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The canonical vacuum energy density

Under the condition, that the free-space vacuum energy vanishes, we obtain the angular integrated form of the canonical vacuum energy density $\left\langle \tilde{\theta}^{00}\right\rangle $ as
$\displaystyle \left\langle \tilde{\theta}^{00}(r)\right\rangle$ $\textstyle \equiv$ $\displaystyle 4\pi\ \left\langle \theta^{00}(r)\right\rangle$ (1)
  = $\displaystyle \frac{1}{2\ \pi^{1/2}} \int_{1/\lambda^2}^{\infty}dz \ \frac{1}{z^{3/2}}$  
    $\displaystyle \times \Big[\sum_{\kappa} \frac{1}{2}\sum^{n_\lambda}_{n}\frac{1}{R^3}\ {\cal N}^2_{n,\kappa}\ (2J+1)$  
    $\displaystyle \phantom{\times \Big[\sum_{\kappa}} \left((j_{l}(\vert\varepsilon... ...repsilon _{n,\kappa}\vert r))^2\right)\ \mbox{e}^{-z \varepsilon _{n,\kappa}^2}$  
    $\displaystyle -\sum_{l}\ \frac{4}{\pi}\ (2l+1) \int_0^{\lambda}dk\ k^2 (j_l(kr))^2\ \mbox{e}^{-zk^2}\,\Big]\ ,$  
J = $\displaystyle \vert\kappa\vert-\frac{1}{2}\ ,\ \ l=\vert J\vert+\frac{1}{2}\ {\rm sgn}\ \kappa\ ,\ \ \bar{l}=l-\mbox{sgn}\ \kappa\ .$  

Thereby, jl denotes the spherical Bessel function, and the subscripts n, $\kappa$, and $\mu$ stand for the radial quantum number, the Dirac quantum number, and the angular momentum projection, respectively. The radial quantum number $n_\lambda$ labels the mode energy closest to $\lambda$, and ${\cal N}^2_{n,\kappa}$ is a normalization constant (see Ref. [21]). In Eq. (1) the integral over k corresponds to the free-space subtraction. Hard fluctuations are excluded by distinguishing two cases: 1) hard fluctuations with $\omega,\ \varepsilon _{n,\kappa}>\lambda$ or $\omega\le\lambda$, $\varepsilon _{n,\kappa}>\lambda$ are omitted by truncation of the mode sum, and 2) hard fluctuations with $\omega>\lambda$, $\varepsilon _{n,\kappa}\le\lambda$ are discarded by restriction of the z-integration. The canonical vacuum energy E is given by $E=\int_{0}^{R}{\rm d}r\ r^2\ \left\langle \tilde{\theta}^{00}(r)\right\rangle $.
next up previous
Next: The fermionic bag constant Up: Calculation Previous: Calculation
Marc Schumann
2000-10-16