By Michael Bordag, Galina Leonidovna Klimchitskaya, Umar Mohideen, Vladimir Mikhaylovich Mostepanenko
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 good thing about the reader, the publication assembles field-theoretical foundations of this phenomenon, purposes of the final concept to actual fabrics, and a accomplished description of all lately played measurements of the Casimir strength with a comparability among scan and thought. there's an pressing desire for a ebook of this kind, given the rise of curiosity in forces originating from the quantum vacuum. various new effects were received within the previous couple of years which aren't mirrored in prior books at the topic, yet that are very promising for primary technology and nanotechnology. The e-book is a different resource of data proposing a serious evaluate of all of the major effects and techniques from enormous quantities of magazine papers. It additionally outlines new principles that have now not but been universally accredited yet that are discovering expanding aid from scan.
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Extra info for Advances in the Casimir Effect (International Series of Monographs on Physics)
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.
The operators an and a+ n are the annihilation and creation operators of a scalar particle with quantum number n. They obey the standard commutation relations an , a+ n = δnn , + [an , an ] = a+ n , an = 0. 57) The vacuum state |0 of the ﬁeld on an interval is deﬁned by an |0 = 0. 58) The energy density operator of the scalar ﬁeld in two-dimensional space–time is given by the 00-component of the energy–momentum tensor (0) T00 (t, x) = c 2 1 m 2 c2 2 2 [∂t ϕ(t, x)] + [∂x ϕ(t, x)] + 2 ϕ2 (t, x) .
T˜µν = T˜νµ . (1) By adding to T˜µν terms having the form of the 4-divergence ∂β (Fµ β Aν ), it is Field equations for ﬁelds of various spins 37 possible to obtain a gauge-invariant, symmetric energy–momentum tensor of the electromagnetic ﬁeld (Itzykson and Zuber 2005) (1) = Tµν 1 4π 1 Fµβ F β ν + gµν Fβγ F βγ 4 . 15), we obtain the familiar expression in terms of the electric ﬁeld and the magnetic induction E2 + B2 . 29) 8π At the end of this section, we mention brieﬂy the main facts related to a spinor ﬁeld of mass m with spin one-half.