Space/time noncommutative field theories and causality

TL;DR

This paper investigates the causality and computation methods in space/time noncommutative quantum field theories, proposing a consistent approach that preserves Lorentz covariance in certain calculations despite causality violations.

Contribution

It clarifies the correct interpretation of time-ordering in noncommutative field theories and computes the one-loop self-energy, showing Lorentz covariance in the results.

Findings

Causality is explicitly violated inside the interaction region.

The proposed method is consistent and related to the interaction picture.

Final results for the self-energy are Lorentz covariant and match naive calculations.

Abstract

As argued previously, amplitudes of quantum field theories on noncommutative space and time cannot be computed using naive path integral Feynman rules. One of the proposals is to use the Gell-Mann--Low formula with time-ordering applied before performing the integrations. We point out that the previously given prescription should rather be regarded as an interaction point time-ordering. Causality is explicitly violated inside the region of interaction. It is nevertheless a consistent procedure, which seems to be related to the interaction picture of quantum mechanics. In this framework we compute the one-loop self-energy for a space/time noncommutative \phi^4 theory. Although in all intermediate steps only three-momenta play a role, the final result is manifestly Lorentz covariant and agrees with the naive calculation. Deriving the Feynman rules for general graphs, we show, however, that such a picture holds for tadpole lines only.