Journal
JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL
Volume 34, Issue 14, Pages 3025-3035Publisher
IOP PUBLISHING LTD
DOI: 10.1088/0305-4470/34/14/309
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Explicit formulae for the zeta functions zeta (alpha)(s) corresponding to bosonic (alpha = 2) and to fermionic(alpha = 3) quantum fields living on a non-commutative, partially toroidal spacetime are derived. Formulae for the most general case of the zeta function associated with a quadratic + linear + constant form (in Z) are obtained. They provide the analytical continuation of the zeta functions in relation to the whole complex s plane, in terms of series of Bessel functions (of fast, exponential convergence), thus being extended Chowla-Selberg formulae. As is well known, this is the most convenient expression that can be found for the analytical continuation of a zeta function; in particular, the residua of the poles and their finite parts are explicitly given. An important novelty is the fact that simple poles show up at s = 0, as well as in other places (simple or double, depending on the number of compactified, non-compactified and non-commutative dimensions of the spacetime) where they had never appeared before. This poses a challenge to the zeta-function regularization procedure.
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