A non-diagrammatic method for perturbative calculations in field theory

Feynman's diagrammatic approach to perturbative quantum field theory is not easily applied unless the interactions have simple power series expansions. In particular, when the interaction involves non-integer powers of the field, as happens when carrying out the so-called delta -expansion, the diagrammatic approach must be supplemented with some prescription for the analytic continuation of the exponents. Here the authors propose a new approach to perturbative field theory when bypasses Wick's theorem, and uses instead the fact that the joint probability distribution function for the fields at a finite set of points can be determined exactly from their expectation values, variances and mutual covariances. One can then calculate expectation values for products of operators at these points, or at least express them as finite-dimensional definite integrals. This technique is illustrated by calculating expectation values for non-polynomial O(n)-invariant operators.