Generalized Free Fields

Abstract

We now turn to representations of the Poincare group corresponding to scalar generalized free fields and their euclidean realizations by representations of the euclidean motion group. We start in Sect. 8.1 with a brief discussion of Lorentz invariant measures on the forward light cone \(\overline{V_+}\) and turn in Sect. 8.2 to the corresponding unitary representations. Applying the dilation construction to the time translation semigroup leads immediately to a euclidean Hilbert space \(\mathscr {E}\) on which we have a unitary representation of the euclidean motion group. In Sect. 8.3 we characterize those representations which extend to the conformal group \(\mathrm O_{2,d}(\mathbb {R})\) of Minkowski space. Then the euclidean realization is a unitary representation of the Lorentz group \(\mathrm O_{1,d+1}(\mathbb {R})\), acting as the conformal group on euclidean \(\mathbb {R}^d\).