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Everything posted by joigus

  1. Again, imprecise. Non-factorizable pure n-particle states that, once an ideal measurement has been performed on them, become strict mixtures of maximum entropy. I think I got it right now. It is much shorter to say 'entangled,' but dangerously vague. I think I've read 'maximally entangled' somewhere... Otherwise, for pure states, you pick your state as making up the first vector of a basis, and then, the entropy is clearly identically zero. For the GHZ experiment, e.g., if you pick the GHZ state as the first vector of your basis, \[\rho=\left(\begin{array}{ccc} 1 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{array}\right)\] so that, \[s\left(\rho\right)=-1\ln1=0\] Instead, if you perform and ideal measurement, coherences are erased and, \[\rho=\left(\begin{array}{ccc} \frac{1}{3} & 0 & 0\\ 0 & \frac{1}{3} & 0\\ 0 & 0 & \frac{1}{3} \end{array}\right)\] so that, \[s\left(\rho\right)=-3\textrm{tr}\left(\frac{1}{3}\ln\frac{1}{3}\right)=\ln3\]
  2. Silly me, forgot: -Entropy: \[s\left(\rho\right)=-\textrm{tr}\rho\ln\rho\] When most physicists talk about entangled states, what they really mean is maximum-entropy non-factorizable pure n-particle states. They are something more than just entangled --non-separable.
  3. I watch science documentaries quite often. I find most other kind of TV quite unbearable. Overall I think it's a positive experience, because you get a lot of visual information that otherwise wouldn't be accessible to you. Plus you get to hear the researches, you learn some of the story of the scientific ideas... But I've noticed it's a bit dangerous to take some statements from them too literally. There are narrative strictures that sometimes have the unfortunate effect of adulterating the message with a pinch of sensationalism. And why is it that you never get the chance to hear a complete argument by any scientist that's being interviewed? It's like a collage of sentences.
  4. Hi, Mordred. I don't think this should be addressed to me. I was just thanking the moderators for sparing me the trouble to read all that nonsense. It really is a thankless job. That's all.
  5. I agree. And this observation has led me to further reflection. One should also consider the context too. Take, e.g., the term 'conspiracy theorist.' In a context like these forums, such tag would more than likely be pointing at logical flaws in someone's argument, or the lack of one such argument. But taken in a wider context, it would be very easy to dismiss just about any suspicion or reasonable case for conspiracy, the latter being a concept that sometimes makes perfect sense, given the context. I am firmly convinced, e.g., that Elizabeth I of England faced a real Catholic conspiracy led by the Pope and Phillip II of Spain to murder her. Same goes with 'alarmists,' 'bleeding hearts,' and many others.
  6. 🤓 ... 🤔 ... 🤭 "But... what happens at zero?," asked the GR disciple. "Zero is not zero," the GR master replied. "How could that be?," said the disciple. "Coordinates are meaningless," the great master said. "I don't understand," the student declared, utterly puzzled. "Every answer has a question in it; every question departs from an answer, they both build on each other," was the master's reply, and hit him with a tensor on his head. The student was immediately enlightened.
  7. Just to contribute another example of how so-called 'fundamental science' pays off: Einstein-Bose discovery of stimulated emission. Originally was just motivated by studying the quantum properties of photons. Went on to completely change the face of the world when Maiman came up with LASER. It may take decades, but pays off by mega-factors.
  8. (my emphasis) This is a very old post, I realize. But I think it may be worth keeping the fire burning. I'd love to have a go at getting into some details. But I don't think anybody can make it simple. The mathematics is not very hard, compared with, e.g., GR, but the connection with the physical intuition is bizarre at best. It would be nice to be able to dispel some common misconceptions. Very interesting. One of my old teachers --now retired-- changed his mind twice in 48 hours! I changed mine circa 1999 --old geezer-- and never went back. Suddenly realized that you don't need to collapse anything if you keep track of mixed states instead of pure states. Nobody would listen back then, though the real savvies knew about it. Life went on and I licked my wounds. I learnt about Ballentine's approach very similar to my view, although something was wanting: What happens to pure states? Are they a figment of the physicist's imagination? Hopefully, someone might catch me in any possible mistakes, imprecision, ambiguity or digression, misquotation. I've compiled a basic dictionary of everything that might be needed: 1) Dirac bra/ket notation. If not, the row/column complex vector would suffice. 2) Spin eigenstates and eigenvalues, Pauli matrices. For example, does this ring a bell? --no pun intended, \[\left(\begin{array}{cc} n_{z} & n_{x}-in_{y}\\ n_{x}+in_{y} & -n_{z} \end{array}\right)\] 3) The multi-particle formalism of quantum mechanics, otherwise known as "tensor products of states". 2-particle will do. 4) The postulates of quantum mechanics (states, observables, eigenvalues and eigenstates, probability amplitudes as Hermitian products, commutators & incompatibility, Hamiltonian or linear+isometric = unitary evolution, projection postulate.) 5) Yes/No observables, otherwise known as projectors (P linear & P2=P). That is, observables of simple bi-valued spectrum {0,1}. Plus useful lemma: \[A^{2}=I\Rightarrow P_{\pm}\stackrel{{\scriptstyle \textrm{def}}}{=}\frac{1}{2}\left(I\pm A\right)\:\textrm{are mutually orthogonal projectors}\] 6) Completeness relation or resolution of the identity for orthogonal projectors, In a nutshell: as much of the basics of the formalism as possible. To really understand the fundamentals, I would also strongly advise anybody to learn about: -The position/momentum representations of the wave function -The concept of a complete set of commuting observables (CSCO) -Superselection observables (those for which superposition cannot be applied) -The density matrix (distincion between pure and mixed states) -Decoherence -Local conservation principles, in particular, local conservation of probability densities Just for hairy details of measurement, QM of open systems, how states actually evolve in space and time, etc. I do have a feeling of impending doom, though. This topic has brought me unbearable pain in the past. I can no longer feel pain, so I'm thinking what the hell. My intellectual appetizer would be the statement of this common misunderstanding: Quantum mechanics is an out and out local theory. The conundrum is more to do with: systems that look to my rational mind as pairs of things (triplets: GHZ-M) are really one thing in some strange, uniquely quantum sense, because they're internally connected. It's non-separability that's at the root of all this, not non-locality.
  9. Forgot to say hello. I'm Joss, I teach Physics, Maths, Chemistry and English @ some academy in Spain. They sometimes make me teach Bio and Spanish, because they somehow assume I must know everything. I'm a theoretical physicist. My alter ego is Sisyphus. PD: I love Yogi Berra quotes
  10. No offence taken. I gave you an apology too, and I owe you an explanation. I thought this was probably just about a Physics student working on a physical problem looking for mnemonic/algorithm. Looks like the magnetic moment of a given current density. The integrand comes from the geometry of the circuit. The way I see it, physical problems in terms of different variables are just useful parametrizations of something 'real.' If the integrals are not manifestly divergent and simple symmetry arguments tell me it better be zero, the maths probably are telling me I must assume those integrals not to be of physical relevance. There are two peaks in the integrand, one at \varphi = \pi and the other at \varphi = \fraq{3}{2}\pi, with opposite signs. In such a way that, if I further change the variables \cos x = u, the integral formally reduces to a \[\int_{1}^{1}\] That tells me it must be zero. I wasn't going for any kind of mathematical rigor. I'm not cut out for that. And I haven't the faintest idea what the Henstock-Kurzweil integral is.
  11. The expression is familiar to me because I've seen it in lattice problems in physics. Here's a test run of your expression with, \[f\left(x\right)=\sin x\] It should give, \[f''\left(x\right)=-\sin x\] Let's see: \[\frac{\sin\left(x+2\varepsilon\right)+\sin x-2\sin\left(x+\varepsilon\right)}{\varepsilon^{2}}\] Using, \[\sin\left(x+2\varepsilon\right)=\sin x\cos\left(2\varepsilon\right)+\sin\left(2\varepsilon\right)\cos x\] \[\sin\left(x+\varepsilon\right)=\sin x\cos\varepsilon+\sin\varepsilon\cos x\] \[\cos\left(2\varepsilon\right)=\cos^{2}\varepsilon-\sin^{2}\varepsilon\] We get, after simplifications, to, \[-\sin x-\varepsilon\cos x+\frac{\varepsilon^{2}}{4}\sin x\] Works for powers of x, works for sines and cosines, therefore exponentials and products of them. Works quite well I think when things are smooth.
  12. This is what people want you to say: \[f''\left(x\right)=\frac{f\left(x+2\varepsilon\right)+f\left(x\right)-2f\left(x+\varepsilon\right)}{\varepsilon^{2}}+o\left(\varepsilon\right)\] Where, \[\lim_{\varepsilon\rightarrow0}o\left(\varepsilon\right)=0\] Can't be false because it's an identity as long as f(x) is twice differentiable at x. Utter the words and fall on your knees. Sorry, were it not for CoVid-19 I would be sleeping like a baby.
  13. During the first months of LHC running its first tests, I got wind that micro BH formation tested negative. It was an informal conversation, but the source was reasonably reliable. We seem to be stuck at Higgs...
  14. That makes sense. Half the viruses is not so good.
  15. What part of this couple of sentences you didn't understand, taeto? I wasn't talking to you when I said it, but anyway, as you seem to care so much... And sorry for 'patronizing' you. Really.
  16. There are more integrals in this world. Have you ever heard of the Lebesgue measure? I often think of integrals in terms of the Lebesgue measure. The trick is looking at the domain and the range of the function. "The range is infinite" becomes no more worrying than "the domain is infinite." Another name for it is mathematical power. No matter how you want to do it with Riemann, you're gonna have to face the singularities, because those are in the domain. If I've found them with my method, believe me, you're gonna find them too. No integral handbook in the world will get around that. That's a property of the integral, not of my method. I may take a look at how rigorously it is defined with the Lebesgue measure. Maybe Cauchy's theorem. But not now. Most integrals in QFT are Lebesgue or Cauchy complex, not Riemann. Think just a little bit outside the Riemann box, please. I said "this reduces to" about the simple algebraic steps, not the convergence arguments. I honestly think I should go to sleep now, but who knows. I'm finding it difficult lately. And oh, please, believe you me: If you find an integral with singularities in the domain that has been solved in a handbook. Some mathematician has proven convergence way ahead of the handbook going to press. Same goes for Wolfram or the like.
  17. I've found a pedestrian way to do it without looking at tables. If you're not interested, don't pay attention. First you reorder your integral by taking care of the $r$ part, while factoring the angular part to the left. You integrate by parts the r factor, \[\int_{0}^{2\pi}d\varphi \frac{1}{\cos\varphi}\int_{0}^{a}dr r\frac{\partial}{\partial r}\left(\log\left(a+r\cos\varphi\right)\right)\] After some algebraic steps you get to, \[\int_{0}^{2\pi}d\varphi\frac{1}{\cos\varphi}\left(-a\cos\varphi\log\left(a\left(1+\cos\varphi\right)\right)+a\cos\varphi+a\log a\right)\] This reduces to, \[-a\int_{0}^{2\pi}d\varphi\log\left(a\left(1+\cos\varphi\right)\right)+a\int_{0}^{2\pi}d\varphi+a\log a\int_{0}^{2\pi}d\varphi\frac{1}{\cos\varphi}\] Both the first and the last integrals have singular integrands, but you can argue that they're convergent. And the value they converge to is precisely zero, as they are periodic functions of $\varphi$ between zero and $2\pi$. The only thing you're left with is, \[a\int_{0}^{2\pi}d\varphi\] Check me for mistakes, I'm sleepy.
  18. Swansont, here's a link to recent study I've seen: https://medicalxpress.com/news/2020-04-sunlight-coronavirus-quickly-scientists.html?fbclid=IwAR2ZQLaVm5X1N--fJQit5xyxBzSTHp8Ow4vsUfQsNjLKi7VF0HNpEf_Ys2A They combine virus in aerosol with high diffusion; as I recall. Humidity is also involved. UV exposure as well. Conditions try to emulate outdoors rather than what Enthalpy suggests for money. But maybe you guys can get something else from it. With all factors together half-life seems to be cut down to 1.5 min. There are separate data with just radiation, just diffusion, just UV, if I remember correctly. For money disinfection, radiation intensity would be crucial, though, so maybe not so useful this study...
  19. I am sorry but your thinking is not good question! Not sense make it now, not sense make it ever! Shut down or up, as you please!
  20. Completely agree. That problem calls for numerical GTR. I was thinking along same lines when I said, On the other hand, linear accelerations for stellar objects are very rare phenomenologically speaking, I surmise. Plus any linear acceleration field would lead to accretion rather that inducing threshold trespassing, my intuition tells me, concurring with you, perhaps. But there could be an experimental/astrophysical context to do the trick perhaps. A neutron star approached by a heavy stellar interloper which induces strong tidal forces in it, as well as very intense centrifugal potentials. See if any such event of BH formation can be detected. But the acceleration field we're talking about would be very different, of course.
  21. Thank you for this pointer. I didn't know about geons. I'm very interested in self-consistent classical solutions.
  22. There are so many speculations about them. You're right. There's nothing common-sense about BHs. I think angular momentum may play a key part and, although Schwarzschild's solution is an exact one, only the Kerr-Newman maybe makes sense, the rotating one. It's in one of his video lectures. Don't know where it is though. I'll look it up. The thing about Susskind is he's so intuitive and pictorial. Even in the most abstract and difficult topics. Although he always discusses the Schwazschild solution. His lectures on supersymmetry, even though he's clearly unhappy at the end, are amazing. SS I think must be correct in some sense we haven't understood. A basic exposition like Susskind's I think is perfect for anybody young, without prejudice, that would like to have a go at a possible re-interpretation. But that's off-topic.
  23. Once it's fallen, there's no way back.That's what the theory says (Hawking radiation aside.) In fact, there is a very, simple, very nice Gedanken experiment explained by Lenny Susskind of a sphere of photons converging to a point, in such a way that a BH is bound to form, but even though you have the illusion that something could be done about it, there is a point past which the photons are doomed. I don't remember the details, but maybe I could find it. But I wouldn't bet my life on what a BH is actually going to do. On a rather more speculative note, I think gravitational horizons have something very deep to do with the problem of the arrow of time that we haven't understood very well at all.
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