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Introduction and Outlook

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Book cover Ten Physical Applications of Spectral Zeta Functions

Part of the book series: Lecture Notes in Physics ((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.

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Notes

  1. 1.

    A hint for Spanish speaking colleagues. In Spanish (and some other languages) the Greek letter ζ is phonetically transcribed as dseta (to mimic its original Greek pronunciation). In Greek the letter which is actually pronounced as the Spanish zeta is θ.

  2. 2.

    With some incredible exceptions, however, as the current Wikipedia article on “Zeta function regularization”, where no mention to Dowker and Critchley is done!

  3. 3.

    We will give more precise specifications at the end of this chapter.

  4. 4.

    An important portion of this book will be devoted to obtain such convenient forms of this reflection formula in different situations of physical interest.

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  1. Institute of Space Science, Higher Council for Scientific Research, Bellaterra (Barcelona), Spain

    Prof. Dr. Emilio Elizalde

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Elizalde, E. (2012). Introduction and Outlook. In: Ten Physical Applications of Spectral Zeta Functions. Lecture Notes in Physics, vol 855. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29405-1_1

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