Space/time non-commutative field theories and causalityH. Bozkaya1, P. Fischer1, H. Grosse2, M. Pitschmann1, V. Putz1, 3, M. Schweda1 and R. Wulkenhaar3
1 Institut für Theoretische Physik, Technische Universität Wien, Wiedner Hauptstraße 8-10, 1040 Wien, Austria
2 Institut für Theoretische Physik, Universität Wien, Boltzmanngasse 5, 1090 Wien, Austria
3 Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstraße 22-26, 04103 Leipzig, Germany
(Received: 11 December 2002 / Revised version: 20 March 2003 / Published online: 23 May 2003 )
As argued previously, amplitudes of quantum field theories on non-commutative space and time cannot be computed using naïve 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 non-commutative theory. Although in all intermediate steps only three-momenta play a rôle, the final result is manifestly Lorentz covariant and agrees with the naïve calculation. Deriving the Feynman rules for general graphs, we show, however, that such a picture holds for tadpole lines only.
© Società Italiana di Fisica, Springer-Verlag 2003