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Path integrals for the telegrapher's and dirac equations ; the analytic family of measures and the underlying poisson process

The Cauchy problem for a free relativistic particle in two space-time dimensions is considered with -im, m being the mass of the particle, replaced by an arbitrary z∈C. The expansion of the solution in powers of z leads to the probabilistic representation in terms of the Poisson process. A family of path-space measures indexed by z is constructed, and for each z a path integral representation of the solution is obtained. It is proved that the family of measures is analytic with respect to z, which is in agreement with the idea of Kac that there is a connection by analytic continuation between the two-dimensional Dirac and telegrapher's equations