By Bordag M., Klimchitskaya G.L., Mohideen U., Mostepanenko V.M.

The topic of this ebook is the Casimir impression, a manifestation of zero-point oscillations of the quantum vacuum leading to forces performing among heavily spaced our bodies. For the advantage of the reader, the ebook assembles field-theoretical foundations of this phenomenon, functions of the final conception to genuine fabrics, and a entire description of all lately played measurements of the Casimir strength with a comparability among test and concept. there's an pressing desire for a ebook of this kind, given the rise of curiosity in forces originating from the quantum vacuum. quite a few new effects were received within the previous few years which aren't mirrored in prior books at the topic, yet that are very promising for basic technology and nanotechnology. The publication is a different resource of data providing a serious evaluate of the entire major effects and ways from hundreds of thousands of magazine papers. It additionally outlines new rules that have now not but been universally authorized yet that are discovering expanding help from scan.

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**Example text**

19), the Casimir force acting between the boundary points of the interval is F (a) = − ∂E(a) π c =− . 1) between two parallel, ideal-metal planes. 18) is a typical example of what are commonly referred to as subtraction procedures, used in quantum ﬁeld theories in order to remove inﬁnities The Abel–Plana formula and regularization 21 from divergent expectation values of physical quantities. Usually, the subtraction of any inﬁnite quantity is interpreted as a renormalization of some physical constant in the bare eﬀective action (see Chapter 4 for more details).

E. to the direction of the Casimir force), k⊥ = |k⊥ | = (kx2 + ky2 )1/2 , and the oscillator frequencies are given by ωJ = ωk⊥ ,n = c 2 + k⊥ πn a 2 . 72) to ∞ E0 (a) = 2 0 k⊥ dk⊥ 2π ∞ ωk⊥ ,n − ck⊥ 2 S. 76) and a factor of 2 has been used to take into account the two polarizations of the electromagnetic ﬁeld. 75) can be rearranged to E0M (a) = ∞ a π 0 ∞ k⊥ dk⊥ 2π dkz ωk S. 78) ∞ dkz ωk . 79) . 34) and A ≡ k⊥ a /π 2 . 80) where the dimensionless variable y = k⊥ a/π has been introduced instead of k⊥ .

7). 7), we also obtain the normalization condition for the solutions ΦJ (r) of the boundary problem, V dr Φ∗J (r)ΦJ (r) = δJJ . 52) J where aJ and a+ J are the annihilation and creation operators of a particle with quantum numbers indicated by the collective index J. The summation over J may also mean integration if some (or all) of the quantum numbers are continuous. The annihilation and creation operators satisfy the commutation relations aJ , a+ = δJJ , J + = 0. 53) The vacuum state of the ﬁeld is deﬁned by aJ |0 = 0.