Dynamical Origin of the Lorentzian Signature of Spacetime

TL;DR

This paper proposes that the Lorentzian signature of spacetime emerges dynamically from quantum fluctuations, with the phase of the metric's signature field stabilizing at the Minkowski value in four dimensions.

Contribution

It introduces a model where spacetime signature is a quantized field, and shows that in four dimensions, the Lorentzian signature is dynamically favored by the effective potential.

Findings

The effective potential $V(\theta)$ is minimized at $\theta=\pi$ in 4D.

The Lorentzian signature is dynamically selected due to quantum effects.

The model interpolates between Euclidean and Minkowski signatures via a phase field.

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.