  We compare the canonical quantization and the effective action method to
derive expectation values of the stress energy tensor for scalar fields
conformally coupled to a 2D Schwarzschild black hole spacetime. Particular
attention is devoted to the thermal equilibrium Hartle-Hawking state where the
striking disagreement of the results may be reconduced to the incomplete
knowledge of the effective action. We show how to reconcile the two procedures
and find physically meaningful analytical approximate expressions for the
stress tensor in the Hartle-Hawking state.
