Field theory of pattern identification.

Abstract

Based on the psychological experimental fact that images in mental space are transformed into other images for pattern identification, a field theory of pattern identification of geometrical patterns is developed with the use of gauge field theory in Euclidean space. Here, the ``image'' or state function \ensuremath{\psi}[\ensuremath{\chi}] of the brain reacting to a geometrical pattern \ensuremath{\chi} is made to correspond to the electron's wave function in Minkowski space. The pattern identification of the pattern \ensuremath{\chi} with the modified pattern \ensuremath{\chi}+\ensuremath{\Delta}\ensuremath{\chi} is assumed to be such that their images \ensuremath{\psi}[\ensuremath{\chi}] and \ensuremath{\psi}[\ensuremath{\chi}+\ensuremath{\Delta}\ensuremath{\chi}] in the brain are transformable with each other through suitable transformation groups such as parallel transformation, dilatation, or rotation. The transformation group is called the ``image potential'' which corresponds to the vector potential of the gauge field. An ``image field'' derived from the image potential is found to be induced in the brain when the two images \ensuremath{\psi}[\ensuremath{\chi}] and \ensuremath{\psi}[\ensuremath{\chi}+\ensuremath{\Delta}\ensuremath{\chi}] are not transformable through suitable transformation groups or gauge transformations. It is also shown that, when the image field exists, the final state of the image \ensuremath{\psi}[\ensuremath{\chi}] is expected to be different, depending on the paths of modifications of the pattern \ensuremath{\chi} leading to a final pattern. The above fact is interpreted as a version of the Aharonov and Bohm effect of the electron's wave function [A. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959)]. An excitation equation of the image field is also derived by postulating that patterns are identified maximally for the purpose of minimizing the number of memorized standard patterns.