Lorentzian Quantum Cosmology

TL;DR

This work advocates a Lorentzian, Picard-Lefschetz-based formulation of quantum cosmology, arguing that deforming the contour over metrics into the complex plane yields an unambiguous, causally coherent semiclassical expansion for minisuperspace models with a positive cosmological constant. By applying Lefschetz thimbles to the lapse-integral and enforcing no-boundary initial conditions, the authors identify the contributing saddle points and show the dominant semiclassical weight is the inverse of the Hartle-Hawking factor, $e^{-12\pi^2/(\hbar\Lambda)}$, in agreement with tunneling-type descriptions but derived from a distinct Lorentzian framework. They connect the path-integral results to the Wheeler-DeWitt equation, presenting an Airy-function representation of the Feynman propagator that matches the Lorentzian saddle-point analysis in various boundary regimes. The study concludes that the Lorentzian approach avoids the conformal-factor problem and yields a well-defined quantum cosmology with clear boundary-condition implementation, while opening avenues for including matter fields and more complex cosmologies. Overall, the work provides a rigorous, causality-preserving alternative to Euclidean quantum gravity and reinforces the utility of Picard-Lefschetz theory in 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 {\it inverse} of the famous Hartle-Hawking result.

Figures (8)

• Figure 1: Left panel: From a saddle point $\sigma$ emanate upward (${\cal K}_\sigma$) and downward (${\cal J}_\sigma$) flows, which are located in the wedges $J_\sigma$ (in green) and $K_\sigma$ (in red) respectively, defined as the regions where the Morse function $h$ is lower (higher) than its value at the saddle, respectively. The arrows along the flows indicate the direction of descent, and the downward flow ${\cal J}_\sigma$ is known as a Lefschetz thimble. The wedges are separated by blue lines along which $h$ is constant and equal to the value at the saddle point $h(p_\sigma.)$Right panel: Along a Lefschetz thimble the real part $h$ of the exponent decreases as fast as possible, ensuring an absolutely convergent integral.
• Figure 2: A pictorial description of the Feynman propagator, with $\mathcal{G}_0$ and $\mathcal{G}_1$ the initial and final three-geometry. Left: an expanding phase. Right: a contracting phase.
• Figure 3: A sketch of the wedges and flow lines emanating from the saddle points in the complex $N$ plane, for classical boundary conditions $q_1>q_0>\frac{3}{\Lambda}$. The Lefschetz thimbles ${\cal J}_\sigma$ reside within the green wedges $J_\sigma$ (within which the magnitude of the integrand is smaller than at the corresponding saddle point), while the contours of steepest ascent ${\cal K}_\sigma$ reside within the red wedges $K_\sigma$ (within which the magnitude of the integrand is larger than at the corresponding saddle point). The arrows indicate the direction of steepest descent. The original integration contour along the positive real axis is shown in orange, and runs through two saddle points in this case. The deformed contour along which the integral is absolutely convergent comprises the thimbles ${\cal J}_1$ and ${\cal J}_2$: the dashed orange line indicates how the original contour is deformed onto to these thimbles. Note that neither the flow lines, nor the original integration contour, include the point at $N=0.$
• Figure 4: For this numerical example we have chosen $k=1, \Lambda = 3, q_0=0, q_1=10.$ The saddle points then lie at $\pm 3 \pm i.$ Shown in the present figure are both the boundaries of wedges (lines of constant real part of the integrand/imaginary part of the action -- light blue lines) and the flow lines (lines of constant real part of the action -- red/green lines). More specifically, the plot shows both $Abs[{\rm Im}(S(N) - S(N_s))]$ and $Abs[{\rm Re}(S(N) - S(N_s))]$, where lighter colours correspond to smaller values. The four saddle points are located at the intersections of the flow lines. More details are provided in Fig. \ref{['fig:upper']}.
• Figure 5: A sketch of the wedges and flow lines emanating from the saddle points in the complex $N$ plane, for "no-boundary" conditions $q_0=0, q_1 > \frac{3}{\Lambda}.$ The loci of the steepest ascent/descent flows (in black) and of the boundaries between wedges (in blue) were determined numerically in Fig. \ref{['fig:saddleS']}. Here the arrows indicate the direction of steepest descent. We have coloured the wedges such that regions $J_\sigma$ with a lower value of the magnitude of the integrand than the corresponding saddle point are green, and regions $K_\sigma$ with a higher value are red, with the exception of the yellow regions which have a value intermediate between the two saddle point values. Comparing with the adjacent colours then avoids any ambiguity. Notice that, due to the symmetry explained above equation (\ref{['eq:intersection']}), there are 'degenerate' ascent and descent flows that link saddle points. This degeneracy is broken by adding an infinitesimal perturbation to the action, as shown in Fig. \ref{['fig:perturbation']}. The original integration contour along the positive real axis is shown in orange, and the deformed contour which Picard-Lefschetz theory picks out as the preferred integration cycle is marked in dashed orange. Again neither the flow lines, nor the original or final integration contours, include the point at $N=0.$ Only saddle point $1$ in the upper right quadrant can be linked to the original integration contour via an upward flow, and this implies that the (orange-dashed) downward flow from this saddle point is the correct Lefschetz thimble along which the path integral should be performed.