  Static traversable wormhole solutions of the Einstein equations require
``exotic'' matter which violates the weak energy condition. The vacuum
stress-energy of quantized fields has been proposed as the source for this
matter. Using the Dewitt-Schwinger approximation, analytic expressions for the
stress-energy of a quantized massive scalar field are calculated in five static
spherically symmetric Lorentzian wormhole spacetimes. We find that in all
cases, for both minimally and conformally coupled scalar fields, the
stress-energy does not have the properties needed to support the wormhole
geometry.
