×

zbMATH — the first resource for mathematics

Noncommutative Grassmannian \(U(1)\) sigma model and a Bargmann-Fock space. (English. Russian original) Zbl 1144.81475
Theor. Math. Phys. 153, No. 3, 1643-1651 (2007); translation from Teor. Mat. Fiz. 153, No. 3, 347-357 (2007).
Summary: We consider a Grassmannian version of the noncommutative \(U(1)\) sigma model specified by the energy functional \(E(P) = \| [a, P]\| _{HS} ^{2}\) , where \(P\) is an orthogonal projection operator in a Hilbert space \(H\) and \(a: H \rightarrow H\) is the standard annihilation operator. With \(H\) realized as a Bargmann-Fock space, we describe all solutions with a one-dimensional range and prove that the operator \([a, P]\) is densely defined in \(H\) for a certain class of projection operators \(P\) with infinite-dimensional ranges and kernels.
Editorial remark: The paper is almost identical to the author’s paper [J. Geom. Symmetry Phys. 10, 41–49 (2007; Zbl 1143.81019)].

MSC:
81R60 Noncommutative geometry in quantum theory
58B34 Noncommutative geometry (à la Connes)
81S05 Commutation relations and statistics as related to quantum mechanics (general)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] W. J. Zakrzewski, Low Dimensional Sigma Models, Adam Hilger, Bristol (1989). · Zbl 0787.53072
[2] J. A. Harvey, ”Komaba lectures on noncommutative solitons and D-branes,” arXiv:hep-th/0102076v1 (2001).
[3] O. Lechtenfeld and A. D. Popov, JHEP, 0111, 040 (2001); arXiv:hep-th/0106213v3 (2001).
[4] A. V. Domrin, O. Lechtenfeld, and S. Petersen, JHEP, 0503, 045 (2005); arXiv: hep-th/0412001v2 (2004).
[5] L. Hörmander, The Analysis of Linear Partial Differential Operators: III. Pseudodifferential Operators (Grundlehren Math. Wiss., Vol. 274), Springer, Berlin (1985).
[6] V. Bargmann, Comm. Pure Appl. Math., 14, 187 (1961). · Zbl 0107.09102
[7] F. E. Burstall and J. H. Rawnsley, Comm. Math. Phys., 110, 311 (1987). · Zbl 0627.58029
[8] D. J. Newman and H. S. Shapiro, ”Fisher spaces of entire functions,” in: Entire Functions and Related Parts of Analysis (Proc. Sympos. Pure Math., Vol. 11), Amer. Math. Soc., Providence, R. I. (1968), p. 360. · Zbl 0191.41501
[9] N. I. Ahiezer, Lectures in the Theory of Approximation [in Russian], Nauka, Moscow (1965).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.