bolbteppa discussing integrability

bolbteppa

May 29, 2015 20:29

@Semiclassical I know that when you decompose the partition function in statistical mechanics into a trace of products of transfer matrices, as in the Ising model, if they are commutative then the problem can be solved. The condition for commutativity is that they satisfy the quantum Yang-Baxter equation. Vaguely I think it's like finding an oasis of commutativity in a non-commutative algebra, i.e. a solvable subalgebra or something...

@Semiclassical so that is the Stat Mech link to the quantum Yang Baxter, and the quantum Yang Baxter is like a deformed Lie algebra Jacobi identity, so a quantum group is basically a deformed Lie algebra which reduces to a Lie algebra (modulo some technicalities). Also, note the partition function in stat mech can be set up for any integrable non-linear PDE, i.e. any non-lin integrable pde can be written as a partition fun, so you can basically set up a quantum group and Yang baxter for solitons

Semiclassical

funnily enough, i'm trying to learn about that connection to nonlinear DE's

mainly from knowledge of how it works in the 2D ising model, where stuff like diagonal correlation functions can be expressed in terms of solutions to the nonlinear Painleve equation

bolbteppa

@Semiclassical the insane thing is that this crazy formality even shows up in elementary quantum mechanics when you solve 1-D scattering problems
https://books.google.ie/books?id=db0dTvNKWFgC&lpg=PA211&ots=QoqznPIiZY&dq=Ising%20Model%20Yang-Baxter&pg=PA211#v=onepage&q=Ising%20Model%20Yang-Baxter&f=false
(The pages around that page are insane, they give intuition for needing Weierstrass/Jacobi elliptic and theta functions as being obvious!)

Semiclassical

snerk. ah, the joy of reflectionless potentials

bolbteppa

Yeah so I think the real thing going on is that they all come from non-linear integrable pde's, i.e. the partition function is basically the 'time-ordered exponential solution' to a non-linear pde, and there are tons of conjectures saying non-linear integrable pde's are solvable in terms of Painleve functions,

Semiclassical

nod

hard to parse that stuff, though

bolbteppa

May 29, 2015 20:40

Yeah

@BalarkaSen I don't understand it well enough to sum it up sadly :D

Semiclassical

the direction i'm trying to understand is from hierarchies (KP, for example) of integrable equations

which isn't an easy nut to crack

bolbteppa

@Semiclassical that is not a good place to start tbh

@Semiclassical basically, view integrable non-linear pde's as though it was literally quantum mechanics, it has a Schrodinger formulation, a Heisenberg formulation, a Hamiltonian system formulation, a path integral formulation, a spinor formulation, (wtf)

nullgeppetto

May 29, 2015 20:58

Hi ppl! I have started a bounty for this question, if you like please take a look! Thanks and apologies for spamming the chatroom!

bolbteppa

@Semiclassical you have a non-linear PDE $F(V) = 0$ and assume you can associate some linear operator $L$ to it whose expected value does not change with time, i.e. $<L,L>$ is time-independent which means $L$ can be written as an eigenvalue equation with a time-independent eigenvalue. This is like writing the Schrodinger equation in implicit $F(V) = 0$ form (where $V$ is the potential btw!), and considering the position operator or something.

From this you re-write $L$ in the Hamiltonian formulation and set up the Heisenberg equations. In non-linear pde's the exact same process gives you the Lax-matrix method of solution. Usually this method is presented backwards in pde's books so you don't see the connection.

Semiclassical

the thing i should point out is that i'm not tryingt to understand the Painleve stuff from a terribly high-level perspective

bolbteppa

So the key to integrability is being able to take your non-linear pde (i.e. your 'Schrodinger equation') and re-writing it in Lax form (i.e. as 'Heisenberg's equations'). If you can do this you have an integrable pde.

Semiclassical

what i'm really after is to understand how the (Toeplitz) determinants which show up in the 2D ising model (and more generically under the rubric of Fisher-Hartwig singularities) can be understood as solutions to the KP hierarchy

bolbteppa

Stay with me for a moment, I am basically just telling you things you already know but with new words

Semiclassical

May 29, 2015 21:06

which as I understand it amounts to showing that those determinants are tau-functions

bolbteppa

Tau functions come up naturally in a moment if I can formulate this right, you already know what they are

You will be able to see it yourself if we do this right hha

@Semiclassical So to sum up again, in a non-linear pde you are trying to find $V$, so you try to assume it is like Schrodinger's equation where your $V$ is actually just the potential not the wave function, then rewrite your pde as a Heisenberg equation, if it can be done, and it can't always be done, you have an integrable pde, the only difficult thing is doing the factorization, for which no general method exists, so this is why it's difficult.

But since we have established the existence of a Hamiltonian, the non-linear pde can also be written as a Hamiltonian system, i.e. you can set up an analogue of Hamilton's equations for the pde too, but you get an infinite system of them

@Semiclassical In fact you get an infinite number of conserved charges too, it's like conservation of energy for each Hamilton's equations. So thus far from a non-linear PDE we have a Schrodinger formulation (the pde itself), a Heisenberg formulation (Lax), a Hamiltonian formulation (Hamiltonian system).

It can also be written like Dirac's equation, factored using spinors, to give the Zero-curvature formalism. It can also be written using 'time-ordered exponentials' (a standard ode/pde method) to give an analogue of the path integral solution of Schrodinger's equation. Now, the 2-D ising model partition function is just a path integral, so already you see we expect some integrability formulation,

The KP Heirarchy is just the Hamilton's equations formulation of a non-linear pde, the Kadomtsev-Petviashvili equation, which can be factored into Lax form

Semiclassical

May 29, 2015 21:24

if ii'm remembering right, in the case of KdV the point regarding Schrodinger's equation with $V$ as potential is actually literally true

bolbteppa

Yeah so KdV is the canonical example because the factorization into Lax form is so easy

Now my (not 100%) understanding is that Tau functions are just the non-linear pde analogue of the QM/QFT partition function, so when you say you want to see why Toeplitz determinants are solutions of the KP hierarchy I think you are equivalently asking why Toeplitz determinants give partition functions which are the path integral solution to the KP equation

To quote a book "Mikio Sato discovered that the totality of solutions of the KP equations form an infinite dimensional Grassmannian, and established the algebraic structure theory of completely integrable systems" (Don't see how this fits in yet)

It's crazy how Toeplitz matrices are connected to convolution (and fourier series), topics so closely related to this whole discussion...

Semiclassical

busy, sorry

bolbteppa

@Semiclassical so if we can figure out exactly how KP relates to Toeplitz and justify determinants of Toeplitz matrices give partition functions I think the whole thing makes sense and is conceptually unified... This pdf arxiv.org/pdf/hep-th/0110125v1.pdf and the first book in the references will give far more detail on what I've said

No problem, back in a while

and a clue is that you said the Ising model uses these ideas

pjs36

May 29, 2015 21:49

@bolbteppa I'm about as far from applied/physics as can possibly be, but I've come into contact with the combinatorics of some KP-related stuff in the past. How odd!

Mike Miller

@lenticcatachresis If you're ever online, I have a spectral sequence question I'd like to ask you.

Semiclassical

@bolbteppa: that bit about the infinite grassmannian is something i'd like to understand a lot better myself, since I think that should connect with the free fermions description of the tau-function

i've got some sources myself, but uh

heavy lifting, to put it lightly

@bolbteppa: the source which would probably resolve things, if i could actually digest it, is

arxiv.org/pdf/1212.6049v3.pdf

bolbteppa

May 29, 2015 22:41

@Semiclassical Page 33 of your pdf:
"As it has been established in the works of the Kyoto school, the expectation values of group-like elements are τ -functions of integrable hierarchies of nonlinear differential equations."
Hopefully you see how this makes sense from what I said. I'm glad they said this haha :D The grassmannian thing is interesting, I'm kind of shocked your paper is restricted to discussing Fermions, that's like an awesome sign this is so coherent

@pjs36 weird, my guess is that the link is via moment-generating functions which I think are like Taylor expansions of random variables which are basically operators acting on states in QM linking all this to integrability so in some perverted sense that might be it

pjs36

@bolbteppa Beats me, I just know I was supposed to read a section of this paper for a reading course on combinatorial polytopes.

bolbteppa

@pjs36 wow cool, the set up of that paper is exactly what we're discussing, there is a lot of this finite field grassmannian combinatorics swimming around this theory too for some reason

pjs36

It's a small world after all! :P I think the triangulations of $n$-gons were what I was supposed to focus on

The pictures of KP waves are crazy, by the way. Here, and here, and here

bolbteppa

May 29, 2015 23:08

@Semiclassical From the intro and chapter 9 of Babelon's integrability book, it says the KP equation comes up because it is the abstract formulation of an integrable PDE with one singularity, i.e. the Lax matrix will have one singularity. The way to avoid worrying about singularities is to go to projective space. Also, this makes sense because the path integral, representing probabilities, cannot be singular, so the KP path integral must be in projective space...

idk if you can see page 299 but they basically show a path integral books.google.ie/… I mean it's crazy

@pjs36 do you think the middle of the X in your second picture is the one singularity?

pjs36

@bolbteppa It certainly seems like a distinguished point! But unfortunately I don't know enough about any of this to have even a guess.

Semiclassical

i've got a copy of Babelon's book, but alas I find it pretty impenetrable

all in all, it's the developments of the kyoto school which i probably need to understand. but easier said than done

bolbteppa

May 29, 2015 23:23

@Semiclassical do you see that the motivation is to just set up the non-linear PDE integrability form of the path integral for the KP equation? However because the Lax matrix you get from KP has a singularity we cannot do this directly, because probabilities cannot be singular, so we must go to projective space to avoid worrying about singularities. So we have a path integral in projective space. Nobody mentioned Grassmannian so far, and your paper doesn't mention it

Semiclassical

that one doesn't, no, but it does talk about plucker coordinates. so while they don't talk in terms of grassmannians i think they're stil there

back later

bolbteppa

As Babelon says, "Grassmannians are projective algebraic varieties", so here is our first link

I'm guessing that grassmannians arise just because you want to take subspaces of the fermionic coordinates at a time in summing the path integral, that's all

So "if the tau functions are interpreted in terms of infinite Grassmanians (fermionic approach) then the Hirota relations sometimes boil down to Pluecker relations", ncatlab.org/nlab/show/Hirota+equation , which is just some condition, and Schur functions are like functions that come up in partitions, so I feel the whole paper you linked to is making some sense :D

@Semiclassical okay cool

This Hirota Bilinear actually has a physical analogue too, wtf... Vertex operators!???

Overall summary so far:
Non-linear PDE = Schrodinger equation in Schrodinger formalism
Lax matrix method = Heisenberg equation form of Schrodinger equation
Hamiltonian system = Hamilton's equations arising from existence of Hamiltonian in Lax method
Zero curvature formalism = Spinor decomposition of non-linear PDE mimicking Dirac equation coming from Klein-Gordon
Tau function = Path integral form of non-linear pde ('time-ordered exponential solution')
KP Equation = General Non-Linear PDE Coming From Assuming Lax Matrix Has One Singularity, Path Integral formulation must be on a projective s…

$Mathematics$ Mathematics

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