We have found a new computer solution to the aesthetic field equations. This solution describes a two-particle system with more structure than previously found. The contour lines show an arm structure. We have observed four arms around the maximum center. The location of the maximum (minimum) center is not along a straight line as a function of time. This is the first time that such an effect has been observed for any kind of nonlinear partial differential equation, so far as (...) we know. A further discussion of the aesthetic principles leading to the field equations is given. (shrink)

We have studied aesthetic field theory in the case where all invariants constructed from Γ jk i and involving g ij are zero. We studied such a “null” theory in 1972, but the cases we cited were plagued with singularities. By introducing complex fields the situation with respect to singularities improved. Complex fields are consistent with the basic “aesthetic principles” we outlined earlier. Within our null theory we see in two-dimensional spacetime a scattering of particles that was more involved than (...) what we had seen before (regardless of dimensions). We see creation and annihilation of particles out of the vacuum. We also see a three-particle system within a small region of spacetime. In three spacetime dimensions we see a bound two-particle system. Another solution suggests a bound three-particle system. As well as we can tell the particles stay together (confinement) and do not give problems with attenuation. We observe in three dimensions one of the bound systems moving along a definite path in time. The four-dimensional spacetime results are not clear at this point. Whether “topological” bound systems of three particles exist has yet to be determined. A map in the four-dimensional case indicates a planar three maxminima confluence and the suggestion of a second such confluence. (shrink)

We have again studied a null theory within the complex aesthetic field theory. This time we required that the spatially inverted origin point data represent the imaginary part of the complex origin point data. This was not the case in our previous studies of the null aesthetic field theory. However, this procedure did not lead to effects not previously observed as far as we could tell. Adding an additional term to the origin point data that presented the null character of (...) the theory also did not lead to new effects. We also investigated a real null theory that led to constant fields. This theory was then made complex by a procedure discussed in the early part of the paper. We found that the resulting complex theory remained trivial nevertheless. Again, we found that it was not necessary for the theory to be null to find “confluence” type solutions. (shrink)

We continue the program of looking for increased complexity within aesthetic field theory. We study a solution with five planar maxima and minima. Another solution in which we counted 19 planar maxima and minima is also studied. This latter solution was obtained by modifying our previous principles by allowing for an arbitrariness associated with the integration path in conjunction with the equation Γ jk:1 i =0.

We are able to incorporate an antisymmetric second-rank tensor into null aesthetic field theory. There are some changes in the solutions due to the introduction of this antisymmetric second-rank tensor, which we discuss. We are not able to find an acceptable bounded particle system in four space-time dimensions.

We discuss the structure of a particle system obtained in “aesthetic” field theory and study the evolution of this system in time. We find the particle system to have more structure than particles found by other authors investigating particlelike behavior in nonlinear field theories. Our particle system has a maximum center in proximity to a minimum center. Thus, we can interpret our system as being constructed of two bodies. We find that the maximum center and the minimum center move in (...) straight lines, to computer accuracy. Thus, we have not found any nontrivial force laws. This suggests that the situation with respect to basic principles be kept fluid. So far as we know, we are the first investigators to study the trajectories of a two-body system which arises as a consequence of nonlinear field equations. (shrink)

We continue our study of the Lorentz-invariant field theory based on the equations Γ jk;l i =0 and gij;k=0. To first order in a perturbation expansion, we find Γ jk;l i =0 reduces to the wave equation. In orders higher than the first, we find that Γ jk;l i =0 cannot be linearized. We also find that the simple wave-type equation gij∂2g/∂xi∂xj=0 is contained in the theory when an appropriate choice is made for the parameters at the origin point.

The aesthetic field equations do not resemble the wave equation, nor was the motivation behind them the wave equation. Nevertheless, we show that there exists a solution to the field equations that satisfies the wave equation. Integrability is also satisfied by this solution. Previously we showed that the Aesthetic Field Equations have particle solutions. Now we see that the equations also have sinusoidal solutions.