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Particles 2019, 2(1), 14-31; https://doi.org/10.3390/particles2010002

Regularization versus Renormalization: Why Are Casimir Energy Differences So Often Finite?

School of Mathematics and Statistics, Victoria University of Wellington, P.O. Box 600, Wellington 6140, New Zealand
Received: 17 August 2018 / Revised: 18 November 2018 / Accepted: 14 December 2018 / Published: 24 December 2018
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Abstract

One of the very first applications of the quantum field theoretic vacuum state was in the development of the notion of Casimir energy. Now, field theoretic Casimir energies, considered individually, are always infinite. However, differences in Casimir energies (at worst regularized, not renormalized) are quite often finite—a fortunate circumstance which luckily made some of the early calculations, (for instance, for parallel plates and hollow spheres), tolerably tractable. We will explore the extent to which this observation can be made systematic. For instance: What are necessary and sufficient conditions for Casimir energy differences to be finite (with regularization but without renormalization)? Additionally, when the Casimir energy differences are not formally finite, can anything useful nevertheless be said by invoking renormalization? We will see that it is the difference in the first few Seeley–DeWitt coefficients that is central to answering these questions. In particular, for any collection of conductors (be they perfect or imperfect) and/or dielectrics, as long as one merely moves them around without changing their shape or volume, then physically the Casimir energy difference (and so also the physically interesting Casimir forces) is guaranteed to be finite without invoking any renormalization. View Full-Text
Keywords: Casimir energy; quantum field theory; renormalization; regularization; finiteness; heat kernel expansion; Seeley–DeWitt coefficients Casimir energy; quantum field theory; renormalization; regularization; finiteness; heat kernel expansion; Seeley–DeWitt coefficients
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Visser, M. Regularization versus Renormalization: Why Are Casimir Energy Differences So Often Finite? Particles 2019, 2, 14-31.

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