@MarkMitchison , @ACuriousMind do you have any idea whether for the classical e.m. field there is a Lagrangian formalism? It seems to me that the Lagrangian formalism id for particle, and alike, but not for waves. What you say?
What astounded me when I first heard it is that, if you restrict the Lagrangian to really only depend on the coordinates, first derivatives and time, then the inverse Lagrangian problem is solved
We actually know exactly what the conditions on a second-order differential equations system are such that it has a Lagrangian.
But it makes sense, right? Dissipation arises when you trace out degrees of freedom. So there is no reason why there should be such a "fundamental" description for a phenomenological model
The Legendre transform is not always well-behaved if you choose ugly Lagrangians, but the Hamiltonian is constrained by the symplectic geometry of the phase space to be a "nice" function, while the Lagrangian has no such geometrical constraint.
If you choose to help her, so be it. She is lucky, and it's up to her whether to take advantage of that luck by communicating with you reasonably.
@Sofia I don't want to get into this discussion with you, it seems to have been coming up a lot recently
I agree with what appears to be the majority community view, namely that this site exists as a repository of good questions and answers.
Not as a help-station for everyone who has a problem
If that user is not satisfied with the answers on the duplicated question, and you think that you can help her, then you should write a new answer to the old question, and direct her there.
That means that she gets help, and the website does not get cluttered with duplicates.
What we want to avoid is the following situation: someone has a question that has been asked before, they search on this site, and find 20 different identical questions.
@Sofia I really don't understand your problem. She wants an answer to a question. You gave it to her.
@Sofia Well, I'm sorry that it bothers you that much.
@Sofia No, but I think this user may not have realized that the email addresses in the profiles are only visible to the users themselves, and no one else.
@StanShunpike: I don't want to fill the comment section with this because I'm not quite sure what the issue is. Do you realize that, if you take a position basis $\lvert x \rangle$ of the Hilbert space (if we ignore the subtleties for a moment), the wavefunction of a state $\lvert \psi \rangle$ is just $\psi(x) = \langle x \vert \psi \rangle$?
(This is why I hate wavefunctions, so much is so much clearer in the abstract Hilbert space)
I guess I should study the Born Rule more carefully and then reconsider the questions I have. Griffiths didn't really harp on that much from what I read, but it sounds like I need to make sure I understand that more clearly.
@StanShunpike I prefer to answer you here. Is it O.K.? I saw your question "1.What was the wave function like prior to normalization? Why did it need to be normalized in the first place?".
@StanShunpike The Born rule is nothing more than the statement that, given $\psi$ and $\phi$ as states, the probability to find one state in the other is the ugly expression in my answer on the question I linked to you question. It's a postulate- Unfortunately, many texts choose to present it only in the normalized form, and then introduce the idea that $\psi$ and $c \psi$ are the same states in some other way
@StanShunpike Yes, the Born rule is a postulate. Given two states $\psi$ and $\phi$, the probability to find one in the other is $ P(\psi,\phi) = \frac{\lvert \langle\psi\vert\phi\rangle \rvert ^2}{\lvert \langle\phi\vert\phi\rangle \rvert \lvert \langle\psi\vert\psi\rangle \rvert }$
This probability is perfectly normalized, if you take one of the states to be any multiple of normalized position states $\lvert x\rangle$ and integrate it over the position range, you get one
@ACuriousMind That makes so much more sense. I kept feeling like I was missing something behind proving that notion. But now I see why the Born Rule is an important postulate. That's obviously a key idea. A key assumption.
@ACuriousMind before teaching integrals, you teach simple algebraic calculus. Let's see him first understand simple things, and only then you make philosophy.
@Sofia Well, let me be more blunt: Stop assuming everyone is stupid. I prefer to assume everyone is smart and knows much and then turn down the sophistication of my arguments until they understand.
@Sofia No, and this is why I talk to them as I talk to everyone else. I told you, I hate lies-to-children. And this whole normalization business has come up various times, and the issue is always that some idiot teacher thought they wouldn't understand the Born rule fully, and instead declared normalization a sacred principle
@Sofia Again, the Born rule that gives the probabilities is $P(\psi,\phi) = \frac{\lvert \langle\psi\vert\phi\rangle \rvert ^2}{\lvert \langle\phi\vert\phi\rangle \rvert \lvert \langle\psi\vert\psi\rangle \rvert }$. This perfectly sums/integrates to 1 no matter how you normalize $\phi$ and $\psi$
Which you can check in the finite-dimensional case with simple linear algebra
@ACuriousMind what you said in the long formula, is that you work with $\psi\rangle$ and $|\phi\rangle$ normalized. You didn't say anything else. So, about what you argue with me?
@Sofia No, I divide by the norm of the state precisely because they need not be normalized. If they were normalized, the denominator is 1 and not there at all.
I think understanding the Born Rule's importance was important, just for me as the OP. I often find if I misunderstand a basic, fundamental principle, it makes it hard to see how things fit together. I didn't get why @ACuriousMind had raised that in the thread he linked me to originally, but now I see why.
I love it, but I don't want to be a physicist. Most of my friends and family don't understand any of this stuff. They are all lawyers and economists. I just do this because I enjoy it.
@Sofia That's one reason why I've made much less effort to learn string theory and emphasized learning fundamentals of GR, QM, and QFT first.
@Sofia Everyone seems to either believe string theory must be true or has reservations about it. And I can't tell what is true and what isn't. So I study what is known to be true based on evidence first before anything else.
@Sofia Like for instance, I feel like there should be a canonical argument for why string theory doesn't need to be background dependent. Like, if GR is background independent, doesn't that mean any consistent theory of gravity has to be? If that's true, how can you take string theory seriously.
But then I watch Susskind and he believes it. So I just assume I don't know enough to judge at this point. But I find it suprising there can be so much disagreement.
Then you have to divide every projector by the scalar product of the basis vector, i.e. $1 = \sum_i \frac{\lvert e_i \rangle \langle e_i \rvert}{\langle e_i\vert e_i\rangle}$
@0celo7 What's the analogy here? Does that mean just because GR is doesn't mean the deeper theory must be necessarily because GR is an approximation of some sort?
I think the answer is $1 = \sum_{ij} (G^{-1})_{ij} \lvert e_i\rangle\langle e_j\rvert$, where $G_{ij} = \langle e_i\rvert e_j\rangle$ is the Gram matrix.
@MarkMitchison I never know whether to feel proud or stupid when I do some two page calculation all by myself to get the right answer and then someone comes along and says, "Oh, if we apply the well-known and really easy theorem X, then the answer just comes out in one line, you know?"
To be fair, my proof demonstrated to me why that identity works more fundamentally than the standard trick, which appears to be "basis-dependent". Of course it eventually turns out not to be. So it was worth doing, I mean.
@MarkMitchison I'm so used to working in orthonormal bases that I don't know what familiar formulas are right and which ones aren't. Apply $\mathbb{1}$ to $|v\rangle$: $$\mathbb{1}|v\rangle=\sum_{ijk}(G^{-1})_{ij}|e_i\rangle\langle e_j| v_k|e_k\rangle=\sum_k v_k|e_k\rangle=|v\rangle$$
But if you want to see how to do it the other way, note that for any set of linearly independent vectors $\lvert e_i\rangle$, the set $\lvert f_i \rangle = \sum_j (G^{-1/2})_{ij} \lvert e_j\rangle$ is orthonormal.
This identity allows you to do pretty much whatever you want with non-orthonormal bases.
And it is one way of seeing why the Gram matrix pops up everywhere.
That's exactly the question I asked when I started thinking about this.
I think so, in a way.
Or at least there are very strong analogies
But here there is only one Gram matrix for a given basis, whereas in geometry there is a metric field for every point on the manifold in a given coordinate system.
@ACuriousMind That Born Rule thing explains what you and @0celo7 were saying the other day about why the probability amplitude had more degrees of freedom and is more fundamental. Because if you assume quantum states follow the Born Rule, that's in other words assuming such a degree of freedom exists effectively. Does that (what I just said) make any sense?
@0celo7 Sure you can do that. I'm not sure what this would really mean though. In QM you just need one Hilbert space of states for functions on the manifold (which is actually the multi-particle configuration space).
@0celo7 You can actually look at a QFT as a thing that assignes to spatial slices of a spacetime Hilbert spaces, and the path integral is the thing that gives the evolution maps between them in the time direction
@StanShunpike I'm not sure I understand what you mean. Do you mean that, by defining the probability to depend on the complex scalar product instead of some real one, we get two d.o.f. instead of one?
@Sofia viXra is an online repository, like arXiv, but they have no standards for accepting submissions, and so they have become the standard place where people with crackpot theories and awful science/math/etc post garbage papers. I'm sure not all papers are garbage on their own, but a delicious piece of chocolate cake, when found among garbage, becomes garbage
@tpg2114 The comment advice says: "If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful."
The "we" in papers with a single author (and also in some with multiple ones) is actually meant to be "we" as in "you and I", and an eternal clash between people who say it sounds arrogant and people who find it nice to include the reader and "take them along on the ride".
@0celo7 Ah, yeah. But like @ACuriousMind just said, I think on our site that's okay -- it's more like a "you and me" thing. But I find it oddly out of place in a paper with a single author.
Nosism, from the Latin nos, "we", is the practice of using the pronoun "we" to refer to oneself when expressing a personal opinion.
Depending on the person using the nosism different uses can be distinguished:
== The royal "we" or pluralis majestatis ==
The royal "we" (pluralis majestatis) refers to a single person holding a high office, such as a monarch, bishop, or pope.
== The editorial "we" ==
The editorial "we" is a similar phenomenon, in which an editorial columnist in a newspaper or a similar commentator in another medium refers to themself as we when giving their opinion. Here, she or...
Although "It is known, in line with the classical theory, that a magnetic field is created by the moving charges and electric currents." is not wrong, for example
A bit of a pleonasm there with currents and moving charges, but okay
http://www.nature.com/nature/journal/v473/n7348/full/nature10104.html I also know the lead author and he's a great physicist. But it's potentially a misleading title... Got the paper into nature though!
Yeah. It's a pretty esoteric field of experimental physics but the people who are into it understand immediately what is meant by shape. So it's not actually that bad as a title. It's just that "shape" has to be interpreted carefully: it's the shape of the vacuum polarisation distribution.
@ACuriousMind This was really quite inspired. I said that wealthy people define Killing fields of the linguistic metric tensor because their language has a smooth evolution.
@0celo7 lol imagine if u had an alterego who wrote racist, off topic trash papers just for vixra and a real self on arxiv. The physcist version of Jekyll and Hyde.
@ACuriousMind physics is vast & agreed its hard to keep track of it all. however, you also seem to appreciate how interconnected it is... seems there are unusual "synchronicities"... eg believe there is a (as yet undiscovered) significant connection between string theory & fluid dynamics... would that make it more interesting? so, anyway, what is your favorite area of physics?
also, on the other hand, scientific compartmentalization/ reductionism can have downsides...
@vzn Uh, my favourite area? Probably quantum field theory, particularly non-perturbative aspects, although I also really like the simplicity of classical mechanics from the Hamiltonian viewpoint.
yes also quite interested in the remarkable complexity inherent in "mere" classical mechanics. which possibly is still not fully/ completely known/ understood & retains some major surprises...
As to fluid dynamics and string theory...well, I don't really know either of the subjects well enough to say whether that would make it more interesting