On quantum field theories with homogeneous interactions

Abstract

Among the conventional renormalizable scalar fields only ~3 or ~0 ~ interactions are amenable to standard perturbation treatment. One could envisage making progress in relativistic quantum field theory by extending these interactions to ~o~-interactions, where 2 takes on real or even complex values. We shall therefore propose a perturbation theory for homogeneous interactions ~ . We call these interactions homogeneous because they give rise in the massless case to Green's functions which are homogeneous functions of space-time. The term dilational covariant may be used synonymously for homogeneous. We shall derive formulae for the Green's functions in perturbation theory which arc valid for all the complex values of 2. By the generalization of the field theory from the positive integer points 2 = 2, 3, 4 to complex values of ~ one obtains an interpolation of field theories and an analytic representation for the n-point Green's functions in the degree of homogeneity 2. The Green's functions in the 2g-th-order perturbation theory occur for N ~ 4 as Sommerfeld-Watson-type spectral transforms. These transforms in the degree of homogeneity have remarkable properties. For example, they allow one to sum N-th-order perturbation graphs of the ~a or ~o 4 interactions into one expression under Mellin-Barnes integrals: the usual Feynman-diagram perturbation contributions for polynomial theories occur as simple poles under these integrals. This work is demonstrated in detail in the fourth-order of perturbation theory at the end of this paper. All the secondand the third-order perturbation contributions can be written down for all complex values of ~ explicitly. I t can be seen that all the N-th-order perturbation contributions with N ~ 4 are functions of harmonic ratios of propagators; this is true also for the ~3 or ~4 interactions. Similar techniques may be used for interactions with spin (see ref. (1)). I t should be noted that several authors (L2) have proposed Green's functions similar in form under the requirement of conformal covariance. The ~ Green's functions resemble those Green's functions found by SVMANZlK in ref. (~). The Green's functions in the