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 TMF: Year: Volume: Issue: Page: Find

 TMF, 2004, Volume 141, Number 1, Pages 60–79 (Mi tmf114)

Rigorous Formulation of a $2D$ Conformal Model in the Fock–Krein Space: Construction of the Global $\operatorname{Op}J^*$-Algebra of Fields and Currents

S. S. Horuzhy

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We use the formalism of the $2D$ massless scalar field model in an indefinite space of the Fock–Krein type as a basis for constructing a rigorous formulation of $2D$ quantum conformal theories. We show that the sought construction is a several-stage procedure whose central block is the construction of a new type of representation of the Virasoro algebra. We develop the first stage of this procedure, which is to construct a special global algebra of fields and currents generated by exponential generators. We obtain a system of commutation relations for the Wick-squared currents used in the definition of the Virasoro generators. We prove the existence of Wick exponentials of the current given by operator-valued generalized functions; the sought global algebra is rigorously defined as the algebra of current and field, Wick and normal exponentials on a common dense invariant domain in a Fock–Krein space.

Keywords: Fock–Krein space, conformal theory, field algebra

DOI: https://doi.org/10.4213/tmf114

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English version:
Theoretical and Mathematical Physics, 2004, 141:1, 1381–1397

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Revised: 18.03.2004

Citation: S. S. Horuzhy, “Rigorous Formulation of a $2D$ Conformal Model in the Fock–Krein Space: Construction of the Global $\operatorname{Op}J^*$-Algebra of Fields and Currents”, TMF, 141:1 (2004), 60–79; Theoret. and Math. Phys., 141:1 (2004), 1381–1397

Citation in format AMSBIB
\Bibitem{Hor04} \by S.~S.~Horuzhy \paper Rigorous Formulation of a $2D$ Conformal Model in the Fock--Krein Space: Construction of the Global $\operatorname{Op}J^*$-Algebra of Fields and Currents \jour TMF \yr 2004 \vol 141 \issue 1 \pages 60--79 \mathnet{http://mi.mathnet.ru/tmf114} \crossref{https://doi.org/10.4213/tmf114} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2124510} \zmath{https://zbmath.org/?q=an:1178.81236} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2004TMP...141.1381K} \transl \jour Theoret. and Math. Phys. \yr 2004 \vol 141 \issue 1 \pages 1381--1397 \crossref{https://doi.org/10.1023/B:TAMP.0000043855.38368.3e} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000225149500004}