The Kallen-Lehmann Representation

Abstract

This chapter deals with an important application of the non-perturbative RQFT, namely, the representation of two-point functions. This was established by Gunnar Kallen for QED, and for generic field theories by Lehmann. This is also called a spectral representation. It expresses the exact two-point function of an interacting quantum field theory as an integral of free two-point functions with positive weights, called spectral functions. The positivity of the spectral functions is shown to lead to the crucial result that exact propagators of interacting QFT can never fall off faster than free propagators. The important result that the integral of the spectral function has to lie between 0 and 1 for physically meaningful theories is explicitly demonstrated. It is close to 0 for weakly interacting theories and close to 1 for very strongly ones. It is pointed out that when it is exactly 1, single particle excitations completely disappear. This could lead to the possibility that a Higgs field could have non-vanishing vacuum expectation value without a Higgs boson in its spectrum. The chapter ends with some lessons for the analytic S-matrix.