From configuration to dynamics: Emergence of Lorentz signature in classical field theory

The Lorentzian metric structure used in any field theory allows one to implement the relativistic notion of causality and to define a notion of time dimension. This article investigates the possibility that at the microscopic level the metric is Riemannian, i.e., locally Euclidean, and that the Lorentzian structure, that we usually consider as fundamental, is in fact an effective property that emerges in some regions of a four-dimensional space with a positive definite metric. In such a model, there is no dynamics nor signature flip across some hypersurface; instead, all the fields develop a Lorentzian dynamics in these regions because they propagate in an effective metric. It is shown that one can construct a decent classical field theory for scalars, vectors, and (Dirac) spinors in flat spacetime. It is then shown that gravity can be included but that the theory for the effective Lorentzian metric is not general relativity but of the covariant Galileon type. The constraints arising from stability, the equivalence principle, and the constancy of fundamental constants are detailed and a phenomenological picture of the emergence of the Lorentzian metric is also given. The construction, while restricted to classical fields in this article, offers a new view on the notion of time.

Figure 1

Example of a spatial configuration of the clock field. Locally, one can define regions such as ${\mathcal{M}}_0$, ${\mathcal{M}}_0^{\prime }$, and ${\mathcal{M}}_0^{\prime \prime }$, in each of which a time direction emerges. Indeed this direction does not preexist at the microscopic level and can be different from patches to patches.

Figure 2

Example of a field configuration with fluctuations on scales larger than $M_y^{-1}$ and shorter than $m_y^{-1}$ (left). When smoothed on scales $m_x^{-1}$ (middle) and $10m_x^{-1}$ (right), the distribution of the clock field is such that ${\partial }_{\mu }\varphi$ can be considered as constant on scales smaller than $m_x^{-1}$.