@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.
> 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...
One 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...
an 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
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
> 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
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,...
@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
@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
@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
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
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
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
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
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
@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
@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
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."
@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
@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
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
@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 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
@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
@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
@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
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...
@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
@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"
@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
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
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...
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?
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 $\pi_i = -\dot{\phi}\partial_i \phi$
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
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
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
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
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
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
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
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
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
@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
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. :)