Abstract
Although some examples and applications of the formulas for analytical continuation have already been given in the preceding chapters, in the present one we properly start with the discussion at length of the applications of the explicit zeta function regularization method. In this sense, the Casimir effect is introduced, along with the Casimir–Polder version, and its relation with the van der Waals forces, the London theory, and the generalization of the Casimir setup in terms of the very far reaching Lifshitz theory are discussed. Also the “mistery” of the Casimir effect, its local formulation and the definition of the Casimir energy in terms of the fluctuations of the quantum vacuum. All the cases we consider here precisely correspond to the situation to which we have restricted ourselves in this work, namely the case when the spectrum of the Hamiltonian of our physical system is known explicitly. Already this startpoint, that might be considered as rather particular, gives rise to enormously interesting situations from the physical viewpoint (and from the mathematical one, too), that are addressed in this chapter.
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- 1.
One should note that in the standard situations of QFT when no coupling of the vacuum diagrams is present, one can get rid of this term by a simple determination of the origin of energies using, e.g., the normal ordering prescription. An absolutely different case is when general relativity is considered in a quantum context, provided quantum vacuum fluctuations are taken to be a ‘legal’ form of energy (see later).
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Elizalde, E. (2012). Physical Application: The Casimir Effect. 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_5
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