An Approach to SU_q(2)p Gauge Theory

In the usual approach to q-deformed gauge theories, the gauge fields are required to be non-local or non-commutative one's. If we introduce, however, an extended product, which we call `` $\star$-product\rq\rq, among the generators of a q-deformed Lie group, the deformed group can be reduced to a ordinary Lie group under the $\star$-product. According to this line of approach, we try to construct a $[SU_q(2)\times U(1)]_\star$, a $SU(2)\times U(1)$ analogue under the $\star$-product, gauge theory. In this gauge theory with the $\star$-product, the U(1) symmetry is naturally incorporated into the SU(2) symmetry. We also study the symmetry breaking by the Higgs mechanism associated with $J=1/2$ and J=1 representations of $SU_q(2)$ algebra, and show that the mixing angle between the SU(2) and U(1) gauge fields is determined uniquely in a tree level.