A new regularization scheme for the wave function of the Universe in the Lorentzian path integral

TL;DR

The paper tackles the conditional convergence of the Lorentzian path integral for the wave function of the Universe by introducing a regulator that enforces an initially vanishing size via a narrow Gaussian in the initial scale factor, $q_0$, with width $σ$. By exchanging the order of integration, the lapse integral becomes absolutely convergent on the real axis for fixed $σ$, allowing a purely Lorentzian treatment that yields the tunneling wave function and naturally suppresses perturbations when initial perturbations are included; the Hartle–Hawking wave function can be recovered with a modified contour that remains in ${ m Im}N\le0$. The regulator corresponds to preparing the universe in a narrow wave packet for its initial size and induces a Robin-type boundary condition at $t=0$, avoiding the conformal problem of Euclidean formulations while maintaining control over perturbations. The framework extends to scalar and tensor perturbations, where initial positive-frequency (BD-like) states ensure suppressed fluctuations, and it provides a consistent route to HH-type results via contour choice, thereby offering a robust, largely Lorentzian approach to quantum cosmology.

Abstract

The Lorentzian path integral for the wave function of the Universe is only conditionally convergent and thus requires a well-defined prescription. The Picard-Lefschetz approach ensures convergence through contour deformation, but it has been argued that this leads to unsuppressed perturbations due to relevant saddle points residing in the region ${\rm Im}N>0$. As an alternative, we propose a simple regulator for the lapse integral in minisuperspace. Specifically, we impose the vanishing initial size of the Universe via a delta function, represented as a narrow Gaussian of width $σ$, and take the limit $σ\to 0$ only after performing the functional integrations. This regulator has a clear physical interpretation: it corresponds to a vanishingly small quantum uncertainty in the initial size of the Universe. For any fixed $σ> 0$, the lapse integral is absolutely convergent along (or slightly below) the real axis, and no excursion into the region ${\rm Im}N>0$ is required. We further argue that the initial wave function for scalar and tensor perturbations should be incorporated in the Lorentzian path integral formalism, and we show that these perturbations are then appropriately suppressed. A purely Lorentzian path integral thus yields the tunneling wave function with suppressed perturbations. We also demonstrate that the Hartle-Hawking wave function can be reproduced by choosing a contour for the lapse integral extending from $-\infty$ to $+\infty$ that passes below the singularity near the origin.

Figures (4)

• Figure 1: $\mathrm{Re}\!\left[i S_g'\right]$ versus $\mathrm{Re}\,N$ along $\operatorname{Im}N=0$ (blue solid), $-0.1$ (pale green solid), $-0.2$ (yellow solid), and $\operatorname{Im}N=-3\pi^2\sigma^2$ (red dashed), shown for $\sigma=0.1$. The large-$|\mathrm{Re}\,N|$ suppression holds for all lines with $\operatorname{Im}N>-3\pi^2\sigma^2$, in agreement with \ref{['eq:asymp_line']}.
• Figure 2: Results of the Lorentzian path integral as a function of $\sigma^{-1}$ for $H=1$ and $q_1 = 2$ (blue curve). The yellow curve represents the estimation obtained by the saddle-point approximation, and the dashed line indicates its asymptotic value in the limit $\sigma \to 0$.
• Figure 3: Convergence diagram in the complex $N$-plane. Green (red) shading indicates asymptotic regions of convergence (divergence). Orange dots denote the four saddle points $N_\star^{(\pm,\pm)}$ and the white dot marks the singularity at $N=-\,i\,3\pi^2\sigma^2$. Black curves are steepest–descent/ascent lines (arrows indicate descent). The HH wave function is obtained by considering an integration contour from $-\infty$ to $+\infty$ that passes below the singularity (red dotted path), which remains entirely in $\operatorname{Im}N\le 0$ and is absolutely convergent. The gray dot-dashed curves represent contours along which the scale factor temporarily vanishes at a finite time.
• Figure 4: Real and imaginary parts of the solutions $N_p^{(i)}(t)$ of \ref{['eq:branch_cubic']} for $H=1$, $q_1=2$, and $\sigma=0.1$.