There are techniques of the fields theory that can be applied to study the dynamics of a production system described from a Lagrangian, which is a function of the variables (q,q), coordinates of a given configuration space. These techniques allow a more general and abstract application than those used by other formalisms from physics, thus giving the possibility of studying the dynamics of various models. In addition, they are an effective tool to obtain more detailed information of the model. Today many of these techniques are applied in classical mechanics, quantum mechanics, particle physics, condensed matter, and in statistical physics models. In this paper they will be presented as an alternative geometric formalism for the study of the dynamics of an economic productive system.

The main features of the different linear gravity theories are reviewed. In particular, the supersymmetric extension of the JackiwTeitelboim (1 + 1) linear gravity is considered in detail within the canonical exterior formalism. In this context the role of the several fields are analyzed. The constraints and the field equation are found. Finally, this supergravity model is treated in the second order formalism.

In the present work a path-integral formalism in which the Hubbard X-operators are used as dynamic field variables is analyzed. The same formalism to the t-J model case is also discussed. Moreover, and by means of arguments coming from the Faddeev-Jackiw symplectic method, a family of first-order Lagrangians for the t-J model is constructed, and it is shown how the corresponding correlation generating functional can be mapped into the slave fermion or slave boson representation. Since 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. So, 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. It is clear that the constraint conditions must be introduced in the classical Lagrangian formulation. Finally, in order to define the propagation of fermions and bosons, we discuss two alternative ways to treat the fermionic and bosonic sector in the path-integral formalism for the t-J model.