@misternobody ah, you should have been more specific about which formula in the answer you mean. I thought we were still talking about the one involving the humidity :P
nope, sorry, ig ive been reading these posts for hours and couldn't zoom out
it's quite nice that the speed of sound (in dry air etc.) can be calculated from such a simple formula of air properties en.wikipedia.org/wiki/Speed_of_sound
do you guys know an example of when integration and limits and differentiation do not commute and it matters in a physics circumstance
i think an example for two limits not commuting occurs in statistical mechanics when taking the limit of an order parameter and the thermodynamic limit. but i am not sure about other situations
hm is convergence too strong a property to ask for? isn't it sometimes sufficient to "truncate-converge", i.e., converge to a truncation of the True Value (expanded in some series).
hey guys, in the von Weiszacker correction, i keep seeing $\frac{1}{2} \langle \nabla | \nabla \rangle$ or $\frac{1}{2} | \nabla |^2$, is this equivalent to writing the KE as the Laplacian? I believe it is, since you can treat the gradient as the momentum operator (if you take out the -i) and since its Hermitian, move it over with the other gradient to get the Laplacian
@Allie Yes. Both those are essentially notations for an inner product, and the Laplacian is (at least morally) the inner product of the gradient with itself
i wonder if anyone could give some feedback before posting an answer, or maybe advice not to publish it
it's in details for calculating the speed of sound from sensor values, for this question which i think it's incomplete without an example physics.stackexchange.com/questions/46705/…
also just no one really explained several intermediate calculations
@misternobody Then the answer should have stated that. Just putting a random formula into an answer without any indication of where it comes from or why it would apply is not an answer.
If you think you can give a better answer than those already present, feel free to post it. But generally we do not expect answers to contain worked examples, and I think tpg's answer is reasonably complete already.
See abiogenesis (literally the first Google hit if you search "origin of life"). Why throw around half-remembered things or trust random chatters if you can just look it up?
also we dont' know what life is, so you can only speculate
my point was a bit like: how one defines a living system, is independent of the exact elements, and more informational; which may be why many end up in complexity metrics; another spin is how life appeared on earth which is more abiogenesis.
many chemists for example, focus on autocatalytic sets.
@misternobody this sort of jargon gives me headache :P. would take ages to be familiar with all of this
@misternobody so they look for molecules which can physically realise some informational structure that they postulate?
sorry. i think what you mean is that the information structure is what matters. and the physical realisation of it in terms of specific molecules doesn't matter and can be left for later study @misternobody
this means that these theories expect alien life to have the same informational structure but hosted by different molecules
yeah, all your comments above are correct imho. it's a mix: if it's more like earth, biologists are fine (but, why?); otherwise you may need to shift to those (and are also a research topic.)
some of them (the research) just stir a pot with a mix of molecules and track complexity
i am seeing parallels with theories of consciousness. they too make it a matter of complexity and information processing, and the leave the physical realisation of it irrelevant
e.g. they speculate if the internet could be conscious or if the cosmos could be conscious. it is called integrated information theory
Chalmers defines something like a phi function to measure complexity. it would be interesting to compare this to similar ideas like assembly theory. consciousness and life are related after all. Chalmers believes consciousness can be based on silicon too, so the information processing structure is all that matters
i think information processing can explain life in a general way, and may help to create life from scratch; i doubt it explains much about how life happened on earth. But can consciousness be explained by information processing? I think part of it yes, but not its nature i.e why it is accompanied by experience.
i think the origin of life is what Chalmers would call an easy problem. so i don't see any fundamental limitations to info processing explaining the entirety of that process. but the subjective experience is a hard problem, yes. there must be an explanatory gap there and Chalmers recognises that. his program is more like correlates of consciousness. he is just trying to correlate consciousness to information
but the other people who work on iit can believe in different philosophies of consciousness. so they may or may not accept the hard problem as genuine
The IIT pushed by Chalmers there is - outside of flashy pop-sci articles - widely regarded as pseudoscience by many other cognition researchers, see this recent letter signed by more than 100 researchers.
@ACuriousMind then I feel it is a controversial theory. i have seen many researchers on youtube talking about it seriously. these 100 researchers must be part of the other camp
besides many of those signing only sign because of disagreement though, like Dennett. And it's important to note that the main competitor (Dehaene) didn't sign.
any proposal to what consciousness is must be controversial. Chalmers' idea is probably wrong, and so is anyone elses'. but that doesn't mean they are doing pseudoscience like what researchers in the other camps would say
@RyderRude "Youtube" is precisely part of the undue media attention the people who wrote that letter are concerned about. Did you even read the letter?
You really need to let go of the idea that by watching science popularization on Youtube (even the good ones) you can get an accurate understanding of what the field actually looks like
@ACuriousMind i haven't read it yet... i am just saying that these people who signed belong in rival camps. iit is anything but pseudoscience and there is genuine effort in trying to model consciousness, whether it is right or wrong is another matter. but calling is psuedoscience would be like calling string theory psuedoscience, for instance
@RyderRude What I mean is that the letter is explicitly concerned with the matter of empirical support for the theory rather than saying the theory possesses no merit at all. Based on your reply you seem to think they're saying it's fraudulent or not a "genuine effort", which is not a claim that's being made.
@ACuriousMind then it seems they are more honest about their criticism than I thought. i personally can't evaluate their claims but like @misternobody said, there is a counter-letter and their claims are addressed by the other camp
i personally can't evaluate these claims because I am not an expert in this. but I would read the letter. thanks for linking it @ACuriousMind
Perhaps the "pseudoscience" label here is harsher than it deserves, but I find this immediate jump to pretending we as outsiders to the field can somehow judge this better unwarranted.
@RyderRude In particular, I find this assertion incredibly arrogant, as it is based on no expertise on your end at all (as you just admitted). How can you judge people as "just trying to discredit" someone and then admit you have no expertise to evaluate their claim at all?
I'm having trouble with a lagrange multiplier. Let me simplify the problem (of course it's not a calculus problem but I will express it as one). Let's say I want to extremize $f(x,y)=2x^2-xy$ with the constrant $x^2+y^2=1$. So, I have to extremize: $$2x^2-xy-\lambda(x^2+y^2)$$ but if I write it in a symmetric fashion (using $2x^2=x^2+x^2=x^2-y^2+1$, the constraint) $$x^2-y^2+1-xy-\lambda(x^2+y^2)$$ I get a... different $\lambda$
@ACuriousMind it is what rival camps do generally. like people who say string theory is not science. people are unreasonably harsh in academia as an effort to discredit other camps and maybe defund them
@HerrFeinmann You can't use the constraint to manipulate the Lagrangian and expect to get the same $\lambda$, since before enforcing the E-L equations, the constraint does not hold (i.e. you're manipulating an off-shell quantity with an on-shell equality here)
@RyderRude The topic of why string theory generates such ire from certain people is much subtler than that (and to some degree very much caused by the string theorists overpromising and underdelivering in their PR), and you are equally unqualified to judge that.
Say we have two bodies that move with velocities $\vec v_1$ and $\vec v_2$ with respect to another inertial frame S. Now relativistic vector addition law in general gives $\vec v_{12} \neq \vec -v_{21}$ ,does this not violate principle of relativity that all inertial frames are identical?
this problem has been bugging me for quite a while
@HerrFeinmann but generally I would also be concerned about why the value of $\lambda$ is relevant - the value of the Lagrange multiplier is not really physics unless there's something else going on (like for temperature, which I still am not sure how to think about correctly in the multiplier approach)
I have been reading wikipedia article on wigner rotations and it is quite confusing,I saw the above inconsistency in that article en.wikipedia.org/wiki/Wigner_rotation
@HerrFeinmann Okay, so you're right, my statement needs to be a little more careful: The value of the Lagrange multiplier can have a meaning, but in that case, if you change the rest of the Lagrangian, the meaning changes, too
@ACuriousMind I'm inclined to spend time looking for some interpreations but, regardless of the meaning of the multiplier, am I wrong to think that using the constraint before invalidates the results?
@HerrFeinmann It depends on what you do with the results, I'd say. Since modulo the constraint the two Lagrangians are the same, they lead to the same solutions on-shell (except for the value of the multiplier)
It makes sense: you found the same extrema, since after all you're on the constraint surface but something else changes. I'll double check this with math fellows too
@RyderRude But you also don't go around and discuss one of the sides as if you could pass any kind of judgement! You can just not have an opinion at all! But what you did was come in here and discuss a contentious speculative theory of a non-physics field, then get upset at me when I point out it's contentious. You can't have it both ways:
Either you are unable to judge, then why are you engaging with this topic at all, or you can, in which case why are you trying to talk about this with a bunch of people who can't?
@Arjun So in what case would these two be different?
If you think you've found a paradox in relativity (or anywhere else, really) it usually helps to first write down the simplest version of it you can with all the steps explicit
@Allie there is essentially just a giant pile of stuff I want to read but never get around to sitting in a folder :P
@ACuriousMind it came up in the flow of conversation, and I don't think I have either claimed it is the correct theory or that I know the technical details of it. i have said the opposite above. but also, this does not mean that one is not allowed to discuss things they don't master in. @misternobody is more familiar with it than me. it was a two-way discussion
@HerrFeinmann the Lagrange multiplier, in general, has a very intuitive meaning: It gives you the rate of change if you change the value of the constraint
@ACuriousMind Then the wiki article says that "composites are not opposites but velocities are opposites" it confused me en.wikipedia.org/wiki/Wigner_rotation
@ACuriousMind but ur comment only presented one side of the debate. the way i would interpret ur comment is not that this theory is controversial, but that this theory is widely considered to be pseudoscience, which is a wrong thing to say
so i think ur comment was sort of unwarranted. but I want to end the discussion now
@Arjun Ah, I see. The article is indeed phrased strangely (I think there's some attempt at a passive interpretation of the Lorentz transformations here), but it's right: As defined here, $\vec v_{12}\neq -\vec v_{21}$. But where is the paradox supposed to be?
You claim this violates some principle that all inertial frames are "identical", but I don't know what you mean by "identical" or why that would be contradicted by the Wigner-rotating velocities
@ACuriousMind when we use relativistic velocity addition rule we clearly obtain $\vec v_{12}\neq -\vec v_{21}$ ,but the wikipedia page says these "components" of velocites are not opposite,but the velocity vectors themselves are opposite, what other velocities is the article talking about here?
@Arjun The whole article is a bit strangely formulated, but what they mean is this: The reason why $v_{12} \neq -v_{21}$ is because the composition of the two boosts introduces a rotation of the observers' coordinate systems relative to each other compared to the lab frame they start in. But that doesn't mean the vectors are not opposite to each other.
Perhaps a simpler example helps:
We consider the standard basis of $\mathbb{R}^3$ and a vector $\vec v = (1,0,0)^T$ in that basis. This is the lab frame. Now consider the same basis, but cyclically permuted (i.e. the first basis vector is now the third, the third the second, and the second the first). This basis would express $\vec v$ as $(0,0,1)^T$. This is observer A
Finally, cyclically permute once more, then that basis would express $-\vec v$ as $(0,-1,0)^T$. This is observer B.
Obviously, the coordinate expression of A for $\vec v$ is not the negative of the coordinate expression of B for $-\vec v$. Yet as abstract vectors, $\vec v$ and $-\vec v$ are still negatives of each other. That's what the article is talking about - the expressions there are for the vector and its negative, but in two different coordinate systems, and the coordinate expressions don't have to be negatives of each other
@TobiasFünke No, I think your point illustrates why one should expect the multiplier to change nicely - if you change the objective function, its rate of change off the constraint surface will also be different, even if it is the same as the previous function on the surface
Now I'm shifting the question too: if the Lagrange multiplier is not god-given, how can $E$ be a Lagrange multiplier in the SE (variational principle)?!
@HerrFeinmann What change to the objective function do you think is allowed in that case?
the constraint is just that the state is normalized
I don't see what terms that vanish for the constraint you're going to add to $\langle \psi\vert H\vert \psi\rangle$ to change the objective function as you did in your example
Yes, exactly. It is indeed uses in my case (superconductivity). Basically, without getting into the details of the calculation, the quantity to extremize is a function of the coefficients $c_n$ which determine the BCS ground state. The normalization is a condition on such coefficients, so you can plug it in the average value you want to extremize and "mess things up" like I did
but isn't it rather easy in the variational SE to just directly derive that the multiplier value is the energy? can't you just mimick that derivation with your changed Lagrangian and see if it still works out to be the energy or not?
Well, that's how I realized something was wrong in the first place: one path gave me the Bogoliubov spectrum $E_k=\pm\sqrt{\xi_k^2+|\Delta|^2}$, the other gave me $E_k=\xi_k\pm\sqrt{\xi_k^2+|\Delta|^2}$
I mean, even without knowing the context you can see that they are different
yeah, but I thought we had now arrived at a state of mind where this is not worrisome and you just need to pay attention to which of these is actually the energy
there is nothing wrong here except the expectation that the L.m. should keep its meaning as the energy when you change the Lagrangian
Great question. I don't know. The second one is what I get without doing nasty stuff, while the first is what books usually call energy but get using the constaint.
@ACuriousMind I understand that when components might not be equal when written in different basis,but $\vec v_{ab}$ and $\vec v_{ba}$ are written in terms of vectors $v$ and $u$ and some other invariants like the inner-products and clearly they don't seem to match.They should match when one writes these expressions in terms of abstract vectors,no?
@Arjun No, as the article says, $\vec v_{12}$ and $\vec v_{21}$ are expressions for the vectors in the respective coordinate systems, and the $\vec u$ and $\vec v$ are the coordinate expressions in the original lab frame
that's why I say the article is strangely written - usually the notation used there would denote abstract vectors, but they don't, they use it to denote coordinate expressions for vectors
well, $\vec u$ and $\vec v$ are written in the same system - the lab frame - but $\vec v_{AB}$ is written in A's system and $\vec v_{BA}$ is written in B's system
@ACuriousMind Hmm,but if I try to break let's say $\vec v_{AB}$ into it's components as written in A's frame.On the LHS, should'nt I consider $\vec v$ and $\vec u$ in the basis of A frame when I decompose the LHS into components?
The problem is really that we're trying to express the fundamentally 4-dimensional geometry in terms of several different 3-dimensional geometries here
resources I would consider "nice" would not try to express it in this form at all :P
@ACuriousMind Oh,I know a little bit about 4-vectors,was the article trying to say the velocity 4-vectors will remain opposites but the 3-velocity vectors won't be or something in those lines?
I would have to think about the most elegant formulation, actually.
But I think the mucb less confusing way is to do what the article does later on and express this in terms of decomposing the product of two boosts into a boost and a rotation
You do $B(u)B(v) = B(x)R$ and if you solve for $x$ you get that expression for $u\oplus v$ (and conversely $v\oplus u$ if you solve $B(v)B(u)$ like that)
(perhaps the order is a bit different, I haven't checked this, maybe we should decompose $B(u)B(-v)$ or something)
but anyway if you decompose the correct product that represents "go from the frame of A to the frame of B" in the lab frame, you get a boost w.r.t to a different velocity, and this then has to be the velocity that A "sees"
there are two boosts because you do "boost from A to the lab, then from the lab to B"
I think the "issue" is about the constraint not being imposed just by the condition that the functional is stationary: you get the Lagrange multiplier from those condition, but up to that point you only imposed that $x^2+y^2=constant$. It is only at the end that you impose $const=1$ and that should fix everything (not that is solves my doubt about energy :P)
As you know, the SE can be derived from a variational principle; the functional that is extremized is $E[\Psi]=\langle\Psi\lvert H\rvert\Psi\rangle$. The extremization is not free and we should impose the normalization of $\Psi$. The Lagrange multiplier is the energy eigenvalue. So we extremize $$\langle\Psi\lvert H\rvert\Psi\rangle-E\langle\Psi\rvert\Psi\rangle$$
Now, if we manipulate $\langle\Psi\lvert H\rvert\Psi\rangle$ and use the constraint equation (e.g. in 2D $|c_1|^2+|c_2|^2=1$) in $\langle\Psi\lvert H\rvert\Psi\rangle$ (which is a function of $c_i$ and for example replace $c_1$ somewhere using the constraint equation) this changes the eigenvalue
@HerrFeinmann I mean - this operation changes your $H$, or rather, the resulting objective function can no longer be written as the expectation value of some linear self-adjoint $H$, right?
so then of course the multiplier $E$ loses its meaning as an eigenvalue of $H$
but now instead it is, as Tobias said earlier, related to the rate of change of this new objective function
I agree with you. It should only be considered as energy when it's in the standard form; in the other cases you are actually extremizing other functions which have the same extrema but different multipliers. All of this was caused by a book about SC doing such manipulations on the functional but still calling the multiplier energy
Although I'm starting to feel a little guilty to be tormenting you guys with this :P
This is the felony
It doesn't say in this part (only later) about the physical significance, but the Bogoliubov dispersion relation $E_k=\pm\sqrt{\xi_k+|\Delta|^2}$ (that you get this way) is well-known, my result (proceeding in the standard way is not. $E_k=\xi_k\pm\sqrt{\xi_k+|\Delta|^2}$
in case smone wants to criticise or tell me about anything unclear, i did the simple answer i talked earlier physics.stackexchange.com/questions/46705/…, hopefully adds important information, as I tried to suggest here.
There is not really much about classical electromagnetism to say, other than introducing the gauge connection and its exterior derivative as a curvature
@DIRAC1930 You don't have to read it all, just the relevant sections
fast to read and you get the gist
Incidentally, @TobiasFünke what is the definition of coherence? I mean, I kind of know what it is, but if someone asked me to define I would feel cornered!
@DIRAC1930 What exactly are you looking for? If you're hoping for this curvature to be manifested as literal curvature (i.e. some space that is curved in the ordinary sense when you embed it into $\mathbb{R}^n$), that's not what's going on, it's just called that because you can describe both the Levi-Civita connection (that gives "actual" curvature) and gauge connections in the same framework (as connections on principal bundles with different groups $G$)
Well I got learning about Affine connections from Schrodingers book and then I heard about the Ehresmann connection which I heard has its uses in gauge theory
On a high level, there is not much more going on than, if you interpret the Riemann tensor $R$ as a 2-tensor valued in $\mathrm{GL}(n)$ and the Christoffel symbols as a $\mathfrak{gl}(n)$-valued 1-form $\Gamma$, you can write $R = \mathrm{d}\Gamma + [\Gamma,\Gamma]$. You can replace $\mathrm{GL}(n)$ by arbitrary groups and this still makes sense, electromagnetism is the case $G = \mathrm{U}(1)$ where the $\Gamma$ is then just $\mathbb{R}\cong\mathrm{u}(1)$-valued.
Although I find the geometrical formulation of gauge theories beautiful, for electromagnetism specifically there is not a lot of benefit
the most concrete effect would be that the Aharanov-Bohm effect can be phrased in terms of general properties of the connection form and non-trivial bundles, i.e. mathematically rigorously
@HerrFeinmann What exactly is the traumatizing answer here except that the magnetic field is the spatial components of $F^{\mu\nu}$ which you could already just say in the covariant formulation without the geometrical formulation of gauge theory?
