Logan M
Reputation
1,542
Next privilege 2,000 Rep.
 Apr 7 comment Heisenberg's uncertainty principle for mean deviation? Note that this is also true for the original derivation. $\hat A$ and $\hat B$, as you've defined them, are not linear operators. But they don't need to be for anything in the derivation of the uncertainty principle, and indeed they definitely can't be because uncertainties are not observables. Apr 7 comment Heisenberg's uncertainty principle for mean deviation? I don't think it's possible to find an "operator which squares to $\hat x- \bar x$", since $\hat x - \bar x$ is not itself a linear operator. In a given state $| \psi \rangle$ we have $(\hat x - \bar x) | \psi \rangle = (\hat x - \langle\psi | \hat x | \psi \rangle )|\psi\rangle$, which is not a linear expression in $| \psi \rangle$, and so (like $\Delta x$) it can't be a (linear) operator at all. To take a simpler example, consider $\hat S_z - \bar S_z$ for a spin-1/2 particle. This gives $0$ when acting on both $\hat S_z$ eigenstates, but not when acting on (say) an eigenstate of $\hat S_x$. Feb 24 comment What is the difference between leptons and baryons? When cosmologists talk about "baryons", what they really mean (usually) is all standard model particles i.e. everything that isn't dark matter. For part of that source, they use that terminology, and the other part uses the more precise particle physics terminology in which a baryon is a composite state of 3 quarks. It's understandable that this would lead to confusion. Oct 17 comment Hopf Algebras in Quantum Groups The keyword to look for here is Tannaka-Krein duality (e.g. on nLab), a natural extension of Pontryagin duality. Perhaps someone else here is capable of giving some simple explanation of it, but I don't think I can do better than what's already on nLab or Wikipedia. Jul 24 comment Is speed of light and sound rational or irrational in nature? I'm not sure in what sense a system with $c=\pi$ would have "very complicated and perhaps even inconsistent behavior under Lorentz transformations". A Lorentz transformation is simply a linear map on $\mathbb R^4$ preserving the quadratic form $(ct)^2-x^2-y^2-z^2$. Even leaving $c$ as a formal parameter, the theory is exactly what we teach in introductory courses. One can just as easily rescale $t$ such that $c=1$ or $c=\pi$ or any other positive real number; the same theorems in dimensional analysis ensure these are all equiconsistent. Feb 21 comment How can stars make up 0.5% of whole universe? Minor terminology quibble: neutrinos aren't baryonic matter. They are, however, ordinary matter. Dec 3 comment Operator-state correspondence in QFT I'm not saying that we never do things like this in physics, but that it isn't exactly what is meant by "local" (at least to me). For example, in the standard Penrose diagram for Minkowski space, past timelike infinity is mapped to a single point, but if you want to talk about local processes occurring in a neighborhood of that point you really have to blow it up to resolve that. There's also a technical issue as to in what sense the limits converge. On AdS this would be a whole different story of course, but in flat space it doesn't make much sense to regard past infinity as a single point. Dec 3 comment Operator-state correspondence in QFT @Axion I agree with Prahar's comment. Perhaps another way to say this is that in a CFT, 0 is literally a point, in that we can compute correlation functions between fields at 0 and other points. In an ordinary QFT, past infinity isn't such a thing. The idea of contracting past infinity to a single point seems inherently nonlocal, in that if I send two wave packets back in time in opposite directions in flat space, I expect them to be getting farther away from each other, not converging to the same point... Sep 25 comment How to deal with the notation of a function $f$ vs its value $f(x)$ in Physics? Are you familiar with the implicit function theorem? Many of the "functions" you describe are probably better understood by mathematicians as relations, which can be converted locally into functions via this if you need to do so. Sep 22 comment How deep can my knowledge of particle physics go without the maths? "Physics, by definition, is the subset of Mathematics which pertains to our universe." I disagree with this. There's plenty more to physics than just pure math. I tend to agree more with Vladimir Arnold that the containment is in the other direction: "Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap." Sep 8 comment What are the physical dimensions (units) of the elements in a Hilbert space of a QM system? @REX Thanks for catching that. I've now corrected it. I make silly typos like that all the time. Feb 12 comment What are the justifying foundations of statistical mechanics without appealing to the ergodic hypothesis? @josephf.johnson As for your answer, I haven't yet had a chance to read it, unfortunately. This question has not had any activity for the better part of a year, and the answers do a pretty good job at least at the level I was looking for, so to be honest I had totally forgotten about it. Your answer seems to be at a more advanced level and does explain things in more detail. I appreciate it, even if I don't get a chance to look at it any time soon. Feb 12 comment What are the justifying foundations of statistical mechanics without appealing to the ergodic hypothesis? @josephf.johnson Alas, while I used the words "justifying foundations," I must admit that particular turn of phrase is not my own, and I can't comment on the intent contained therein. The title of this question was copied from a question posed on the Area 51 proposal of the now-defunct Theoretical Physics site. I agree with you that the phrase "justifying foundations" is a bit strange, but it seemed imprudent to copy the idea for the question but change the title; instead I tried as best I could to maintain the intent of the original asker and cited the location which I had found it. May 16 comment Does dark energy affect asymptotic freedom? It seems to me the idea you are talking about is the Big Rip Hypothesis. In this scenario, quarks do eventually de-hadronize and quark confinement fails. As far as I know the idea is a bit out of favor, but not ruled out for our universe. May 4 comment Will a stone thrown in space move forever? I didn't -1, but this answer has issues even if it is technically correct. For me, black holes are incompatible with the OP's statement that "gravity is equal zero" (admittedly poorly phrased, but the idea is to ignore gravity, i.e. work in Minkowski Spacetime). Answering the question in a GR context is also tricky because of strange things like closed timelike geodesics, making both "forward" and "forever" difficult to define. But the basic idea is that "this is mostly still true in general relativity once we make sense of what that even means" which I think you mostly captured. Apr 26 comment If randomness doesn't exist, how come the universe isn't a perfect sphere with predictable distribution of matter? Back to probability, I don't think there should be a consistent notion of a "random pick from [0,1]" in the way you are claiming. Such a thing is an abuse of language, as are random bit sequences. A Vitali set isn't an event, so there's no reason to expect it to have a probability. I don't see any paradox with that. It is true that this notion of probability and the physical one are somewhat different, but I don't see a problem with that. If you want to continue this discussion, let's move it to chat, because it's become entirely irrelevant to the topic of the question. Apr 26 comment If randomness doesn't exist, how come the universe isn't a perfect sphere with predictable distribution of matter? It is definitely not true that any ring I am interested in is countable; take for instance $\mathbb{C}^n$ with the direct product. Most of the explicit examples I'm interested in could be dealt with by countable choice, but for both categorical and practical reasons it it's far easier to just deal with all rings. Apr 25 comment If randomness doesn't exist, how come the universe isn't a perfect sphere with predictable distribution of matter? I don't totally understand what you mean, and I definitely like AC (otherwise how will I know that my rings have maximal ideals?). Personally, I'm of the opinion that random numbers/sequences are essentially abuse of language, but if I translate what you are saying into my language then I think what you call 'pure randomness' is what I would just call randomness, which in my book is a fundamentally physical (i.e. nonmathematical) concept. But in any case I don't think there's any more need for discussion of it here. Apr 25 comment If randomness doesn't exist, how come the universe isn't a perfect sphere with predictable distribution of matter? @Anixx I don't see how the two are asking the same question. They seem different both in focus and in the OPs' levels of understanding. I believe that the standard test is something like 'if the answers on a previous question would also constitute complete answers to this question'. If I take, say, Joe's answer on that question and apply it here, it doesn't seem to answer this question or be at an appropriate level. Apr 25 comment If randomness doesn't exist, how come the universe isn't a perfect sphere with predictable distribution of matter? This is a nice answer. It addresses the heart of the question succinctly, though my answer is more in depth on some tangential issues.