Associativity Anomaly in String Field Theory

We give a detailed study of the associativity anomaly in open string field theory from the viewpoint of the split string and Moyal formalisms. The origin of the anomaly is reduced to the properties of the special infinite size matrices which relate the conventional open string to the split string variables, and is intimately related to midpoint issues. We discuss two steps to cope with the anomaly. We identify the field subspace that causes the anomaly which is related to the existence of closed string configurations, and indicate a decomposition of open/closed string sectors. We then propose a consistent cut off method with a finite number of string modes that guarantees associativity at every step of any computation.