On the nonlinear properties of retarded products

Abstract

We start from a set of functions t( p 1, … p n ) that satisfy the unitarity equations appropriate for time-ordered functions of n current operators, n = 4, 5, …. In terms of functions t, we define r α ( p 1, … p n ) of each order n which correspond to the generalized retarded functions of axiomatic local quantum field theory. Using the technique of generating functionals, the corresponding “products of operators” T and R are defined. The t( p 1, … p n ) and r α ( p 1, … p n ) are well defined as distributions in momentum space, possibly not tempered; and the time-ordered and retarded operators are also distributions in momentum space, and as operators in Fock space have reasonable domains of definition. We find that the commutator [ R α , R β ] between any two retarded operators can be expressed as sums and differences between retarded operators of order n + m, where n is the order of R α and m that of R β . The various r α -functions coincide in some region of momentum space, and any given r α coincides with the t-function in the region of momentum space appropriate to the function r α . The R α -operators trivially satisfy the Steinmann relations. Thus many of the “linear” properties of the retarded operators that follow from causality and Lorentz invariance also follow from unitarity, when no locality is assumed.