Witten's Open String Field Theory in Constant B-Field Background

In this paper we consider Witten's bosonic open string field theory in the presence of a constant background of the second-rank antisymmetric tensor field $B_{ij}$. We extend the operator formulation of Gross and Jevicki in this situation and construct the overlap vertices explicitly. As a result we find a noncommutative structure of the Moyal type only in the zero-mode sector, which is consistent with the result of the correlation functions among vertex operators in the world sheet formulation. Furthermore we find out a certain unitary transformation of the string field which absorbs the Moyal type noncommutative structure. It can be regarded as a microscopic origin of the transformation between the gauge fields in commutative and noncommutative gauge theories discussed by Seiberg and Witten.