Complex-Distance Potential Theory and Hyperbolic Equations

An extension of potential theory in Rn is obtained by continuing the Euclidean distance function holomorphically to Cn. The resulting Newtonian potential is generated by an extended source distribution δ̃(z) in Cn whose restriction to Rn is the point source δ(x). This provides a possible model for extended particles in physics. In C, interpreted as complex spacetime, δ̃ acts as a propagator generating solutions of the wave equation from their initial values. This gives a new connection between elliptic and hyperbolic equations that does not assume analyticity of the Cauchy data. Generalized to Clifford analysis, it induces a similar connection between solutions of elliptic and hyperbolic Dirac equations. There is a natural application to the time-dependent, inhomogeneous Dirac and Maxwell equations, and the ‘electromagnetic wavelets’ introduced previously are an example. 1 Motivation and Preliminaries Most fundamental theories of physics are based on the concept of potentials and fields generated by point sources, which presupposes that objects or “particles” can, in principle, be localized within arbitrarily small regions of space and/or time. This is a vast extrapolation from empirical evidence, and it should perhaps not come as a surprise if such theories experience some fundmental difficulties. In Newtonian mechanics, the problem of N point “bodies” interacting through gravitation has, in general, no solution due to the possibility of collisions. This becomes a serious difficulty for N ≥ 3, where the set of initial conditions leading to collisions is nontrivial [AM78, Chapter 10]. In classical electrodynamics, difficulties arise where the field produced by a point charge unavoidably acts back on the same charge, leading to infinite selfenergies and run-away particle trajectories [J99]. A way out of this dilemma was proposed by Wheeler and Feynman [WF45, WF49], but their action-at-a distance theory, apart from being highly counter-intuitive [F64, p. 28-8], has resisted quantization and is not generally regarded as being a fundamental description of Nature. In quantum electrodynamics and other quantum field theories, point particles cause divergences which necessitate infinite “renormalization” procedures, a subject of some contraversy [C99]. String theory [P98] does, in fact, not need infinite renormalization because its basic objects (strings) are extended in space rather than mathematical points. This is one of the reasons it is regarded with great hope as a possibility for unifying physical theories. However, a full development of (super)string physics is rather difficult and not expected to near completion for many years. The ideas developed here began many years ago, motivated in part by the hope that an extension of physics to complex spacetime, justified at the foundational level, might give a way to circumvent the problems associated with point sources by applying residue methods. After some years of study and research this led to papers [K77, K78, K80, K87] and books [K90, K94] whose main thrust has been to develop a direct physical interpretation of the complex spacetime as an extended phase space, with the imaginary spacetime parameters carrying directional information while the real spacetime parameters describe (approximate!) localization. A major stumbling block in this program has been the construction of extended sources, since it seemed that they tend to spoil the holomorphy of the theory globally rather than just locally. Here we propose a natural solution to this problem. It turns out that although holomorphy enters the theory at the