@ACuriousMind I'm not sure what that means, as a scalar quantity (the units of) one is a multiple of the other, as vectors the obvious distinction is that Torque involves pseudovectors while momentum is a vector so comparing them is comparing two different things...
@bolbteppa Why did you put "the units of" in brackets there? It's a fundamentally different thing to say that $E/c$ and $p$ have the same units and that there is a constant $\lambda$ such that $p = \lambda E/c$.
@ACuriousMind Because you seem to be bickering over whether $[p] = [E/c] = [\frac{mv^2}{2v}]$ or $[p] = [E/c] = [\frac{mv^2}{v}]$, if someone wants to know "how to intuitively recognize" whether $E/c$ is a momentum they can look at the units, seeing the units do not give units of length or time or energy or electricity is certainly one way to recognize you're dealing with a momentum and not electric charge.
That last part is what I still disagree with. That it has units of momentum tells you it's not a length or energy, but nothing tells you it is actually a meaningful momentum of something.
@bolbteppa You agreed that having the same units does not imply $p\propto E/c$, did you not?
And now you're saying it's ridiculous to not accept that $p\propto E/c$ on the basis of having the same units? Or does "being momentum" somehow not mean to you that the quantity X that "is momentum" should be proportional to actual momentum?
The question was "But given $E/c$, $E$ being the photon's energy, how do I interpret that - directly - as the momentum? Why should it be the momentum? What's an obvious physical - not symbolic - reason? ... I was asking how to intuitively recognize that as momentum."
I answered by trying to say: just look at the units $[E/c] = [E]/[c] = [p]$, this is the main way to see it is momentum because it has the right units. If you want to be sure of anything you justify it rigorously, if you know the answer and just want to understand it intuitively, try dimensional analysis, now we know it's not…
@ACuriousMind honestly actually not really following DSs original point apparently linking human heredity/ evolution/ DNA & art or science or the point being debated/ overall topic/ assertion etc, there seems to be some "talking past one another". but there is possibly some connection in some areas eg music in particular. anyway this all seems to be triggered by a rather provocative/ argumentative/ sacred cow-goring n00b...
@ACuriousMind what does Moretti mean by "cyclic vector for $X$ and $P$", is it a vector $v$ such that its images $X^n(P^m(v))$ span the whole Hilbert space (or a dense subspace)? If yes, what would be an example for such a vector, for example in $L^2(\mathbb R)$?
On the topic of the level of this site, and comparing/contrasting with Math, it would be interesting to reflect on whether anything has changed since I asked them about it 3 years ago.
I've just voted to close this post, which I will slightly paraphrase because I'm sure it will soon be deleted:
I would like to know in simple words why time slows down and lengths change when I am in my private rocket traveling near the speed of light.
Although I think that almost everyone wi...
@Bass Well, since $X$ and $P$ don't commute you should probably write that as $X^{n_1}P^{m_1}\dots X^{n_q} P^{m_q}v$, but yes, that's what being cyclic means. an example of such is every function in $C_c^\infty(\mathbb{R})\subset L^2(\mathbb{R})$, since if there was any non-cyclic function in that, you would have found a true invariant subspace, showing the standard representation is not irreducible.
@ACuriousMind you know, the n00b in the room aka troll p*ssing on science/ physics & other misc sacred cows etc (who some others think it nec to respond/ counter)
@DanielSank there are a lot/ many motivations, a spectrum/ rainbow, its not blanket/ "one-size-fits-all", live & let live :)
I've just voted to close this post, which I will slightly paraphrase because I'm sure it will soon be deleted:
I would like to know in simple words why time slows down and lengths change when I am in my private rocket traveling near the speed of light.
Although I think that almost everyone wi...
