joigus

You guys are drifting dangerously from "wise" to "cute". My own heuristics on the question is that kids are suckers for reactions in the adults, and what they do there is kind of a mixture between combinatorics of mouthfuls of meaning they've picked up somewhere and real striving to understand. But it's hunger for feedback what really drives them into bold new semantic territory. But, for kids that age, I think it's the emotional feedback that's the bonus. I bet next thing that goes on in the kid's mind is 'wow, I've been funny' rather than 'so that's what this is about'. Nothing wrong with that.

People will give you hints, not solve the problem for you. That's against the rules.

@MigL: I meant 'that's quite common among men my age.' Never a bad choice of preposition was as open to misinterpretation as this once.

I join my voice to the chorus. And please tell me what a 'virtual reference system' is.

In my experience it's been more cleverness than wisdom. One sample (from my younger English students about 11) is: 'Say, teacher, how did people come up with language before there was any language?' The thing with kids is you cannot be completely sure whether they're mimicking things they've heard, re-processing it in their own language, or genuinely coming up with a brilliant point from scratch. Even if the former were the case, it's quite impressive. It's been scientifically proven that communities of kids isolated from adult language can create build up their own language. So something relational is going on there, I surmise. https://en.wikipedia.org/wiki/Nicaraguan_Sign_Language Edit: Very interesting topic.

Excellent answers. Yes, from what I know, most peculiarities of Jupiter's moons are due to extremely high tidal forces. Another example is Io's anomalous volcanism. Jupiter is squashing it like silly putty.

Ok, so here's the mathematical scoop. Copenhagen's interpretation of QM has 8 postulates (presentations may vary depending on the particular version). Two of them are concerned with how states change with time. One of them is called "evolution postulate" or "Schrödinger's equation"; the other is about so-called "measurements". Suppose there are only two states, \( \left|1\right\rangle \) and \( \left|2\right\rangle \). An arbitrary system in this simplified version of the world can be in an arbitrary superposition of both, \[\left(\begin{array}{c} c_{1}\\ c_{2} \end{array}\right)=c_{1}\left|1\right\rangle +c_{2}\left|2\right\rangle\] where \( c_1 \) and \( c_2 \) are complex numbers such that \( \left|c_{1}\right|^{2}+\left|c_{2}\right|^{2}=1 \). Whenever you write a state like this, a certain observable \( Q \) is implied, and \( c_1 \) and \( c_2 \) give you the probabilities that measuring this observable, the outcomes are either \( q_1 \) or \( q_2 \): \[\mathcal{P}\left(q_{1}\right)=\left|c_{1}\right|^{2}\] \[\mathcal{P}\left(q_{2}\right)=\left|c_{2}\right|^{2}\] A convenient way to express this observable \( Q \) is in term of its projectors, \[ Q=q_{1}P_{q_{1}}+q_{2}P_{q_{2}}=q_{1}\left(\begin{array}{cc} 1 & 0\\ 0 & 0 \end{array}\right)+q_{2}\left(\begin{array}{cc} 0 & 0\\ 0 & 1 \end{array}\right) \] But other projectors exist, because other observables exist. Projectors, in this simple case, are Hermitian --essentially real-- \( 2\times2 \) matrices satisfying, \[P^{2}=P\] For example, \[P_{r_{1}}=\left(\begin{array}{cc} \frac{1}{2} & \frac{1}{2}\\ \frac{1}{2} & \frac{1}{2} \end{array}\right)\] \[P_{r_{2}}=\left(\begin{array}{cc} \frac{1}{2} & -\frac{1}{2}\\ \frac{1}{2} & -\frac{1}{2} \end{array}\right)\] These ones correspond to a different observable, say \( R \) , incompatible with \( Q \) , \[R=r_{1}\left(\begin{array}{cc} \frac{1}{2} & \frac{1}{2}\\ \frac{1}{2} & \frac{1}{2} \end{array}\right)+r_{2}\left(\begin{array}{cc} \frac{1}{2} & -\frac{1}{2}\\ \frac{1}{2} & -\frac{1}{2} \end{array}\right)\] Observables can be seen as fundamental questions on the system, like 'what is the value of \( Q \)?' Projectors can be seen as observables themselves; answers to the questions 'is the value of \( Q \) equal to \( q_1 \)?' They are yes/no questions. What Copenhagen's QM measurement postulate tells you is that, when you \(Q\)-measure ('ask a \(Q\)-question') to the system, and the output happens to be, say, \( q_1 \), the salient state is, \[\left|\psi'\right\rangle =\frac{P_{q_{1}}\left|\psi\right\rangle }{\left\Vert P_{q_{1}}\left|\psi\right\rangle \right\Vert }=\frac{1}{\left|c_{1}\right|}\left(\begin{array}{c} c_{1}\\ 0 \end{array}\right)=e^{i\alpha}\left(\begin{array}{c} 1\\ 0 \end{array}\right)\] OTOH, what the Schrödinger equation tells you, in a nutshell, is that the most general form of evolution is given by, \[\left(\begin{array}{c} c_{1}'\\ c_{2}' \end{array}\right)=\left(\begin{array}{cc} u_{11} & u_{12}\\ u_{21} & u_{22} \end{array}\right)\left(\begin{array}{c} c_{1}\\ c_{2} \end{array}\right)\] where, \[u_{11}u_{22}-u_{12}u_{21}=e^{i\alpha}\] \( e^{i\alpha} \) is the way you parametrize a complex number of modulus \( 1 \). This is the complex-numbers equivalent to a rotation. This \( U \) matrix is not an observable; it's an evolution operator. And it doesn't have to be real; it must be 'unitary' –conserve probability–. Now, it is impossible to equate the change of state posited in the projection postulate to any \( U \). That is, in a nutshell, the problem of measurement. Very mathematically, if very simply --I hope--, stated.

Can you, please, address the objections about redshift, clearly identifying a misconception of yours, before opening another thread to cast doubt on expansion? Appreciated.

Why?

Thanks a lot. Today has been a good day for beauty and wonder.

That was amazing.

That's quite common between men my age. You just made me a primate, @MigL.

Neither have I, but it's just because of the lack of information about that lovely fellow.

can be argued to be true- as long as you take a figurative interpretation of "an incredible distance away". But having only one sentence that might be correct is, as you say, something of an achievement. Indeed.

Utter nonsense.

I tried to make them smaller or link to thumbnail versions... I would've thought you'd like them, given how fond you are of science fiction.

Ok. I have no idea what a cerafe is, but: (My emphasis.) The rest, I suppose, can be explained by the behavioural traits of flies.

Treehoppers!! (from Wikipedia.) https://en.wikipedia.org/wiki/Treehopper Oak treehopper: https://insider.si.edu/2017/08/beautiful-bizarre-treehoppers-suck-sap-can-spread-disease/8577969459_49866036ff_b/ (from Smithsonian Magazine online.) https://www.smithsonianmag.com/science-nature/treehoppers-bizarre-wondrous-helmets-use-wing-genes-grow-180973713/

Nothing. You're going in circles and then playing a naming game. The definition of the fine structure constant, \( \alpha \) in SI units is, \[\alpha=\frac{e^{2}}{4\pi\epsilon_{0}\hbar c}\] \( e \), \( \hbar \), and \( c \) are measured quantities. \( \epsilon_{0} \) is a convention. \( \mu_{0} \) is another convention coming from convention \( \epsilon_{0} \), and measured \( c \), \[\left(\epsilon_{0}\mu_{0}\right)^{-1}=c^{2}\] Then you play with the identity, \[\hbar=\sqrt{\epsilon_{0}\mu_{0}}\frac{e^{2}}{4\pi\epsilon_{0}\alpha}=\sqrt{\frac{\mu_{0}}{\epsilon_{0}}}\frac{e^{2}}{4\pi\alpha}\] And then you name, \[Z_{0}=\sqrt{\frac{\mu_{0}}{\epsilon_{0}}}\] the impedance of the vacuum. You have no operational definition for that as an impedance.

Was in response to: Not in response to @Anchovyforestbane. Sorry. It was the quote of a quote, and the quote function doesn't, or didn't, embed the quotes.

I agree with you, basically, that the Copenhagen interpretation is not satisfactory, and neither it is the many-worlds interpretation. But the Copenhagen interpretation works like a dream. That's the problem, actually. It works like a dream and mathematically, it cannot be the whole story. As Bell said, Copenhagen's interpretation is good FAPP (for all practical purposes.) As Bell also said, Mind you: He didn't mean classical-mechanical arguments; he meant quantum-mechanical arguments. I'm working on a miniature of explanation in 2-dimensional quantum mechanics, if you're interested. The many-worlds interpretation is not a corollary of the Copenhagen version. It's more like what @Sriman Dutta says: I totally agree with this. Conjugate variables are certainly peculiar. Their properties cannot be simulated by any finite-dimensional space of states and thereby cannot be completely understood with discrete mathematics. They are the domain of transcencental mathematics. Unlike the famous \( J_x \), \( J_y \), \( J_z \) that people use in all the completeness theorems, they always pair in couples, one of which is conserved, the other is not.

Yes, that's what I would expect from the acolytes. I suppose people elbowing for position are calculating their moves. That checks with what I was thinking. You're right that I don't know the American political landscape very well. But if I remember correctly Mitt Romney has presidential aspirations...

No serious Republicans taking positions to save face when all this pissing in the wind blows over? I'm sure it must be embarrassing for somebody in their ranks. Just curious.

You're quite right. There is a caloric part in \( G \) that is usable for doing work. Thank you.