Convolutions and Product Representations

Abstract

The Feynman integrals in special relativistic quantum field theories involve convolutions of energy-momentum distributions. The on-shell parts for translation representations give product representation coefficients of the Poincaré group, i.e., energy-momentum distributions for free states (multiparticle measures, discussed ahead). The off-shell interaction contributions (“virtual particles”) are not convolutable; this is the origin of the “divergence” problem in quantum field theories with interactions.With respect to Poincaré group representations, the convolution of Feynman propagators embedding the point- wise product of interactions, e.g. $({\frac{e^{-mr}}{2}}^2)$, makes no sense.

The pointwise product algebra of the essentially bounded complex functions of a real Lie group, $L^\infty \left( G \right) = L^\infty \left( G \right) \cdot L^\infty \left( G \right)$, characterizes its representations. The cone of positive-type functions $d = \hat d \in L^\infty \left( G \right)_ +$ induces the scalar products for cyclic Hilbert representations (see Chapter 8). Its conjugation property $d \leftrightarrow d^ - = \bar d$ connects dual representations. The point-wise product of two positive-type functions is a positive-type function for the product representation