The theory of asymptotic behavior

Abstract

The Green’s functions of renormalizable quantum field theory are shown to violate, in general, Euler’s theorem on homogeneous functions, that is to say, to violate naive dimensional analysis. The respective violations are established by explicit calculation with Feynman diagrams. These violations when incorporated into the renormalization group, then provide the basis for an entirely new approach to asymptotic behavior in renormalizable field theory. Specifically, the violations add new delta-function sources to the usual partial differential equations of the group when these equations are written in terms of the external momenta of the respective Green’s functions. The effect of these sources is illustrated by studying the real part, Re Γ(6)(λp), of the six-point 1PI vertex of the massless scalar field with quartic self-coupling—the simplest of renormalizable situations. Here, Γp is symbolic for the six-momenta of Γ(6). Briefly, it is found that the usual theory of characteristics is unable to satisfy the boundary condition attendant to the respective dimensional-analysis-violating sources. Thus, the method of characteristics is completely abandoned in favor of the method of separation of variables. A complete solution which satisfies the inhomogeneous group equation and all boundary conditions is then explicitly constructed. This solution has the following interesting properties: 1) It possesses Laurent expansions in the scale λ of its momentum arguments for all real values of λ2 except λ2=0, where it has a delta-function singularity in λ2 in addition to its Laurent expansions for λ2=0+ and λ2=0−. For |λ2|→∞ and |λ2|→0, the solution’s leading term in its respective Laurent series is proportional to λ−2, so that it behaves canonically in both limits! 2) The limits λ2→0+ and λ2→0− of λ2 RE Γ(6) are both nonzero and unequal. Further, on account of the delta-function singularity at λ2=0, the value of the solution at λ2=0 is not simply related to the value of either of these limits! Apparent divergences in the coefficients in the respective Laurent expansions are removed by the use of Heisenberg’s uncertainty principle. In this way, the new approach would appear to be operationally established.