SQUARE-ROOT ACTIONS, METRIC SIGNATURE, AND THE PATH-INTEGRAL OF QUANTUM-GRAVITY

We consider quantization of the Baierlein-Sharp-Wheeler form of the gr avitational action, in which the lapse function is determined from the Hamiltonian constraint. This action has a square root form, analogous to the actions of the relativistic particle and Nambu string. We argu e that path-integral quantization of the gravitational action should b e based on a path integrand exp[root iS] rather than the familiar Feyn man expression exp[iS], and that unitarity requires integration over m anifolds of both Euclidean and Lorentzian signature. We discuss the re lation of this path integral to our previous considerations regarding the problem of time, and extend our approach to include fermions.