Simultaneous "Partial-Wave" Expansion in the Mandelstam Variables: Crossing Symmetry for Partial Waves

The amplitude for the elastic scattering of two spinless particles of equal mass ½ is expanded in terms of eigenfunctions which form a complete set for a certain class of functions of the Mandelstam variables $s,t,u(s+t+u=1\mathrm{})\mathrm{}$ and which display the threshold behavior of the partial-wave amplitudes. The eigenfunctions are generated by a partial differential operator which commutes with the total angular momentum in any of the three channels and which is invariant under $s$, $t$, $u$ permutations. An infinite number of finite-dimensional crossing relations for the partial-wave amplitudes which are necessary and sufficient for the crossing symmetry of the total amplitude are derived, as well as an explicit form for the corresponding crossing matrices. It is shown that the Fourier coefficients of the expansion satisfy a Froissart-Gribov integral representation whose kernel is determined by the imaginary parts of the partial-wave amplitudes.