02.10.v Logic, set theory, and algebra
 02.10.Ab Logic and set theory
 02.10.De Algebraic structures and number theory
 02.10.Hh Rings and algebras
 02.10.Kn Knot theory
 02.10.Ox Combinatorics; graph theory
 02.10.Ud Linear algebra
 02.10.Xm Multilinear algebra
 02.10.Yn Matrix theory
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Infinite infrared regularization and a state space for the Heisenberg algebra
(2003)

Andreas U. Schmidt
 We present a method for the construction of a Krein space completion for spaces of test functions, equipped with an indefinite inner product induced by a kernel which is more singular than a distribution of finite order. This generalizes a regularization method for infrared singularities in quantum field theory, introduced by G. Morchio and F. Strocchi, to the case of singularites of infinite order. We give conditions for the possibility of this procedure in terms of local differential operators and the GelfandShilov test function spaces, as well as an abstract sufficient condition. As a model case we construct a maximally positive definite state space for the Heisenberg algebra in the presence of an infinite infrared singularity. See the corresponding paper: Schmidt, Andreas U.: "Mathematical Problems of Gauge Quantum Field Theory: A Survey of the Schwinger Model" and the presentation "Infinite Infrared Regularization in Krein Spaces"