Nuclear Physics B316 (1989) 5%79 North-Holland, Amsterdam THE MIDPOINT TRANSFORMATION IN WITTEN'S STRING FIELD THEORY Robertus POTTING* and Cyrus TAYLOR** Department of Physics and Astronomv, Rutgers University, Piscatawav, NJ, 08855-0849, USA Received 29 February 1988 (Revised 12 August 1988) We study the transformation to midpoint coordinates in Witten's string field theory using a ~'-function regularization. We find that the vertex is well defined and local (in that it makes no explicit reference to the midpoint coordinate) as the regulator is removed. This is in contrast to the heat-kernel regularization previously studied by others. We interpret the locality and unitarity anomalies of the heat-kernel regularization in terms of the associativity anomaly of the vertex. We also study the BRST charge in the same limit, verify that it is well defined, and propose a sense in which it is nilpotent. We then argue that the transformed vertex is BRST invariant. We discuss the structure of the space of states in the transformed theory; physical states are singular as the regulator is removed, suggesting that they be interpreted in a distributional sense. 1. Introduction Some time ago, Witten proposed a field theory of interacting strings [1]. The quantization of this theory has been studied, and Feynman rules for calculating on-shell scattering amplitudes have been derived [2-4]. This is not sufficient, however, to ensure that the theory has an interpretation in terms of quantum mechanical states and operators. To that end, Witten proposed studying the canonical structure of the classical theory [5]. In particular, he observed that the interaction term of the field theory is local with respect to the midpoint of the string, and argued that this implied that the theory has a canonical structure. This was the basis for the work of Potting, Taylor, and Velikson, who, under the assumption that the transformation to midpoint coordinates could be implemented, began the study of the constraint algebra of the interacting field theory in a hamiltonian framework [6]. * Supported by a Marion Johnson Graduate fellowship. Address after Oct. 1, 1988:Universit6 Libre de Bruxelles, Service de Physique Throrique, Boulevard du Triomphe, Brussels, Belgium. ** Supported in part by National Science Foundation grant PHY-84-15534. Address after Aug. 15, 1988: Department of Physics, Case Western Reserve University, Cleveland, OH 44106, USA. 0550-3213/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)