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

The topic of this publication is the Casimir impression, a manifestation of zero-point oscillations of the quantum vacuum leading to forces appearing among heavily spaced our bodies. For the good thing about the reader, the publication assembles field-theoretical foundations of this phenomenon, functions of the overall idea to genuine fabrics, and a complete description of all lately played measurements of the Casimir strength with a comparability among test and conception. there's an pressing want for a booklet of this sort, given the rise of curiosity in forces originating from the quantum vacuum. a variety of new effects were got 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 e-book is a special resource of data proposing a serious evaluation of the entire major effects and ways from countless numbers of magazine papers. It additionally outlines new principles that have no longer but been universally permitted yet that are discovering expanding help from test.

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34) L(1/2) = 2 Here, ψ¯ = ψ + γ 0 is the Dirac conjugate bispinor. 35) V where V is the quantization volume (the whole space or some ﬁnite region restricted by boundaries). The energy–momentum tensor of the spinor ﬁeld is given by i ¯ (1/2) ¯ µψ . 36) 2 It is notable that all of the energy–momentum tensors obtained above in the absence of sources satisfy the conservation law (s) ∂ µ Tµν = 0, where s = 0, 1/2, or 1 is the spin of the ﬁeld. 2 Field quantization and vacuum energy in the presence of boundaries Various boundaries and boundary conditions In Part I of the book, we consider the Casimir eﬀect in regions of space restricted by ideally smooth boundaries.

31) and ψ(x) is a 4-component bispinor. e. 32) and the σk are the Pauli matrices (I is a 2 × 2 unit matrix). 33) where the Lagrangian density is i ¯ µ ¯ µ ψ − mψψ. 34) L(1/2) = 2 Here, ψ¯ = ψ + γ 0 is the Dirac conjugate bispinor. 35) V where V is the quantization volume (the whole space or some ﬁnite region restricted by boundaries). The energy–momentum tensor of the spinor ﬁeld is given by i ¯ (1/2) ¯ µψ . 36) 2 It is notable that all of the energy–momentum tensors obtained above in the absence of sources satisfy the conservation law (s) ∂ µ Tµν = 0, where s = 0, 1/2, or 1 is the spin of the ﬁeld.

As a result, the vacuum energy of the electromagnetic ﬁeld between the idealmetal planes can be presented in the form ∞ E0 (a) = 2 −∞ dkx 2π ∞ −∞ dky 2π ∞ ωk⊥ ,n S. 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.