Excerpted here are

Proposal Title | Chaotic Field Theory |

NSF Division | Physics |

Program Name | Mathematical Physics |

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.