De Sitter Bra-Ket Wormholes

TL;DR

This work develops a bra-ket wormhole framework in de Sitter Jackiw-Teitelboim gravity to describe the universe's initial state via a Wigner distribution on classical phase space. It demonstrates semiclassical saddles for pure gravity and for models with CFT and inflaton matter, showing the connected geometry can dominate over the Hartle-Hawking saddle for large universes but does not yield a normalizable global phase-space measure. The analysis reveals that matter content sharpens the distribution around classical trajectories while introducing non-normalizable directions, and it explores probabilistic interpretations, local observables, and the spectrum of fluctuations, including a thermal-like regime for certain modes. The paper also discusses one-loop determinants, the comparison to HH, and potential extensions to four-dimensional cosmology and entanglement entropy via islands.

Abstract

We study a model for the initial state of the universe based on a gravitational path integral that includes connected geometries which simultaneously produce bra and ket of the wave function. We argue that a natural object to describe this state is the Wigner distribution, which is a function on a classical phase space obtained by a certain integral transform of the density matrix. We work with Lorentzian de Sitter Jackiw-Teitelboim gravity in which we find semiclassical saddle-points for pure gravity, as well as when we include matter components such as a CFT and a classical inflaton field. We also discuss different choices of fixing time reparametrizations. In the regime of large universes our connected geometry dominates over the Hartle-Hawking saddle and gives a distribution that has a meaningful probabilistic interpretation for local observables. It does not, however, give a normalizable probability measure on the entire phase space of the theory.

Figures (8)

• Figure 1: Here we see two potential contributions to the density matrix as defined in \ref{['eq:formalexpressiondensitymatrix']}. The green region denotes a gravitating region, which is glued at the reheating surface to a flat space region (yellow), which we think of as a toy approximation of a weakly gravitating FLRW cosmology. As we are working with JT gravity, only constant positive curvature metrics contribute to the gravitational path integral. (a) The Hartle-Hawking prescription demands that only those complex metrics are considered that are regular and do not exhibit a further boundary in addition to the late time boundary. Note that bra and ket are separately defined objects, such that this is a disconnected contribution to \ref{['eq:formalexpressiondensitymatrix']}. (b) A bra-ket wormhole exhibiting a connection in the past. Bra and ket are now connected via a complex contour, which avoids any singularities. The geometries analysed in Chen:2020tes are also traced in the future and therefore correspond to a torus.
• Figure 2: Depiction of the Wigner boundary conditions corresponding to \ref{['eq:formalexpressionwignertransform']}. Both bra and ket future infinity are now free to oscillate and the difference is integrated over. Without any further prescription there are disconnected and connected contributions, which correspond to integral transforms of the geometries pictured in figure \ref{['fig:densitymatrixcontribution']}.
• Figure 3: The perturbed $\pi$-contour in FLRW coordinates a) and conformal coordinates b) and $v$ coordinates c). The original contour analysed in Chen:2020tes is depicted in green dashed lines, deviations are depicted as blue and red solid lines. The bra and ket endpoints of the new contour are depicted as red and blue dots respectively. The singularities in the metric are denoted by black dots. The new set of boundary conditions used for the Wigner distribution allow for complex shifts of the contour endpoints.
• Figure 4: Illustration of the bra-ket wormhole with Wigner boundary conditions and its geometric parameters. The two cyles of the torus are depicted as dashed, coloured lines. The blue cycle along the time direction sets the temperature for matter fields and corresponds to the Euclidean difference between bra and ket. While for the basic contour of Chen:2020tes it is set to $\pi$ it can receive corrections with the new boundary conditions. The second cycle exhibits a minimal length geodesic of size $2 \pi b$ but grows with the expansion of the universe towards the two boundaries. The asymptotic bra and ket endpoints of the contour may now fluctuate, which is depicted as a red double-arrow.
• Figure 5: (a) The integration contour of the Hartle-Hawking wave function in terms of the global coordinates defined in \ref{['eq:globalmetric']}. (b) The contour for the Hartle-Hawking density matrix. Bra and ket are shown in blue and red respectively. The bra and ket future boundaries are denoted by blue and red dots, the poles of the half-spheres are instead denoted by circles. Note that there is no connection between bra and ket in the past. On the bra side time goes in the opposite direction because of complex conjugation. For the Wigner distribution of this quantity we integrate over the full domain of possible differences between bra and ket contour endpoints.