TITLE: Some General Properties of the Floquet States for the Two Dimensional
             Interacting Fermion Sistems with Quadratic Form Hamiltonians

ABSTRACT:
 

A fermion 2-dimensional interacting sistem that is coupled with an external classical field having a time periodic dependence is considered.

In the absence of the external field, the one particle Hamiltonian is quadratic and linear in respect to the canonical operators and the particles have static, scalar, two-body self-interactions; in addition, each particle interacts with an external clasical field and the coupling functions with the canonical operators (both the momenta and the position coordinates) are time periodic.

This model is a generalization of the two-dimensional electron gas in the presence of a monochromatic linear or circular polarized electomagnetic field. Using the Second Quantization version of the Floquet formalism, we obtain the solution of the eigenvalue problem for the Floquet Hamiltonian with the time-reducing transformation method.

We construct an unitary transform that produces a transformed Floquet Hamiltonian that is not time-dependent; then, the transformed eigenvalue equation can be resolved and this solution is close related to the solution of the energy eigenvalue equation of the same system in the absence of the external field.

This solution of the Floquet problem has the following important consequences:

- Green functions and the correlation density functions of this system are related to the corresponding quantities of the conservative system, so it is possible to develop a diagramatic method for the perturbed evaluation of these quantities in a similar manner to the conservative situation;

- when the system is invariant in respect to space translations in the absence of the external field, then the diagramatic analysis can be performed using a space-time Fourier transform, and this property leads to great simplifications and close correspondences to the conservative theory;

- it is posible to construct a result similar to the Pauli theorem, i.e. the quasi-energy eigenvalue of the interacting system (when the classical time-periodic field is present) differs from the quasi-energy of the free system (that is, the same sistem without self-interactions) by an integral over a constant-coupling of the time average of the expectation value of the self-interaction potential.

We remark that the Floquet version of the Pauli theorem is not a strait transposition of the conservative form, because there are some complications due to the fact that extension in the Floquet space needs the use of singular state vectors for the scalar product; however, if the system is homogeneous when the external field is absent, then this theorem becomes very similar to the corresponding conservative version.

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