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The Strong-Coupling Expansion and the Ultra-local Approximation in Field Theory

We discuss the strong-coupling expansion in Euclidean field theory. In a formal representation for the Schwinger functional, we treat the off-diagonal terms of the Gaussian factor as a perturbation about the remaining terms of the functional integral. We first study the strong-coupling expansion in the ϕ^4 theory and also quantum electrodynamics. Assuming the ultra-local approximation, we examine the analytic structure of the zero-dimensional generating functions in the complex coupling constants plane. Second, we discuss the ultra-local generating functional in two idealized field theory models. To control the divergences of the strong-coupling perturbative expansion two different steps are used. First, we introduce a lattice structure to give meaning to the ultra-local generating functional. Using an analytic regularization procedure we discuss briefly how it is possible to obtain a renormalized Schwinger functional associated with these scalar models, going beyond the ultra-local approximation. Using the strong-coupling perturbative expansion we show how it is possible to compute the renormalized vacuum energy of a self-interacting scalar field, going beyond the one-loop level.