Many panelists found the proposal overly ambitious and with no clear standard of success. The PI motivates the project as an approach to turbulence, as well as a re-examination of the path integral formulation and the role of classical solutions in quantization of nonlinear field theories. Very novel and powerful ideas would be needed to make rigorous progress on either. The PI proposes to use periodic orbit theory to make progress. One idea is to use trace formulae relating periodic orbits and spectra to study classical and quantum field theory by analogy with classical and quantum mechanics. This study is mathematically very difficult, and very little detail is given on how the PI intends to make progress on it or to validate the results against other methods. The extension to quantum field theory seems overly ambitious given the mathematical limitations of the periodic orbit theory in finite dimensions, and the presence of features in the quantum theory not captured by semiclassical expansions.

Some panelists found the idea interesting as a speculative, exploratory project, though others could not find explicit goals that could be achieved. The proposal involves difficult dynamical issues, but does not reference any of the mathematical work in the field aside from Rugh, e.g. the work of Pollicott, Baladi, Ruelle.

It was felt by the panel that the NSF panel in applied dynamical systems may have been more appropriate for evaluating this proposal.

Rating: E,E,E,V,G,F
Recommendation: The panel recommends that this proposal be funded if possible

The PI seems to be able to generate a great deal of enthusiasm for his ideas.

It is at the interface between mathematical and physical work, though mainly addressed to physicists and with no references to the related mathematical work.

It is difficult to believe that the PI can achieve the goals set forth in this proposal. Even in the more precisely defined parts, such as sections 3 and 4, it is difficult to make out what statement Cvitanovic aims to make. He does not even attempt to describe the relevant mathematical results known at this time. The more speculative parts on chaotic field theory strain credulity.

The idea that chaotic dynamics is built upon unstable periodic orbits is hardly new. In classical chaos theory, it has been known for many years that for hyperbolic systems, ergodic averages associated with natural invariant measures can be expressed as weighted summations of the corresponding averages about the infinite set of unstable periodic orbits embedded in the underlying chaotic set. For nonhyperbolic systems, there is no rigorous assurance of the validity of the periodic-orbit theory, although recent success on explicit enumerations of unstable periodic orbits in low-dimensional maps leads to confidence in the applicability of the theory. In semiclassical quantum mechanics, Gutzwiller taught us more than thirty years ago that the density of states, upon which most quantities of physical interests build, can be expressed as an infinite sum in terms of classical periodic orbits.

The PI's idea to apply the periodic-theory to spatiotemporal chaotic systems may be interesting but clearly not original. The key questions are thus what new understanding would one possibly gain about spatiotemporal chaotic systems and turbulence, and how useful and feasible such an approach could be.

First of all, what new insights into spatiotemporal chaos or turbulence can the periodic-orbit theory provide? For instance, in turbulence there is already vast knowledge about the various scaling laws dealing with energy, velocity, vorticity, and wavenumber statistics. Are the predictions of the PI's "chaotic field theory" consistent with the existing understanding and more importantly, what NEW results can one expect from such a theory? Unfortunately these issues were not addressed clearly in the proposal.

The second issue concerns the difficulty to compute unstable periodic orbits of chaotic systems (the PI should be very well aware of this). Although, in recent ten years or so there were progresses in enumerating periodic orbits of low-dimensional chaotic systems (mostly two-dimensional invertible maps or three-dimensional flows), how to obtain a relatively complete set of orbits for systems in higher dimensions remains to be an open problem. One difficulty concerns the high values of the topological entropies typically seen in high-dimensional chaotic systems: the number of unstable periodic orbits increases so extremely rapidly with the period that not many orbits of even low periods can be computed reliably. In the case of spatiotemporal systems which are much higher-dimensional than, say, two-dimensional maps, it is not clear how unstable periodic orbits can be computed in general (in some specific setting, there was success to compute a small set of these orbits a few years ago by Greenside and Zoldi). To this reviewer the issue of ACTUALLY computing unstable periodic orbits is more important than applying known formulae for statistical averages to spatiotemporal systems. A detailed description of the method to compute these orbits and justification for its feasibility for spatiotemporal chaotic systems would be much more useful than a show-off of fancy formulae or diagrams from existing papers. Unfortunatly the PI just mentioned very briefly that "variational methods for determining recurrent patterns are currently under development."

For people who practice and/or apply nonlinear dynamics, the burning issue in terms of any unstable-periodic-orbit based theory is how to compute the orbits from measurements, when the system equations are not known. For researchers in physical or biomedical sciences, systems of interest are usually such that explicit descriptions of their equations are not available. Periodic orbits are interesting only when they can be computed from time series, typically under noise. Works and partial success in the past demonstrate how difficult this task can be. There is no mentioning of this important issue in the proposal. It is difficult to see how the proposed research could be of interest even to a researcher in nonlinear dynamics, let alone people from other disciplines.

The narrow scope and the possible lack of appeals of the proposal even to researchers indicate that it may be difficult to attract talented students or postdocs to the proposed research.


Summary Statement

Professor Cvitanovic (PI) proposed a field theory to describe spatiotemporally chaotic or turbulent dynamical systems based on unstable periodic orbits (whether these are called "unstable recurrent patterns" or "unstable coherent structures" does not matter) embedded in the underlying dynamical invariant sets. According to PI, this is purely a dynamic theory because "the dynamics over large space and time scales is built up from small,computable patches of periodic solutions, without recourse to statistical assumptions." The theory will yield "global averages characterizing the chaotic dynamics, as well as a starting semiclassical approximation to the quantum theory". In terms of the quantum aspect the PI particularly proposed to implement nonlinear field transformations with the promise to yield "perturbative corrections in a form more compact than the Feynman diagram expansions." To carry out these tasks, the PI will devote a summer month, and plan to hire a postdoc, a graduate student, and an undergraduate student.

While it may be possible to have a chaotic field theory to describe some idealized dynamical systems, from the standpoint of feasibility, the proposed approach may be less interesting because it is unlikely to apply such a theory to any realistic systems including low-dimensional, nonhyperbolic chaotic systems of physical interest, let alone complex, spatially extended dynamical systems. The proposal is thus weak in terms of its Intellectual Merit. On the other hand, the difficulty to obtain unstable periodic orbits of chaotic systems from computations or from experiments raises some serious question with implications to possible Broader Impacts of the proposed research.