Novel Use of Legendre Transformation in Field Theory and Many Particle Systems

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Abstract
The Legendre transformation is one of the ways to rewrite the theory by small number of selected variables. It leads naturally to the generalized action functional called the effective action and everything starts from this generating functional. It is applied to various problems of physics ranging from the relativistic field theory to the condensed matter physics. The main techniques used here are the on-shell expansion and the inversion method. The scheme presented in this article is thus a generating functional formalism of the many particle systems.