The Wave Function of the Universe by the New Euclidean Path-integral Approach in Quantum Cosmology

The wave function of the universe is evaluated by using the Euclidean path integral approach. As is well known, the real Euclidean path integral diverges because the Einstein-Hilbert action is not positive definite. In order to obtain a finite wave function, we propose a new regularization method and calculate the wave function of the FriedmannRobertson-Walker type minisuperspace model. We then consider a homogeneous but anisotropic type minisuperspace model, which is known as the Bianch type I model. The physical meaning of the wave function by this new regularization method is also examined. Quantum cosmology is one of the most fascinated subjects in modern physics. [1,2] In quantum theory, the wave function characterizes each theory. Therefore the main purpose in quantum cosmology is to evaluate the wave function of the universe. As is well known, the wave function of the universe is obtained as the solution of the Wheeler-DeWitt second-order functional differential equation. This equation, however, is very difficult to solve even in minisuperspace models, although in minisuperspace models the Wheeler-DeWitt equation is reduced to an ordinary differential equation. In addition, we have to know the initial condition of the universe which is hard to understand completely. Another approach, which gives the wave function by the Euclidean path integral, has been proposed by Hartle and Hawking [3,4] . They claimed that the path integral should be performed over regular and closed four-dimensional manifolds. This idea is called the boundary condition of no-boundary. The four-dimensional manifold must be Euclidean to avoid the cone singularity at an initial point. Their proposal solves the initial condition problem but the real Euclidean path integral does not converge because the Einstein-Hilbert action is not positive definite. The question is therefore whether we can construct the Euclidean path integral approach which is free from a divergence problem. A first way to avoid this difficulty was proposed by Gibbons, Hawking and Perry [5] . They suggested that the path integral could converge if we performed the conformal rotation. This proposal, however, has several objections [6] . Hartle proposed another path integral approach. He suggested that the path integral should be taken along the steepest-descent path in the space of complex fourmetrics. Following the Hartle’s alternative proposal, we can make the Euclidean path integral convergent by changing an integration contour from a real axis to a curve on a complex plane. Halliwell and Louko calculated the wave function of the universe following this method [8,9,10] . The integral along this complex curve is so difficult to perform that they evaluated the wave function in a saddle point approximation [8] . Moreover the physical interpretation of this wave function is hard because the path integral is defined on the complex four-manifold [11] .