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 explicity 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. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 114. No. 2, pp. 250–270, February 1998.