« first day (4858 days earlier)      last day (369 days later) » 

01:27
Lol the first sentence reads as if it was not written by a human
@SillyGoose And if you understand it, then we can point out that it is a nuisance that people insist upon old mathematics. It is extremely clear to anybody who has understood things, that momentum is properly a covariant quantity, not a contravariant one like velocities are. However, when we want to talk about the direction of a momentum, we have to convert back to contravariant version of it just to see which direction it is going, and this can easily cause confusion.
right I believe I've gotten it cleared now :) thakns
 
5 hours later…
06:20
@ACuriousMind There's some appeal to have the same formalism in both cases I guess
And since the tensor product is almost identical to the cartesian product in the classical case, it seems like a good idea maybe?
 
2 hours later…
ah yes classic rational epistemic agent
@bolbteppa @Slereah would I get more credibility as a practitioner of the dark arts of crackpottery if I read some LQG?
08:16
Did you study GR - if you study LQG without studying GR then you will get more credibility in that specific dark art for sure, vixra is your next step
loopity loop
LQG appears 17 times on their qgst section
> The paper "Oops for the Loops" pointed out and amplified the arguments against the LQG approach encountered so far. The original new contribution, of this paper, illustrates how the LQG quantization scheme does not extremize the Hilbert Einstein Action, even under the classical expectations of (pseudo)Riemannian geometry. Hence the need for amendments to the approach that the LQG community must be investigated, before LQG can be trusted as a suitable model for many aspects of quantum gravity.
I don't see a Hilbert Einstein action written in the underlying paper, I do see some vague equations...
There are probably crazier theories for QG
pick one
this one looks excellent but it's in French
This one vixra.org/pdf/2402.0112v1.pdf has real equations
ohmy what is vixra
where are all these people writing these coming from xD
08:30
@bolbteppa I mean, I've been talking about GR here for a month straight :P
@bolbteppa my crackpot index is not enough for ViXra
@SillyGoose the crackpot version of ArXiV
You just haven't yet tried to derive the fine structure constant
Dec 17, 2023 at 15:34, by Mr. Feynman
Oh my, what is Vixra? The anti Arxiv?
@SillyGoose lol same reaction
Is there a user named "cinahcemq" featuring ViXra in their bio?
This one vixra.org/pdf/2312.0075v1.pdf starts so strong in the abstract, then goes south fast (still in the abstract)
Why are you searching on Vixra and not Arxiv?
Don't tell me LQG is so non-mainstream it has no place on Arxiv
I said doing LQG without knowing GR will land you on vixra
08:37
@bolbteppa is the strong part "Maxwell wrote"
It starts off as if it's going to re-do Maxwell's equations in a fast manner via quaternions, then it veers off the deep end linking it to gravity
@bolbteppa oooh I didn't read the whole message. Now I see what you did there
What about Rovelli?
There is only so much time in a day, if these warning signs are not convincing that this would be a waste of time
30
A: Why is Standard Model + Loop Quantum Gravity usually not listed as a theory of everything

Urs SchreiberOne can pinpoint the technical error in LQG explicitly: To recall, the starting point of LQG is to encode the Riemannian metric in terms of the parallel transport of the affine connection that it induces. This parallel transport is an assignment to each smooth curve in the manifold between point...

24
A: Can Loop Quantum Gravity connect in any way with string theory?

Luboš Motlan equivalence between LQG and string theory - or an LQG-like description of string theory physics - has surely been an attractive idea for many physicists (myself included) but it is impossible because of fundamental differences in virtually all general features and predictions of both framework...

and you want to see for yourself, there are a few sets of lectures on youtube that would probably speed things up
Apparently there are Poisson maps between systems and subsystems
So maybe that is the nuance I was missing
09:17
shopping for groups is a lazy way to construct QG theories
@bolbteppa Lubos is extremely biased here
Lubos says "LQG implies there cant be forces other thn gravity"
What he's saying is that on the most fundamental level at the microscopic scale, they are starting by ignoring the more dominant forces even at the microscopic scale and basing everything on a correction, which is not what string theory does
Why do people want QG to be a unified field theory
is a QG theory not enough
Who are you, the king of Siam?
09:48
@Slereah Rovelli always says LQG is not intended to be a ToE
lots of people will judge it on that ground though
@bolbteppa i dont think string theorists should be the ones to call any other theory unsatisfactory
all they have is an S matrix and they go around saying they have a ToE
@Slereah I saw a debate between him and Gross and the latter kept criticizing LQG as a defective ToE, so Rovelli said it is not a ToE
And Gross wouldn't believe him I guess
Sidenote, what an amazing quote:
> To give an example, I read Faraday's Chemical History of a Candle, a set of six Christmas lectures for children. The point of Faraday's lectures was that no matter what you look at, if you look at it closely enough, you are involved in the entire universe.
I've stupidly been going around saying watching paint dry can be the most fascinating thing in the universe, when I could have whipped out Faraday's candle
Feynman's favorite example was about a glass of wine though
FLP I, one of the first lectures. Possibly the very first?
10:03
i dont understand what he is saying
The Chemical History of a Candle was the title of a series of six lectures on the chemistry and physics of flames given by Michael Faraday at the Royal Institution in 1848, as part of the series of Christmas lectures for young people founded by Faraday in 1825 and still given there every year. The lectures described the different zones of combustion in the candle flame and the presence of carbon particles in the luminescent zone. Demonstrations included the production and examination of the properties of hydrogen, oxygen, nitrogen and carbon dioxide gases. An electrolysis cell is demonstrated,...
10:25
@SillyGoose Yes, I'm going to have to think about this some more
Can anyone sit these exams?
11:15
@DIRAC1930 Think you have to go there in person, not sure though
11:36
@SirCrackpot i always thought LQG, whether or not it describes our universe, is still a beautiful theory, especially the spinfoam approach. I wouldnt trust Lubos too much on his criticism because he is famous for being biased on this. The problem with polymer quantization is definitely there though, at least in the canonical theory, not so sure with spinfoams
however LQG is way different than any other stuff you study and it wasnt too useful to read about it when im now doing something else
on the other side studying string theory will be of some help even if you end up doing condensed matter
the book Covariant Loop Quantum Gravity and Spinfoam Theory by Rovelli and Vidotto is a good introduction to the spinfoam approach
@RyderRude this sounds funny but its a bit diminishing of some other merits of string theory
12:29
@ekardnam_ There is also a previous book by Rovelli alone
"Quantum Gravity"
Although I still don't have elements to have an opinion about either theory, I still don't understand the stance of most people on either side. Sure, it's not surprising to have fierce debates and strong opinions but... I don't think an entire theory of such a great extent could be confuted by a single message on SE (I'm not specifically talking about LQG or ST here)
I mean, were it so easy there wouldn't be people concerned with such theories. Sure, one may say that would also hold for a crackpot's argument but I don't think it's the case here (I mean, a field may attract crackpots but that doesn't make the field itself crackpottery)
In debates STheorists and LGQists make the opposite side look like flat-earthers lol
I think people are way too serious about all of this rly
I don't think it's that interesting to debate which theory is worth something
12:47
Honestly, I think it's far too premature to make any statements about the physical correctness of any thing beyond QM and GR
No, I think it is always necessary to debate about whether resources should be spent on which proto-theories. Resources are finite.
Please, like it's that expensive to pay a theorist :p
I'm not one for austerity
I think the bigger problem is that most people working on fundamental theory are mathematicians and not physicists
I think it depends on where you live but nearly all the people doing a PhD in fundamental theory where I live have a degree in mathematics not physics
As it should be
12:57
Depends what you mean by "fundamental" I guess
Mathematical physicists have background on both mathematics and physics, correct?
It depends on the mathematical physicist
I find it hard to believe mathematical physicist is completely unaware of basics of electrodynamics, quantum mechanics and relativity.
It is not impossible of course, but sounds like a minority
The training is just completely different
Just take a look on the archive for mathematical physics
@DIRAC1930 another reason for me to take a math major ig
@DIRAC1930 well the"training" is what they do with their time, so I don't see why they can't be competent with both math and physics to a reasonable level
13:08
@nickbros123 That is true
I do think the first degree effects the way you think for the rest of your life
By that I mean the questions you find interesting
For example, I only care about the photon, proton, neutron, electron and the non-rel limit lol because that's all that I really knew about when I did my physics degree
And then how those physically manifest
I don't rly believe "degree" channels your thought, but rather what you sink your teeth into more, which can be anything really, not degree dependent (although it's understandable you'd be doing X subject more in X degree)
What if someone (hypothetical) studies AG hartstone in the morning, then Landau in the evening
@DIRAC1930 nah, it's just a matter of personal preference
It really depends
@DIRAC1930 that's true even within physicists themselves
Take ACM as an example. His stance is very close to a mathematician's in many things
13:16
Have you ever opened an experimental book
@Slereah like hell I am
I would class what ACM is interested in as physics
Since he can answer physical questions
idk have you ever seen him tell someone what happens if you throw a ball
Who is ACM again?
If you look at the stuff mathematical physicists do in QM, all you will recognize is that they have a Hilbert space and then they will start introducing some weird group that has no relation to reality and then just continue going in some esoteric direction for another 50 pages
13:20
I mean, QM is pretty much understood on physical ground. I'm not astonished by the fact all that remains is technical stuff
And then if you ask them to solve the Schrodinger equation for an infinite potential well, they'll start whipping out the energy stress tensor, asking you in how many dimensions
@Zeick @ACuriousMind
Thanks
It's PSE's official AI
Just kidding, he's a mod
He's a PSE mod created by PSE modding community
13:22
@SirCrackpot Yes but fundamental theory is not
I'm starting to realise the importance in proper experiment
How would u define a mathematical physicist
@nickbros123 If you put your preprint in arXiv at hep-th instead of hep-ph you are a mathematical physicist
Stuff at hep-th is almost completely detached from experiments
Anyone working on fundamental theory should understand most of L&L
In India the case as far as I've seen is that almost all theorists are physics majors through and through. In fact it's impossible to get into any (but 1) institute for a PhD physics program with a math degree
@SirCrackpot i think that one is also on the canonical theory
13:31
In India, are the theoretical physicists more interested in physicality compared to in other countries?
@Zeick i dont really agree with this
I think trying to distinguish mathematical physicists from physicists is a bit too coarse of a distinction even :p
@DIRAC1930 as fas as I've seen, yes, a lot of them are expt high energy, or expt cond matter, th-cond matter etc. but a good chunk are also string theorists
Mathematical physics is anything that is completely devoid of any sort of reality. Hep-th contains things that are at least somewhat motivated by answering some physical question (e.g. ST etc.). However the issue is that ST and anything classed as fund theory is now moving into the former category
@DIRAC1930 Is the study of symplectic geometry mathematical physics or physics
You can rattach it to physical reality more easily than string theory
13:37
To be honest, the few people I've seen who outright call themselves mathematical physicists are in the math dept, with a math PhD
I feel like everything would be sorted if we followed the climate of physics in Russia in the 60s
@DIRAC1930 Forbidding jews from entering?
Except that
Forbidding the Copenhagen interpretation and cosmology?
They forbade most of QFT
Fadeev couldn't get his paper published in Russia so he had to get it done in America
QFT techniques applied to cond matter systems became everything though
13:50
@DIRAC1930 the climate of physics in Russia in the 60s launched Sputnik and ignighted the math wars in the US.
14:02
Russia really produced some very strong cond matter theorists
Agreed.
@SirCrackpot this (written by string theorists) attempts to be an objective evaluation, and it just finds multiple problems and basic issues, this would be a good thing to use if you wanted to get into it
Reviews some of the jist of it along with a more general discussion
This lecture is very good, e.g. the whole 'quantizing EH to some would seem like trying to find out about quantum theory by quantizing Navier-Stokes' point
@DIRAC1930 Most people in that list did around 3 exams I guess because they are entrance exams to a certain course, I really doubt anybody can just do it without being there
14:21
Yeah you're probably right
14:42
In van Nieuwenhuizen's notes on gauge field theory, it is said that if we start with a given action e.g. Yang Mills and derive a set of transformation laws such that the action remains invariant under them then that method is "dynamic" and if we write the terms i in an action by postulating that under some given transformation law the action remains invariant, then this approach is "kinematic".
What might be the reason for choice of these names dynamic and kinematic?
He uses the "dynamic" approach to derive BRST transformation laws
@bolbteppa I downloaded it from libgen. You can search for "Advanced quantum gauge field theory"...it is on pg 214-215
> "We shall now derive the remaining BRST transformation rules from the requirement that the quantum action be BRST invariant. This is thus a dynamical approach. Afterwards we shall check the nilpotency of these transformation rules. We could instead have started with a kinematical approach, namely by requiring that the BRST transformation rules be nilpotent, and then afterwards construct a quantum action that is invariant under these rules. The results of both approaches are the same."
@Sanjana do you teach English in HS, right?
@SirCrackpot yes
How does it feel to know more Physics than your colleagues who teach Physics? :P
I'd go insane (in a positive sense) if my English teacher knew gauge theory :P
14:54
I don't think that is a serious distinction, I wouldn't take what he's saying seriously
@SirCrackpot Lol
@SirCrackpot tbh they mostly don't care if somebody knows gauge theory...all they care about whether someone can solve complicated Atwood machine and wedge block problems :p
@bolbteppa hmm
@SirCrackpot ur new pic is scaring me
@Sanjana I mean, when I asked my hs prof what this weird symbol $\nabla$ was I got "don't get into this stuff, physics is a practical science"
I would have asked my English teacher, had she known this stuff :P
@RyderRude are you scared of... pots?
15:14
lol...but in the school, we have a geography teacher who knows spaceflight mechanics on one hand and complex analysis on another.
@SirCrackpot idk. it looks like something from.horror movies
@Sanjana I had a math and physics teacher who knew neither lmao
i think physics and math r more about specialisations. u can know like algebraic geometry but not know complex analysis maybe
15:38
I wish I knew more statistical physics
@bolbteppa Saved it
Just finished rewatching that, gets pretty devastating for LQG around min 30-33
15:57
@Sanjana what kind of school is this xD
@RyderRude Oh no...you had to choose algebraic geometry and complex analysis for giving an example?
Griffiths-Harris, one standard book on algebraic geometry starts with complex analysis :p
@SirCrackpot my teacher in 10th grade taught us that if a particular determinant (basically, the curl) is 0, force is conservative. This was one of the "tricks" in JEE, but no-one knew what curl was in 10th, and when someone asked if it was the curl, he said yes, but he also said don't think too much about this, you'd be wasting your time xD
@Sanjana oh
@nickbros123 Last year 4 people cracked JEE Advanced! I came to know about physics stack exchange through one of them...
16:13
maybe i shud say someone can know QM but not know classical
i have a really nice problem
suppose with 50% chance, u get cloned rn and then faint and wake up tomorrow
tomorrow, what r the chances that u hav a clone
u wake up not knowing if u urself r the clone
and u wud faint and wake up tomorrow even if u didnt get cloned
so now u dont know if u got cloned and u dont know if u r the clone
@Sanjana thats nice!
though, even for a school with JEE advanced students (mine had a few, including yours truly :p), its rather unusual for geo teachers to know complex analysis and english teachers to know string theory
@nickbros123 It has a name? It is still a "trick" in JEE, I assure you.
@Stuti its the curl $\nabla \times \vec F$. Im still semi glad its still a trick in JEE
Yeah, I was just looking at where it comes from right now
16:29
@Stuti I was also taught about a matrix (now I know it as the moment of inertia matrix), for angular momentum calculation for non fixed axis rotation. One of the brighter students in the class asked the teacher if it was a tensor. My teacher gave a big sighh and said yes, and proceeded to give a long lecture on why we should not be thinking about these things etc
I shouldnt say "taught", rather, I was told about
Yeah, they do that a lot. Didn't stop me from learning tensors on my own though, lol
Just out of interest, was everyone always interested in maths even in high school?
I don't think it was until university till I was interested in maths in the service of physics
I was definitely not interested in math in high school:P
Speaking as a representative of my friends, they run away screaming at the mention of math, but are somehow, still interested in physics.
16:43
I’m still not really that interested in the math i originally was not interested in during high school (calculus, though now analysis) xD
i like linear algebra
I think the first time I thought something very weird was going on between maths and physics was when we did an experiment to determine the mass-to-charge ratio of an electron. IIRC correctly, we had to calculate the deviation of an electron beam in a magnetic field using Maxwell's equations (which was crazy that that even worked) and then we somehow determined the mass to charge ratio of an electron which was completely crazy that it worked
@DIRAC1930 this also happened to me. But i think sometimes i end up just liking doing math more than the physics Lol
Linear algebra is cool Indeedo
@nickbros123 this is also what i have seen mostly. Although i think someone like Arthur Jaffe probably is considered a mathematical physicist, and he is in the physics department of his university
@Sanjana Id have to point out that spaceflight mechanics and complex analysis kinda go together.
@nickbros123 probably spent more time lecturing yall not to waste time on it than just properly teaching it and proving it right then and there.
17:02
It would seem like a mathematician could understand topological quantum field theories without knowing an ounce of physics
and this topic is considered mathematical physics
@DIRAC1930 in school I was interested in physics, and math was a requisite tool, so I learnt it as such, (and it was taught as such), in highschool. Ofcourse, math was interesting then, but only in my 1st year did I develop an independent liking towards theoretical math as strong as (or bigger than) the liking I have for physics
Well it’s also possible that most physicists would not understand what a topological field theory is, too
@nickbros123 at least he knew curls
I need to ask a question like "do you know any place in the literature following the same path as this paper?" but I'm afraid that's not ok here
@naturallyInconsistent I kinda understand his sentiment though the proof is not easy and would've flown right past my head then
@nickbros123 Nah, you can very easily motivate and derive the result of Green's theorem in a plane, which is the 2D analogue, and then assert that this is the 3D version of the same
17:20
@naturallyInconsistent he can perhaps motivate greens and stokes, and hand wave his way to telling that the integrand should go to 0, but the actual proof that the integrand goes to 0 might actually kill some students
@nickbros123 why would that be? If they are already at the level to learn Green's, the zeroing of Green's around a square should be a rather tolerable proof
in HS we definitely werent at the level of learning greens tbh
I miss that period in which GF were just GF to me
Fucking propagators
I still don’t get why they’re called green functions 🤔
Or perhaps as one might say: a gunction
17:42
@SillyGoose you have this guy to thank
George Green (14 July 1793 – 31 May 1841) was a British mathematical physicist who wrote An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism in 1828. The essay introduced several important concepts, among them a theorem similar to the modern Green's theorem, the idea of potential functions as currently used in physics, and the concept of what are now called Green's functions. Green was the first person to create a mathematical theory of electricity and magnetism and his theory formed the foundation for the work of other scientists such as James Clerk...
He's not even a martian
@SirCumference everyones asking whether someone with a math or physics degree qualifies as a mathematical physicist, whilst this chad had neither when he dropped his magnum opus
@Slereah Sure - but I like universal formalisms where I can explain to myself why they are the correct ones in all cases ;) my "functor of states" idea is closer to that in my mind than just postulating that the tensor product is how you combine states
18:06
@naturallyInconsistent Oh yes...both are related to geography too...He studied complex analysis while studying geodesy and cartography and spaceflight mechanics also seems like related to geography cz it is the study of sending artificial satellites instead of natural ones.
@SirCumference Maybe SillyGoose was pointing at the fact that the functions are called "Green functions" not "Green's functions"
@Sanjana they are eco-friendly of course
(that was dumb)
18:51
I found out there Green Michelin stars for restaraunts
What about green hodge stars
@SirCumference oh i meant why propagators are called green functions
@ACuriousMind I've seen people declare unified theories of physics for less
@SillyGoose see physics.stackexchange.com/a/20812/50583, there's a bit of confusion between a Green's function and kernels, but they're "morally" quite similar
@Slereah I'm not in the business of declaring unified theories just for the sake of it :P
@ACuriousMind That's how you make the big bucks
People will basically declare some unified theory anytime any theory produces a vector or tensor field really
Okay so definitively the propagator is not a green function
Or i guess if i mean the propagator in non rel textbook quantum mechanics
So the way propagator is used is just a poor choice of naming convention
19:09
@SillyGoose we don't ask that here
well I guess it's not unusual to call the same "sort of thing" the same name across different physical theories. like observables. so, I suppose calling all these things propagators is sensible. but for sakurai to call the non-rel textbook quantum mechanics propagator both a green's function and a kernal is strange
Also you have to remember that mathematicians are just as guilty of misusing physicist words
ie
In mathematics, a Higgs bundle is a pair ( E , φ ) {\displaystyle (E,\varphi )} consisting of a holomorphic vector bundle E and a Higgs field φ {\displaystyle \varphi } , a holomorphic 1-form taking values in the bundle of endomorphisms of E such that φ ∧ φ = 0 {\displaystyle \varphi \wedge \varphi =0} . Such pairs were introduced by Nigel Hitchin (1987), who named the field φ...
it does seem true that mathematicians use physics words for strange reasons indeed, but i guess maybe things come from good intentions :P
like the quantum dimension of a graded module
this is kinda sudden but does anyone else find it insane that JWST has been out for 2 years already?
like already 10% of its lifespan
it feels like it just launched yesterday
I've already been out for more than 10% of my lifespan
Much more terrifying
19:20
@Slereah well when you're 26 you've already lived a third of your expected lifespan
that kinda sucks
I'm way over that
with relativity we found out all these cool things about the flow of time
why couldn't we have found a way to go back in time and give me some more years
Lack of funding
@Slereah how dare you
Mathematicians are flawless beings
Well they're better than physicists. But that's not really a high bar to clear
19:49
it's interesting that the EPR paper has no section for citations. today, are papers ever written without citing other work in the literature?
It wasn't that long ago that citations were considered mostly optional
It was generally considered nice but it wasn't quite the huge faux pas that it is today to not cite anyone
also on the flipside
Intellectual property used to not be a thing
So some very old book would just cite gigantic swaths of other books
Or just rewrite them
Up until the 19th century writing science papers was mostly just a hobby for rich people
No need to be so formal
20:17
honk
20:28
@Slereah what matters is truth and math is the highest form thereof
By the way, some people self cite their own unpublished work
Without even the title, what the hell?
Not uncommon
> The gravitational field of vacuum is 7$^a$
$[a]$ Sir Crackpot (unpublished)
That's not better than "God told me in a dream"
yet, "truth" is what mathematicians like Frege struggled with the most
 
1 hour later…
21:36
I thought something about "active" and "passive" in field theory, but really it's just bookkeeping. It's not about what these mean or anything, just the names: thinking of it in terms of differential geometry, I considered two possibilities: changing coordinates (which I would call passive) and moving points with a diffeomorphism (which I would call active). I noticed that the naming used in QFT books is the opposite, though. Do you know why?
I mean, relabeling coordinates should be considered passive
$\phi'(x)=\phi(\Lambda^{-1}x)$ is really the condition that $\Phi(p)$ (the original function) is unchanged
OTOH, if I act with a diffeomorphism $\tilde{\Lambda}$, $\Phi(p)\mapsto\Phi(\tilde{\Lambda}p)$ and the local representation would yield $\phi'(x)=\phi(\Lambda x)$
So TL;DR should we swap the names of active and passive transformations?
god index notation makes me sad
how do i systematically keep track of minus signs when doing index notation computations involving components of the stress energy tensor?
We start out with quantities like $\pi^i = T^{i0}$. Then, we might take, say, the cross product $r \times \pi$, but I don't really understand what to do with the indices when doing such things ._.
because I am getting a wrong minus sign in a computationo im currently doing
Like for a real scalar field, we might have $\pi^i = \dot{\phi}\partial^i \phi$
but then books will do computations using $\pi_i$ which is presumably equal to $\pi_i = -\dot{\phi}\partial_i \phi$
...why would you presume that with a minus?
i thought if you raise or lower space indices you pick up a minus sign
oh maybe it is $\pi^{i} = -\dot{\phi}\partial_i\phi$
raising and lowering is so named because you literally just raise or lower the index and do nothing else :P
but isn't $\pi^\mu = \eta^{\mu\nu}\pi_\nu$, so $\pi^1 = \eta^{11}\pi_{1} = -\pi_1$ with signature $(+, -, -, -)$
21:48
and there's not really any "lowering space indices" - you have $\pi^\mu = \dot{\phi}\partial^\mu \phi$ and $\pi_\mu = \dot{\phi}\partial_\mu \phi$ and then you can look at both of these equations for any specific value of $\mu$ you want, in particular restricting it to the spatial values
I guess for the problems I am doing, I was told that $\partial_\mu$ differentiates functions of $x^\mu$ as you would expect. And, conventionally we are writing everything as functions of $x^\mu$. So, if I have an expression containing $\partial^\mu$ it is systematic to lower the index by means of the metric tensor and then compute the derivative and then get the end result
you're saying things that are not wrong but I struggle to figure out how you think they are related to each other :P
well also maybe the problem is the book my class follows uses mixed notation it seems
so for example i want to compute $(\vec{r} \times \vec{\pi})_z$ where $\vec{r}$ is i guess position vector and $\pi$ is the momentum density vector, resulting in the component of "angular momentum density" in the z direction.
yes, "$\pi^i = -\pi_i$" in Minkowski space and if we violate the rules of index notation - remember the two only rules: 1. No index appears more than twice, and not more than once upper and once lower. 2. All free indices on the two sides of an equation must have the same position.
it's clear what "$\pi^i = -\pi_i$" is supposed to mean but if you ever arrive at this "naturally", chances are you've made a mistake because if you properly manipulated all the indices you cannot end up with the $i$ in different positions on the two sides
i see okay i guess i will try to just keep indices in the proper place
but is the cross product taken to be an operation on covectors $\times$ covectors to vectors?
like should i write $\epsilon^{ijk}v_iw_j$?
21:57
In the sense that it's a covariant mess, quoting Ryan Unger
@SillyGoose yes, that's your problem (although that's just a "numerical" equality)
It is a little tricky since the cross product is typically written in vector calculus, where we often don't differentiate between vectors, covectors and bivectors
@SillyGoose we usually look at the cross product on Euclidean space where the distinction is mostly meaningless
@SillyGoose there is also the problem of the convention for the levi Civita symbol and the metric
And then the notation may gey confused with levi civita tensor
i mean suppose i have $l^3 = \epsilon^{ij3} r_i\pi_j$. Contracting this and wanting to get an actual value out, we have $l^3 = r_1\pi_2 - r_2\pi_1$, where the levi cevita has been evaluated and now I'm left with mismatched indices and confusing to evaluate covector components
@SillyGoose you're not left with mismatched indices
21:59
like if $x^i = (x, y, z)$ is $x_i = -(x, y, z)$?
the two golden rules are only for when you keep the indices abstract/unspecified
as soon as you plug something like $k = 3$ into a free index and start evaluating, you just get expressions in components
Although it's true that in euclidean space the position of indices is irrelevant, it is very common in QFT books to define Levi-Civita simbol with upper indices instead of lower indices
well, the upper/lower notation is a useful way to keep track of what's summed how regardless of the specific meaning
22:01
so in the above case my cross product is on euclidean space?
.-.
@SillyGoose what does it matter?
the minus sign you might incur cancels anyway because the r.h.s. contains only even products
i mean if it is using the metric to raise and lower indices I get extra negative signs if i have time and space components in the same term
@ACuriousMind But then they sum over spacetime indices with Einstein convention and over space indices having them repeated both upper or lower :P
That to avoid confusion with the sign
which is the problem i am running into in the ccomputation i am actually doing
I guess it's the only drawback of the holy signature
22:02
@SillyGoose it's all just convention and you need to learn how to track yours :P
i cannot tell if the problem statement is contradicting itself in notation though :P
Oh @SillyGoose just to warn you for the future. Weinberg uses the evil person signature $-+++$
@SirCrackpot Lol
If you happen to read that book remember it
i think srednicki also uses that signature sad
22:04
like, instead of worrying about the minus sign here as coming "from the metric" you might equally well consider that the cross product is determining a vector normal to the two vectors you're cross-producting, and there are two of these vectors of equal length, only distinguished by sign - this isn't "plus-metric-vs-minus-metric", it might also be right-handed-vs-left-handed
I once wasted a couple of hours thinking Weinberg was bad with indices
well i get the idea that if i fix a convention then everything will be consistent and good
What if I choose to use a compromise $--++$ signature
I'm scared of how pathological that would be, damn
index computations really are just something you need to work through often enough so they become second nature; if you stop at every step and try to think through it in abstract terms you're throwing away the one thing this notation excels at, namely making many computations tedious but rote if you only adhere to the rules
it's like learning long division
you just do the algorithm, not think about what it's doing
You can't hide it from us. I know that you did all of your GR course without writing an index
Damn phone sending messages twice
22:07
@ACuriousMind i tried to ;-;, but i have an obvious sign error at the end
well, that's normal :P
sign errors are universal
but im trying to see if the convention as set in the problem is inconsistent
Where is you sign error?
the problem is essentially computing $l_z$ (angular momentum density in z direction) for a complex scalar field as obtained from noether current. then showing that $(\vec{r} \times \vec{\pi})_z = l_z$
but i think i am confused about the mixed notation and the meaning of using subscript $z$ on these quantities
among other things
@SillyGoose some people use $x,y,z$ instead of $1,2,3$
if it's in place of an index it's obviously just "the z-component"
22:13
but would $r_1 = r^1 = x^1 = x_1 = x$?
depends on your signature convention :P
given we start with $x^\mu = (t, x, y, z)$, I think I am not understanding what $x_i$ would then be.
also probably not $=x$ unless you're in 1d
with signature $(+, -, -, -)$
i thoughtt $x_i = - (x, y, z)$
@SillyGoose $x_i$ is just $x_\mu$ but $i$ runs only over the spatial indices
@SillyGoose also not great notation, but yes, that's correct in that signature
22:19
Incidentally, is there a way to fix the index mess in equations like? $v_i=-v^i$? I guess we can't do better than $v_i=-\delta_{ij}v^j$ and leave it like that
I prefer to express this more like $v^\mu = (v^0,v^i)$ (this fixes which index position we consider the "natural" one) and then writing $v_\mu = (v_0,-v_i)$; this avoids violating the rules of index notation and is unambiguous
22:54
i found the sign error...
@SirCrackpot On myow part miao litters the expressions with $\eta^{tt}=\eta_{tt}$ so that myow expressions work for all choices of metric signature. Also, it is simply a fact that it is time that is different from space, and so it is -+++ that is natural. +--- is the one that is evil
23:45
@SirCrackpot Greg Egan wrote a novel set in a --++ universe.
Dichronauts is hard science-fiction novel by Australian author Greg Egan. The novel was published by Night Shade Books on 11 July 2017. The novel describes a universe with two time dimensions, one of which correcponds to the time perception of the characters while the other influences their space perception, for example by rotations in this directions to be impossible. Hence a symbiosis of two life forms is necessary, so that they can even see in all directions. Furthermore, many fundamental laws of physics are altered crucially: Objects can roll uphill or not fall over any more when orient...
I haven't read the novel, just the free excerpt on his site. He also has articles on the physics of that world, including a simple interactive app. gregegan.net/DICHRONAUTS/02/Interactive.html
I'm a Greg Egan fan, but I felt from the excerpt that Dichronauts is a bit too contrived, and too alien to visualise what's going on. Still, I'm glad that he at least tried to write a story in a --++ world. :)

« first day (4858 days earlier)      last day (369 days later) »