Lorentzian quantum cosmology

We argue that the Lorentzian path integral is a better starting point for quantum cosmology than its Euclidean counterpart. In particular, we revisit the minisuperspace 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.

Figure 1

Left panel: From a saddle point $\sigma$ emanate upward (${\mathcal{K}}_{\sigma }$) and downward (${\mathcal{J}}_{\sigma }$) 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 ${\mathcal{J}}_{\sigma }$ 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 $q_1>q_0>\frac{3}{\mathrm{\Lambda }}$. The Lefschetz thimbles ${\mathcal{J}}_{\sigma }$ 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 ${\mathcal{K}}_{\sigma }$ 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 ${\mathcal{J}}_1$ and ${\mathcal{J}}_2$: the dashed orange line indicates how the original contour is deformed onto 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 $k=1$, $\mathrm{\Lambda }=3$, $q_0=0$, $q_1=10$. 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 $\text{Abs}[\mathrm{Im}(S(N)-S(N_s))]$ and $\text{Abs}[\mathrm{Re}(S(N)-S(N_s))]$, where lighter colors correspond to smaller values. The four saddle points are located at the intersections of the flow lines. More details are provided in Fig. 5.

Figure 5

A sketch of the wedges and flow lines emanating from the saddle points in the complex $N$ plane, for no-boundary conditions $q_0=0$, $q_1>\frac{3}{\mathrm{\Lambda }}$. The loci of the steepest ascent/descent flows (in black) and of the boundaries between wedges (in blue) were determined numerically in Fig. 4. Here the arrows indicate the direction of steepest descent. We have colored 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 colors then avoids any ambiguity. Notice that, due to the symmetry explained above Eq. (6), 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. 6. 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.

Figure 6

Due to the reality of the action, discussed above Eq. (6), and as shown in Fig. 5, a curve of steepest ascent from saddle point 1 coincides with a curve of steepest descent from saddle point 4. This degeneracy can be lifted by adding a small complex perturbation to the action, in this example $\mathrm{\Delta }{S}_0=iN/100$. The new contours of steepest ascent and descent, as well as the level sets of the magnitude of the integrand are shown for saddle point 1. In the presence of the perturbation, the lower steepest ascent contour from saddle point 1, instead of joining saddle point 4, now runs left to join the origin $N=0$ from below. Hence the formula for the intersection number (6) may now be used to unambiguously determine that saddle point 1 is relevant to the Lorentzian path integral. Note that the two lower curves, one green/red and one blue, do not now pass through a saddle point and should be ignored.

Figure 7

A sketch of the wedges and flow lines emanating from the saddle points in the complex $N$ plane, for the boundary conditions $q_1>q_0=\frac{3}{\mathrm{\Lambda }}$. The colors and arrows are as described in the caption of Fig. 3. For these boundary conditions, the saddle points are degenerate, and there are three lines of steepest ascent and descent emanating from them.

Figure 8

A sketch of the wedges and flow lines emanating from the saddle points in the complex $N$ plane, for nonclassical boundary conditions $\frac{3}{\mathrm{\Lambda }}>q_1\ge q_0$. The lines, arrows and colors are as described in the caption of Fig. 5. Here we again have yellow wedges that are both higher than one saddle point and lower than another one, $J_2=K_1$ and $J_4=K_3$. The original integration contour can be deformed into the dashed preferred contour by flowing down. Note that the preferred, dashed contour is chosen to follow the paths of steepest descent near the saddle points, so as to facilitate a saddle point approximation of the propagator.