1998/05/31 by Valter Moretti · 2 citations
Mathematics · Physics and Astronomy · #Noncommutative and Quantum Gravity Theories #Quantum Electrodynamics and Casimir Effect #Quantum and Classical Electrodynamics #gr-qc #hep-th #math-ph #math.MP
paper · pdf · doi:10.1007/s002200050558
published as Commun.Math.Phys. 201 (1999) 327-363 · 40 pages, latex, no figures, shortened version, some previous Comments and minor errors corrected, final version accepted for publication in Commun. Math. Phys
arxiv created 1998/08/27 · openalex publication_date 1999/03/01 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/29
Some general properties of local ζ-function procedures to renormalize some quantities in D-dimensional (Euclidean) Quantum Field Theory in curved background are rigorously discussed for positive scalar operators -Δ+ V(x) in general closed D-manifolds, and a few comments are given for nonclosed manifolds too. A general comparison is carried out with respect to the more known point-splitting procedure concerning the effective Lagrangian and the field fluctuations. It is proven that, for D>1, the local ζ-function and point-splitting approaches lead essentially to the same results apart from some differences in the subtraction procedure of the Hadamard divergences. It is found that the ζ function procedure picks out a particular term w0(x,y) in the Hadamard expansion. Also the presence of an untrivial kernel of the operator -Δ+V(x) may produce some differences between the two analyzed approaches. Finally, a formal identity concerning the field fluctuations, used by physicists, is discussed and proven within the local ζ-function approach. This is done also to reply to recent criticism against ζ-function techniques.