Quantum field theory on the noncommutative plane with E-q(2) symmetry

We study properties of a scalar quantum field theory on the two-dimensional noncommutative plane with E-q(2) quantum symmetry. We start from the consi deration of a firstly quantized quantum particle on the noncommutative plan e. Then we define quantum fields depending on noncommutative coordinates an d construct a field theoretical action using the E-q(2)-invariant measure o n the noncommutative plane. With the help of the partial wave decomposition we show that this quantum field theory can be considered as a second quant ization of the particle theory on the noncommutative plane and that this fi eld theory has (contrary to the common belief) even more severe ultraviolet divergences than its counterpart on the usual commutative plane. Finally w e introduce the symmetry transformations of physical states on noncommutati ve spaces and discuss them in detail for the case of the E-q(2) quantum gro up. (C) 2000 American Institute of Physics. [S0022-2488(00)02004-1].