By Noah Graham, Markus Quandt, Herbert Weigel

This concise textual content introduces innovations from quantum mechanics, in particular scattering concept, to compute the consequences of an exterior history on a quantum box typically, and at the homes of the quantum vacuum particularly. This method may be succesfully utilized in an more and more huge variety of occasions, starting from the examine of solitons in box thought and cosmology to the choice of Casimir forces in nano-technology.

The process brought and utilized during this e-book is proven to provide an unambiguous connection to perturbation concept, enforcing average renormalization stipulations even for non-perturbative backgrounds. It either provides new theoretical insights, for instance illuminating longstanding questions concerning Casimir stresses, and likewise offers a good analytic and numerical device compatible to sensible calculations. final yet now not least, it elucidates in a concrete context the various subtleties of quantum box concept, comparable to divergences, regularization and renormalization, via connecting them to extra widespread leads to quantum mechanics.

While addressed basically at younger researchers coming into the sphere and nonspecialist researchers with backgrounds in theoretical and mathematical physics, introductory chapters at the theoretical points of the strategy make the publication self-contained and hence compatible for complex graduate students.

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L. Jaﬀe, M. Quandt, and H. Weigel, Phys. Rev. Lett. 87 (2001) 131601. 32 3 Quantum Field Theory and the Spectral Method The techniques of quantum ﬁeld theory are by now a standard topic in theoretical physics, but their realization in the spectral method is rather subtle. In this chapter, we begin with a review of standard results of quantum ﬁeld theory and then show how spectral techniques apply in this context. We develop a general set of tools applicable both to calculations of the Casimir energy and to more general observables, such as the energy density, relevant for general relativity, or the interface tension of a brane-like background conﬁguration.

We compute them in momentum space ˆ)|k = ei(k −k)x k 0 k 0 + k · k + m2 , k |Tˆx(0) δ d (x − x k |Tˆx(1) δ d (x − x ˆ)|k = σ(x)ei(k −k)x . 47) The Feynman series consists of all graphs with a single φ-loop and arbitrary (0) (1) insertions of Tˆx , Tˆx , and σ(x). To order σ 1 , for instance, we have the two diagrams shown in Fig. 2. The graph in the right panel is a total derivative and there is no counterterm for it. This diagram therefore should be ﬁnite in all dimensions d ≤ 4, which is conﬁrmed by explicit calculation [10].

Hence for = −1.