Relativistic Partial-Wave Analysis in Two Variables and the Crossing Transformation

The properties with respect to the crossing transformation of previously suggested relativistic two-variable expansions are considered. Signature amplitudes with definite symmetries are introduced, and it is shown that if the total amplitude is an analytic function satisfying a subtractionless Mandelstam representation, and if the signature amplitudes are square-integrable functions (with a suitable measure) in the physical region of each channel, then two-variable expansions of one type can be continued into expansions of another type, from one channel into the other. The expansion coefficients, called Lorentz amplitudes, in both physical regions, carrying all the dynamics, are shown to be two separate pieces of a general analytic function of the complex angular momentum $l$. It is suggested that these analyticity properties should be made use of to relate low-energy and high-energy scattering data.