Dynamical Origin of the Lorentzian Signature of Spacetime

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 $g_{\mu \nu }=e_{\mu }^a{\eta }_{ab}{e}_{\nu }^b$ is introduced, where ${\eta }_{ab}=diag\{e^{i\theta },1,\mathrm{.}\mathrm{.}\mathrm{.},1\}$. 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.