Wave propagation according to higher order field equations I

A detailed study is made of wave propagation according to a sixth-order partial differential equation with complex masses proposed by Swieca and Marques, which presents a kind of generalized Klein-Gordon equation. The choice of definite Green's functions in the corresponding Yang-Feldman integral equation corresponds to a certain choice of boundary conditions for the allowed solutions of the corresponding partial differential equation. The advanced and retarded Green's functions used possess the anomalous feature of having non-zero values in the neighbourhoods of those, past or future parts of the light cone, for which traditional advanced and retarded Green's functions are zero. However, it is shown that a suitable averaging procedure provides the possibility of defining sets of functions, such that solutions of the Yang-Feldman equations belonging to this set possess the property that the future behaviour of the solution is determined by its asymptotic initial conditions. Certain features of the wave propagation, according to the equations considered, can be usefully compared with the properties of the solutions of the ordinary differential equation - and corresponding integral equation - which represents the equation of motion of a charged particle including the force for radiation reaction. The particle then has a certain “size”. Analogously the “non-local field equations” have solutions characterized by a certain “fundamental length” indicating the space-time distances for which averaging occurs. The admitted solutions of the field equations seem to represent a relativistic field with a “finite a number of degrees of freedom” within a finite volume.