The origin of chiral anomaly and the noncommutative geometry

We describe scalar and spinor fields on a noncommutative sphere starting fr om canonical realizations of the enveloping algebra A=U(u(2)). The gauge ex tension of a free spinor model, the Schwinger model on a noncommutative sph ere, is defined and the model is quantized. The noncommutative version of t he model contains only a finite number of dynamical modes and is nonperturb atively UV regular. An exact expression for the chiral anomaly is found. In the commutative limit the standard formula is recovered. (C) 2000 American Institute of Physics. [S0022-2488(00)03905-0].