Gauge invariance of parametrized systems and path integral quantization

Gauge invariance of systems whose Hamilton-Jacobi equation is separable is improved by adding surface terms to the action functional. 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 un iverses, which are quantized by imposing canonical gauge conditions in the path integral; in the case of empty models, we first quantize the parametri zed system called "ideal clock," and then we examine the possibility of obt aining the amplitude for the minisuperspaces by matching them with the idea l clock. The relation existing between the geometrical properties of the co nstraint surface and the variables identifying the quantum states in the pa th integral is discussed.