The X-Operator Formalism

A key element for the reduction to two-component form is the analysis of the relationship between the large and small components of exact eigenfunctions of the Dirac equation, which we have already encountered in section 5.4.3. This relationship emerges because of the (2 x 2)-superstructure of the Dirac Hamiltonian, see, e.g., Eq. (5.135), which turned out to be conserved upon derivation of the one-electron Fock-type equations as presented in chapter 8. Hence, because of the (2 x 2)-superstructure of Fock-type one-electron operators, we may assume that a general relation. 

A state or orbital index has been dropped for the sake of clarity. With V = 0 this equation reduces to the free-particle Dirac equation of chapter 5. For V = Ir we have the equation for Dirac hydrogen-like atoms for point- 

If is a solution of Eq. (11.3) — and V does not contribute off-diagonal terms that would have to be added io c p — an expression for X can easily be 

For an energy-independent X-operator it is necessary to employ the Dirac equation in a different form. Multiplication of the upper of the two Dirac equations in split notation, Eq. (5.80), by X from the left produces a right-hand side that reads in stationary form XEip. From Eq. (11.2) we understand that this is identical to XEtp = Etp, and hence the two left-hand sides of the split Dirac equation become equal. 

X must satisfy this equation for all possible choices of the large components ip, and is therefore determined by the nonlinear operator identity [603] 

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Physics described by the model with so many parameters is very rich and the model is able particularly to treat heavy fermion systems. To study the model many approaches were suggested (see reviews [2-5]). They are successful for particular regions of the parameter space but no one is totally universal. In this paper we apply to PAM the generating functional approach (GFA) developed first by Kadanoff and Baym [6] for conventional systems and generalized for strongly correlated electron systems [7-10]. In particular it has been applied to the Hubbard model with arbitrary U in the X-operators formalism [10]. The approach makes it possible to derive equations for the electron Green s function (GF) in terms of variational derivatives with respect to fluctuating fields. 

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