Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

The abstract Hilbert space generalization of Feenberg’s perturbation theory—A new method of quantum field theory

Samuel P. Bowen

Published Mar 1, 1975• 24 citations
Publisher

Abstract

A rigorous generalization of Feenberg’s perturbation theory to the calculation of thermodynamic Green’s functions is outlined. This method offers an algorithm for calculating correlation functions and propagators for any Hamiltonian, bounded or unbounded, and is exact and tractable in the thermodynamic limit. The method is defined and its use is illustrated by several brief examples.

DOI
10.1063/1.522562

Related papers

Recursive Green's function approach to Feenberg perturbation theory

Ishida, K.

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.

arXiv:1904.00957•Apr 1, 2019•0 citations

Electron Correlation in a Narrow Band Metal a New Method of Calculation

Samuel P. Bowen

A generalization of Feenberg's Perturbation Theory is shown to give exact expressions for the temperature dependent retarded greens functions for a many body problem. The theorem establishing this relationship is sketched and the result briefly applied to the Hubbard model.

Annual Conference on Magnetism and Magnetic Materials (MMM)•Jan 1, 1974•0 citations

Convergent methods for calculating thermodynamic Green functions

Samuel P. Bowen, Clayton D. Williams, Jay D. Mancini

A convergent method of approximating thermodynamic Green functions is outlined briefly. The method constructs a sequence of approximants which converges independently of the strength of the Hamiltonian's coupling constants. Two new concepts associated with the approximants are introduced: the resolving power of the approximation, and conditional creation (annihilation) operators. These ideas are illustrated on an exactly soluble model and a numerical example. A convergent expression for the scattering rate in a field theory is also derived.

Physical Review A•Aug 1, 1984•5 citations

From the density functional to the single-particle Green function

Viktor Borisovich Bobrov

An analysis shows that the ground state of the inhomogeneous system of interacting electrons in the static external field, which satisfies the thermodynamic limit, can be consistently described only using the Green function theory based on the quantum field theory methods (perturbation theory diagram technique). In this case, the ground state energy and inhomogeneous electron density in such a system can be determined only after calculating the single-particle Green function.

arXiv:1409.0094•Aug 30, 2014•0 citations

Shortest derivation of time-independent perturbation theory

Boris Blankleider, Alexander N. Kvinikhidze

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.

American Journal of Physics•Jul 23, 2025•2 citations

Stochastic Quantization Method in Operator Formalism

M. Namiki, Y. Yamanaka

The stochastic quantization method is developed in a possible “Heisenberg” operator theory of stochastic processes so designed as to keep a formal analogy to quantum mechanics. The general theory is first formulated for stochastic processes of the Wiener type and then its application to quantization of boson fields is presented. It seems that the operator formalism is convenient to examine implications of the stochastic quantization method. Within the theoretical framework we develop the “Feynman-Dyson” approach to field propagators in perturbation theory. This approach enables us to simplify and systematize practical calculations in comparison with the Langevin equation method. In the Appendix we formulate the Green function approach to field propagators using the characteristic functional.

Progress of Theoretical Physics•Jun 1, 1983•20 citations
A rigorous generalization of Feenberg’s perturbation theory to the calculation of thermodynamic Green’s functions is outlined. This method offers an algorithm for calculating correlation functions and propagators for any Hamiltonian, bounded or unbounded, and is exact and tractable in the thermodynamic limit. The method is defined and its use is illustrated by several brief examples.
A convergent method of approximating thermodynamic Green functions is outlined briefly. The method constructs a sequence of approximants which converges independently of the strength of the Hamiltonian's coupling constants. Two new concepts associated with the approximants are introduced: the resolving power of the approximation, and conditional creation (annihilation) operators. These ideas are illustrated on an exactly soluble model and a numerical example. A convergent expression for the scattering rate in a field theory is also derived.
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.
The stochastic quantization method is developed in a possible “Heisenberg” operator theory of stochastic processes so designed as to keep a formal analogy to quantum mechanics. The general theory is first formulated for stochastic processes of the Wiener type and then its application to quantization of boson fields is presented. It seems that the operator formalism is convenient to examine implications of the stochastic quantization method. Within the theoretical framework we develop the “Feynman-Dyson” approach to field propagators in perturbation theory. This approach enables us to simplify and systematize practical calculations in comparison with the Langevin equation method. In the Appendix we formulate the Green function approach to field propagators using the characteristic functional.