Gross-Neveu Models, Nonlinear Dirac Equations, Surfaces and Strings

Recent studies of the thermodynamic phase diagrams of the Gross-Neveu model (GN2), and its chiral cousin, the NJL2 model, have shown that there are phases with inhomogeneous crystalline condensates. These (static) condensates can be found analytically because the relevant Hartree-Fock and gap equations can be reduced to the nonlinear Schr\"odinger equation, whose deformations are governed by the mKdV and AKNS integrable hierarchies, respectively. Recently, Thies et al have shown that time-dependent Hartree-Fock solutions describing baryon scattering in the massless GN2 model satisfy the Sinh-Gordon equation, and can be mapped directly to classical string solutions in AdS3. Here we propose a geometric perspective for this result, based on the generalized Weierstrass spinor representation for the embedding of 2d surfaces into 3d spaces, which explains why these well-known integrable systems underlie these various Gross-Neveu gap equations, and why there should be a connection to classical string theory solutions. This geometric viewpoint may be useful for higher dimensional models, where the relevant integrable hierarchies include the Davey-Stewartson and Novikov-Veselov systems.Comment: 27 pages, 1 figur

Geometric approach to sampling and communication

Relationships that exist between the classical, Shannon-type, and geometric-based approaches to sampling are investigated. Some aspects of coding and communication through a Gaussian channel are considered. In particular, a constructive method to determine the quantizing dimension in Zador's theorem is provided. A geometric version of Shannon's Second Theorem is introduced. Applications to Pulse Code Modulation and Vector Quantization of Images are addressed.Comment: 19 pages, submitted for publicatio

Magnetic Field and Curvature Effects on Pair Production II: Vectors and Implications for Chromodynamics

We calculate the pair production rates for spin-$1$ or vector particles on spaces of the form $M \times {\mathbb R}^{1,1}$ with $M$ corresponding to ${\mathbb R}^2$ (flat), $S^2$ (positive curvature) and $H^2$ (negative curvature), with and without a background (chromo)magnetic field on $M$. Beyond highlighting the effects of curvature and background magnetic field, this is particularly interesting since vector particles are known to suffer from the Nielsen-Olesen instability, which can dramatically increase pair production rates. The form of this instability for $S^2$ and $H^2$ is obtained. We also give a brief discussion of how our results relate to ideas about confinement in nonabelian theories.Comment: 24 pages, 9 figure

Subjectively interpreted shape dimensions as privileged and orthogonal axes in mental shape space

The shape of an object is fundamental in object recognition but it is still an open issue to what extent shape differences are perceived analytically (i.e., by the dimensional structure of the shapes) or holistically (i.e., by the overall similarity of the shapes). The dimensional structure of a stimulus is available in a primary stage of processing for separable dimensions, although it can also be derived cognitively from a perceived stimulus consisting of integral dimensions. Contrary to most experimental paradigms, the present study asked participants explicitly to analyze shapes according to two dimensions. The dimensions of interest were aspect ratio and medial axis curvature, and a new procedure was used to measure the participants' interpretation of both dimensions (Part I, Experiment 1). The subjectively interpreted shape dimensions showed specific characteristics supporting the conclusion that they also constitute perceptual dimensions with objective behavioral characteristics (Part II): (1) the dimensions did not correlate in overall similarity measures (Experiment 2), (2) they were more separable in a speeded categorization task (Experiment 3), and (3) they were invariant across different complex 2-D shapes (Experiment 4). The implications of these findings for shape-based object processing are discussed