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The parametrized system called "ideal clock" is turned into an ordinary gauge system and quantized by means of a path integral in which canonical gauges are admissible.Then the possibility of applying the results to obtain the transition amplitude for empty minisuperspaces, and the restrictions arising from the topology of the constraint surface, are studied by matching the models with the ideal clock. A generalization to minisuperspaces with true degrees of freedom is also discussed.

Gauge invariance of systems whose Hamilton-Jacobi equation is separable is improved by adding surface terms to the action fuctional. The general form of these terms is given for some complete solutions of the Hamilton-Jacobi equation. The procedure is applied to the relativistic particle and toy universes, which are quantized by imposing canonical gauge conditions in the path integral; in the case of empty models, we first quantize the parametrized system called "ideal clock", and then we examine the possibility of obtaining the amplitude for the minisuperspaces by matching them with the ideal clock.The relation existing between the geometrical properties of the constraint surface and the variables identifying the quantum states in the path integral is discussed.