Why is Spacetime Lorentzian?

Abstract

We expand on the idea that spacetime signature should be treated as a dynamical degree of freedom in quantum field theory. It has been argued that the probability distribution for signature, induced by massless free fields, is peaked at the Lorentzian value uniquely in D=4 dimensions. This argument is reviewed, and certain consistency constraints on the generalized signature (i.e. the tangent space metric $\eta_{ab}(x)=\mbox{diag}[e^{i\theta(x)},1,1,1]$) are derived. It is shown that only one dynamical "Wick angle" $\theta(x)$ can be introduced in the generalized signature, and the magnitude of fluctuations away from Lorentzian signature $\delta \theta = \pi - \theta$ is estimated to be of order $(l_P/R)^3$, where $l_P$ is the Planck length, and $R$ is the length scale of the Universe. For massless fields, the case of D=2 dimensions and the case of supersymmetry are degenerate, in the sense that no signature is preferred. Mass effects lift this degeneracy, and we show that a dynamical origin of Lorentzian signature is also possible for (broken) supersymmetry theories in D=6 dimensions, in addition to the more general non-supersymmetric case in D=4 dimensions.