2D black holes and effective actions

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.