Edge-of-the-wedge theorem
Encyclopedia
In mathematics
, Bogoliubov's edge-of-the-wedge theorem implies that holomorphic function
s on two "wedges" with an "edge" in common are analytic continuation
s 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 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.
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 
.
on the real axis and holomorphic functions 
defined in 
and 
satisfying
for some non-negative integer N, the boundary values
of 
can be defined as distributions on the real axis by the formulas
Existence can be proved by noting that, under the hypothesis,
is the 
-th complex derivative of a holomorphic function which extends to a continuous function on the boundary. If f is defined as 
above and below the real axis and F is the distribution defined on the rectangle 
by the formula
then F equals
off the real axis and the distribution 
is induced by the distribution 
on the real axis.
In particular if the hypotheses of the edge-of-the-wedge theorem apply, i.e.
, then
By elliptic regularity it then follows that the function F is holomorphic in
.
In this case elliptic regularity can be deduced directly from the fact that
is known to provide a fundamental solution
for the Cauchy-Riemann operator
.
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 space
s on the circle. Indeed let
and 
be holomorphic functions defined exterior and interior to some arc on the unit circle such that locally they have radial limits in some Sobelev 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. By the argument for continuous functions, F and G therefore patch to give a holomorphic function near the arc and hence so do f and g.
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.
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.
s. A hyperfunction is roughly a sum of boundary values of holomorphic function
s, and can also be thought of as something like a "distribution of infinite order". The analytic wave front set
of a hyperfunction at each point is a cone in the cotangent space
of that point, and can be thought of as describing the directions in 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 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.
The connection with hyperfunctions is described in:.
For the application of the edge-of-the-wedge theorem to quantum field theory see:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Bogoliubov's edge-of-the-wedge theorem implies that holomorphic function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
s on two "wedges" with an "edge" in common are analytic continuation
Analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...
s of each other provided they both give the same continuous function on the edge. It is used in quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
to construct the analytic continuation
Analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...
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
Freeman Dyson
Freeman John Dyson FRS is a British-born American theoretical physicist and mathematician, famous for his work in quantum field theory, solid-state physics, astronomy and nuclear engineering. Dyson is a member of the Board of Sponsors of the Bulletin of the Atomic Scientists...
(1958), H. Epstein (1960), and by other researchers.
Continuous boundary values
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 planeComplex planeIn mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
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
Morera's theorem
In complex analysis, a branch of mathematics, Morera's theorem, named after Giacinto Morera, gives an important criterion for proving that a function is holomorphic....
. 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.
Distributional boundary values on a circle
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 expansionsabsolutely 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 .Distributional boundary values on an interval
In general given an open interval on the real axis and holomorphic functions defined in and satisfyingfor some non-negative integer N, the boundary values
of can be defined as distributions on the real axis by the formulasExistence can be proved by noting that, under the hypothesis,
is the -th complex derivative of a holomorphic function which extends to a continuous function on the boundary. If f is defined as above and below the real axis and F is the distribution defined on the rectangle by the formula
then F equals
off the real axis and the distribution is induced by the distribution on the real axis.In particular if the hypotheses of the edge-of-the-wedge theorem apply, i.e.
, thenBy elliptic regularity it then follows that the function F is holomorphic in
.In this case elliptic regularity can be deduced directly from the fact that
is known to provide a fundamental solutionFundamental solution
In mathematics, a fundamental solution for a linear partial differential operator L is a formulation in the language of distribution theory of the older idea of a Green's function...
for the Cauchy-Riemann operator
. Using the Cayley transform
Cayley transform
In mathematics, the Cayley transform, named after Arthur Cayley, has a cluster of related meanings. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. In complex analysis, the Cayley transform is a conformal mapping in...
between the circle and the real line, this argument can be rephrased in a standard way in terms of Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
and Sobolev space
Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space...
s on the circle. Indeed let
and be holomorphic functions defined exterior and interior to some arc on the unit circle such that locally they have radial limits in some Sobelev space, Then, lettingthe 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. By the argument for continuous functions, F and G therefore patch to give a holomorphic function near the arc and hence so do f and g.
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
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.
Application to quantum field theory
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.)Connection with hyperfunctions
The edge-of-the-wedge theorem has a natural interpretation in the language of hyperfunctionHyperfunction
In mathematics, hyperfunctions are generalizations of functions, as a 'jump' from one holomorphic function to another at a boundary, and can be thought of informally as distributions of infinite order...
s. A hyperfunction is roughly a sum of boundary values of holomorphic function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
s, and can also be thought of as something like a "distribution of infinite order". The analytic wave front set
Wave front set
In mathematical analysis, more precisely in microlocal analysis, the wave front WF characterizes the singularities of a generalized function f, not only in space, but also with respect to its Fourier transform at each point...
of a hyperfunction at each point is a cone in the cotangent space
Cotangent space
In differential geometry, one can attach to every point x of a smooth manifold a vector space called the cotangent space at x. Typically, the cotangent space is defined as the dual space of the tangent space at x, although there are more direct definitions...
of that point, and can be thought of as describing the directions in 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 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.
Further reading
.The connection with hyperfunctions is described in:.
For the application of the edge-of-the-wedge theorem to quantum field theory see: