Use of commutation relations in no integrability aesthetic field theory

摘要

We work with a simple set of no integrability field equations obtained from the aesthetic field theory. We show that with the use of commutation relations we can calculate the field at any point by integrating only along a single path, together with a correction term which is different at each point. This approach is superior, from a numerical point of view, to the summation over paths technique which we developed in a previous paper, because we can better avoid the path proliferation problem. Furthermore, we can make use of considerably smaller grid sizes, which allows for greater accuracy. The resulting map obtained for the simple system of equations considered was similar to the summation over paths results. We found no sign of any two dimensional maxima or minima in the region studied. Combining this result with our previous studies, we conclude that the type of result that we get, when we consider the summation over paths degree of freedom, depends on the kind of lattice solution we are working with. In particular, the summation over paths degree of freedom appears consistent with multiparticles for a “loop” lattice. A loop lattice is illustrated in the Appendix. It appears when we specify an integration path.