Complex-Distance Potential Theory and Hyperbolic Equations

Details Complex-Distance Potential Theory and Hyperbolic Equations Complex-Distance Potential Theory and Hyperbolic Equations

1999-08-31

arxiv.org/abs/math-ph/9908031v2

Clifford Analysis, J. Ryan and W. Sprossig, eds., Birkhauser Progress in Physics, Vol. 19, 2000.

Physics

Mathematical Physics

25 pages, submited to Proceedings of 5th Intern. Conf. on Clifford Algebras, Ixtapa, June 24 - July 4, 1999

Scientific paper

An extension of potential theory in R^n is obtained by continuing the Euclidean distance function holomorphically to C^n. The resulting Newtonian potential is generated by an extended source distribution D(z) in C^n whose restriction to R^n is the delta function. This provides a natural model for extended particles in physics. In C^n, interpreted as complex spacetime, D(z) 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.

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