Symmetry properties, tests, and reduction of the crossing matrix

We derive symmetry properties of the crossing matrix from general symmetry arguments. These properties can be used 1. (a) To test the known crossing matrices. We find that the crossing matrix of C-TMN [1] satisfies these conditions only if its overall phase factor is modified. We also show that the crossing matrix of Trueman and Wick [2], as we interpret it, is equivalent to that of C-TMN if one takes into account the different continuation paths. 2. (b) To reduce the crossing matrix into two or more submatrices, such that one submatrix connects "symmetry-conserving" amplitudes of the s and t channels with each other, whereas the other submatrix connects only the "symmetry-breaking" amplitudes of the two channels with each other. This fact may be useful for bootstrap calculation. In addition, we derive in a simple way the crossing matrices for the crossing of any two particles in terms of that for particles 1 and 4. Furthermore, we derive in the appendices the exact symmetry relations of the c.m. helicity amplitudes under T, CPT, E₁₂, E₃₄, and E for general reactions using consistent intrinsic phases, as these relations are not available in the literature. Our method is simpler than that of Jacob and Wick [3], since it doesn't involve any partial wave amplitudes.