  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
consideration of a firstly quantized quantum particle on the noncommutative
plane. Then we define quantum fields depending on noncommutative coordinates
and construct a field theoretical action using the $E_q(2)$-invariant measure
on the noncommutative plane. With the help of the partial wave decomposition we
show that this quantum field theory can be considered as a second quantization
of the particle theory on the noncommutative plane and that this field theory
has (contrary to the common belief) even more severe ultraviolet divergences
than its counterpart on the usual commutative plane. Finally, we introduce the
symmetry transformations of physical states on noncommutative spaces and
discuss them in detail for the case of the $E_q(2)$ quantum group.
