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[1][2] 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.
[3] Further proofs and generalizations of the theorem were given by Res Jost and Harry Lehmann (1957),[4] Freeman 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.
In this example, the two wedges are the upper half-plane and the lower half plane, and their common edge is the real axis.
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.
[5][6] The more general case is phrased in terms of distributions.
[7][8] This is technically simplest in the case where the common boundary is the unit circle
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
satisfying for some non-negative integer N, the boundary values
can be defined as distributions on the real axis by the formulas[9][8] Existence can be proved by noting that, under the hypothesis,
In this case elliptic regularity can be deduced directly from the fact that
[10] Using the Cayley transform between the circle and the real line, this argument can be rephrased in a standard way in terms of Fourier series and Sobolev spaces on the circle.
be holomorphic functions defined exterior and interior to some arc on the unit circle such that locally they have radial limits in some Sobolev space, Then, letting the equations can be solved locally in such a way that the radial limits of G and F tend locally to the same function in a higher Sobolev space.
For k large enough, this convergence is uniform by the Sobolev embedding theorem.
be an open cone in the real vector space
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.
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 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!
(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.)
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 like a "distribution of infinite order".
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.
The connection with hyperfunctions is described in: For the application of the edge-of-the-wedge theorem to quantum field theory see: