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EspañolEDGE-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.
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.
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''(''z''1, ..., ''z''''n'')
depending on variables ''zi'' in the complexification of Minkowski spacetime. They are defined and holomorphic in the wedge where the imaginary
part 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 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.)
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''. 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.
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
| Contents |
| The one-dimensional case |
| The general case |
| Application to quantum field theory |
| Connection with hyperfunctions |
| Further reading |
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
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''(''z''1, ..., ''z''''n'')
depending on variables ''zi'' in the complexification of Minkowski spacetime. They are defined and holomorphic in the wedge where the imaginary
part 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 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 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''. 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.
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
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