Dynamical Origin of the Lorentzian Signature of Spacetime

Abstract

It is suggested that not only the curvature, but also the signature of spacetime is subject to quantum fluctuations. A generalized D-dimensional spacetime metric of the form $g_{μν}=e^a_μη_{ab} e^b_ν$ is introduced, where $η_{ab} = diag\{e^{iθ},1,...,1\}$. The corresponding functional integral for quantized fields then interpolates from a Euclidean path integral in Euclidean space, at $θ=0$, to a Feynman path integral in Minkowski space, at $θ=π$. Treating the phase $e^{iθ}$ as just another quantized field, the signature of spacetime is determined dynamically by its expectation value. The complex-valued effective potential $V(θ)$ for the phase field, induced by massless fields at one-loop, is considered. It is argued that $Re[V(θ)]$ is minimized and $Im[V(θ)]$ is stationary, uniquely in D=4 dimensions, at $θ=π$, which suggests a dynamical origin for the Lorentzian signature of spacetime.