Lorentzian quantum cosmology

Abstract

We argue that the Lorentzian path integral is a better starting point for quantum cosmology than the Euclidean version. In particular, we revisit the mini-superspace calculation of the Feynman path integral for quantum gravity with a positive cosmological constant. Instead of rotating to Euclidean time, we deform the contour of integration over metrics into the complex plane, exploiting Picard-Lefschetz theory to transform the path integral from a conditionally convergent integral into an absolutely convergent one. We show that this procedure unambiguously determines which semiclassical saddle point solutions are relevant to the quantum mechanical amplitude. Imposing "no-boundary" initial conditions, i.e., restricting attention to regular, complex metrics with no initial boundary, we find that the dominant saddle contributes a semiclassical exponential factor which is precisely the inverse of the famous Hartle-Hawking result. * Electronic address: jfeldbrugge@perimeterinstitute.ca † Electronic address: jlehners@aei.mpg.de ‡ Electronic address: nturok@perimeterinstitute.ca

Gravitational decoupling and the Picard-Lefschetz approach

“…Fortunately, this has already been proven in [21] for cases such as the one presented here. 12 The reader may note that this potential is unphysical for α < 0 since it is not bounded from below.…”

“…A very similar discussion has unfolded in the recent literature where the authors [21][22][23] have claimed that the Hartle-Hawking wave function [24] is not smooth due to unsuppressed fluctuations in the Euclidean action, while [25] has countered that a proper choice of path-integration contour eliminates these issues. Our point is slightly orthogonal.…”

“…The choice C GHY ¼ 1=3 appears to be special since this eliminates the first term in (21). In fact, this choice has been proposed for HT singularities for orthogonal reasons.…”

“…However, in our case the singularity of _ a causes a _ aδa to be finite at the boundary. This gives us the expression (21). The subtle change in the variation of S g is the essential reason decoupling fails for HT-type singularities.…”

“…8 Fortunately, this problem may be surmounted with PL theory [26]. PL theory has already found applications for the sign problem [39], resurgence [40][41][42][43], and quantum cosmology [21][22][23]25]. While Euclideanization is based on complexification of the time coordinate and contour deformation of the action, PL theory instead complexifies the fields themselves.…”