Hamiltonian diagonalization, as a tool to define the vacuum state of a massless field, is studied in two-dimensional space-time. The Hamiltonian definition depends on the time notion, i.e., on the manner in which space-time is foliated to separate space and time. According to which foliation is chosen, conformal or nonconformal vacua are obtained, and all these vacua result to be renormalizable. The formalism is applied to the eternal black-hole geometry, where the foliation attached to Unruh’s vacuum definition is found, and the quantization in geodesic reference systems is studied. In four dimensions it is shown that the Unruh vacuum can also be introduced as the state diagonalizing the Hamiltonian, when ‘‘Unruh’s time’’ goes to minus infinity.

• Received 19 June 1990

DOI:https://doi.org/10.1103/PhysRevD.43.2610