In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Bogoliubov's edge-of-the-wedge theorem implies that
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
s on two "wedges" with an "edge" in common are
analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new ...
s of each other provided they both give the same continuous function on the edge. It is used in
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
to construct the
analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new ...
of
Wightman function
In mathematical physics, the Wightman axioms (also called Gårding–Wightman axioms), named after Arthur Wightman, are an attempt at a mathematically rigorous formulation of quantum field theory. Arthur Wightman formulated the axioms in the e ...
s. The formulation and the first proof of the theorem were presented
by
Nikolay Bogoliubov
Nikolay Nikolayevich Bogolyubov (russian: Никола́й Никола́евич Боголю́бов; 21 August 1909 – 13 February 1992), also transliterated as Bogoliubov and Bogolubov, was a Soviet and Russian mathematician and theoretic ...
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.
The one-dimensional case
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 plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
that is
holomorphic
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...
on the
upper half-plane
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0.
Complex plane
Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to t ...
, 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
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poi ...
. This result can be proved from
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.
Morera's theorem states that a continuous, complex-valued function ''f'' defined ...
. 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