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 $g_{\mu \nu}=e^a_\mu \eta_{ab} e^b_\nu$ is introduced, where $\eta_{ab} = diag\{e^{i\theta},1,...,1\}$. The corresponding functional integral for quantized fields then interpolates from a Euclidean path integral in Euclidean space, at $\theta=0$, to a Feynman path integral in Minkowski space, at $\theta=\pi$. Treating the phase $e^{i\theta}$ as just another quantized field, the signature of spacetime is determined dynamically by its expectation value. The complex-valued effective potential $V(\theta)$ for the phase field, induced by massless fields at one-loop, is considered. It is argued that $Re[V(\theta)]$ is minimized and $Im[V(\theta)]$ is stationary, uniquely in D=4 dimensions, at $\theta=\pi$, which suggests a dynamical origin for the Lorentzian signature of spacetime.