By Michael Bordag, Galina Leonidovna Klimchitskaya, Umar Mohideen, Vladimir Mikhaylovich Mostepanenko

The topic of this booklet 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 publication assembles field-theoretical foundations of this phenomenon, purposes of the overall concept to genuine fabrics, and a finished description of all lately played measurements of the Casimir strength with a comparability among test and concept. there's an pressing want for a booklet of this sort, given the rise of curiosity in forces originating from the quantum vacuum. quite a few new effects were acquired within the previous few years which aren't mirrored in past books at the topic, yet that are very promising for basic technology and nanotechnology. The publication is a distinct resource of data offering a serious evaluation of all of the major effects and techniques from 1000's of magazine papers. It additionally outlines new principles that have no longer but been universally authorised yet that are discovering expanding aid from scan.

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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 a J , a+ = δJJ , J + [aJ , aJ ] = a+ = 0. 53) The vacuum state of the field is defined by aJ |0 = 0. 54) The states with particles are obtained by applying the creation operators to the vacuum state. For example, the state with one particle is Canonical quantization and the vacuum energy as a mode expansion |1 = a+ J |0 . e.

These are similar to boundary conditions for classical material boundaries. The simplest example is provided by the interval 0 ≤ x ≤ a whose initial and end points are identified by means of the following periodic conditions: ϕ(t, 0) = ϕ(t, a), ∂x ϕ(t, x)|x=0 = ∂x ϕ(t, x)|x=a . 42) is a circle of circumference a. Both manifolds are flat, but their topologies are different. Interval I [Fig. 42) possesses the topology of a circle S 1 [Fig. 1(b)]. 2). 3), new solutions are allowed, such that ϕ = 0 at the points x = 0, a.

5) in this chapter, which is devoted to simple models. A configuration of two parallel planes of very large area S spaced a distance a apart is shown in Fig. 2. Mathematically, the area S of each plane is supposed to be infinitely large. However, the results obtained are applicable for the condition √ a S. From classical electrodynamics, the electric field and the magnetic induction, of both polarizations, satisfy the following boundary conditions on the surface of an ideal metal: E t (t, r)|S = B n (t, r)|S = 0.

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