A regularization of the Hartle–Hawking wave function

The paper puts forward a modification of the no-boundary Hartle–Hawking wave function in which, in the general case, the Euclidean functional integral can be described by an inhomogeneous universe. The regularization of this integral is achieved in arbitrary canonical calibration by abandoning integration over the lapse and shift functions. This makes it possible to ‘correct’ the sign of the Euclidean action corresponding to the scale factor of geometry. An additional time parameter associated with the canonical calibration condition then emerges. An additional condition for the stationary state of the wave function's phase after returning to the Lorentzian signature, serving as the quantum equivalent of the classical principle of the least action, was used to find this time parameter. We have substantiated the interpretation of the modified wave function as the amplitude of the universe's birth from ‘nothing’ with the additional parameter as the time of this process. A homogeneous model of the universe with a conformally invariant scalar field has been considered. In this case, two variants of the no-boundary wave function which are solutions of the Wheeler–DeWitt equation have been found.