Locality conditions on form factors

Abstract

We derive a set of integral equations which are necessary and sufficient conditions on the form factors of local field theory, i.e. on the matrix elements of local operators. The basic idea is that out of the set of all (distribution-valued) functions defined on the boundary of an analyticity domain, in general only a subset are boundary values of functions which are analytic within the domain. The form factors are boundary values of a vertex function which, due to the general assumptions of locality, reasonable energy, and mass spectrum and Poincaré covariance, is analytic at least in the domain constructed by Källén and Wightman. The characteristic boundary of the domain (``the distinguished boundary'') is the set of physical values of the arguments of the form factors, and the integral equations in that way only involve such values. The main advantages in formulating the locality conditions in this way are that (1) only the physical quantities of the field theory, i.e., the matrix elements between the field operators, enter into the equations and (2) the frustrating complications which are met in the construction of the domains of analyticity for n-point functions with n > 3 might hopefully be avoided because the distinguished boundary can be constructed even if the whole domain is not known. The integral equations have naturally no unique solutions, because, e.g., all perturbation-theoretical form factors must, of course, fulfill them. The equations may, however, function as a convenient starting point for approximations and ``model building'' for form factors outside the presently used perturbation theories. The integral equations are straightforward generalizations of the notion of ``weak local commutativity'' for the two-point function. This condition means that the two spectral functions connected to two locally commuting operators should be equal. The conditions on the form factors (which are the generalizations of the two-point spectral functions) are that the difference should vanish when integrated over certain physical sets of mass space.