Instantons and unitarity in quantum cosmology with fixed four-volume

Abstract

We find a number of complex solutions of the Einstein equations in the so-called unimodular version of general relativity, and we interpret them as saddle points yielding estimates of a gravitational path integral over a space of almost everywhere Lorentzian metrics on a spacetime manifold with topology of the "no-boundary" type. In this setting, the compatibility of the no-boundary initial condition with the definability of the quantum measure reduces reduces to the normalizability and unitary evolution of the no-boundary wave function ψ. We consider the spacetime topologies R^4 and RP^4 # R^4 within a Taub minisuperspace model with spatial topology S^3, and the spacetime topology R^2 x T^2 within a Bianchi type I minisuperspace model with spatial topology T^3. In each case there exists exactly one complex saddle point (or combination of saddle points) that yields a wave function compatible with normalizability and unitary evolution. The existence of such saddle points tends to bear out the suggestion that the unimodular theory is less divergent than traditional Einstein gravity. In the Bianchi type I case, the distinguished complex solution is approximately real and Lorentzian at late times, and appears to describe an explosive expansion from zero size at T=0. (In the Taub cases, in contrast, the only complex solution with nearly Lorentzian late-time behavior yields a wave function that is normalizable but evolves nonunitarily, with the total probability increasing exponentially in the unimodular "time" in a manner that suggests a continuous creation of new universes at zero volume.) The issue of the stability of these results upon the inclusion of more degrees of freedom is raised.

32 pages, REVTeX v3.1 with amsfonts. (v2: minor typos etc corrected.)

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• The wave function discord
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Cited by in corpus (19)

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• Discrete Space-Time Volume for 3-Dimensional BF Theory and Quantum Gravity
• The Landscape of String Theory and The Wave Function of the Universe
• The no-boundary proposal in biaxial Bianchi IX minisuperspace
• Everpresent Lambda - II: Structural Stability
• Unimodular conformal and projective relativity
• Is the spacetime metric Euclidean rather than Lorentzian?
• Unimodular Hartle-Hawking wave packets and their probability interpretation
• Unitary and Vilenkin's wave functions
• Quantum Schwarzschild-(A)dS Black Holes: Unitarity and Singularity Resolution
• Vortices, topology and time
• Instanton Tunneling for De Sitter Space with Real Projective Spatial Sections
• Quantum wormholes at spatial infinity
• Unimodular time in JT gravity: a holographic clock
• Unimodular Proca Theory: Breaking the U(1) gauge symmetry of unimodular gravity via a mass term