@ChrisWhite I figured it out... Turns out only some of the terms can be written in divergence form (the ones I knew how to do) and others cannot (the ones I was stuck on).
@tpg2114 Ah, that is a reason for the tag to exist (or, at least, these are potentially good questions I could see tagged with it). Is there something else? The tag sounds really unspecific to me.
@bolbteppa In a sense, yes, but you can't "define" the operator before you have defined the space it acts on, so you must first postulate the existence of the vacuum and then observe that the operator-state correspondence maps the vacuum to the identity.
My book says "it can formally be introduced by an insertion of the unit operator I at the origin of the z-plane, since this corresponds to the 'infinite past'"
@bolbteppa No, there is a one-to-one correspondence between states/rays and operators, that's what the state-field/operator correspondence is all about
@DanielSank You don't get the specific state though unless you specify the space the group is represented on
"The operator-state correspondence says that all states in the theory can be created by operators which act locally in a small neighborhood of the origin. That is to say that the entire Hilbert space of a CFT can be thought of as living at a single point. The key here is that for CFTs we have radial quantization, and states evolve radially outwards unitarily from the origin."
The operator-state correspondence says that all states in the theory can be created by operators which act locally in a small neighborhood of the origin. That is to say that the entire Hilbert space of a CFT can be thought of as living at a single point. The key here is that for CFTs we have radi...
But, as I said, the correspondence is based on the existence of the vacuum, since you identify the operators with their action on the vacuum, so this view is really bit backwards
@bolbteppa That's true, but a tautology, because an operator times the identity is just the operator itself. The crucial feature of a CFT is that the result of the action on the vacuum state uniquely determines the operator
@ACuriousMind If the actual moment of inertia of a pulley is less than what I calculate, is that accounted for in the stretching of the string and friction against the axle?
@bolbteppa I would not really think about the poles or the plane in that argument. The $L_n$ are the operators that lower conformal weights, and negative conformal weights spoil unitarity, so for a unitary rep you need the action of $L_n$ on the vacuum to be zero, and not an actual state of lower conformal weight.
Okay because $L_0$ is a derivative the $h$ goes to $h-1$ (giving eigenvalue $h$), so when $n > 1$ all the derivatives end up such that nothing turns into $1/z$ so Cauchy's theorem makes them disappear since it's all holomorphic?
I don't get why you want to drag the complex analysis into this - it follows from unitarity of the representation and the commutation relations of the Virasoro algebra.
@bolbteppa A state with negative conformal weight would have negative norm
The $L_n$ lower the conformal weight
The vacuum has conformal weight 0
Hence, the action of the $L_n$ must not produce a state in the space for the representation to be unitary/for the representation space to be a Hilbert space, hence they give zero
Eeek. I was hoping would notice I'm here. To tell you the truth, I'm profoundly embarrassed by the incompleteness, untidiness and sloppiness of my book-in progress. But I hope there are one or two useful "good bits".
By the way, I started off academically doing mathematical physics in the olden days, but I had to drop out and do mathematics instead because I couldn't make sense of any of the mathematics in physics. Now I realise that most of the mathematics in physics is really incomprehensible even if you've done a PhD in mathematics and have researched it for many years. The math in physics is just really, really hard. That's why I started writing a book about differential geometry.
Yes, it too me about 30 years to finally work out what tangent bundles really are. The trouble with differential geometry is that it has been continually rewritten for the last 150 years to try to simplify it. But every simplification is inconsistent with all earlier simplifications. So a hug body of inconsistent formalisms has arisen.
I think actually the best thing about my book-in-progress is the table of contents. The actual contents are a total mess. But I am only partly to blame for that. The subject of differential geometry is itself a mess. You have to read the whole history of its development to work out how all the definitions arose from other definitions.
I really don't have time right now to talk about this but I definitely will sometime soon, your book looks like the treasure trove I've been waiting for, I mean you reference Allendoerfer lol
Did you see his proof of the inverse function theorem? It's insane, he gives two and one is crazy
Yes, Allendoerfer and Oakley is a stunningly good book. We had it in the olden days for first year math. It is a really crystal clear book, and deep, and broad and fascinating. Someone (like Dover) should republish it.
Yes, and 1477 pages is all in A4. So when reduced to normal book dimensions, that will be about 1800 pages. And when it's finished, it will be over 2000 pages. It all started with an attempt to solve a 3rd order Jacobi field, which should have been an hour's work. But so far, I still can't solve this problem. All I wanted to do was extend my PhD results from flat space to curved space. But that has required a complete overhaul of mathematics to try to find out what jacobi fields really mean.
Anyway, I have a lot of work to do on Lie algebras, connections on differentiable fibre bundles (related to Gauge fields), curvature, holonomy etc. today. Thanks for the chat.
In both my QFT books I have seen $\phi = \frac{\phi_1 + i \phi_2}{\sqrt{2}}$ and $\phi^* = \frac{\phi_1 - i \phi_2}{\sqrt{2}}$. I am supposed to be able to rewrite the Klein Gordon Lagrangian so instead of $\mathcal{L} = \partial^2 \phi - \frac{1}{2} m^2 \phi^2$, I should be able to write $\mathcal{L} = (\partial_\mu \phi)(\partial^\mu \phi^*) - m^2 \phi^*\phi$.
But that's not what I get
I get $\phi^*\phi = \frac{\phi_1^2 + \phi^2_2}{2}$. And that doesn't equal the first Lagrangian.
I think I am confused about the complex part vs real part.
Then the question should say that. "How do I use Numerov's method to calculate eigenvalues...? I've tried X but it doesn't work because Y. I'm specifically confused about Z" (there is where the subquestions come in)
Numerov's method, as exposed on wikipedia, is an iterative method for solving initial value problems. So using wikipedia as a source to solve Schrödinger's equation is a bit hopeless because the vast majority of solutions you find won't respect the boundary conditions.
Specifically, it says " From what I've seen Numerov's method is the way to solve Schrödinger's equation but I don't see how to use it to solve an eigenvalue problem."
The main question is how to use Numerov's method to solve a Schrödinger type equation.
The following question: "Wouldn't numerically solving the DE just give one solution for a given value of the energy? " <- the asker is saying here that solving a diff eq iteratively typically just gives a unique solution, which is a true statement, and indicates why the asker is confused about this
The one after: "And wouldn't finding those solutions require knowing the energy eigenvalues to begin with?" <- here the asker is saying that in order to use an iterative method such as RK to solve Schrödinger's equation they can't have any undetermined constants such as "E" in there. They have to be known beforehand, or the solutions you find won't respect the boundary conditions.
And the one after this: "I've seen some mention of "tridiagonal matrices" being generated somehow, but am not sure what the elements of that matrix would be or how that applies to the problem." <- here the asker is indicating what other information they've heard about solving a Schrödinger type equation via Numerov's method, and that they think it applies but don't know how exactly. This is indeed the case.
Where the asker is going wrong is in assuming that they're supposed to solve the diff eq iteratively, when in fact the discretization of the Schrödinger equation maps a differential eigenvalue problem to a matrix eigenvalue problem, and the fact that there is a second order operator means the matrix to be diagonalized will be tridiagonal.
I understand, but I really think your use of moderator powers to close this question outright was excessive. A simple vote to close would have been enough.
As written, the question is clear enough to me and I suspect to others as well.
I thought the question was obviously enough unclear that I was comfortable closing it with only one other close vote on it.
And I still think it's unclear to the extent that I don't feel inclined to vote to reopen it. If other people do so, then that's fine, it can get reopened by the community or another moderator. But there has been very little support for reopening so far (from anyone other than you, of course).
Views are distinct IP addresses, if I remember correctly.
Also the comments are split.
And the recommendation of the comments is really not my point. I'm not voting to reopen the question in its current form because I don't believe it's clear, and the comments don't change that.
Well, I can guarantee many of those views are mine. There's no guarantee the other IP addresses belong to people with high enough reputation to cast votes to reopen
I saw it's not just a page views number, because, if I press F5 several times, it won't increase the number of views.
They could store my IP address in a table, but wouldn't that make it slow? They would need to query a database one more time for each request.
"This information is saved in an expiring cache entry for about 15 minutes. If a subsequent hit sees the entry is still there it discards the new hit. If it is already gone it allows for a new record."
I am reading Shilov's book linear algebra. He explains how to compute determinants. Basically, for the plus terms you write
\begin{equation}
x_{a1}x_{b2}x_{c3}x_{d4}x_{e5} x_{f6}
\end{equation}
and then permute the left side indices, giving
\begin{align}
&x_{a1}x_{b2}x_{c3}x_{d4}x_{e5} x_{f6}\\
&
I don't know if the following post is appropriate for Meta Physics SE; in case it is not, please let me know.
Recently I've extended the scope of an old question of mine on nonlinear analysis asked on Mathematics SE. Since the question is very deeply connected to mathematical physics, I thoug...
@StanShunpike Note that the complex scalar field Lagrangian does not have the usual factor of 1/2 present in the real scalar field, cf. e.g. Srednicki Quantum Field Theory Eq. (3.37) or Kaku Quantum Field Theory Eq. (3.35).
@Danu Exactly. I think Stan was plugging a complex field into a Lagrangian with a factor of 1/2, thus giving him an extra 1/2 at the end.
(I can't reference the post because I'm on mobile.)
@Danu Do you know if it is possible to picture a 3-chain in 3 dimensions? Nakahara says nonchalantly that the torus is not the boundary of a 3-chain. To me that's not obvious.
I'm sorry that I don't have years of time to read countless other books that may or may not help me. I just figured you'd know a little algebraic topology.
But I don't know what happened. All I got was the canned response of "Our applicant pool is of an extremely high calibre, and unfortunately we are only able to accept a small number of students each year....."
@alarge Nah, he wouldn't have found out about lack of money until this month. I'd hope he'd let me know, but maybe he didn't. Onthe other hand, his silence could be because he's trying to get the committee to reconsider me and doesn't want to say anything until he hears something
Time will tell
Until then, it'll have to be a bank job. They love hiring physicists
@Danu Yes. Physicists are used to applying complicated math to solving practical problems. Most other professions don't do that nearly so well. And they need people who can solve complex differential equations and stuff
I mean, not to put you down or anything, but I don't think banks really need that many physicists anymore (as there's been a lot of change in the industry after 2007/2008). Being a physicist most certainly does not disqualify you, but you'll probably face tough competition from MFEs, say.
@alarge You say that, but I've looked on a few banks' websites and they have FAQs under the career section that says "I have a degree in physics/applied mathematics. Are there positions for me? A: Yes! We're always happy to....."
@Jimnosperm I mean sure, they hire physicists, but the competition is tougher than it used to be, and their roles have also changed. Like 10 years ago you didn't have people who specifically learned the craft.
@Danu IMO: spending 5-8 years getting a PhD to get a career teaching at a salary equivalent (sometimes less) than what people with a BS can make is selling ones soul
I think the top software companies (like Google) pay similar money as the big IBs (although, hedge funds and prop shops will get you more, but they are much, much more selective as well), so your third choice of becoming an engineer (software, I'm assuming) is not a bad choice either.
But really, no offense intended. I think a lot of my friends studying mathematical physics are thinking of getting out of academics before they've "wasted too much time" as well
Go do management consulting. A friend of mine did that (at one of the big three) after spending some time in academia (computational physics) then switched jobs and now is up in the ivory tower of a big engineering company.
Correct me if I'm wrong, but GRE stands for Graduate Readiness Evaluation. If I already have a graduate degree, isn't that proof enough that I'm ready for graduate studies?
@KyleKanos I think especially the famous ones at least have sort of a cut-off (say ~75th or maybe 80th percentile) so they just stop considering you if you go below it
I was quite bummed when I scored 80th percentile on mine :\
Should've done more brainless cramming and stuff instead of trying to actually relearn the material haha
@KyleKanos Not very widespread. They require it at least for formality. Even if they ignore it, you have to include your score to even finish the application. It costs me $2000 to write the test. I can't afford that unless I'm guaranteed acceptance
True story: one of the professors at my MS university said Well if you don't get it, you can always just move to the university you really want to go and sit in on the lectures. If you do well enough there, they might just let you in
@KyleKanos This is actually super common in Asia: you go for one year as a "research student", and then the next year apply for the PhD course. Now the problem is of course money, so people there usually tap into their parents' wallets, but the labs do sometimes pay something for their time.
Some countries in Europe have PhD as a salaried position where you basically write a pretty regular-looking work contract, and you can start almost any time of the year. Restrictions may apply (never looked into it) if you're not a EEA citizen.
@Danu Yeah, the Netherlands was one of the countries I was thinking about, actually, as a friend of mine did his PhD there. I remember the job posting looked almost like any regular job advert. I think there was no admission test, but rather when the professor took him in (and it was his lab who paid him, I think), that was it.
Sausages universally split parallel to the length of the sausage. Why is that?
I saw it's not just a page views number, because, if I press F5 several times, it won't increase the number of views.
They could store my IP address in a table, but wouldn't that make it slow? They would need to query a database one more time for each request.
I am reading Shilov's book linear algebra. He explains how to compute determinants. Basically, for the plus terms you write
\begin{equation}
x_{a1}x_{b2}x_{c3}x_{d4}x_{e5} x_{f6}
\end{equation}
and then permute the left side indices, giving
\begin{align}
&x_{a1}x_{b2}x_{c3}x_{d4}x_{e5} x_{f6}\\
&
I don't know if the following post is appropriate for Meta Physics SE; in case it is not, please let me know.
Recently I've extended the scope of an old question of mine on nonlinear analysis asked on Mathematics SE. Since the question is very deeply connected to mathematical physics, I thoug...
