Microcausality without Lorentz invariance

A bstract Microcausality — the vanishing of commutators outside the lightcone — is a fundamental property of relativistic quantum field theories. We derive its implications for two-point functions of scalar operators on Lorentz-breaking states. We restrict to spatially homogeneous and isotropic states, at zero and finite temperature, such as finite-density states of matter and primordial inflationary states. In a mixed ( t , $$ \overrightarrow{k} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>k</mml:mi> <mml:mo>→</mml:mo> </mml:mover> </mml:math> ) representation, we find certain analyticity and exponential boundedness conditions, which we verify in a variety of examples. Crucially, we discuss how our conditions can be tested within the regime of validity of Lorentz-breaking low-energy effective field theories, clarifying the role of the group velocity of low-energy excitations. In the cosmological case, we derive a positivity condition on an EFT coefficient in an inflationary background. Lastly, we comment on how microcausality can be used to constrain higher-point correlation functions, via suitable nested commutators.