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