Quantum vacuum energy in a closed universe

The problem of defining the vacuum energy and pressure of quantized fields in the static Einstein universe is considered. A regularization procedure which utilizes a wavelength cutoff in the mode sum is discussed and applied to the cases of the massive conformally coupled scalar field, the electromagnetic field, and the neutrino field. In all cases a positive vacuum energy density and pressure are obtained. In the case of the massive scalar field it is possible for the vacuum pressure to exceed the vacuum energy density, thus violating the dominant energy condition. For the electromagnetic and neutrino fields the energy density and pressure are of the form $\rho =\gamma \overline{h}\frac{c}{a^4}$ and $P=\frac{\rho }{3}$, respectively, where $\gamma =\frac{11}{240{\pi }^2}$ for the electromagnetic field and $\gamma =\frac{17}{1920{\pi }^2}$ for the neutrino field, and where $a$ is the radius of the universe.