Edge-of-the-wedge theorem

In mathematics, the 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 one-dimensional case

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.

The general case

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$E\; imes\; iC$ in the complex vector space "Cⁿ",and write "W' " for the opposite wedge $E\; imes\; -iC$.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$W\; cup\; Ecup\; W\text{'}$ 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 extendedto 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.

Application to quantum field theory

In quantum field theory the Wightman distributionsare boundary values of Wightman functions "W"("z"₁, ..., "z"_"n")depending on variables "z_i" in the complexification of Minkowski spacetime. They are defined and holomorphic in the wedge where the imaginarypart of each "z"_"i"−"z"_"i"−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 containingall "n"! wedges. (The equality of the boundary values on the edgethat we need to apply the edge-of-the-wedge theorem follows from the locality axiom of quantum field theory.)

Connection with hyperfunctions

The edge-of-the-wedge theorem has a natural interpretation in the language of hyperfunctions. A hyperfunction is roughly a sum of boundary values of holomorphic functions, and can also be thought of as something likea "distribution of infinite order". The analytic wave front setof a hyperfunction at each point is a cone in the cotangent space of that point, and can be thought of as describing the directionsin which the singularity at that point is moving.

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 functionon a wedge, then its analytic wave front set lies in the dual of thecorresponding 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, sothe 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.

Further reading

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