Modified Microcausality from Perturbation Theory

TL;DR

This work investigates how microcausality manifests in relativistic scalar theories with derivative interactions, notably $P(X)$ models, on backgrounds that spontaneously break Lorentz boosts. By developing a perturbative, operator-valued commutator calculation in the interaction picture, the authors show that first-order corrections involve derivatives of $\delta$-functions centered on the Minkowski light-cone; on Lorentz-breaking states these terms act to approximate a modified sound-cone via a Taylor expansion. Linear response to an extended source reveals that deviations from standard causality appear at leading order in $\alpha\mu^2$, with negative $\alpha$ yielding superluminal propagation, while thermal states do not activate the derivative terms. The results illuminate how Lorentz invariance and microcausality interact in derivative-interacting EFTs, expose potential UV-cutoff obstructions for $\alpha<0$, and point toward broader implications for gravity and curved spacetime causality in similar effective theories.

Abstract

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 $P(X)$ 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 $λφ^4$ theory, we apply it to a $P(X)$ model. There, we show that the perturbative corrections to the free-theory commutator contain derivatives of delta 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.

Figures (2)

• Figure 1: Pictorial representation of the commutator in a $P(X)$ theory. Delta functions and their derivatives are regularized using Gaussians with finite variance. The blue and red shaded regions denote the interior and exterior of the light-cone, respectively. The free commutator is modeled as a Gaussian sharply peaked on the light-cone. In contrast, the first-order correction appears as the derivative of a Gaussian—antisymmetric and vanishing exactly on the light-cone. Its interference with the free commutator depends on the sign of the coupling: for positive coupling, the correction enhances the free Gaussian inside the light-cone and suppresses it outside; for negative coupling, the opposite occurs. In the limit of vanishing variance, the sum approximate a delta-function supported on the peak of the new distribution.
• Figure 2: The plot shows the response of the expectation value of the field over the coherent state $|\mu\rangle$ to the external Gaussian source \ref{['eq:source']}, evaluated at the origin. The response is given in function of the time $t_0$, normalized to the value $|\vec{x}_0|$. On the y-axis, we plot the canonically rescaled field $\tilde{\phi}(x)=J^{-1}_0\sqrt{2\pi}\sigma |\vec{x}_0| \phi(x)$, and we fixed $|\vec{x}_0|^2=20 \sigma^2$. The Red, Blue, and Purple curves are the response for a negative, positive, and null coupling respectively. When the theory is free ($\alpha=0)$, the response is maximized for a Gaussian source centered on the past light cone of Minkowski. If the coupling is positive/negative, the response is maximized for a source centered inside/outside the past light cone. We conclude that for $\alpha<0$ we have superluminal propagations.