Modified microcausality from perturbation theory

Relativistic microcausality is the statement that local field operators commute outside the light cone. This condition is known to break down in low-energy effective theories, such as <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>P</a:mi><a:mo stretchy="false">(</a:mo><a:mi>X</a:mi><a:mo stretchy="false">)</a:mo></a:mrow></a:math> models with a derivative interaction term of the “wrong sign.” Despite their Lorentz-invariant form, these theories can exhibit superluminal propagation on Lorentz-breaking backgrounds. We approach this phenomenon by computing the full operator-valued commutator in position space, perturbatively in interaction picture. After testing this formalism on a <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>λ</e:mi><e:msup><e:mi>ϕ</e:mi><e:mn>4</e:mn></e:msup></e:math> theory, we apply it to a <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>P</g:mi><g:mo stretchy="false">(</g:mo><g:mi>X</g:mi><g:mo stretchy="false">)</g:mo></g:math> model. There, we show that the perturbative corrections to the free-theory commutator contain derivatives of <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>δ</k:mi></k:math> functions with support on the standard Minkowski light cone. While these corrections vanish on Lorentz-invariant states, they become “activated” on states where Lorentz symmetry is spontaneously broken. In this case, they approximate the new “sound cone” by means of a Taylor expansion. By applying linear response theory to an extended source, we show that deviations from standard causality are already present at first order in this expansion. Finally, we try to understand what goes wrong with the standard argument according to which Lorentz invariance implies microcausality.