Grassmann algebra in statistical mechanics: from the foundations to partition function

Abstract

Several methods for the study of materials were developed from formalism of the quantum many-particle mechanic in terms of Grassmann algebra. The dynamical mean-field theory is an example. Taking aim to aid pos-graduation students and researches at general on the study of these methods we elaborated this work, where the conceptual and mathematical structure of the Grassmann algebra are presented with detailed deduction. The grand partition function in this formalism is obtained using the Feynman path integrals. As an example for a Hamiltonian specific we wrote the grand partition function in terms of Grassmann variables for the Hubbard model. In order to do this work more auto-explicated we prepare an appendix where the creation and annihilation operators are defined and their anticomutation relations are deduzed of a manner physically intuitive.