Edge-of-the-wedge theorem

In mathematics, Bogoliubov's edge-of-the-wedge theorem implies that holomorphic functions on two "wedges" with an "edge" in common are analytic continuations of each other provided they both give the same continuous function on the edge. It is used in quantum field theory to construct the analytic continuation of Wightman functions. The formulation and the first proof of the theorem were presented by Nikolay Bogoliubov at the International Conference on Theoretical Physics, Seattle, USA (September, 1956) and also published in the book Problems in the Theory of Dispersion Relations. Further proofs and generalizations of the theorem were given by R. Jost and H. Lehmann (1957), F. Dyson (1958), H. Epstein (1960), and by other researchers. In one dimension, a simple case of the edge-of-the-wedge theorem can be stated as follows. Suppose that f is a continuous complex-valued function on the complex plane that is holomorphic on the upper half-plane, and on the lower half-plane. Then it is holomorphic everywhere. In this example, the two wedges are the upper half-plane and the lower half plane, and their common edge is the real axis. This result can be proved from Morera's theorem. Indeed, a function is holomorphic provided its integral round any contour vanishes; a contour which crosses the real axis can be broken up into contours in the upper and lower half-planes and the integral round these vanishes by hypothesis. The more general case is phrased in terms of distributions. This is technically simplest in the case where the common boundary is the unit circle in the complex plane. In that case holomorphic functions f, g in the regions and have Laurent expansions absolutely convergent in the same regions and have distributional boundary values given by the formal Fourier series Their distributional boundary values are equal if for all n. It is then elementary that the common Laurent series converges absolutely in the whole region .