The index of scattering operators of Dirac equations (Q1098143)

The author considers a quantized Dirac field on Minkowski-space, minimally coupled to an external gauge field with gauge group U(N). It is shown that, under some technical conditions, the index of the scattering operator: (i) vanishes in the massive case; (ii) equals the instanton number \((-1/8\pi^ 2)\int Tr(F\wedge F)\) in the massless case, where F denotes the curvature two form of the gauge field. The aforementioned conditions are such that the unitarity of S and the compactness of [P,S] are guaranteed (where \(S=scattering\) operator and \(P=projection\) onto the positive spectral subspace of the free Dirac operator).