Quantized Equations of Motion and Currents in Noncommutative Theories

Abstract

We study quantized equations of motion and currents, that means equations on the level of Green's functions, in three different approaches to noncommutative quantum field theories. At first, the case of only spatial noncommutativity is investigated in which the modified Feynman rules can be applied. The classical equations of motion and currents are found to be also valid on the quantized level, and the BRS current for NCQED is derived. We then turn to the more complicated case of time-space noncommutativity and consider the approach of TOPT. Additional terms depending on $θ^{0i}$, which are not present on the classical level, appear in the quantized equations of motion. We conclude that the same terms arise in quantized currents and cause the violation of Ward identities in NCQED. The question of remaining Lorentz symmetry is also discussed and found to be violated in a simple scattering process. Another approach to time-space noncommutative theories uses retarded functions. We present this formalism and discuss the question of unitarity, as well as equations of motion, and currents. The problems that emerge for $θ^{0i}\neq 0$ are seen to arise from a certain type of diagrams. We propose a modified theory which is unitary and preserves the classical equations of motion and currents on the quantized level.