Infinite infrared regularization and a state space for the Heisenberg algebra

  • 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 Gelfand-Shilov 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"

Download full text files

Export metadata

Author:Andreas U. SchmidtGND
ArXiv Id:
Document Type:Article
Year of Completion:2003
Year of first Publication:2003
Publishing Institution:Universit├Ątsbibliothek Johann Christian Senckenberg
Release Date:2005/06/21
Tag:Gelfand-Shilov space; Heisenberg algebra; Infrared singularity; Krein space; indefinite inner product space
Page Number:18
First Page:1
Last Page:18
Aktualisierte Fassung, zuerst erschienen in: Journal of mathematical physics, 43.2002, Nr. 1, S. 243-259, Erratum erschienen in: Journal of mathematical physics, 43.2002, Nr. 6, S. 3412
Source:Version of 03 March 2003, Orig. published in Journal of Mathematical Physics 36 no. 1 (January 2002) 243-259; Erratum:ibid. 43 no. 6 (June 2002) 3412 ,
Institutes:Informatik und Mathematik / Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
MSC-Classification:46-XX FUNCTIONAL ANALYSIS (For manifolds modeled on topological linear spaces, see 57Nxx, 58Bxx) / 46Cxx Inner product spaces and their generalizations, Hilbert spaces (For function spaces, see 46Exx) / 46C20 Spaces with indefinite inner product (Kreïn spaces, Pontryagin spaces, etc.) [See also 47B50]
46-XX FUNCTIONAL ANALYSIS (For manifolds modeled on topological linear spaces, see 57Nxx, 58Bxx) / 46Fxx Distributions, generalized functions, distribution spaces [See also 46T30] / 46F05 Topological linear spaces of test functions, distributions and ultradistributions [See also 46E10, 46E35]
47-XX OPERATOR THEORY / 47Bxx Special classes of linear operators / 47B50 Operators on spaces with an indefinite metric [See also 46C50]
81-XX QUANTUM THEORY / 81Txx Quantum field theory; related classical field theories [See also 70Sxx] / 81T05 Axiomatic quantum field theory; operator algebras
PACS-Classification:00.00.00 GENERAL / 02.00.00 Mathematical methods in physics / 02.10.-v Logic, set theory, and algebra
00.00.00 GENERAL / 02.00.00 Mathematical methods in physics / 02.30.-f Function theory, analysis / 02.30.Tb Operator theory
10.00.00 THE PHYSICS OF ELEMENTARY PARTICLES AND FIELDS (for experimental methods and instrumentation for elementary-particle physics, see section 29) / 11.00.00 General theory of fields and particles (see also 03.65.-w Quantum mechanics and 03.70.+k Theory of quantized fields) / 11.10.-z Field theory (for gauge field theories, see 11.15.-q)
Licence (German):License LogoDeutsches Urheberrecht