A structurological consideration on a unified field

Abstract

A certain kind of unified field (i.e. the (x, ~p)-field) has been constructed in the previous papers (1) by attaching to each point (x) of the base field (i.e. the (x)-field) the four-component spinor (~) chosen as the internal variable. This unified field has been regarded as an interaction field between the (x) and (~) fields, the former is likened to the gravitational field governed by general relativity, while the latter to the spinor field governed by quantum mechanics. However, from another viewpoint, this unified field may also be regarded as a non local field obtained by (~ nonlocalizing ~) the (x)-field. In the ordinary nonlocal field theory (2), a certain kind of directional vector is taken as the internal variable, so that this is geometrically grasped by Finslcr geometry (3). On the contrary, since the spinor (~) is adopted in our case, the (x, ~)-field cannot be embedded in Finsler space. Now, as is well known (4), the independent variables in Finsler space are the line elements (x, ~), where the x's denote the co-ordinates of position and the ~'s the direction co-ordinates, the latter being considered from our viewpoint the internal degrees of freedom associated with each point. Furthermore, the physical meaning of ~ is somewhat arbitrary, so that it can be compared to the electromagnetic potential, which has been introduced into our field theory (~) through Weyl's gauge invarianee (5). Therefore, a certain kind of unified field (i.e. the (x, ~)-field) between the gravitational and electromagnetic fields can be considered by means of Finsler geometry, as has been actually done by several authols (e.v). If we want to consider a Finslerian generalization of the (x, ~)-field, we had better construct first such a Finslerian unified field as the (x, 9)-field and consider secondly