On an Almost Complex Manifold Approach to Elementary Particles

Abstract

An almost complex manifold description of elementary particles is proposed which links the approaches given independently by J. Ławrynowicz and L. Wojtczak, and by C. von Westenholz. This description leads to relations between the curvature form of an almost complex manifold, which accounts for the symmetry classification schemes within the frame of principal fibre bundles, and a curved Minkowski space-time via induced smooth mappings characterizing nuclear reactions of type N+π⇄N, where N is some nucleon and π the virtual π-meson of this reaction. Both approaches follow the same main idea of D. A. Wheeler developed in a different way by A.D. Sakharov.

Fields Connected with Particles in Terms of Complex-Riemannian Geometry

“…in the theory of Hartree. Thirdly, in contrast to the usual procedure of solving the equations a fcM = 0 f°r k = 3,0 and c j (e, m) = 0, c l (e) 4> 2 = m <fi 2…”

“…The functions # = <P k (y 1 + iy 2 , e, 0), k = 1,3, and $ = <P k (y 1 -i y 2 , e, 9), k = 2, 4, are the partial Fourier transforms with respect to the variable s of arbitrary four functions x ¥ k : (C 2 -»(C, holomorphic for k odd: *P k = 'F k (y 1 + iy 2 , e + iß), k = l,3, and antiholomorphic for k even: <F k = *F k (y 1 -iy 2 , e-i9), k = 2,0. The transformation is determined by substituting the expressions (15) into (13) and, similarly, 7 2 by substituting the expressions (16) into (14).…”

“…Such a result gives an additional information on the transformations v ? and v n of [2,3,11], describing the correspondence between the space of the particle and the space of observations. Another characterization of this correspondence has been determined in the preceding section in terms of the transformation S in (2).…”

“…The SO-hyperbolic and SO-elliptic cases in the scheme (3) correspond to the space of observations and the space of the particle, respectively. The transformation S in (2) fully determines the transformation between these spaces, defined non-effectively in [2]…”

“…Following the previous papers by two of us [1,2] concerning the description of elementary particles in terms of real-and complex-Riemannian manifolds, here we deal with the construction of fields connected with particles and regarded as a deformation of the particle space, thus providing a natural continuation of [3]. By analogy to the general relativity, where the gravitational field and the space curvature are tightly related, the appearance of a particle determines the space geometry whose properties reflect the particle properties and describe the fields produced by them.…”