Exact DeWitt kernels in Lorentzian quantum cosmology

Abstract

The DeWitt boundary condition can be implemented in the Lorentzian path integral of minisuperspace quantum cosmology by summing only over non-singular paths. We give a general definition of the resulting DeWitt kernels, together with explicit rules for constructing them, and evaluate them exactly. The lapse integral over the positive half-line yields the Dirichlet Green function of the Wheeler--DeWitt operator, whereas the integral over the whole real line yields the Dirichlet group-averaging kernel; both vanish when either endpoint lies at the singularity. Since the minisuperspace actions are at most quadratic, all lapse integrals are performed in closed form without saddle-point approximations. We obtain the kernels for the closed universe without cosmological constant, where the whole-real-line kernel vanishes identically, for the flat de Sitter universe, and, in terms of Airy functions, for the closed and open de Sitter universes. Throughout, the DeWitt condition is imposed as the boundary condition at the singularity, which selects a self-adjoint constraint operator on the half-line and fixes the kernels uniquely, including their normalization.