Crossing, Hermitian Analyticity, and the Connection between Spin and Statistics
Authors: Stapp Henry P. 
Published: February 2004

The analytic S-matrix framework is further developed. First some results of earlier works are collected and the physical-region analyticity properties recently derived from macroscopic causality conditions are described. These entail that scattering functions are analytic at physical points except on positive-α Landau surfaces, and that there they are iε limits of analytic functions from certain well-defined directions, except possibly at certain points where four or more positive-α surfaces intersect. A general iε rule that also covers these exceptional points is then stated. It is then shown that the scattering function defined by analytic continuation is either symmetric or antisymmetric under interchange of variables describing identical particles and that the sign induced by the interchange is independent of the particular scattering function in which the variables appear. The physical-region analyticity properties of bubble-diagram functions are then derived from the general iε rule. These functions are products of scattering functions and conjugate scattering functions integrated over physical internal-particle variables, as in the terms of unitarity equations. They are shown to be analytic in the physical region except on Landau surfaces and, more specifically, except on those Landau surfaces that correspond to Landau diagrams that are supported by the bubble diagram in question, with the further restriction that the Landau α's must be positive or negative for lines lying within positive or negative bubbles, respectively. Also, the basic rule for continuation around these singularities is derived. A new general derivation of the pole-factorization theorem is given, which is based on slightly weaker assumptions than earlier proofs. Particular attention is paid to the over-all sign. A general derivation of the crossing and Hermitian analyticity properties of scattering functions is then given. On the basis of the deduced general rules for constructing the paths of continuation that connect the crossed and Hermitian conjugate points, the various related points are found to be boundary values of a single physical sheet. In particular, a certain sequence of continuations is shown to take one back to the original point. From this fact it follows that abnormal statistics are incompatible with simultaneous unitarity in both the direct and crossed channels. The proof given here does not depend on the notion of interchange of variables other than those of identical particles. Earlier proofs depended on the unphysical notion of interchange of variables representing conjugate particles. Finally it is shown that the analytically continued M functions with normal-ordered variables are precisely the scattering functions: no extra signs are needed or permitted. Aside from the general i rule, the analyticity assumptions are these: (1) The discontinuity around a singularity of a bubble diagram B has no residue at a physical-particle mass value (in an appropriate variable) unless the singularity corresponds to a diagram that is supported by B and has the single-particle-exchange form that corresponds to a pole at that mass value. (2) The residue just described has the pole-factorization property. (3) Confluences of infinite numbers of singularity surfaces do not invalidate the results established by assuming that this number is locally finite. Assumption (1) entails that all relevant singularities of scattering functions lie on Landau surfaces. That is the basic assumption.