We study the mathematical structure of finite-dimensional parametrized systems. We find that the systems must possess a so-called global time, if they are to be reducible. We list all conditions on the topology of the Hamiltonian vector field, which follow from the existence of a global time. The topological obstructions to the existence of a global time are analogous to the Gribov effect. We consider the canonical quantization and show that each known method of constructing a unitary quantum theory is based on the use of a global time. Studying cosmological models we show how choices of wrong candidates for time, as well as extensions of the configuration spaces, lead to violations of unitarity. We give simple examples of configuration spaces in which there is no global time. The interpretation of these results is that we are quantizing in wrong coordinates. Indeed, there is a class of global times for each parametrized system. These times cannot be functions of the internal and external three-geometry and of the instantaneous states of all fields only. We give an example of the transformation from geometrodynamical variables to variables containing a global time; its inverse is a sort of covering map and the topology of the Hamiltonian vector field is changed.

• Received 24 March 1986

DOI:https://doi.org/10.1103/PhysRevD.34.1040