I knew about the residual theorem, that if your contour encircles some singularities of a function, then the value is just the values of the singularities added together (or smoething like that)
@DanielSank Well, if you find what you're searching for please show me! I have a feeling we're talking slightly past each other here, but I don't think what you're asking for has currently any chance to exist.
@ACuriousMind Well, it is good to know. From now I will only do this, if I think I can make an enough important change to significantly shift the review decisions.
The proof of residual theorem strongly reminds of green's theorem because in one complex analysis lecture notes they started drawing circles around the singularities and that reminds of circulations
What is physical interpretation of residues of poles (of any order) of a complex function? Poles represents the points where a complex function cease to be analytic and residues are calculated to solve complex integration but I am curious about it's physical interpretation, if any?
@DanielSank Again, you might be misunderstanding me: I'm saying the generic QFT path integral is not understood well enough to write what you're asking for. It's not that no one has bothered to write that down, it's that we mostly just stumble our way through specific applications of a general case where we have no idea what the general case actually does (beyond being an integral).
@ACuriousMind Yes but you can usually do something like lattice-ify your space, work things out, wave your hands at the limit as the lattice spacing goes to zero, and get perfectly happy answers. If there are divergences it's rather clear why they're there, etc.
@DanielSank Oh, you haven't seen what the lattice QFT people have to deal with, have you? If I remember correctly, no one even knows whether the continuum limit of 4D lattice $\phi^4$ theory really is the continuum $\phi^4$ theory. In some dimensions it is known to not be the case - the continuum limit just becomes the free field.
@ACuriousMind I mean that isolated oscillators/fields/whatever often behave poorly, but that behave well if they're coupled to something else which can wash out singularities.
@DanielSank ahahaha, well, then you're in the realm of thermal field theory or even non-equilibrium theory, which requires different techniques still. Most non-equilibrium simulations are fine in this sense, although they might only be because we have no chance in hell to compare them to an analytical result, anyway...
But dude, isolated oscillators are pathological even in the absolute simplest case.
Try analyzing a driven simple harmonic oscillator. It's totally pathological. However, if you add damping everything is fine, and it's almost always the case that you can take the damping to zero at the end of your calculation and get a reasonable result.
@DanielSank I laugh because you kinda remind me of this guy. Yes, the things you say work, but they spawned entirely new subfields instead of fixing the original problem.
@ACuriousMind Based on my experience, which is limited, your position that this is all very complicated is caused by having learned the techniques in the context of complicated systems. I guess the techniques themselves are not so hard.
I have this nitpick about proving Cauchy's theorem by applying the Green's theorem though, but I am not sure if a physicist would appreciate it. If you apply Green's theorem, you're assuming the real and imaginary parts are differentiable, hence that $f$ is $C^1$. That's slightly stronger than assuming $f$ is holomorphic, in a way.
This is the reason most complex analysis book introduces the proof differently, by Goursat's lemma (Cauchy-theorem on a triangle) and then constructing an antiderivative explicitly.
Why geometrically a complex integral of this form $\oint \frac{1}{z}=2\pi i$ nonzero. We knew that a closed contour encircling a singularity cannot be continously deformed into one that is located in a simply connected region, but
@DanielSank The point I made at the beginning was that in this case the technique doesn't really generalize. The QM path integral is well-defined, usually well-behaved, and well-understood, and you can in principle directly evaluate it to compute stuff. The QFT path integral is horribly ill-defined, usually assumed to behave similarily to the QM path integral, and a purely formal object that you always need to translate into something else to even have a chance at computing it.
what geometric properties of that loop determines the value you get for a singularity. E.g. if I have two loops that are encircling a pole of 1st and 2nd order respectively, then it is easy to find out what they integrate to, but why are these loops different, because they both seemed to be just circling a hole of genus one?
I give you all your points about the shifting of poles or the other things where you dislike that the QFT people act as if they invented it, but this is a different thing: We're desperately trying to generalize a well-known technique and are still, in general, failing at making it precise.
@DanielSank How general is the technique? Basically what you're doing there is adding a damping term so the Fourier integral will converge, but if memory serves, you do in some cases have to choose the damping terms carefully (I forget the details, but working out some periodic Fourier transforms, basically). I guess you could use distributions to do things more rigorously?
@EmilioPisanty Sure it is. Adding damping to Schrodinger's equation means that the system has some probability to disappear. If you're working with bound states, then you're giving the system a probability to escape the bound state. You can take that probability to zero later if you want.
But until you actually do the tracing out, and show that there were no ugly singularities in there to begin with, I'm not sold that this fixes the problem.
@EmilioPisanty You can look at it either way. Throwing a damping term in there for the sake of regularization is at least physically motivated by the fact that in reality you probably do have some coupling to the environment there.
Daniel, I cannot wait that your physical visualisation technique will allow us to visualise general ring algebraic structures, cause that will be really cool (and rigorous enough to make layperson to understand rings in the most general sense)
@ACuriousMind @alarge @EmilioPisanty Somewhere I have a nice writeup of how the Caldeira Leggett model works. I should share that in a self Q&A. The model shows how an infinite set of lossless oscillators looks like a lossy element which can add regularization.
@DanielSank en.wikipedia.org/wiki/Ring_(mathematics) sorry I was talking about this, I added the words "algebraic structure" so that we won't be mistaken for other ring like things e.g. superconductor rings (sorry mind mind tend to do such things to avoid any possibility of misconception)
@0celo7 Crap, another mistake! I really have to organise my mind a bti more
In the chapter on perturbation theory, there's a homework problem asking you to do perturbation theory in the case where the unperturbed system has degeneracy, and where the perturbation does not lift the degeneracy to first order.
@Secret What do you mean "what is it doing there"? It appears when you compute the creation/annihilation operators in terms of the field.
@DanielSank I'm not sure why exactly I should be interested in that model, but I usually enjoy reading your posts once written, so go ahead and write it ;)
@ACuriousMind It's just an example of a very "complicated" topic, originally explained via path integration, being easy to understand in a relatively simple system.
@ACuriousMind in the article, f and g are the flat waves of the field, thus there should be a way to interpret what $\overleftrightarrow{\partial}_0$ is doing to the flat waves?
I also noticed that $\overleftrightarrow{\partial}_0$ look very similar to a term in the probability current of QM, i.e. $\psi^*\nabla \psi - \nabla \psi \psi^*$