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 Tr. Mat. Inst. Steklova, 2000, Volume 228, Pages 90–100 (Mi tm493)

Generalized Functions for Quantum Fields Obeying Quadratic Exchange Relations

H. Grossea, M. Oberguggenbergerb, I. T. Todorovc

a Institute for Theoretical Physics
b Institut für Mathematik, Universität Innsbruck
c International Erwin Schrödinger Institute for Mathematical Physics

Abstract: The axiomatic formulation of quantum field theory (QFT) of the 1950's in terms of fields defined as operator valued Schwartz distributions is re-examined in the light of subsequent developments. These include, on the physical side, the construction of a wealth of (2-dimensional) soluble QFT models with quadratic exchange relations, and, on the mathematical side, the introduction of the Colombeau algebras of generalized functions. Exploiting the fact that energy positivity gives rise to a natural regularization of Wightman distributions as analytic functions in a tube domain, we argue that the flexible notions of Colombeau theory which can exploit particular regularizations is better suited (than Schwartz distributions) for a mathematical formulation of QFT.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2000, 228, 81–91

Bibliographic databases:

UDC: 530.1
Language: English

Citation: H. Grosse, M. Oberguggenberger, I. T. Todorov, “Generalized Functions for Quantum Fields Obeying Quadratic Exchange Relations”, Problems of the modern mathematical physics, Collection of papers dedicated to the 90th anniversary of academician Nikolai Nikolaevich Bogolyubov, Tr. Mat. Inst. Steklova, 228, Nauka, MAIK «Nauka/Inteperiodika», M., 2000, 90–100; Proc. Steklov Inst. Math., 228 (2000), 81–91

Citation in format AMSBIB
\Bibitem{GroObeTod00} \by H.~Grosse, M.~Oberguggenberger, I.~T.~Todorov \paper Generalized Functions for Quantum Fields Obeying Quadratic Exchange Relations \inbook Problems of the modern mathematical physics \bookinfo Collection of papers dedicated to the 90th anniversary of academician Nikolai Nikolaevich Bogolyubov \serial Tr. Mat. Inst. Steklova \yr 2000 \vol 228 \pages 90--100 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm493} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1782574} \zmath{https://zbmath.org/?q=an:0986.81071} \transl \jour Proc. Steklov Inst. Math. \yr 2000 \vol 228 \pages 81--91 

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This publication is cited in the following articles:
1. Oberguggenberger M., “Generalized functions in nonlinear models – a survey”, Nonlinear Analysis–Theory Methods & Applications, 47:8, Part 8 Sp. Iss. SI (2001), 5029–5040
2. “On the Foundations of Nonlinear Generalized Functions I and II”, Memoirs of the American Mathematical Society, 153:729 (2001), 1–93
3. Khrennikov A.Y., Shelkovich V.M., Smolyanov O.G., “Locally convex spaces of vector–valued distributions with multiplicative structures”, Infinite Dimensional Analysis Quantum Probability and Related Topics, 5:4 (2002), 483–502
4. Nagamachi S., Bruning E., “Hyperfunction quantum field theory: Localized fields without localized test functions”, Letters in Mathematical Physics, 63:2 (2003), 141–155
5. Droz-Vincent Ph., “Scalar products of elementary distributions”, Journal of Mathematical Physics, 49:6 (2008), 063501
6. Hoermann G., “The Cauchy problem for Schrodinger-type partial differential operators with generalized functions in the principal part and as data”, Monatsh Math, 163:4 (2011), 445–460
7. Sarizadeh A., “Non-Removable Term Ergodic Action Semigroups/Groups”, Proc. Amer. Math. Soc., 143:8 (2015), 3445–3453
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