A Wick rotation for EPRL spin foam models

TL;DR

This paper demonstrates a mathematical connection between Euclidean and Lorentzian EPRL spin foam models via a Wick rotation of the Immirzi parameter, involving analytic continuation of the underlying algebraic structures.

Contribution

It introduces a novel analytic continuation method linking Euclidean and Lorentzian models in Loop Quantum Gravity through a Wick rotation of the Immirzi parameter.

Findings

Euclidean and Lorentzian EPRL amplitudes are related by a Wick rotation.

Analytic continuation of gauge groups and representations establishes the connection.

The approach applies to the decomposition into $SU(2)$ invariants and booster functions.

Abstract

We show that the Euclidean and Lorentzian EPRL vertex amplitudes of covariant Loop Quantum Gravity are related through a ``Wick rotation'' of the real Immirzi parameter to purely imaginary values. Our result follows from the simultaneous analytic continuation of the algebras, group elements and unitary irreducible representations of the gauge groups $Spin(4)$ and $SL(2,\mathbb{C})$, applied to the decomposition of the two models in terms of $SU(2)$ invariants and booster functions.