Self sustained traversable wormholes and the equation of state

We compute the graviton one loop contribution to a classical energy in a $\mathit{\text{traversable}}$ wormhole background. The form of the shape function considered is obtained by the equation of state $p=\omega \rho$. We investigate the size of the wormhole as a function of the parameter $\omega$. The investigation is evaluated by means of a variational approach with Gaussian trial wave functionals. A zeta function regularization is involved to handle with divergences. A renormalization procedure is introduced and the finite one loop energy is considered as a $\mathit{\text{self-consistent}}$ source for the traversable wormhole.The case of the phantom region is briefly discussed.