Ten Physical Applications of Spectral Zeta Functions

Abstract

The book starts with a nice historical introduction to the subject of zeta functions, zeta function regularization and its applications in physics. The author emphasizes some of the advantages of the zeta function regularization method. In particular, the zeta function regularization can conveniently be applied in a curved spacetime. Anybody who wants to get a quick overview of the state of the art in the zeta function regularization method is recommended to read this introduction. Apart from the introduction the main virtue of this book is to provide help to any physics student and/or researcher who encounters zeta functions in his/her work. For this purpose the book provides the reader first of all with many technical details about zeta functions and some of their most important properties. Crucial amongst these properties is the existence of the so-called zeta function regularization theorem. This theorem is discussed and explained in the first part of the book. At a more advanced level, the author emphasizes the importance of a non-trivial extension of the celebrated Chowla--Selberg formula. This extension was invented by the author and plays an important role in some of the physics applications discussed later in the book. The second and main part of the book is devoted to illustrate the zeta function regularization method by giving ten physical applications. These applications cover such a wide range of topics as the Casimir effect, Kaluza--Klein compactification, two- and three-dimensional quantum gravity, extended objects like strings and membranes, critical behaviour of a field theory at non-zero temperature and topological mass generation. It is needless to say that all these applications give an impressive overview of the universality and power of the zeta function regularization method. I like in particular the discussion of the Casimir effect which is done in quite some detail. Apart from the technical discussion the author also explains why the Casimir effect only received so much attention many years after its discovery. Finally, I should note that it is interesting that the book discusses an application to the theory of membranes and, more generally p-branes, which has received such renewed interest in the past year. In short, 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.