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Introduction and Outlook

  • Emilio Elizalde
Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 855)

Abstract

In this introductory chapter, an overview of the method of zeta function regularization is presented. We start with some brief historical considerations and by introducing some of the specific zeta functions that will be used in the following chapters in physical situations, as the Riemann, Hurwitz (or Riemann generalized), and Epstein zeta functions. We summarize the basic properties of the different zeta functions. We show explicitly how to regularize the Casimir energy in some simple cases in a correct way, thereby introducing the zeta-function regularization procedure. We compare it with other regularization methods and point out to some missuses of zeta regularization. These fundamental concepts are both extended and made much more precise in the last section, where examples of recent developments on powerful applications of the theory are discussed. We define the concept of zeta function associated with an elliptic partial differential operator, and point towards its uses to define ‘the determinant’ of the operator in the zeta regularized sense. We discuss the multiplicative anomaly or defect of the zeta determinant and finish with further perspectives of this regularization method, as the so-called operator regularization.

Keywords

Zeta Function Dimensional Regularization Riemann Zeta Function Casimir Energy Operator Regularization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institute of Space ScienceHigher Council for Scientific ResearchBellaterra (Barcelona)Spain

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