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Ten physical applications of spectral zeta functions. 2nd ed. (English) Zbl 1250.81004
Lecture Notes in Physics 855. Berlin: Springer (ISBN 978-3-642-29404-4/pbk; 978-3-642-29405-1/ebook). xiv, 227 p. (2012).
The purpose of these lecture notes is twofold: to develop some basic ideas of zeta-function regularization in physics and to summarize the basic properties of different zeta functions as defined by Riemann, Hurwitz and Epstein. Still, the level of the book is elementary, intended for physics students with little knowledge of the mathematical subject. It is the second edition of the book (see Zbl 0855.00002 for a review of the first ed.) which now includes in Chapter 1 a definition of zeta functions corresponding to second order elliptic (sometimes pseudo-) differential operators. In Chapter 4, a view on the Chowla-Selberg series is included. In Chapter 5, a thorough discussion of experiments concerning the Casimir effect is avoided. In Chapter 6, another physical application has been added: the treatment of scalar and vector fields on a spacetime with a noncommutative toroidal part. In Chapter 7, the reader finds a new section on zeta and Hadamard regularizations related to the Casimir effect. Chapter 10 is entirely new and discusses the application of the zeta function regularization technique in the context of cosmology: vacuum fluctuations and cosmological constants. And so the number of physical application has now been raised from ten to twelve.
I find this book extremely useful and important, because it signifies the beauty of a mathematical technique in physics in general.

MSC:
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
81T75 Noncommutative geometry methods in quantum field theory
11Z05 Miscellaneous applications of number theory
81T55 Casimir effect in quantum field theory
83C45 Quantization of the gravitational field
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