Dynamical Model and Path Integral Formalism for Hubbard Operators

TL;DR

This paper investigates the feasibility of constructing a path integral formalism using Hubbard operators as dynamical variables, revealing fundamental constraints and differences from the full algebra, and discusses implications for perturbative theory.

Contribution

It demonstrates the limitations of classical dynamics for Hubbard operators and clarifies the constraint conditions needed for a consistent path integral formalism.

Findings

Classical dynamics cannot fully replicate Hubbard $X$-operator algebra.

Weaker constraints are sufficient in the Lagrangian formulation than in the full algebra.

Perturbative diagrammatic rules are derived for the model.

Abstract

In this paper, the possibility to construct a path integral formalism by using the Hubbard operators as field dynamical variables is investigated. By means of arguments coming from the Faddeev-Jackiw symplectic Lagrangian formalism as well as from the Hamiltonian Dirac method, it can be shown that it is not possible to define a classical dynamics consistent with the full algebra of the Hubbard $X$-operators. Moreover, from the Faddeev-Jackiw symplectic algorithm, and in order to satisfy the Hubbard $X$-operators commutation rules, it is possible to determine the number of constraint that must be included in a classical dynamical model. Following this approach it remains clear how the constraint conditions that must be introduced in the classical Lagrangian formulation, are weaker than the constraint conditions imposed by the full Hubbard operators algebra. The consequence of this fact is analyzed in the context of the path integral formalism. Finally, in the framework of the perturbative theory, the diagrammatic and the Feynman rules of the model are discussed.