Talk:Supersymmetric theory of stochastic dynamics

Supersymmetry, Chaos, and Wikipedia

This page is about a theory that establishes a close relation between the two most fundamental physical concepts, supersymmetry and chaos. The story of this relation has two major parts. The first is the well celebrated Parisi-Sourlas stochastic quantization of Langevin SDEs. The second is the more recent generalization of this procedure to SDEs of arbitrary form. At the first sight, it may look like it is too early for the second part to be on a wikipage. On the other hand, without this part there is no supersymmetry-chaos relation because Langevin SDE are never chaotic. Their evolution operators have real and non-negative spectra. As a result, partition functions of Langevin SDEs never exhibit exponential growth in time that would signify the key feature of chaotic behavior - the exponential growth of the number of closed trajectories.

Needless to say that notability for a general audience is one of the wikipedia requirements for a theory to have its own wikipage and that it is the connection of supersymmetry to the ubiquitous chaotic behavior in Nature that makes STS notable for a reader that has no background in mathematical/theoretical physics.

To assure that the supersymmetry-chaos relation is suitable for wikipedia, creation of this page had to wait until the material had been published a sufficient number of times in Physical Review, Annalen der Physik and a few other scientific peer-reviewed journals. By wikipedia regulations, this material is no longer an “original research” because it is now an opinion of not only a handful of authors recently working on this subject but also (at least partly) of the reviewers and editors of the above journals. This is why the tagging of this page for deletion (see the top of this talk page) was ruled in favor of keeping this page.

By now, I have been almost the sole editor of this page and the presentation is most likely biased. Please help by editing the page or discussing possible ways to improve it on the talk page.

Vasilii Tiorkin (talk) 15:07, 12 December 2021 (UTC)

Cleanup

I'm less than entirely happy with the current state of this article. It's unreadable to those with an ordinary education in mathematics. I'm thinking that it would be a wise idea to split this article into two, or maybe three. with the first preliminary case dealing with just the Parisi-Sourlas N=2 supersymmetry. Just saying "oh la de dah its just BRST quantization" is useless, given the rather poor condition of the current version of the BRST page. I spent all day yesterday, whacking on it, to get at least the informal description mostly coherent. The formal mathematics description is a train wreck. This article then goes on to invoke (-1)F which is currently a freakin stub that got nominated for AfD, and Witten index which is also a stub. Both of those articles need to be fixed first.

I currently have this super hand-wavey sketch. It need to be fleshed out. Start with Stochastic differential equation#Use in physics which currently states

Therefore, the following is the most general class of SDEs:

${\dot {x}}(t)=F(x(t))+\sum _{\alpha =1}^{n}g_{\alpha }(x(t))\xi ^{\alpha }(t),\,$

where $x\in X$ is the position in the system in its phase (or state) space, $X$, assumed to be a differentiable manifold, the $F\in TX$ is a flow vector field representing deterministic law of evolution, and $g_{\alpha }\in TX$ is a set of vector fields that define the coupling of the system to Gaussian white noise, $\xi ^{\alpha }$.

I guess that phase space X can be replaced by a symplectic manifold or a Poisson manifold. The section Stochastic differential equation#SDEs on manifolds is underwhelming as currently written. This needs to be fixed/expanded and the various deficiencies corrected. ... Anyway $F\in TX$ so the Lie derivative ${\mathcal {L}}_{F}$ makes sense. If we freeze time. I don't entirely understand what happens when the gaussian noise is added. Also, since X is supposed to be symplectic, it seems like there should be relationships to either a Poisson bracket or maybe a Schouten–Nijenhuis bracket or something, who knows, a detailed reference is needed. Then, write

$M=\exp -{\mathcal {T}}\int _{t_{0}}^{t}{\mathcal {L}}_{F}$

where ${\mathcal {T}}$ is the time-ordering operator, and the $\exp -{\mathcal {T}}\int {\text{stuff}}$ is explained very poorly in product integral and in State-transition matrix, both of which are in woeful shape, and a slightly better in Magnus expansion and perhaps best explained in ordered exponential and a worthy special case in Dyson series. I attempted to make some minimalist repairs to these five articles in the last 48 hours, but each one requires many days of work to whip into shape. Magnus expansion defines

$\exp -{\mathcal {T}}\int _{t_{0}}^{t}A$

but wants A to be an NxN time-dependent matrix. That's OK, as long as we are careful to then say $A=\left.{\mathcal {L}}_{F}\right|_{p}$ for a point $p\in TX$ (I think this is correct, I'd like to have an actual reference for this.) Thus,

$\left.M\right|_{p}=\exp -{\mathcal {T}}\int _{t_{0}}^{t}\left.{\mathcal {L}}_{F}\right|_{p}$

is well-defined, assuming that ordered exponential is spruced up, and maybe some extensions of Magnus expansion to a suitable manifold setting. All is cool so far. The operator M is being called the "Stochastic evolution operator", it seems.

A few more details are needed... there needs to be some kind of averaging over the gaussian noise. I do not currently understand how to do this correctly and formally. I never read a formal, mathematical treatment of the Langevian eqn, so I have lots of little questions that math people who enjoy rigor would ask. Next, we have to argue that

$M=\exp -(t-t_{0})H$

for some Hamiltonian-like H which is the "Stratonovich interpretation of SDEs" .. something something Stratanovich integral which I don't currently understand. Equivalently, this is a "(bi-graded) Weyl symmetrization" ... I assume that the bi-grading refers to the Gerstenhaber algebra and I assume that Gerstenhaber appears there because everything is being done on a Poisson manifold, but this is unclear. The Poisson superalgebra has the other kind of grading. I bitched about that on Talk:Graded ring a few days ago, too. For Weyl, perhaps we need to point at Moyal product, but maybe instead the deformation quantization is the other article. After getting all this untangled, we can finally write

$W={\text{Tr}}(-1)^{F}M$

which is the Witten index after handwaving that $M=e^{-(t-t_{0})H}=e^{-\beta H}$ but its unclear how to do the handwaving. I'm also not sure how (-1)F got in there, except maybe it has something to do with ... beats me.

Meanwhile, at the bottom of Langevin equation#Path integral we've got the nascent sketch of a path integral formulation. Apparently, the Parisi-Sourlas supersymmetrizes that up to N=2. I guess ??? It goes something like this: treat the Langevin equation as if it were a constraint, create a Lagrange multiplier for

$\delta ({\dot {x}}(t)-F(x(t))-\sum _{\alpha =1}^{n}g_{\alpha }(x(t))\xi ^{\alpha }(t))$

This suffers from all the conventional issues during quantization, because it looks like a gauge fixing term, which is why BRST is invoked. I understand BRST, but I don't understand this particular leap. There is finally one more (one last?) leap; take something that resembles the BRST charge

$Q_{BRST}=c^{i}\left(L_{i}-{\frac {1}{2}}{{f_{i}}^{j}}_{k}b_{j}c^{k}\right)$

and something something something $H=[Q,{\overline {Q}}]$ and claim this is exactly the same H needed to construct the Witten index. Clearly I'm totally lost by here. And if I understand correctly, this is "merely" for the Parisi-Sourlas results. There's a whole lot of connect-the-dots here that remain unconnected, for me. Assuming that I'm mathematically average, then other wikipedia readers will be just as lost. This is the basic problem, here. 67.198.37.16 (talk) 05:42, 30 May 2024 (UTC)

Lorenz attractor

Is it possible to use the Lorenz attractor as a concrete example? Without any noise, each tangent space to the phase space splits into an unstable manifold and a stable manifold and a center manifold. Although one can always integrate along a particular flow, there are going to be saddle points and bifurcations; I don't know how to describe them or think about them correctly. That's without noise. With noise .. ?

Unrelated to the above, I also don't understand what part of all this looks "gauge invariant" and thus why the Langevin eqn looks like a "gauge fixing term". 67.198.37.16 (talk) 02:51, 31 May 2024 (UTC)