[ Quantum field theory | Group theory | Nonlinear dynamics | Research plan | Group | Students ]
Ph.D. work with T. Kinoshita: evaluation of the sixth-order corrections to the electron magnetic moment. At the time it was perhaps the most demanding numerical computation in theoretical physics, and remained QED's most precise theoretical prediction for a number of years. We developed a new perturbative expansion for the electron magnetic moment and new effective methods for dealing with Feynman diagram infrared divergences.
For me the conceptually most striking lesson of this long calculation were the amazing cancellations induced by gauge invariance. The desire to understand and exploit gauge invariance more effectively has motivated much of my subsequent research. The most interesting results of this effort were the mass-shell QCD Ward identities and the construction of the QCD gauge sets. This work motivated the formulation of planar field theory, rediscovered in 1994 by M.A. Douglas, D.J. Gross and R. Gopakumar, D.V. Voiculescu and others.
Other papers in this series are those on the QCD mass-shell infrared singularities and the diagram counting, both attempts to formulate gauge-invariant models computable to high orders, in order to investigate the nature of gauge-invariance induced cancellations.
My understanding of field theory has changed considerably over the years, largely due to advances in condensed matter theory. The field theory that I have used in my research is quite different from the textbook field theory, so I wrote my own textbook.
[ Quantum field theory | Group theory | Nonlinear dynamics | Research plan | Group | Students ]
Group
Theory
The studies of the QCD gauge invariance led to investigations of its group theoretical structure, and to the development of new methods for evaluating group-theoretic weights. The project also led to the magic triangle, a new construction of the Lie algebras which I consider to be my most original work. A part of this has been rediscovered by Deligne in 1995 and is currently an active research topic in group theory. Hope that this construction would account for the global symmetries of extended supergravities seems in retrospect too optimistic. This work led to yet another fascinating discovery: the connections between various groups by continuations to negative dimensions. In some cases, like the spinor representations we introduced, these connections can be understood as a supersymmetry (this has also been implemented on computer). In other cases, they remain magic.
[ Quantum field theory | Group theory | Nonlinear dynamics | Research plan | Group | Students ]
Nonlinear
dynamics
Frustration with the lack of methods for dealing with strongly nonlinear field theories has led me to investigate the general properties of nonlinear systems. I have found this line of research extraordinarily rewarding; from the lonely beginnings in 1976 this research has turned to an explosively growing field. Universality in Chaos, a collection of lectures and articles, has become one of standard references in the subject.
Personally I consider the following contributions as potentially important in the development of a theory of chaotic systems:
(1) In 1976, M.J. Feigenbaum got me interested in his discovery of universality in one-dimensional iterative maps. Following his functional formulation of the problem, I derived the universal equation for period doubling, which has since played a key role in the theory of transitions to turbulence (see M. J. Feigenbaum, J. Stat. Phys. 19, 25 (1978) and 21, 669 (1979)). We have generalized the universal equations to period n-tuplings; since then we have constructed universal scaling functions for all winding numbers in circle maps, and established the universality of the Hausdorff dimension of the critical staircase.
(2) In order to describe the topological structure of "strange attractors" of H'enon type, I introduced the notion of pruning front. This 2-d generalization of the kneading theory for 1-d maps has subsequently made it possible to work out symbolic dynamics for many other systems of physical interest, in particular a variety of billiards and 2-d potentials. Today, pruning fronts are actively investigated by both physicists and mathematicians.
(3) The discovery of phase transitions on "strange sets" was followed up by many other authors; such transitions were subsequently found in a variety of dynamical systems.
(4) The periodic orbit theory has been developed for classical and quantum non-integrable systems, and for renormalization group flows. In particular, the cycle expansions have contributed to the recent progress in the semi-classical theory of chaotic systems and its applications: improved the convergence of periodic orbit expansions, unified the Ruelle's theory for classical flows and Gutzwiller's theory for semi-classical quantization, and set the stage for accurate tests of tunnelling and other corrections to semi-classical approximations. Currently, the most promising directions lie in applying our expertise in "wave chaos" to the new type of acoustics experiments initiated by our CATS experimental group, and developing periodic orbit theory for stochastic nonlinear flows.
[ Quantum field theory | Group theory | Nonlinear dynamics | Research plan | Group | Students ]
Research
plan
Though my recent research focuses on nonlinear dynamics, I perceive myself not as a nonlinear dynamicist, but as a theorist with theoretical and mathematical physics interests in quantum theory, statistical mechanics, dynamical systems theory and computational physics. The actual work spans such seemingly disparate topics as applications of group theory to particle physics, study of topology of hyperbolic flows, or advising experimentalists on analysis of their data.
As is unavoidable in trying to see into the future, I will list below projects which are natural continuations of current work and thus almost guaranteed to bear some fruit. However, in all honesty I hope that what I will actually do will be something altogether different; my hiden risk fraught agenda involves projects such as gauge theories without Feynman diagrams and gauge fixing, thermodynamics of fractionally charged anyons, a geometrically intuitive theory of exceptional Lie groups, classical and semiclassical chaos without Markov partitions or periodic orbits, dynamical rather than statistical theory of turbulence.
In the near future my own research will concentrate on continuing development of theories of classical and quantum chaos, together with their experimental applications. I intend to complete the two large monographs currently in progress: an advanced graduate textbook - Chaos: Classical and Quantum, and Group theory, part II: exceptional Lie groups. I plan to work towards a periodic orbits theory of spatiotemporal "turbulence" of infinite dimensional dynamical systems such as the Kuramoto-Sivashinsky equation, and deterministic theory of far-from-equilibrium processes in settings such as transport in the Lorentz gas and other idealized gases. I hope that incorporation of the h-bar and/or weak noise effects into the periodic orbit theory will suffice to cure problems of "infinite detail" in partitioning generic chaotic flows. Coupled with continuing developments in the theory of pruning fronts for (by now a large class of) low dimensional dynamical systems, this should give us better control of "intermittency", broadly construed as coexistence of regular and chaotic phase space regions for generic dynamical systems. I would also like to see semiclassical quantization in terms of periodic orbit theory fashioned into an alternative exact quantization theory, by systematic inclusion of diffractive, penumbra, creeping, complex periodic orbits and perturbative h-bar into generalizations of the Gutzwiller trace formula and related spectral determinants. I hope that we can turn such theory of "wave chaos" into a general tool for practical applications such as testing shapes of elastic objects by their acoustic spectroscopy. Finally, some recent and quite surprising results on exact sum rules for stabilities of periodic orbits suggest that one might abandon periodic orbits altogether and find more powerful alternative formulas for spectra of chaotic systems.
[ Quantum field theory | Group theory | Nonlinear dynamics | Research plan | Group | Students ]
"Quantum
chaos" group
The periodic orbit theory program is too ambitious for one person, and it has involved many collaborators. The program has generated a number of results in which we take pride: among those were the Dahlqvist-Russberg disproof of the conjectured ergodicity of the (x y)^2 potential, K.T. Hansen's work on pruning fronts, H.H. Rugh's proof of the analyticity of the Fredholm determinants that we had introduced (approximations previously used by Ruelle and others destroyed analyticity, and gave correctly only the leading spectral eigenvalues), G. Vattay's work on corrections to the semiclassical trace formulas, G. Vattay and A. Wirzba's inclusion of diffraction effects into the periodic orbit theory, and my favorite application of the periodic orbit theory, D. Wintgen's and G. Tanner's quantization of Helium.
March 19 2003
