Friedel sum rule, Levinson theorem, and the Atiyah-Singer index

Abstract

With the chiral representation, the Friedel sum rule for chiral fermions in odd space dimensions is given and the Levinson theorem which determines the number of discrete zero modes is obtained. Furthermore, the connections among the Friedel sum, Levinson theorem, and the Atiyah-Singer index are discussed, thereby proving that the Euclidean Atiyah-Singer index for the background field that describes an arbitrary short-range central force is zero.