Operator Product on Locally Symmetric Spaces of Rank One and the Multiplicative Anomaly.pdf

Operator Product on Locally Symmetric Spaces of Rank One and the Multiplicative Anomaly.pdf

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Operator Product on Locally Symmetric Spaces of Rank One and the Multiplicative Anomaly

a r X i v : h e p - t h / 0 3 0 5 0 3 1 v 2 3 0 M a y 2 0 0 3 OPERATOR PRODUCT ON LOCALLY SYMMETRIC SPACES OF RANK ONE AND THE MULTIPLICATIVE ANOMALY A. A. BYTSENKO1, E. ELIZALDE2, M. E. X. GUIMARA?ES3 1. Departamento de F??sica, Universidade Estadual de Londrina Caixa Postal 6001, Londrina–Parana?, Brazil 2. Institut d’Estudis Espacials de Catalunya, Consejo Superior de Investigaciones Cientificas (IEEC/CSIC) Edifici Nexus, Gran Capita? 2–4, 08034 Barcelona, Spain; Departament d’Estructura i Constituents de la Mate?ria, Facultat de F?isica, Universitat de Barcelona, Av. Diagonal 647, 08028 Barcelona, Spain 3. Universidade de Bras??lia, Departamento de Matema?tica CEP: 70910–900, Bras??lia–DF, Brazil February 1, 2008 Abstract The global multiplicative properties of Laplace type operators acting on irreducible rank one symmetric spaces are considered. The explicit form of the multiplicative anomaly is derived and its corresponding value is calculated exactly, for important classes of locally symmetric spaces and different dimensions. 1 Introduction In theories of quantum fields (for example, in higher-derivative quantum grav- ity) one has to deal with the product of two (or more) elliptic differential op- erators. It is natural, therefore, to investigate multiplicative properties of the determinants of differential operators, in particular the so–called multiplicative anomaly [1, 2] (for the definition of this anomaly see Sect. 3 below). The mul- tiplicative anomaly can be expressed by means of the non–commutative residue associated with a classical pseudo–differential operator, the Wodzicki residue [3]. Recently, the important role of this residue has been recognized in physics. The Wodzicki residue, which is the unique extension of the Dixmier trace to the wider class of pseudo-differential operators [4, 5], has been considered within the non–commutative geometrical approach to the standard model of the elec- troweak interactions [6, 7, 8, 10, 9, 11]. This res

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