2(\vec{r})$, then if I want to shift the potential by a constant, then do I do $\phi_1(\vec{r}) + C + \phi_2(\vec{r}) + C$?

I feel like no

@MoreAnonymous He's using weird notation for just a simple matrix. But it seems like he's relating the Newtonian gravity results for tidal forces to the Riemann curvature tensor at that point

Probably because he wants to distinguish regular matrices from tensors

@MoreAnonymous It's basically this matrix when $x=y=0$

@Arjun oh

Maybe he is doing pure math on the side now

or maybe he never truly loved it in the first place

@Amit thanks ... I think it's supposed to represent the relative acceleration of a test particle?

I don't get why one of the components is different

I think I see it now

@MoreAnonymous Look again at the three equations (1.5) in page 29. The idea is that $\ddot{\xi}^i$ is the relative acceleration between the two particles ($\xi$ is a separation vector). When they lie along the same radial direction,which is $z$ in this case, they're getting farther apart because the closer particle gets attracted to earth with a greater acceleration (Newton'ially speaking :)). Same idea with $x$ and $y$, to the degree that their separation is perpendicular to the radial direction...

They're getting accelerated towards each other, due to tidal "squishing", hence the minus sign of these acceleration components.

Apart from sign, I don't see exactly how he explained why the radial direction also has a factor of $2$. But I think it's a simple computation to do (I'm too lazy rn to try... :P)

Thanks