Abstract
This is a very basic and pedagogical review of the concepts of zeta function and of the associated zeta regularization method, starting from the notions of harmonic series and of divergent sums in general. By way of very simple examples, it is shown how these powerful methods are used for the regularization of physical quantities, such as quantum vacuum fluctuations in various contexts. In special, in Casimir effect setups, with a note on the dynamical Casimir effect, and mainly concerning its application in quantum theories in curved spaces, subsequently used in gravity theories and cosmology. The second part of this work starts with an essential introduction to large scale cosmology, in search of the observational foundations of the Friedmann-Lemaître-Robertson-Walker (FLRW) model, and the cosmological constant issue, with the very hard problems associated with it. In short, a concise summary of all these interrelated subjects and applications, involving zeta functions and the cosmos, and an updated list of the pioneering and more influential works (according to Google Scholar citation counts) published on all these matters to date, are provided.
Highlights
The unreasonable effectiveness of mathematics in the natural sciences, as Eugene Wigner put it, is an old and, as of still intriguing question [1]
In a PRL paper with Jaume Haro [160], we developed a consistent approach to the dynamical Casimir effect; in Ref. [161] we constructed a physically sound Hamiltonian formulation of the same and, subsequently, we applied the dynamical Casimir effect with semi-transparent mirrors to cosmology [162]
The method has been used in the paper, for the regularization of physical quantities, such as quantum vacuum fluctuations in several contexts, always taking advantage of the insight one gets from very basic examples
Summary
The unreasonable effectiveness of mathematics in the natural sciences, as Eugene Wigner put it, is an old and, as of still intriguing question [1]. We face up an example of this situation when we calculate the very simple case corresponding to the vacuum to vacuum transition, or zero point energy (or, the related Casimir energy [2]) of a quantum operator, H, whose spectral values are λn, namely Note this is a simplified notation, since the ‘sum’ over n may include a number of different, continuous and discrete indexes. A very elementary, highly pedagogical review of the zeta function method will be given, stressing the basic concepts and the history of this procedure, always with the idea of complementing other works were much more rigorous presentations have been given It will be explained how the method can be used for the regularization of physical quantities, such as quantum vacuum fluctuations in several contexts, with the use of very basic examples.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
