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Working with a toy model whose partition function consists of a discretesummation, we introduce the statistical field-theory methodology bytransforming a partition function via a formal Gaussian integral relation (theHubbard-Stratonovich transformation). We then consider Gaussian type ofapproximations, wherein correlational contributions enter as harmonicfluctuations around the saddle-point solution. The work focuses on how toconstruct a self-consistent, non-perturbative approximation without recourse toa variational construction based on the Gibbs-Bogolyubov-Feynman inequalitythat is inapplicable to a complex action. To address this problem, we propose aconstruction based on a selective satisfaction of a set of exact relationsgenerated by considering a dual representation of a partition function, in itsoriginal and transformed form.

Introduction to statistical field-theory: from a toy model to a one-component plasma