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arXiv:1904.00957

Recursive Green's function approach to Feenberg perturbation theory

Ishida, K.

Published Apr 1, 2019• 0 citations
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Abstract

We propose a new procedure by using the recursive Green's functions which remove all the repetition terms from the time-independent perturbation series for finite-level quantum systems. These Green's functions are introduced as a generalization of the Brillouin-Wigner perturbation theory and the calculations of their diagonal elements can naturally give the effective propagators which are equal to the ones in the Feenberg perturbation theory.

DOI
10.48550/arxiv.1904.00957

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We propose a new procedure by using the recursive Green's functions which remove all the repetition terms from the time-independent perturbation series for finite-level quantum systems. These Green's functions are introduced as a generalization of the Brillouin-Wigner perturbation theory and the calculations of their diagonal elements can naturally give the effective propagators which are equal to the ones in the Feenberg perturbation theory.
A new time-independent theory of Green’s functions is presented, which is based on a Fock space Liouvillean and its resolvent. Unlike current time-independent approaches in this formalism there is no need to introduce a complete operator manifold like that of Manne and Dalgaard for inner projection, nor for invoking a ‘‘killer condition.’’ A perturbative construction of the Green’s functions involves a perturbation expansion of both the resolvent of the Liouvillean and of the wave function. The approach is very general and is by no means limited to a closed-shell reference function. For functions of the latter type a very compact approach is possible in a particle–hole formalism, which automatically leads to the Feynman–Dyson series. An essential point in its derivation is that the perturbation expansion of the resolvent and of the wave function lead to diagrams with the same topology but with different energy denominators but that such diagrams can be added up to a single one, which may contain both ‘‘Rayleigh–Schrödinger’’ and ‘‘Brillouin–Wigner’’ denominators. It is shown that a consistent theory can be based on ‘‘principal-value’’ Green’s functions. The one-particle Green’s function is treated in some detail, the polarization propagator is only briefly discussed.
A perturbation theory is presented which is suitable for the treatment of strong or resonant interactions in quantum systems described by time-independent Hamiltonians. The formulation is exact for finite-level systems and encompasses both nondegenerate and degenerate problems. The derivation is based on the partitioning of the levelshift operator, an operator which occurs naturally through the use of projection operators. The formulation is applied to the eigenvalue problem and to the calculation of the transition amplitude between states of the unperturbed system induced by a time-independent perturbation. The results are expressed in terms of Green's functions involving continued fractions which are truncated for finite-level systems.
We present a new derivation of Time-Independent Perturbation Theory (PT) that is shorter and more general than the usual ones presented in Quantum Mechanics (QM) texts. Expressing the task of PT in terms of Green's functions, the key to our approach is to write the bound-state equation for the perturbed wave function directly in terms of the unperturbed Green's function. Not only does this greatly simplify the derivation of PT expressions for energies and wave functions, but it does so without using the orthogonality or completeness properties of unperturbed Hamiltonian eigenstates, which form the basis of standard derivations of PT. As a result, our approach does not require the potentials to be energy independent as is required in standard derivations, nor does it require the inverse free Green's operator G0−1(E) to be a linear function of energy, as is the case in QM. Consequently, our approach is applicable not only to QM but also to various extensions of QM, including Quantum Field Theory. For students, our approach provides an easy-to-understand introduction to the use of Green's functions in QM while at the same time presenting a description of PT that is shorter and more fundamental than the standard ones currently advocated.
Abstract Brillouin‐Wigner (BW) perturbation formulae can be rearranged into a form first proposed by Feenberg. Feenberg's perturbation formulae also follow from a variational principle. They are successfully tested at two typical problems, for which ordinary perturbation techniques completely fail. The first is to find conditions for the bosonization of the Tomonaga model perturbatively. The second is to clarify whether non‐Fermi liquid behavior of the momentum distribution function of the Luttinger model can be achieved perturbatively.
A formulation of Schr\"odinger perturbation theory is developed that gives a unified treatment of non-degenerate and degenerate cases, is unique, and has a nonzero radius of convergence under very general conditions. Two alternative procedures are given for finding perturbed eigenvectors, one of which is simpler for the nondegenerate case or for small finite degeneracy, the other simpler for infinite or large finite degeneracy. The low-order terms in the perturbation expansions of quantities used in applications are given. The perturbation theory formulated in this paper has the following advantages over the conventional Schr\"odinger and Brillouin-Wigner perturbation theories: (i) When the convergence criterion is satisfied, bounds on the error made in replacing an appropriate infinite perturbation series by its first nnn terms can be obtained. (ii) For the case of degeneracy, the conventional Schr\"odinger perturbation theory can break down under conditions to which the convergence of the perturbation theory developed in this paper are insensitive. (iii) There is no implicit dependence on the eigenvalue, such as appears in the Brillouin-Wigner perturbation theory. (iv) For the case of degeneracy, statistical information about the distribution of certain eigenvalues can be obtained without finding the individual eigenvalues. (v) The theory is applicable to a wider class of problems than the conventional Schr\"odinger and Brillouin-Wigner perturbation theories.