Microcausality in Curved Space-Time
Sergei Dubovsky, Alberto Nicolis, Enrico Trincherini, Giovanni Villadoro
TL;DR
The paper addresses whether microcausality, the vanishing of $[\mathcal{O}_1(x),\mathcal{O}_2(y)]$ for $(x-y)^2<0$, persists in curved space-time where the background breaks Lorentz invariance. It presents two general arguments—one via the functional integral and one via canonical quantization—that the quantum theory inherits the causal structure of the corresponding classical theory, independent of exact Lorentz invariance. A key point is that causality depends on the full off-shell propagator structure, $F(\omega, p)$, rather than on-shell dispersion alone, and the authors illustrate this with kinetic-function analysis and a cylinder example showing no imaginary part develops outside the light-cone at one-loop. The work supports the consistency of quantum field theory in curved space and has implications for gravity-related contexts and QED in curved backgrounds, clarifying that microcausality remains intact even when Lorentz symmetry is explicitly broken by the background geometry.
Abstract
It is well known that in Lorentz invariant quantum field theories in flat space the commutator of space-like separated local operators vanishes (microcausality). We provide two different arguments showing that this is a consequence of the causal structure of the classical theory, rather than of Lorentz invariance. In particular, microcausality holds in arbitrary curved space-times, where Lorentz invariance is explicitly broken by the background metric. As illustrated by an explicit calculation on a cylinder this property is rather non trivial at the level of Feynman diagrams.