Discussion about Hairer's existence theory for stochastic differential equations

Question

The discussion started below this submission creation request, but it is better deserves a separate chat thread.

Answer 1

http://arxiv.org/abs/1303.5113 also http://arxiv.org/abs/1401.3014 , there are also great later articles by Hairer citing these. These are, to my mind, the biggest step in mathematical quantum field theory since Wilson.

Comments

I didn't say it was trivial, he doesn't claim to have solved 4d gauge theory existence, but when I thought about such things, the main obstacle was always that I had to sort out the convergence of the long-distance stuff at the exact same time as I had to sort out the convergence of the short distance stuff. Going to stochastic quantization separates the two problems cleanly, and this gives a way forward for asymptotically free theories, because the short distance stuff is what determines the SPDE behavior, while the long distance stuff is about the equilibration of the fields once you have the SPDE. This is why I am so gung-ho on the approach, it's stochastic quantization 2.0.

---

He didn't even claim to have solved the Minkowski $\Phi^4_3$ QFT. But we discuss it here because of its potential relevance to QFT. 

---

Hairer used stochastically quantized $\phi^4_3$ as the example SPDE in his Columbia lectures, I am going by those, since this is the presentation I know.

---

But this is Euclidean $\Phi^4_3$, which is unrelated to its Minkowski version unless $O(4)$ invariance and reflection positivity are shown.

---

O(4) invariance is really automatic, as it is present in O(4) invariant regulators (his regulator is a smooth function convolved with the noise, you can make it a rotationally invariant bump, he always does in his specific examples anyway), and the convergence was established to be  independent of regulator (this is one of the great advances of his formalism, the form of the regulator clearly doesn't matter in taking the limit). The thing that he constructs is an SPDE solution, with a good (distributional) continuum limit, whose long-time solution, if you take a constant time slice, has statistics that eventually converges to a random pick from $\phi^4_3$ path integral. These are already known to be reflection positive from other work, and it should be easy to show it directly, you asked about it, let an answer come. I believe (not sure, haven't thought at all) that with appropriate reflection positive statistics on the initial conditions, it is extremely easy to show reflection positivity  for all time slices and for all regulators, not just asymptotically at the large t limit.

There is no way reflection positivity is going to stop the program (unlike, say, the superrenormalizability business which requires a real idea advancing this stuff). The program constructs the stochastic quantized field theory in what can only be called 'the right way'.

Any difficulty in turning the SPDE construction into a field theory construction can only lie in establishing that the long-time limit is converging (but it is true that the long time limit will be converging to the appropriate thing). You could, for example, define the SPDE for $\phi^3$ field theory in 4d, and it should be ok as an SPDE, the theory is superrenormalizable, but it would not converge at large times to anything, the solutions would run away to $\phi\rightarrow -\infty$.

It might also have a finite time limit to the SPDE evolution, there are no results that guarantee global solutions, as sometimes they won't exist. The results he proves show local existence of SPDE solutions, and you need a further estimate for global solutions. This one of the places where he focuses his current research, connecting the SPDE to the stationary distribution at long times. Again, he is interested in SPDEs, but the formalism is general enough to apply to any (bosonic/real-Euclidean-action) QFT.

I don't claim that Hairer solved all the problems of constructing QFTs at one stroke, he just solved the main one--- defining a rigorous renormalization procedure that is completely non-perturbative, easy to work with and prove things about, is general enough that it should work precisely in those cases when traditional renormalization does, and corresponds exactly to what physicists know about renormalization (it's essentially a framework for producing convergence from a rigorous version of the OPE--- the OPE defines the possible renormalization terms in products of regularized distributions, and then the analysis parts just show the statistical convergence when the regulator is relaxed).

Because he is doing stochastic quantization, his renormalization for the stochastic time evolution is completely separately from constructing a vacuum state, and it works even when there is no vacuum state. The two problems are decoupled, and this allows you to think about short-distance renormalization without considering long-distance properties, like confinement. This separation property is central, it's what allowed the Glimm/Jaffe program to succeed too, but there the separation was intrinsically due to the superrenormalizability, it wasn't something that could be generalized. Here it can be generalized to any nonlinear SPDE even stochastically quantized Yang-Mills theory.

---

Oh wow! He got recognized so quickly? Great! congrats to him.

---

@ArnoldNeumaier: What Hairer did is for stochastic PDEs, the stochastically quantized versions of quantum field theories. Because they are stochastically quantized, you need to take a long-time limit to find the ground state (but this means that the long-distance ground state statistics are completely decoupled from the renormalization procedure--- this is an advance over the methods of the 1970s, it means you can rigorously renormalize a theory with no regard to what is going on at long distances).

His "regularity structures" both define a rigorous version of OPE in full generality, and also allow you to produce sample distributions from the solution to the SPDEs by a straightforward limiting procedure which is bounded using analysis estimates from the local behavior of the field fluctuation size.

To implement his general program, he did the full renormalization procedure in the stochastic quantized versions of superrenormalizable field theories, in one of which cases (3d \phi^4) he could compare the results to existing Glimm/Jaffe descriptions of the steady-state.The statement of reflection positivity is for the steady-state, for the vaccuum, it is not a property of the SPDE itself, so I don't think he bothered to prove this. But reflection positivity is not a critical feature for proving existence of a statistical theory anyway, there are lots of nonunitary statistical models. I also don't believe it's hard to do, you can ask it as a question "How do you prove refection positivity in a theory defined by a Hairer regularity structure?" The intermediate steps in his regularization are smooth solutions, made by mollifying the noise, which should statistically be reflection positive at any value of the mollifying parameter.

The formalism works whenever you can bound the short-distance multiplication and fluctuation properties of operators, and superrenormalizable theories are an example. Another example is asymptotically free theories, but here you have to work harder, because the multiplication only simplifies in the limit of extremely short distances, and is dependent on a fine logarithmic details of the renormalization--- the sign of the beta function. A trick around this is to modify the propagators to make the 4d theory superrenormalizable (but not unitary), there is no locality requirement in the formalism! You can apply it to crazy nonlocal power-law actions just as well as to standard actions.

The formalism is not so new to physicists, because it reproduces OPE results, but it is new in a sense that these OPE results are done for the SPDE, rather than the steady-state (the QFT vacuum correlations). The method is great, and in some sense is a vindication of Parisi's approach, which is really quite powerful and has never been used before as a method for rigorous construction of QFTs.

I believe the limitations in the old Glimm/Jaffe program were technical too, but here there is a problem in that the oldest results, from 1968-1972 were done using Feynman resummation, which only can possibly work in the superrenormalizable case, and then the results were converted to a path integral approach later. The technical limitations there are much stronger, because the construction is inherently based on perturbative constructions. But the details of Hairer's OPE are also doing these types of perturbative calculations to bound the distributions he is approximating, but in this case, it seems that one can see how to relax the procedure for logarithmic renormalization, unlike in the Glimm/Jaffe methods, which, if they would work in general for logarithmic theories would make QED as well defined as QCD.