The h Bar

Slereah

13:00

That's the form in which general relativity is written [unless you consider a cosmological constant]

You can show it from the EFE by taking the trace

Amit

It doesn't mean that $\text{Riem}$ vanishes, I think

Slereah

$$\mathrm{Tr}(R_{ab} + \frac{1}{2} R g_{ab}) = \mathrm{Tr}(T_{ab})$$

This means that $R + nR/2 = T$, so $R = T / (1 + n/2)$

If the stress energy tensor is zero, so is its trace and therefore the Ricci scalar

Which indeed doesn't mean that the Riemann tensor is zero

Although it means that the only contribution to the Riemann tensor is the Weyl tensor

Relativisticcucumber

i see but then how can gravitational waves exist in vacuum?

Slereah

That's because gravitational waves are purely Weyl tensor entities

Relativisticcucumber

:o

they dont need to manifest in $T$ at all?

Slereah

13:04

that's why when you see the action of a gravitational wave on a circle of particles, it only distort that circle

and doesn't expand or shrink it

Because that's what the Ricci part of the Riemann tensor does

@Relativisticcucumber No they are related to the boundary condition if $T = 0$

ACuriousMind

@Relativisticcucumber the $T$ in the EFE is explicitly the $T$ of the matter part

Slereah

Same way that EM waves for the sourceless case means the EM waves have existed for ever

ACuriousMind

it's not some sort of general "stress-energy", the proper derivation of the EFE form the E-H action shows you that the $T$ in there is the stress-energy of the matter part of the action

Slereah

ACuriousMind

the gravitaitonal waves are just part of the Einstein tensor $G$

Slereah

13:05

^the action of the Weyl tensor

The Weyl tensor can only do shearing and not dilations

Amit

That's what shear said

ACuriousMind

but you can do some approximation for grav. waves where you split $G = G_0 + G_\text{waves}$, then you can put the $-G_\text{waves}$ on the other side of the equation and claim it's the "stress-energy" for gravitational waves :P (see e.g physics.stackexchange.com/a/486124/50583)

Slereah

What do you mean "claim"

I believe

ACuriousMind

I mean it's all just definitions

youcan call this stress-energy

but then you need to be careful that people like cucumber don't get this mixed up with the matter "stress energy" that appears in the normal EFE

I refrain from any ontological statement of whether this "is" stress-energy

asking what things "are" is just asking for trouble

Slereah

Mr. Popper was right about the paradigm but he just didn't notice that people are just taught like 5 different successive paradigms in school

Relativisticcucumber

13:09

@ACuriousMind *vegetables like cucumber

but ok i see thanks for the link

Slereah

Relativisticcucumber

@Slereah ah i just found the things you were mentioning! in gravitational wave solutions section of carroll

i will go read it

ACuriousMind

@Slereah I think it might not have been like that when Popper went to school in like the 1920s?

Slereah

@ACuriousMind possibly so

ACuriousMind

today we can get all the EM and relativity and quantum stuff in school (and also in some weirdly watered down form in chemistry) but I think that probably hadn't made it into school curricula at that point

Slereah

13:13

I'm not up to date on early 20th century German curriculim

Or Austrian

ACuriousMind

also according to Wiki he left school at 16 to just take university courses

Slereah

I have a 19th century French science textbook at home

I should give it a read

Ryder Rude

13:56

do u think Schrodinger equation would become a school.topic by 2100?

what about QFT and GR? I think they would probably be school topics by 2150 (just the most basic intro math)

Amit

Why not tomorrow?

The education system is just run by a gang of over conservative dudes

Ryder Rude

idk why but over decades advanced topics become school topics

maybe it's becuz presentation of the ideas improves

Amit

14:12

@Relativisticcucumber Dirac wrote in his GR book in his concise style "Gravity doesn't disturb the emptiness. Other fields do." (More or less, up to phrasing)

@RyderRude No, they still ruin even the topics that used to be advanced

Ryder Rude

why do u think so

Amit

First of all you have to know, the widespread law of mandatory schooling is quite new

@RyderRude Just experience, when it comes to math and science they intimidate kids

"If only you had the brains you would understand it by now" is the common approach lol

Secondly, every kind of teaching that's not initiated by the student is a problem. Teachers try to shove knowledge down kids throats

Ryder Rude

oh. Teaching is terrible, yes. But what about books? Advanced topics become convey-able in teenager books over decades

because of advancements in notation and presentation

Amit

I don't see any evidence of that, sure notation gets better. But is classical DG really any easier to grasp than it used to be?

Just an example

Ryder Rude

what is DG

Amit

14:19

Diff geometry

Ryder Rude

one example is special relativity. It's much less confusing now than when Einstein discovered it

Amit

Classical, the one with the differentials and voodoo and nice embeddings lol

Ryder Rude

and Einstein's notation for GR was also reportedly very bad

back then, u cudnt take GR as an undergraduate

it was cutting edge. it was still being understood

Amit

@RyderRude I think it's easier to get to the point wherr you think you understand SR lol, because people wrap it in nice words and notations. In reality the subtler points still take a longer time

Ryder Rude

and the ideas about the dual space being its own thing is new

it didnt use to be this way

people had some other replacement ideas for dual vectors

Amit

14:22

I think it asymptotes very quickly. Sure, when it's closer to the time of discovery the pedagogy isn't ironed out

Ryder Rude

then by ur hypothesis, QFT and GR have been ironed out enough

and they will never b school topics

Amit

With GR I can also claim the opposite: students today that want to understand it via modern DG have a lot more hoops to jump through!

Ryder Rude

but history has taught us that they will be school topicd

Amit

@RyderRude No, I say they will but it will be too late lol

Why not tomorrow

Ryder Rude

education syllabus changes very slowly

Amit

14:24

Not to bring high schoolers to the level of expertise, but just the basics

Ryder Rude

it will require a drastic shift

calculus wud need to be shifted to middle school

Amit

Give it to a team of physics experts who have pedagogical talent for 2 weeks, they'll do a beautiful presentation on it for high schoolers

Ryder Rude

and classical physics too

@Amit but high schoolers are already busy enough with advanced topics.u wud also need to push those topics to middle school

Amit

Of course, calculus is taught too late

Ryder Rude

then u can bring QM in high school

ACuriousMind

14:27

@RyderRude What are you talking about, (simple versions of) the Schrödinger equation already appear in advanced physics classes here in school

just because you didn't see it in school doesn't mean no one does

Ryder Rude

@ACuriousMind so it has already started

ACuriousMind

this is less a question of "time" and more a question of what kind of education you're getting

Ryder Rude

soon GR and QFT will be in school

Amit

Finland may be a good example too (I just know they have very advanced schooling there)

Ryder Rude

@ACuriousMind it is very much a question of time too

ACuriousMind

14:28

@RyderRude Why would you think so? I see no reason to expect the SE to occur in curricula in e.g. 2050 if those curricula don't already include it now

Ryder Rude

when new physics gets discovered, the older stuff gets pushed to school

ACuriousMind

not really how it works

Ryder Rude

Newton's stuff used to be a undergraduate topic

but when u got QM, it got pushed to school

ACuriousMind

@RyderRude The nature of schooling in 18** was very different from the nature of schooling today

you can't just ascribe the changes in curricula to "new stuff gets discovered"

Amit

@RyderRude that's wagging the tail by the dog ;-)

Ryder Rude

14:30

i'm just learning from history @ACuriousMind

ACuriousMind

there are much more fundamental shifts in society, its attitude towards learning and the meaning of education involved

@RyderRude no, you're looking at history and drawing completely bonkers conclusions :P

correlation is not causation

Ryder Rude

@ACuriousMind the nature of schooling in 2100 will be very different from the nature of schooling today

ACuriousMind

@RyderRude perhaps, but that does not imply it will involve more advanced physics topics that today's education

Ryder Rude

i'm sure syllabus will get more optimized over time

ACuriousMind

e.g. in the German math curriculum over the last 50 years, you can see topics getting removed rather than added

it is not the case that syllabi just grow and grow monotonically over time

Ryder Rude

14:32

@ACuriousMind one thing that will happen is the optimization of syllabus

Amit

Nothing guarantees that syllabus won't get worse you know. You're up to date on what's happening in Hungary?

ACuriousMind

@RyderRude what does "optimization" mean, and why do you think it will happen?

Ryder Rude

because as history has shown, advanced topics tend to get pushed to school

ACuriousMind

@RyderRude ...and I've been teiling you that's not really what's happening

Ryder Rude

people want children to know more and more by the time they're 18

@Amit what is happening in Hungary?

ACuriousMind

14:34

in many cases you can find that an advanced syllabus from 1950 contains more advanced math and physics than a 2020 syllabus

and even today syllabi are so wildly different between countries that it is ridiculous to claim this would be a universal trend in any case

Ryder Rude

@ACuriousMind are these cases not exceptions?

@ACuriousMind they do differ.but i think syllabus has motonously grown over the past 300-400 years in each country

ACuriousMind

heck, even within a country: when I started study I knew what a Taylor series was and how to differentiate and integrate many things. I had fellow students who had barely seen an integral of anything more complicated than a polynomial before.

Amit

@RyderRude They have a shortage of teachers, teachers are underpaid, lots of protests, etc.

Ryder Rude

@ACuriousMind it is upto choice which topics students want to take up

but we cant deny that calculus got pushed to school

Amit

Apparently they also don't have a separate minister in charge of education anymore

Ryder Rude

14:37

it's a choice.but it got pushed

similary, GR and QFT could get pushed. but it would be a choice
to take them up @ACuriousMind

ACuriousMind

@RyderRude no, this is just patently untrue

e.g. Germany rather recently just reduced the whole duration of advanced schooling by a year

and hence a bunch of topics had to go from every syllabus

Ryder Rude

i mean it has grown more often than not. there may be exceptions

is there a study about it

ACuriousMind

then don't say "monotonously" if you don't mean monotonously!

Ryder Rude

it's not monotonous but there is a correlation with time

ACuriousMind

@RyderRude why would that happen

Ryder Rude

14:40

because people want students to know more and more at an early age

probably because human knowledge is growing too fast

ACuriousMind

literally who thinks it's a good idea to try to teach QFT to 16 year olds if we haven't figured out yet how to teach it to adults!

Ryder Rude

and u need the cutting edge knowledge to do jobs

ACuriousMind

@RyderRude [citation needed]

Ryder Rude

so advanced topics get pushed to school

@ACuriousMind the same could be said about Newtonian mechanics back in 1800

ACuriousMind

generally the humanist position on education as an end in itself is not the main driver of education in most countries; there are already plenty of people complaining about all the "useless" math and science we teach in schools that will not matter for almost all later careers these children take

Ryder Rude

14:42

and no one understood GR for decades

@ACuriousMind yes. this is why i say that the advanced topics would be a choice but they will get pushed

what would happen when th cutting edge topic is quantum gravity

QFT would be pushed to UG

ACuriousMind

Specialization doesn't work like that

generally we don't push subfields down to general education when they grow

it's why we have so many specialists today compared to the generalists of a century ago

Ryder Rude

maybe u r correct but only partially. i still expect the syllabus to be growing more or less monotonously in each country

if we look at the data

and education would change drastically with AI

@ACuriousMind this could also be uses to support my hypothesis. suppose the future education system lets people specialsie very early

Amit

@RyderRude Historicism is the term for trying to characterize history by certain laws. I think in education and other fields today we don't really have evidence that any historicistic claims hold water.. because some changes are simply sudden and unexpected, among other reasons

Ryder Rude

so if they wanna do physics at 12, they can throw away historry

this means they can specialise in QFT early @ACuriousMind

and i think the future education system is indeed progressing toward specialisation

ACuriousMind

Oh, we're doing another sci-fi round of speculation here instead of actually looking at the world we live in. I'm out.

Ryder Rude

14:48

@ACuriousMind no, forget the AI bit. I mean that the education system is progressing toward prioritizing specialisation

if u can choose specialisation early, u can learn advanced topics early

ACuriousMind

@RyderRude again, [citation needed] in the context of school that we're talking about here

university-level education is certainly getting more specialized in some sense, but I don't see any such obvious trend in the mandatory school stages

Ryder Rude

there is increasing outrage about the useless stuff people learn

young people hate general education. it is an outdated thing

and these people will be ministers in a few decades

ACuriousMind

@RyderRude is there? is the outrage increasing? what are you basing this claim on?

students not wanting to learn all the "useless" stuff they're forced to is pretty much a given throughout the history of mandatory mass education

Ryder Rude

i'm just going off by my personal experience. do u not think that schools r progressing toward specialisation

ACuriousMind

@RyderRude No.

Ryder Rude

14:53

i will see if i can find sources

google's not helping

i couldnt find studies about trends in syllabus

ok maybe im wrong. my hypothesis was based on Newtonian mechanics and calculus pushed to school @ACuriousMind

i mean calculus isnt mandatory

but it's still available as a school topic

ACuriousMind

perhaps not where you are

Ryder Rude

i first thought that children were just getting smarter due to better living conditions

so they could learn stuff early

Amit

That's an example of historicism, the statement that "what will be is what there was"

ACuriousMind

I think you're extrapolating from an extremely limited subset of the school system of your country to some sort of general trends without having any reason to believe this extrapolation works

Ryder Rude

@Amit i didnt conclude it with certainty. but u can certainly put some weight to trends

so in conclusion, QFT and GR are objectively too hard for school

ACuriousMind

14:59

do you know anything about the differences between e.g the US and European school systems? Do you know how similar or different the systems e.g in Germany and Japan are? Do you know what their syllabi contain? If the answer to any of these questions is no, how could you ever have any certainty when talking about education and school so generally?

@RyderRude No one said that

Ryder Rude

i still think this chat will age like milk tho. looking at GR being taught in 2100 school :P

ACuriousMind

something that no one said is not a conclusion

4

all we pointed out is that you don't have any actual arguments to expect GR or QFT to be taught in school in the future

showing that one position has no support does not lead to a "conclusion" that the completely opposite position is true

Ryder Rude

right. so we cant say conclusively either way

i still think that u r putting little weight to the argument that caclulus got pushed to school

it is a good argument

Amit

Why do you use the word pushed

Ryder Rude

it got pushed from university

Amit

15:03

You have to really understand why and when and where it started being taught at schools to draw any conclusions

University pushed schools to teach it?

ACuriousMind

@RyderRude when do you think this happened?

Ryder Rude

it happened around 1950s i guess

it is a good argument. i can see many other people agreeing

ACuriousMind

@RyderRude the time frame is actually pretty close to correct for a lot of Western countries, what do you think the argument was?

Ryder Rude

i can think of 1. Calculus had become well understood and formalised using limits 2. People needed to study QM in UG

because u need to do physics research by the time u r like 23. and at that time, QFT was cutting edge

Amit

Why wasn't predicate logic pushed as much?

Ryder Rude

15:08

so calclulus in school was a need

@Amit maybe it wasnt relevant to cutting edge research

ACuriousMind

@RyderRude here's the thing: realistically it was mostly because people thought the Soviets did it and so they had to, too, in order to stay competitive

and then there's the more nebulous argumentation behind New Math in general

Ryder Rude

and Soviets did it because they wanted their people to know cutting edge physics early?

Amit

See, what ACM gave an example of is why historicism doesn't hold water

ACuriousMind

@RyderRude I don't really know why the Soviets did it but I'd suspect it to be more connected to the general "scientific" pretensions of Marxism-Leninism than rational need (in case you think something ideological like that wouldn't happen because it might fuck up the actual science, read up on Lysenkoism)

Ryder Rude

15:23

what about other fields like computer science, history and language

programming has gotten more pushed to school

idk if history or language education has become more pushed. but if they have too, then we cannot ignore the pattern

history is a funny one lol

because it grows monotonously lol

for people in year 10000, they may only study a single line on world war 2

or maybe ignore it altogether :P

Amit

Who says there will still be schools in 100 years

Ryder Rude

many people today would be ignorant about Mongols and that was only 800 years ago

and it was a huge deal at the time.

Amit

Have you heard about The Great Manure Crisis of 1894?

Ryder Rude

u wud get to see mountains of bodies

@Amit no

Amit

Google it ;-)

Just a cautionary tale about over-extrapolation

Ryder Rude

15:31

sounds like horseshit

Amit

Lol

You say it now ;-)

Ryder Rude

this too was a huge deal

probably their equivalent of climate change

ACuriousMind

@RyderRude That's mostly because of industry need: The need for programmers has increased over time as well

Ryder Rude

a similar argument can be made for cutting edge physics too

ACuriousMind

why would we need more cutting edge physicists today?

Ryder Rude

15:35

countries want their people to know the latest stuff

@ACuriousMind to stay ahead of other countries

ACuriousMind

it's already the case that many physicists and other STEM-educated people end up in jobs unrelated to their actual fields of study, i.e. we're overproducing science-educated people from the viewpoint of industry

Ryder Rude

i didnt mean that we need them in huge numbers. i mean that if new physics gets discovered, countries will want people to learn it

and they will be competing like USSR and US

it's still a stretch that this will lead to topics getting pushed to school

ACuriousMind

@RyderRude almost no cutting-edge technology is produced within a single country today; there are no comparable country-level initiatives today on science as the were during the Space Race

Ryder Rude

USA and China are certainly afraid of each other

ACuriousMind

production chains are globalized, and e.g. semiconductor manufacture is so expensive it's out of reach of anyone except already established players

Ryder Rude

15:40

@ACuriousMind and new physics opens to u alternatives to current resources

ACuriousMind

@RyderRude sure, but the competition is mostly economic; I don't see any equivalent to the Space Race

Ryder Rude

the competition is always scientific too. if u r incompenent in scientific research, u wud lose ur superpower status in no time

imagine if Nazis made the bomb first

and now imagine the upcoming bombs and defense mechanisms

ACuriousMind

most industry-relevant scientific research today is by corporations, not countries

Amit

You may be ascribing too much intelligence to the political leaders of the current "superpowers"

ACuriousMind

e.g. the next generation of computer chips will be developed by smart people at NVIDIA or AMD, not in some government lab

the next breakthrough in solar panels will happen in the labs of a solar panel manufacturer, not in a government lab

Ryder Rude

15:43

what about classified tech like in defense and weapons

ACuriousMind

industry-relevant research today is extremely privatized compared to the early Cold War

Ryder Rude

i dont mean industry relevant research.i mean research about weapons of mass destruction

and sci fi tech like invisible people

Amit

Lol

Ryder Rude

and shields to counter nuclear weapons

@Amit invisible people are def real by now

Amit

The term is what I find funny

ACuriousMind

15:45

@RyderRude we've already reached the point of MAD like a thousand times over; sure, classified military research still happens but its importance isn't like in the early nuclear age

and most armies rely largely on technology bought from a few multi-national private manufacturers, too

unless you're an isolated country like North Korea you aren't really dependent on any research of your own

Amit

Once STEM didn't mean high salary jobs like it is today, so governments may needed more propoganda. Today it's not required, just get us some nerds to make us weapons and we'll give them money, lol

Ryder Rude

@ACuriousMind how do we know that MAD is still even a thing

im pretty sure tons of research is about neutralising nuclear weapons

some of that tech is public. US can defend against nukes

and that's just the public tech. imagine the classified ones.

there is some tech about targeting nuclear weapons while they r in the air

it has a great success rate

and this is just the public information

Amit

There's also just hacking the nuke to disable it

Cos apparently the owners can

Ryder Rude

im pretty sure MAD wont even be a thing a century from.now

nukes wud b completely neutralized

and all this relies on scientific research. if a country is incompetent in that department, they r screwed

@Amit i thought that was just Hollywood lolol

arent actual nukes like offline and unhackable

ACuriousMind

@RyderRude this is just false, see e.g. armscontrolcenter.org/issues/missile-defense/…, breakingdefense.com/2022/02/…

Amit

15:52

Look up "permissive action link"

ACuriousMind

no known missile defense system would be "successful" against a large ICBM nuclear strike with modern rocket systems: Shooting down like 80% of the incoming missiles isn't a success if the remaining 20% still suffice to flatten every major population center of your country

Ryder Rude

@ACuriousMind wow. i had read it was 50% success

ACuriousMind

@RyderRude ...and you thought 50% "success" would save anyone attacked with the thousands of rockets one expects in a MAD scenario?

Ryder Rude

@ACuriousMind known is the key term here

imagine the classified defense mechanisms. MAD soon wont be. a thing, if it already isnt

ACuriousMind

@RyderRude given the general track records of government, I would not rely on the idea that there necessarily exists classified tech that performs better

Ryder Rude

15:55

@ACuriousMind no. i knew that this tech was still not sufficient to make a country unafraid of nuclear war

ACuriousMind

remember, classified projects gave us both the atom bomb and the clusterfuck of MKULTRA

Amit

Russell and Einstein worked together on Pugwash that's the only reasonable course of action I think. Well at least until the evil aliens arrive

Ryder Rude

@ACuriousMind but one thing we can be sure of is that both US and China are working on getting rid of MAD

and if they r successful, they wont make the information public

so cutting edge physics still remains the most important thing for a superpower country

ACuriousMind

To be honest I think the idea that technological advancement will save us from the threat of atomic disaster is mostly wishful thinking: When one realizes that a societal solution to the problem seems impossible, of course one turns to hoping that technology will save us

Amit

Just because it's important doesn't mean they understand it is important

We always assume governments make decisions due to a lot of hidden reasons. But I think often the main reason is ignorance/stupidity

Ryder Rude

16:01

@ACuriousMind it is important that a fewscientists in each country must keep that hope. because u dont want to be late in inventing the next equivalent of an atomic bomb

ACuriousMind

there's a not-so-rare theme in Cold War sci-fi (and among actual UFO believers) that aliens with some advanced technology will step in and turn off all the nukes

Ryder Rude

remember that Nuclear bomb was taken to be wishful thinking even by Einstein

@Amit i think they r very competent about tech. the ones who are not, will cease to be superpowers

ACuriousMind

@RyderRude almost everything possible today was once thought to be impossible; this is not an argument to start believing everything believed today to be impossible will one day be possible

Ryder Rude

@ACuriousMind i am making a much weaker argument. im not saying that such a tech will one day be possible. im saying that u shud not give up on trying to make that tech possilble

becuz u dont want to take the risk of the enemies making it possible before u do

Slereah

@ACuriousMind Do you mean Superman III

Oh wait I'm thinking of Superman IV

ACuriousMind

16:07

@Slereah lol, yes, that's one example

Ryder Rude

not just that, it is EXTREMELY urgent to make nuclear bomb neutralisers possible

because MAD wont forever protect the world.

Slereah

I would also like a machine that turns troubles into chocolate

Ryder Rude

"it's not possible, it's necessary" -Interstellar

Slereah

The Power to Turn Beavers Into Gold

►

I wish for this mysterious device

ACuriousMind

turning gold into beavers would be much funnier

have fun dealing with that, Fort Knox!

Slereah

16:10

Not everyone is into it

Freakazoid knows how to handle creeps (Good Quality)

►

Ryder Rude

if a country learns to weaponise a black hole in a controlled way, then MAD is gone for them

and what about dome shaped shields made of pure energy from sci-fi

those shields dont sound too ridiculous for physics to prohibit

by black hole weapons, i didnt mean interstellar black holes used as weapons. but rather, micro ones. there are theories about them

this article mentions them en.m.wikipedia.org/wiki/Micro_black_hole .In the Human-made black holes section

i just realised this is from Spider Man 2 lol

Relativisticcucumber

16:52

@RyderRude modern physics (which very vaguely introduced simple qm problems) was offered at my high school during students' senior year (i didnt take it because i did maths and bio track)

well im not sure how vague it was actually. it was using young and freedman which is the most shit book ever but i guess it can be considered introducing the ideas since may unis use this book

in the states at least

also at my uni most students in maths and sciences take classes at the graduate school starting junior year. this was too hard for me so i only did this for one course but its still quite common somehow. so it will be interesting to see what becomes commonplace as time goes on

sadly it seems like graduate programs expect this :/ at least one emailed me saying so and many say on their websites :/

rip to my chances of a physics career

"competitive applicants have a 4.0 GPA, a couple grad courses under their belt, and 1-2 pubs in their area of interest"

great

can barely find my pants

nickbros123

@Relativisticcucumber damn 1-2 publications just for grad apps?

Relativisticcucumber

yup. and everyone i saw on grad cafe that got in confirmed that statement XD

riiiiiiiiip

@nickbros123 at least in the US. i have heard other countries are more lax

the thing that i hate is the boost you get from having experience in ur area of interest

bc what if u started research in a topic and wanna finish those projects and dont have access/arent advanced enough to do ur actual interest yet

well then u die i guess

one group told me they dont take students who dont have experience in high energy. my uni doesnt even have a high energy group or prof :(

ok enough sob story

Slereah

17:33

I called my affine space A and now I'm not sure how to denote the set of affine transformations on it

Is it A(A)???

Mad

18:14

i asked chat GPT to represent an operator in the energy eigenket base

$\exp\left(-\frac{iHt}{\hbar}\right) = \sum_n \exp\left(-\frac{iE_nt}{\hbar}\right) \vert E_n \rangle \langle E_n \vert
$

he gave this (the operator on the left)

my question is: i thought exp(A+B) = exp A * exp B

why does he write a sum on the right?

IE am i wrong, or is it making a mistake

ACuriousMind

@Mad in this case ChatGPTs answer is correct, but generally you should never trust an answer by that thing you can't yourself verify

Mad

Can you tell me why i am wrong

ACuriousMind

no, because I have no idea what exp(A+B) = exp(A) * exp(B) is even remotely supposed to have to do with this

Mad

defining exp(A)= \sum_k 1/k! (A)^k

ACuriousMind

I know what an exponential is

Mad

18:25

i am not sure what you mean

we defined exp of a matrix as i wrote

operators can be represented as a matrix (if finite dimension)

ACuriousMind

all that is true, still not seeing what that is supposed to have to do with this

the statement here is simply that $\exp(H)$ is diagonal in the basis $\lvert E_n\rangle$ with eigenvalues $\exp(E_n)$, so $\exp(H) = \sum_i \exp(E_i)\lvert E_i\rangle\langle E_i\rvert$

Mad

well

the sum should be inside the EXP

right?

ACuriousMind

why?

Mad

because H = sum E_i ket E_j Bra E_j?

$ H = \sum_j \vert E_j \rangle \langle E_j \vert E_j $

ACuriousMind

(I'm typing)

ACuriousMind

18:46

oh, that's not how we usually derive this, but in any case: all the projectors $P_i = \lvert E_i\rangle\langle E_i\rvert$ commute, their products are zero (orthogonality) and you have $P^k = P$, so for 2 dimensions you'd get $\exp(H) = \exp(E_1P_1 + E_2P_2) = \exp(E_1 P_1)\cdot \exp(E_2 P_2)$ and now $\exp(E_i P_i) = \sum_k \frac{E^k_i}{k!}P_i^k = 1 + \sum_{k >1}\frac{E^k_i}{k!}P_i = 1 + (\exp(E_i) - 1)P_i$. Then
$$ \exp(H) = (1 + (\exp(E_1) - 1)P_1)(1 + (\exp(E_2) - 1)P_2) = 1 + (\exp(E_1) - 1)P_1 + (\exp(E_2) - 1)P_2 = 1 - P_1 - P_2 + \exp(E_1)P_1 + \exp(E_2)P_2$$

Mad

bruh

nickbros123

@ACuriousMind chatgpt will revert it's answer and tell the opposite if u pressure it a little lol

Mad

19:07

Yes

i did exactly that

And then usually it writes something in the lines of "apologies master, you were right"

ask the question again and it repeats the answer lol

Sanjana

19:51

Hello I am new here.
I was wondering why the center of mass of an astrometric binary trace out a straight line in a declination-right ascension plot as shown in Fig 7.1 of Carroll Ostlie.
The center of mass moves with constant velocity which gives $v_\theta=r \dot{\theta}=c , v_\phi= r \cos(\theta)\dot{\phi}=d$ where $c$ and $d$ are constants...which after elimination of $t$ doesn't give a straight line in $\theta-\phi$ plane afterall.

I guess the question has to do with elementary Newtonian mechanics only.

Mad

i understood nothing

god knows what an astrometric binary trace is (i guess this is some astrophysics terms)

Sanjana

I am sorry...I don't think it is relevant here. It is just a kind of system with a center of mass...and it is acted upon by no forces, so by Newton's 1st law it should move in a straight line.

But anyway, astrometric binary is a system of two stars where one is so bright that we can't see the other one directly by observation.

and by trace, I just meant that it "makes a mark" on the declination-right ascension plot...which is like a latitude longitude plot.

Mad

now i may be not informed in your subject, but does an object moving linear in the normal coordinate space correspond to a linear movement in spherical coordinates?

chat gpt says no

Sanjana

20:07

Yes! That's what I am getting---a weird curve...But the book says otherwise.
How can a motion in a generic straight line in cartesian coordinates look like a straight even in equatorial coordinates(almost like the usual spherical polar coordinates)?

Mad

x= r cos phi , y = r sin phi, z= r sin theta

Whereas, r might be taken as a function here.

in your case, x = v_x t... and so on

Have you considered the fact that r = r(t)

Silly Goose

@Mad this statement is not true in general. only when $A$ and $B$ commute. cf. baker-campbell-hausdorff formula which solves for $Z$ in the equation $\exp(X) \exp(Y) = \exp(Z)$

I.e. $Z = X + Y$ only when $X$ and $Y$ commute

Mad

Yea thats true goose!

thanks for letting me know

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