Wick rotation for spin foam quantum gravity

Abstract

Models of spin foam quantum gravity come in different spacetime signatures, either Euclidean or Lorentzian. The choice is reflected in the use of $Spin(4)$ or $SL(2,\mathbb{C})$ as gauge groups for the holonomies of the connection. In this work we show that a rotation of the Immirzi parameter to purely imaginary values maps the Euclidean Engle-Pereira-Rovelli-Livine spin foam model to its Lorentzian version and vice versa. Our methods provide a general recipe for relating spin foam models of different signature through analytic continuation of the gauge groups and their unitary irreducible representations.