General solution to the quantum master equation in the finite-dimensional case

The general solution to the quantum master equation (and its $Sp(2)$ symmetric counterpart) is explicitly constructed in the case of a finite number of variables. It is shown that the finite-dimensional solution is physically trivial and, therefore, cannot be directly extended to a local field theory. Thus, the locality condition is important in obtaining nontrivial physical results when quantizing gauge field theories in the field-antifield formalism.