Study of a three component lattice system

Abstract

There are lattice solutions of different character in the no integrability Aesthetic Field Theory. In this paper we study an example of a three component lattice system which is of three dimensional character. When an integration path is specified we find that the lattice particles behave like solitons. These solitons undergo sinusoidal motion in both x and y directions. When we introduce a four dimensional e α i, to make the system four dimensional we find the lattice is made up of what appears to be an infinite number of loop particles. The summation over path degree of freedom is introduced using the commutator method. The program is to study how the summation over path degree of fredom affects the lattice solution. We find that the additional degree of freedom affects the three dimensional system markedly. We still have evidence for an infinite number of planar maxima and minima but the zero contour lines are now bent into closed curves resembling circles. The solution appears to have a regular pattern although it is not as symmetric as the lattice itself. A conclusion we can reach is that the summation over path degree of freedom affects different lattice solutions in different ways. Although a five component lattice is not studied in this paper we remark that the summation over paths leads to a more disordered system in this case.