Letters in Mathematical Physics 34: 395-406, 1995. 395 © 1995 Kluwer Academic Publishers. Printed in the Netherlands. Renormalization of the Relativistic Delta Potential in One Dimension RHONDA J. HUGHES* Department of Mathematics, Bryn Mawr College, Bryn Mawr, PA 19010, U.S.A. (Received: ! I August 1994) Abstract. Several authors have noted an ambiguity with the Dirac equation in one dimension. In the case of a delta-function potential, the coupling constant is subject to an apparently arbitrary renormalization when the delta function is approximated in different ways. We explain these differences in terms of strong resolvent limits of self-adjoint operators on L2(R), and obtain a precise formula for the renormalized coupling constant in the case of separable potentials. The examples in the literature follow as special cases. Mathematics Subject Classifications (1991). 47N50, 81Q05. 1. Introduction Several authors have noted an ambiguity with the Dirac equation in one dimension I-1, 10, 11]. In the case of a delta-function potential, the coupling constant is subject to an apparently arbitrary renormalization when the delta function is approximated in different ways, and the size of the jump in the boundary condition at the origin seems to depend arbitrarily on the strength of the coupling constant. Our approach involves an interpretation of the formal expression (1/i)(d/dx) + c6 (due to Segal (cf. 1-9])), that systematically explains these ambiguities in terms of strong resolvent limits of self-adjoint operators on L2(R). We show that no renormalization is necessary for local potentials, while for separable potentials, we provide a precise formula for the renormalization as a function of the coupling constant. The examples in [10] and 1-11] then follow as special cases. The free Dirac operator in one dimension is a self-adjoint operator on the Hilbert space L2(R) ® C 2, defined by D0= -i~x®aS +re®a3, where 10) .=(; *Research supported in part by a grant from the National Science Foundation.