Areal Theory ∗

New features are described for models with multi-particle area-dependent potentials, in any number of dimensions. The corresponding many-body field theories are investigated for classical configurations. Some explicit solutions are given, and some conjectures are made about chaos in such field theories. Area-dependent potentials, V (A = r1 ∧ r2), or their mathematical equivalents, appear in several physical problems of contemporary interest. Notable among these problems are Yang-Mills theory (especially for spatially homogeneous configurations, with only time dependence) [3], extended supersymmetric field theories with (pseudo)scalar self-interactions [18, 13, 20], and more recently, membrane models [11, 16]. In the last century, Feynman [14] even assigned to students an exercise involving such potentials: Show the energy spectrum for a quantized areal potential model is discrete and quantum particles cannot escape from such a potential, even though the spectrum for the classical model is continuous and classical particles can escape along special trajectories for which A = 0. Many of us pondered this problem over the intervening years, especially in the field theory context [4]. Meanwhile, Barry Simon worked out five or six solutions to Feynman’s exercise and published them [25]. Classically, such potential models are interesting insofar as they may provide simple examples of chaotic systems [24, 2]. While there are special trajectories for which the motion is quite regular, including those for which particles can escape (such as straight line, free particle motion), for most trajectories this is not the case. The current consensus is such models are not integrable in the Liouville sense, and do not admit the construction of a Lax pair. Is this really so? ∗Based on a talk given at the “Coral Gables” conference, Fort Lauderdale, Florida, 12 December 2001. c © American Institute of Physics, http://proceedings.aip.org/proceedings/#624: Coral Gables Conference on Cosmology and Elementary Particle Physics; B. N. Kursunoglu, Editor.