Double commutator in quantum field theory

by R. F. Streater; J. Mathematical Phys., 1, 231-233, 1960.

ABSTRACT

A representation is obtained for the most general function with the properties of the double commutator of scalar quantised fields in the sense of Wightman, including non-zero mass thresholds but not the Jacobi identities. The thresholds are proved to satisfy triangular inequalities (without using any more information) which are always true physically. The problem of incorporating a discrete level at the mass of a stable particle is not solved.

Note. The results in this paper were obtained in my Ph. D. thesis. Dyson (Phys. Rev., 111, 1717, (1958)) had obtained a representation for the double commutator using his representation (Phys. Rev., 110, 1460 (1958))for the commutator, but had over-simplified the result, leading to a loss of generality, as was demonstrated to me by Symanzik. My version of the double commutator does give the most general function.<>/p>

Later, Kallen said to me that he had known for "some time" that there were examples of functions that violated my claimed result that the thresholds obey a triangular inequality; these examples were sums of functions from the perturbation theory of different processes, each obeying the triangle inequality. In these examples, the support of the vertex function F(p,q) as a function of the four-vector p depends on the four-vector q; physically, this is expressed as the statement that the threshold in the variable p. depends on the process. It is then true that if one defines the threshold in the invariant p² as the infimum of the thresholds in p² for all values of q, (and similarly for the other two thresholds in q² and (p-q)²) then the three thresholds do not need to satisfy the triangle inequalities. However let us define (as in my paper) the thresholds (now depending on the process) to be a triple of numbers m(i)² in the support of the function of the invariants
p², q², (p-q)²,
such that the function is zero in a neighbourhood of any point with a smaller value of any one of the invariants; then the triangular inequalities hold good.