Is vacuum energy infinite? If it is, how and why?

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Vacuum energy is usually not of interest to particle physicists, since it can be thought of as a shift in the ground state energy (usually taken to be zero) of a system.

However, vacuum energy is of interest to cosmologists because of its possible role as a source of the cosmological constant and/or dark energy which drives the expansion of the universe.

Vacuum energy also manifests in the famous so-called Casimir effect.
 
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Vacuum energy is usually not of interest to particle physicists, since it can be thought of as a shift in the ground state energy (usually taken to be zero) of a system.

However, vacuum energy is of interest to cosmologists because of its possible role as a source of the cosmological constant and/or dark energy which drives the expansion of the universe.

Vacuum energy also manifests in the famous so-called Casimir effect.
No, one cannot repeat often enough: The Casimir effect has nothing to do with the vacuum. By definition the vacuum is empty. There's really nothing by definition, and so there cannot be forces.

The Casimir effect is observed (although it's damn difficult to get it quantitatively accurate in experiment), and it doesn't refer to the vacuum at all. There are always overall neutral solids around, and that's far from beeing the vacuum. It consists of a lot of positive and an equal amount of negative charges, and the Casimir force results from a kind of van der Waals force due to quantum fluctuations of these charges.

The usual handwavy derivations in some QED textbooks (for a pretty good one see, e.g., Itzykson+Zuber QFT) are in fact the limit ##\alpha_{\text{em}} \rightarrow \infty##. For a thorough discussion of the facts, see the famous paper by Yaffe:

R. L. Jaffe, The Casimir effect and the quantum vacuum, Phys. Rev. D, 72 (2005), p. 021301.
http://dx.doi.org/10.1103/PhysRevD.72.021301
http://arxiv.org/abs/hep-th/0503158
 
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