Implementing Unitarity in Perturbation Theory

Unitarity cannot be perserved order by order in ordinary perturbation theory because the constraint UU † = 1 is nonlinear. However, the corresponding constraint for K = ln U , being K = −K†, is linear so it can be maintained in every order in a perturbative expansion of K. The perturbative expansion of K may be considered as a non-abelian generalization of the linked-cluster expansion in probability theory and in statistical mechanics, and possesses similar advantages resulting from separating the short-range correlations from longrange effects. This point is illustrated in two QCD examples, in which delicate cancellations encountered in summing Feynman diagrams of are avoided when they are calculated via the perturbative expansion of K. Applications to other problems are briefly discussed. Probability conservation is not maintained order by order in ordinary perturbation theory. This happens because the unitarity relation UU † = 1 (for the time-evolution operator U) is nonlinear, whereas the constraints for the other exact conservation laws, such as energy, momentum, and charge, are linear. Order-by-order probability conservation can be restored if we expand instead K = ln U ≡ ∑n≥1 Kn/n, for then the unitarity constraint becomes the linear constraint K = −K†. As long as every Kn is kept to be anti-hermitean, U = exp(K) will be unitary no matter where the K-expansion is truncated. We shall henceforth refer to