PHYSICAL REVIEW D VOLUME 24, NUMBER 10 15 NOVEMBER 1981 Scale covariance, nonrenormalizable interactions, and high-temperature expansions John R. Klauder Bell Laboratories, Murray Hill, New Jersey 07974 (Received 13 July 1981) Modified (scale-covariant) Heisenberg operator equations of motion, derived on the assumption that local products are based on an operator-product expansion, are shown to exhibit a nonclassical, one-parameter degree of freedom in their functional integral formulations. By a proper choice of this freely adjustable, scale-covariance parameter the suitably normalized, connected (truncated) four-point functions can be changed from a bounded negative quantity to one that is positive and arbitrarily large. For nonrenormalizable, or possibly for nonasymptotically free renormalizable models, which when conventionally formulated as lattice theories violate hyperscaling and exhibit a trivial continuum limit, the previously stated property suggests that the scale-covariance parameter can be chosen to achieve nontrivial results for such models. This conclusion is supported by results of a high-temperature series analysis that incorporates the effects of the additional parameter. A. Operator heuristics According to the arguments of scale-covariant quantum field theory'-3 the usual Heisenberg op- erator equations of motion are modified whenever local products are defined by an operator-product expansion. In particular, for a (GP),, p even, scalar field theory with action functional the modified, scale-covariant, equation of motion is given by the formal expression This equation arises from the stationarity of I un- der infinitesimal scale transformations of the form 6@(x) = 6 s(x)@(x),6s being a c number. In (1) and (2) it is understood that local products are obtained from an operator-product expansion and not from any sort of normal ordering; as a consequence (2) cannot be interpreted simply as @ times the usual operator equation of motion-even in the special case that k - 0. The usual equation of motion arises from stationarity of I under the c -number variations 6@(x) = 612 ( x ) appropriate to normal-or- dered local products, but such variations yield nothing if operator-product expansions are appro- priate.'~~ B. Functional formulation Operator field equations such as (2) may be re- cast into coupled sets of Green's-function equa- tions4 or into other equivalent reformulations such as that provided by functional integrals. In the latter approach one may consider the formal ex- pression for the Green's functional 2 '(h) =X / exp 6 / {h@ ++ [(a,@)' - m'@] where Fn is chosen so that Z1(0) =1 and Sf@ is left unspecified for the moment. If we change the in- tegration variables according to @(x) - S(x)@(x), S(x) > 0, then where in the last line we have assumed that the formal measure fulfills a condition of scale co- variance, WS@=F(S)P1@, (5) and the factor F(S) has been absorbed in %, . Clearly an expression for X, i s given by The expression (4) [with (611 for Z1(h) seems to depend on S but in fact does not, and so a relation that leads in the usual way to a function- al differential equation3s4 given by