In one dimension, a simple case of the edge-of-the-wedge theorem can be stated as follows.
A wedge is a product of a cone with some set.
Let C be an open cone in the real vector space Rn, with vertex at the origin. Let E be an open subset of Rn, called the edge. Write W for the wedge in the complex vector space Cn, and write W' for the opposite wedge . Then the two wedges W and W' meet at the edge E, where we identify E with the product of E with the tip of the cone.
Suppose that f is a continuous function on the union that is holomorphic on both the wedges W and W' . Then the edge-of-the-wedge theorem says that f is also holomorphic on E (or more precisely, it can be extended to a holomorphic function on a neighborhood of E).
The conditions for the theorem to be true can be weakened. It is not necessary to assume that f is defined on the whole of the wedges: it is enough to assume that it is defined near the edge. It is also not necessary to assume that f is defined or continuous on the edge: it is sufficient to assume that the functions defined on either of the wedges have the same distributional boundary values on the edge.
In quantum field theory the Wightman distributions are boundary values of Wightman functions W(z1, ..., zn) depending on variables zi in the complexification of Minkowski spacetime. They are defined and holomorphic in the wedge where the imaginary part of each zi−zi−1 lies in the open positive timelike cone. By permuting the variables we get n! different Wightman functions defined in n! different wedges. By applying the edge-of-the-wedge theorem (with the edge given by the set of totally spacelike points) one can deduce that the Wightman functions are all analytic continuations of the same holomorphic function, defined on a connected region containing all n! wedges. (The equality of the boundary values on the edge that we need to apply the edge-of-the-wedge theorem follows from the locality axiom of quantum field theory.)
In the edge-of-the-wedge theorem, we have a distribution (or hyperfunction) f on the edge, given as the boundary values of two holomorphic functions on the two wedges. If a hyperfunction is the boundary value of a holomorphic function on a wedge, then its analytic wave front set lies in the dual of the corresponding cone. So the analytic wave front set of f lies in the duals of two opposite cones. But the intersection of these duals is empty, so the analytic wave front set of f is empty, which implies that f is analytic. This is the edge-of-the-wedge theorem.
In the theory of hyperfunctions there is an extension of the edge-of-the-wedge theorem to the case when there are several wedges instead of two, called Martineau's edge-of-the-wedge theorem. See the book by Hörmander for details.
For the application of the edge-of-the-wedge theorem to quantum field theory see:
Streater, R. F.; Wightman, A. S. PCT, spin and statistics, and all that. Princeton University Press, Princeton, NJ, 2000. ISBN 0-691-07062-8
The connection with hyperfunctions is described in:
Hörmander, Lars The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis. Springer-Verlag, Berlin, 2003. ISBN 3-540-00662-1
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