Nuclear Physics B357 (1991) 289-307 North-Holland PATH INTEGRALS AND UNITARITY IN QUANTUM COSMOLOGY Noah LINDEN* and Malcolm J. PERRY** Deparmwnt of Applied Matlwmatics and Theoretical Physics, Unicersi~"of Cambridge, Sih'er Street, Cambridge CB3 9EW, UK Received 13 December 1990 (Revised 31 January 1991) We explore a simple cosmological model in the minisuperspace approach. The global structure of the model leads to subtleties in its quantization which are treated using the affine, rather than the Heisenberg, algebra. We develop a path-integral formulation for this algebra, and apply it to our cosmological model. We investigate the Friedmann-Robertson-Walker universe with a scalar field as source and discuss in detail the constraint that the universe have positive radius. 1. Introduction There have been many attempts to understand the quantum mechanics of the entire universe. The first serious attempts appear to have been the efforts of Wheeler [1] and DeWitt [2], who studied the canonical quantization of general relativity, and then applied their results to the universe as a whole. These pioneering works led to a number of papers that concentrated on finding the wave function of the universe, or at least that of some model universe. These wave functions were solutions of quantum mechanical analogs of the diffeomorphism and hamiltonian constraint equations of general relativity. These are now known as the Wheeler-DeWitt equations. It was found that there are infinitely many solutions of the Wheeler-DeWitt equation, even for very restricted cosmological models. This is because the Wheeler-DeWitt equations are essentially hyperbolic, and so must be supplemented by extra assumptions about which solution, or solutions, are physical. The subject was revitalized by Hartle and Hawking [3] who proposed that the correct solution of the Wheeler-DeWitt equation is that found by defining the wave function of the universe as a euclidean path integral. It was shown that such a wave function has a number of rather appealing cosmological consequences * E-mail address: NLI01(, UK.AC.CAM.PHX ** E-mail address: MJPI (, UK.AC.CAM.PttX (155(1-3213/91/$03.50 ~,~ ' ~ 1991 - Elsevier Science Publishers B.V. (North-Holland)