i.e. you make some Markovian assumptions about the environment and you end up with that equation?
I don't really understand what the question is, then - if you could solve the arbitrary problem without making assumptions, then you wouldn't be making assumptions, would you?
maybe you've been tricked by the usual exercises we do in physics: Almost no real-world systems other than the very specific ones on your exercises are exactly solvable :P
also have you heard of octant coordinates for complex projective space?
so the top image shows a particular choice of "local coordinates" (to my understanding) of $\mathbb{C}P^n$. the bottom picture is the Fubini-Study measure in said coordinates.
I am wondering if problems will arise if I just use the "local coordinate" form of elements of $\mathbb{C}P^n$ and $d\Omega_n$ to do computations. I am not really sure when the aforementioned would fail, if at all, to give the correct results.
the text, to my understanding, makes it sound like $\mathbb{C}P^n$ can be covered by two coordinate patches similar to a sphere, so perhaps it is not a problem because we only have one point not covered by the choice of local coordinates? I don't know, I am just guessing here.
i think the text presupposes familiarity with differential geometry generally :P but right now I am just trying to know enough to compute expectation values w.r.t. the Fubini-Study measure
@SillyGoose the answer is: if your coordinates are defined everywhere except on a set of zero measure, then just using the measure in the coordinate expression is fine
you can integrate in spherical coordinates because the only point where they don't work is the origin, which is not a 3d subset and hence has zero measure
@RyderRude no I just googled "chloroform damaging gloves" and found a bunch of people complaining about there being no good gloves for protecting against phenol chloroform mixtures used in DNA/RNA extraction :P
are the rays covering the 0th slot of equation 4.7 a measure 0 subset? To my understanding it is pointing out that we have not "covered the 0th slot" of our rays with one coordinate patch, so if the set of rays covering the 0th slot of equation 4.7 is a measure 0 subset, then all we have not covered is a measure 0 subset.
@SillyGoose you are, because you should have intuition for what it means in $\mathbb{R}^n$ - "measure zero" are just the sets that don't have non--zero volume
concretely, any 2d plane inside $\mathbb{R}^3$ has measure zero, which is the analogue to what you're looking at here
you completely misunderstood the point I was making, namely that you can learn a lot of history - regardless of your silly obsession with dividing it into subsets - by starting by looking it up on Wikipedia
hm so is it important to actually ultimately integrate quantities parameterized by such local coordinates? since it is the act of integrating which makes it okay to miss out on measure zero sets?
@RyderRude No, but it's improving. Slowly. Scott Aaronson mentions any noticeable progress on his blog. Here's something from a few months ago: Staggering toward quantum fault-tolerance
there is a way to write the "average over pure states" linear entanglement entropy of a time evolved state of a 2-qubit system in a (still convenient) 10+ term sum using the Bloch sphere parameterization of states. However, there is another way using the Fubini-Study measure (which is why I have been asking ACM about the Fubini-Study measure stuff) which lets me write down the exact same function but in a much cleaner sum of four terms
and they exactly agree :DD
more importantly, the fubini-study measure way very straightforwardly generalizes to higher qubit systems since there is no clear Bloch sphere analogue for higher qubit systems. and there is a very concrete expression for all the Fubini-Study in full generality
tomorrow i will try to code up something to compute it for $n$-qubit systems in mathematica :P the algebra will get unwieldy quite fast i think even though the $2$-qubit case was okay to do by hand
i wonder why i have not seen this usage of the fubini-study measure in the literature more...maybe I missed it or did not understand it :P. it seems an awfully convenient way to compute fubini-study (or Haar) averages.
@SillyGoose Hm...I haven't seen FS stuff too. Where did you get these from? I myself had a question about Haar averages a few months ago which I posted here.
There is a book "Geometry of Quantum States" by Ingemar Bengtsson and Karol Życzkowski, the latter of which is extremely prolific in terms of entanglement theory in terms of geometry (the former of which I am not as familiar). Chapter 7.6 has the statement of the Fubini-Study measure and chapter 4.7 has the background information on the local coordinatization of $\mathbb{C}P^n$ (which is used to write the Fubini-Study measure concretely).
@PM2Ring I know. I wonder what is the industry's current impression of elliptic curve cryptography.
@DIRAC1930 actually, this is good. Good books will tell you that the concept of both heat and work in thermodynamics is ill-defined. Forgetting them is ok.
@Obliv She is wrong. Instead, Derek is slowly getting people around, e.g. electroboom. AlphaPhoenix's many videos on the topic is actually the one that clarifies much more.
@Obliv no, you are wrong as to what is happening.
@SillyGoose Nobody has ever written down a Hamiltonian suitable for an entire measuring apparatus and environment and did this partial trace exactly. There is never a correct "exact time evolution" for you to compare your results with.
So, using gauge symmetry you can make the worldsheet metric conformally flat, but only locally
Then using Weyl symmetry you make it Minkowski
So a question rises: how can you replace the worldsheet metric in the Polyakov action if this only holds locally?
Mhhh I don't think we are just restricting to the range of validity of conformal flatness, otherwise there would be problems with the boundary conditions and the variation
i think these ideas are not supposed to be understood all that much. if they could be understood, we wouldnt need math and could do everything in english
these ideas r not self evident at the level of human brain. maybe they are to some alien brain.
@Mr.Feynman Why is that a problem? Renormalisation is an irritation that should only be annoying theorists. Experimentalists can make do with just tree level stuff and get a lot of good results. Just the gauge bits would already be a tonne of calculations.
@naturallyInconsistent that's cause no one trains us for that in the entire education system ... I think it is meaningful (yes assuming others get u too)
Is there a way I could pivot to become an educator?
@MoreAnonymous One is usually supposed to absorb by osmosis after seeing a lot of great examples. There is a tonne of such great examples, since the history of science is so colourful. However, one has to first ask you what do you even mean by meaningful, since you have not yet established what it is you are talking about.
@naturallyInconsistent unfortunately, the frequent reference to specific renormalization schemes in the PDG data is not due to the annoyance of theorists ;P
it's true that you can work at tree-level without worrying about renormalization too much, but many hep experiments operate at resolutions where tree-level is not sufficient
@ACuriousMind That is definitely the case; the theoretical headaches obviously impact the entire theory, but experimenters ought to simply hire cheap theorists to help them with it, rather than worry about those details themselves.
also, the independence of the tree-level results from the specific renormalization scheme doesn't mean you can just omit talking about the renormalization scheme - if you don't do any renormalization, the vacuum energy (and the other renormalization constants) remain formally infinite!
you don't need to get deep into RG flows or whatever, but basic counterterm renormalization is necessary to arrive at finite results even at tree level
Mr. F was trying to ridicule the notion that a professor could meaningfully make a SM course module covering only just tree level stuff without renormalisation. I was trying to reply to that. There is no lack of useful physics that ought to be introduced to students without having to grapple with renormalisation right off the bat along with all the new physics. Just covering them in the simplified no-renormalisation manner is perfectly valid pedagogically.
@MoreAnonymous The NS equations are usually investigated as an initial value problem on $\mathbb{R}^3$, not a boundary problem.
of course you will have boundary conditions when you look at them e.g. confined to a pipe, but that's just how it always works when you restrict the domain of an equation, nothing to do with any vector fields or the NS equations specifically
@ACuriousMind oh lol ... But if I solve via integration I would have realistic boundary conditions as integration constants ... Interestingly NS is a 2nd order equation which gives me something to say about the problem?
It's not that obscure that there's renormalization also in other contexts, but as a matter of fact many people will usually encounter renormalization in a standard hep QFT course for the first time, thus impressing on them the idea it's a rel. QFT thing
it's like a sizable fraction of students thinking that "the Hamiltonian" is a quantum concept since many places don't teach classical Hamiltonian mechanics before the first QM course
One of the advisors I once reported to, was an expert in QFT methods in condensed matter, but he admitted that his work did not use any renormalisation; he did not properly learn it himself. However, because he would be writing down very complicated Hamiltonians that already incorporated empirical values or values we supplied from DFT, in actuality he already had some self-energies and whatnot, i.e. there was some renormalisation there too.
there is a trend, and somewhat of a simplification and beauty, covering renormalisation starting with dimensional regularisation though. My prof openly asserted that there was a reason why 't Hooft really deserved his Nobel prize. I don't really like 't Hooft's writings, but that part is really scarily awesome.
@MoreAnonymous yes, that he definitely is. But I don't like the physics he is going for. Some of his teaching work, his choice of presentation irks meow.
@naturallyInconsistent yea ... But I don't blame ppl ... The entire language gives the idea of platonic ideals and objects existing as independent entities
Well that's one way for the next generation to enter the field :p
Okay here's a riddle: I have a dust particle at the surface of the water in my glass. I pour some water at the center (not where the particle is). Does the particle move toward or away from the center and why?
@Mr.Feynman If it is an introductory one that is meant to be followed up by a module that does cover the renormalisation parts, or that there is another module that solely deals with renormalisation, that should be tolerable?
@Mr.Feynman then one is supposed to combine them outside of class. This is almost always the case, actually. Universities these days tend to be so tight on teaching time that they seem to relegate all these important stuff to that.
@Mr.Feynman on shell with dim reg and nothing else?
Be glad that you get QED renormalisation. We only got scalar phi^4 renormalisation and even then, not fully explained. A whole chunk was missing between the introduction of renormalisation basics, and suddenly covering RG flow, and then the module abruptly ended.
I mean, my prof openly complained about a lack of time, and only had time to list that cut-off directly violated Lorentz invariance and is thus very ugly to work with. I forgot what he complained about Pauli-Villars, but he did say that if we really wanted to learn those, we could just look up the course textbook. All those textbooks cover them properly. And then he listed the nice bits of dim reg and just asked us to take it.
@Mr.Feynman sharp cut-off makes for extremely disgusting computations. Soft cut-off is unwieldy but much more physically pleasing. In a sense, Pauli-Villars is somewhat of the soft-cut-off style.
@MoreAnonymous errm, if you noticed that nobody is interested in it, why are you bringing to chat back to it?
In the terminology of quantum field theory, a ghost, ghost field, ghost particle, or gauge ghost is an unphysical state in a gauge theory. Ghosts are necessary to keep gauge invariance in theories where the local fields exceed a number of physical degrees of freedom.
If a given theory is self-consistent by the introduction of ghosts, these states are labeled "good". Good ghosts are virtual particles that are introduced for regularization, like Faddeev–Popov ghosts. Otherwise, "bad" ghosts admit undesired non-virtual states in a theory, like Pauli–Villars ghosts that introduce particles with negative...
"...such as Pauli–Villars ghosts, whose existence allows the probabilities to be negative thus violating unitarity"
@Mr.Feynman I suppose from the PoV of condensed matter, which Wilsonian scheme comes from, sharp cut-off is most physically motivated. However, that is coming from crystalline physics, and spacetime is probably not meant to be treated with such a scheme.
It is important that a regularisation scheme is easy to compute stuff in. Sharp cut-off is not like that.
The reason why OS is called "physical" I think is only because the renormalization conditions are essentially involve "physical" parameters e.g. the pole of the propagator is at the physical mass etc.
@naturallyInconsistent at the end of the day what matters is the final result. My point is that sharp cutoff still looks more meaningful to me than e.g. dimreg
@Mr.Feynman Technically, fractional dimensions are already a known thing from fractals. It is not that far-fetched a concept.
In particular, since you already agreed that all that matters is the final result, then it really does not matter which particular regularisation that you picked, as long as you get the correct final result. Then it becomes an important criteria that the regularisation that you picked can quickly land you upon the correct answer, regardless of its physicality. That is part and parcel of the niceness of having quasiparticles.
The rate at which an approximation scheme converges towards the correct final answer is an important consideration, be it from physics or from mathematics or computer science.
@MoreAnonymous I haven't claimed that it is bad; what do you think is bad about it? I mean, I do think it is bad, but without elaboration, I cannot tell what you absorbed about it.
@DIRAC1930 The quasiparticle concept that is Landau so importantly picked out, tells us that even though a crystal might have a lot of electrons, the way to get the condensed matter physics correct quickly is to use the quasiparticle version of electrons, where the mass of the electrons (and holes) differ from the physical masses, and so on, and the approximation scheme converges much quicker. You can prove a lot of stuff from there.
Like, aren't you one of the few who should be studying Landau religiously?
@Mr.Feynman That is also a known concept; There are no lack of youtube videos covering fractional derivatives, and thus also fractional integration along with that.
Or rather, if you want to assert that we have no good reason to expect that physics, where we are always interested in 3+1D, is supposed to depend upon the number of dimensions smoothly enough that the derivative and limit of the number of dimensions is sensible, then yes, we indeed do not have good reason to expect so. That is part of why 't Hooft's proof that it will never cause problems to all orders, is such a tour de force
@Mr.Feynman No, in maths, it is simply defined as anti-derivation. You can work stuff out that way.
I mean, there is again no lack of good youtube videos covering the fractional derivatives concept which also, in the same video, cover the fractional integrals.
@DIRAC1930 quasiparticles in crystalline physics automatically get the sharp cut-off regularisation. Mr. F and I were just talking about that before you cut in.
@DIRAC1930 I never stated that you start with quasiparticles.
@Mr.Feynman I think I get what it is you are hung up upon. Yes, you begin teaching calculus by defining them differently and proving the FTC as a theorem that asserts that those two things are inverse operations of each other. However, once you get to a certain point, it is common to extend definitions of stuff by having the inverse operation be taken as the definition.
Almost. For small $u$, $\frac{1}{1+u}\approx 1-u$. Also, $(1+u)^2\approx 1+2u$, so $\sqrt{1+u}\approx 1+u/2$. https://puzzling.stackexchange.com/a/122023/36040
So it's a pretty cool way to calculate e to medium high precision. OTOH, it does require arbitrary precision floating-point (or fixed-point) arithmetic. Whereas the "factorial base" algorithm just requires machine integers.
For seriously high precision, you can use the algorithms of Brent and the Borwein brothers which use Jacobi theta functions and the AGM to calculate logarithms. And then use Newton-Raphson to get e or e^x
@Obliv Derek was clear even in his original video that he meant just the earliest little bit of energy delivered is going to light his ideal bulb. He never asserted that it would be bright like in steady state.
@Obliv That you would assert so shows that you did not understand any bit of the correct physics being presented.
I have already pointed you towards the video itself, and the other people who have given correct elaborations on the same topic. Please go and learn. Derek is simply in the correct here, and Kathy is the one that is wrong.
@Obliv No, she is simply completely wrong in refusing to consider what the physics is being presented as is necessary follows from Maxwell's equations. She is not even attempting to discuss in good faith the contradictions in her own arguments (and the various non-scientific non-arguments she had in her earlier videos on the topic). I think I have no other choice but to actually visit her and ask her to sign my copy of her book.
It does not matter if the initial pulse lights up the bulb to anybody's standard or not, the concept that Derek is trying to convey is the real physics that actually explains how energy is transmitted in any electrical circuit. Everything else is an approximation of it.
Also, I am not a fanboy of Derek's. I have followed his work since the very beginning of his channel, i.e. more than a decade ago, but recently he is a bit more clickbait and loose with facts. His video that is a total advertisement of self-driving earned criticism. But this is not one of them.
off topic, why don't FM radios experience interference? Can't I just be a nuisance and create radio waves on the same wavelength in my home to interfere with people's radios
IIRC that is something that can be looked up. I have no intention of guessing nor delving more into this topic than I have to.
Your garage can draw mains power at up to a few kilowatts. I think going for hundreds of miles is more than possible. But without calculations, so dont pin meow to the wall for stating so.
Greg Egan has a darkly humorous short story about physics presented via the "social media circus". It also reveals how much he hates terms like geek and nerd. It's free to read on his site: In the Ruins. It even contains a nice little puzzle from celestial mechanics. (Feynman also liked that particular puzzle).