Dynamical Wick Rotation and the Anderson Localization of the Timelike Dimension

We present a novel result whereby in a quantum gravity path integral, formulated via spectral-geometric variables, the signature of spacetime undergoes a dynamical Wick rotation and the Lorentz signature emerges from an underlying Euclidean signature. The physical mechanism behind it is the Anderson localization of one of the spacelike dimensions. We find that the fluctuations of the bosonic and fermionic quantum fields source a disorder for the background geometry, which then induces the localization of one of the dimensions, and the emergence of a different effective signature. We first describe a toy model of signature change, and then the full quantum gravity path integral, based on previous work with M. Reitz and A. Kempf in Phys. Rev. Lett. 131, 211501 (2023). We also bolster our claim by demonstrating how an elliptical operator, in this case the Laplacian on a Euclidean manifold, can surprisingly change its character to a hyperbolic operator by an addition of an interaction term, via a mechanism similar to one discovered by Velo and Zwanziger (1969). We discuss the application to the no-boundary proposal by Hawking and Hartle.