On Stapp's Theorem

Peter Minkowski, David N. Williams
Rudolf Seiler

Institüt für Theoretische Physik
Eidgenössische Technische Hochschule
Zürich, 1964

Abstract: Stapp's Theorem says that the complex domain of regularity of a function holomorphic on a real domain of several four-vectors, perhaps restricted to the mass shell, and covariant under the connected component of the real homogeneous Lorentz group, is a union of invariant sheets consisting of orbits under the connected component of the complex, homogenous Lorentz group, under which the function is complex covariant. The original proof used a local connectedness property of complex orbits. R. Jost found a counterexample for certain points of degenerate dimension, which invalidated the proof there.

Two of us provide independent proofs that the theorem is nevertheless true at such points, based on a weaker connectedness property, and one of us shows that Jost's counterexample characterizes the points where strong local connectedness fails. Finally, we point out that the proof of the theorem extends to the complex classical groups of any finite dimension GL(n,C), SL(n,C), and O+(n,C). Possibly it works for Sp(n,C), but the weaker connectedness property remains to be checked in that case.

Published in Lectures in Theoretical Physics, vol. VII A, Lorentz Group, (University of Colorado Press, Boulder, 1965), pp. 173–189.

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