Dispersion Relations and Spectral Representations

Abstract

The two response functions _Xkl (t) and Φ _kl (t) by definition [(2.25) and (3.12)] vanish for t < 0. Physically speaking this expresses “causality”: the “effect” Q(t) is influenced only by “causes” f ^e(t′) and f(0), at times t′ and 0, respectively, before t. Mathematically one can use this fact to continue the Fourier transform of the correlation functions [e.g. (3.6)] into the (upper) complex ω half plane without endangering the convergence of the Fourier integrals. The Fourier integrals on the real ω-axis then can be obtained as limits of the integrals in the complex plane from above.