Ten Physical Applications of Spectral Zeta Functions

1. Front Matter

   Pages I-XII

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2. Introduction and Outlook

   Pages 1-20

3. Mathematical Formulas Involving the Different Zeta Functions

   Pages 21-49

4. A Treatment of the Non-Polynomial Contributions: Application to Calculate Partition Functions of Strings and Membranes

   Pages 51-71

5. Analytical and Numerical Study of Inhomogeneous Epstein and Epstein-Hurwitz Zeta Functions

   Pages 73-96

6. Physical Application: The Casimir Effect

   Pages 97-127

7. Four Physical Applications of the Inhomogeneous Generalized Epstein-Hurwitz Zeta Functions

   Pages 129-155

8. Miscellaneous Applications Combining Zeta with Other Regularization Procedures

   Pages 157-178

9. Applications to Gravity, Strings and P-Branes

   Pages 179-192

10. Last Application: Topological Symmetry Breaking in Self-Interacting Theories

    Pages 193-207

11. Back Matter

    Pages 209-224

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This monography is, in the first place, a commented guide that invites the reader to plunge into the thrilling world ofzeta functions and their appli­ cations in physics. Different aspects ofthis field ofknowledge are considered, as one can see specifically in the Table of Contents. The level of the book is elementary. It is intended for people with no or little knowledge of the subject. Everything is explained in full detail, in particular, the mathematical difficulties and tricky points, which too often constitute an insurmountable barrier for those who would have liked to be­ come aquainted with that matter but never dared to ask (or did not manage to understand more complete, higher-level treatises). In this sense the present work is to be considered as a basic introduction and exercise collection for other books that have appeared recently. Concerning the physical applications of the method ofzeta-function reg­ ularization here described, quite a big choice is presented. The reader must be warned, however, that I have not tried to explain the underlying physi­ cal theories in complete detail (since this is undoubtedly out of scope), but rather to illustrate - simply and clearly - the precise way the method must be applied. Sometimes zeta regularization is explicitly compared in the text with other procedures the reader is supposed to be more familiar with (such as cut-off or dimensional regularization).

• Casimir effect
• Regularization methods
• Zeta function
• gravity
• perturbation theory
• phase transition
• string theory

"...I can recommend this book to anybody who encounters zeta functions in research and/or education, Undoubtedly, this book will be of great help in taking away some of the confusion on this topic." Classical and Quantum Gravity

• Camí de Santa Bàrbara, Blanes Center for Advanced Studies, CSIC, Blanes, Spain

  Emilio Elizalde

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