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arXiv:hep-th/0108160 · Physics of Atomic Nuclei

Quantum many-body problems and perturbation theory

A. V. Turbiner

Published Jun 1, 2002• 2 citations
arXivHTMLPDF

Abstract

We show that the existence of algebraic forms of exactly solvable A-B-C-D , G 2 , and F 4 Olshanetsky-Perelomov Hamiltonians allows one to develop algebraic perturbation theory, where corrections are computed by purely algebraic means. A classification of perturbations leading to such a perturbation theory based on the theory of representations of Lie algebras is given. In particular, this scheme admits an explicit study of anharmonic many-body problems. Some examples are presented.

DOI
10.1134/1.1490123

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We show that the existence of algebraic forms of exactly solvable A-B-C-D , G 2 , and F 4 Olshanetsky-Perelomov Hamiltonians allows one to develop algebraic perturbation theory, where corrections are computed by purely algebraic means. A classification of perturbations leading to such a perturbation theory based on the theory of representations of Lie algebras is given. In particular, this scheme admits an explicit study of anharmonic many-body problems. Some examples are presented.
Exactly solvable quantum systems represent a fascinating and critically important class of models in theoretical physics, offering exact solutions to non-trivial problems that often shed light on the behavior of more complex, intractable systems. The remarkable feature of these systems lies not only in their solvability but also in the underlying mathematical structures that facilitate such solutions. This paper explores the concept of a universal algebraic structure that underpins exactly solvable quantum systems. We delve into how diverse models, ranging from one-dimensional spin chains to many-body systems and field theories, exhibit common algebraic symmetries, such as Lie algebras, affine Lie algebras, and quantum groups. These algebraic frameworks provide a unified language for understanding integrability, the existence of an infinite number of conserved quantities, and the ability to construct exact eigenstates. We review the historical development of these ideas, from the Bethe ansatz to the development of the inverse scattering method and quantum groups. Our discussion highlights how these mathematical structures are not merely tools for calculation but fundamental descriptors of the physical reality of these systems, offering profound insights into non-perturbative quantum phenomena. The identification of a universal algebraic architecture suggests a deeper coherence in the landscape of quantum mechanics, paving the way for new discoveries and classifications of quantum models.
The number of exactly solvable eigenvalue problems has been rapidly increasing since the advent of quantum mechanics and there are many efficient ways to treat them. As a general rule, it is possible to write exactly solvable abstract eigenvalue equations in terms of the generators of finite-dimensional Lie algebras and solve them by means of purely algebraic methods. Matrix elements are commonly necessary for the application of perturbation theory and variational principles, among other approximate methods. The chapter discusses examples of three-dimensional Lie algebras with well-known quantum-mechanical realizations. Typically, exactly solvable quantum-mechanical problems are either onedimensional models or lead to separable equations in many dimensions. In the coordinate representation both cases reduce to ordinary differential equations of second order that determine the eigenvalues and separation constants compatible with the physical boundary conditions. In its purest form, a Lie algebraic method is representation-independent. Extensive application of algebraic methods to selected physical problems is already available elsewhere.
>A generalized ford of a perturbation theory for a (nonrelativistic) quantum mechanical many-body Hamiltonian is given that can be useful for problems of quantum chemistry and other problems with a mcderate number of particles. A modification of Watson's t-operator allows a perturbational development of a many body problem in terms of simpler subproblems. Some examples of such cluster approximations are given. The whole theory is formulated in terms of state vectors and there are no assumptions about the degeneracy of the Hamiltonian. The given approximations to the level shift transformation are both unitary and Lie functions in every order of the development. Every step of the calcalation can be done in the domain of a Lie algebra and it is recomdended that full advantage be taken of this fact in practical calculations. The use of the diagram technique is avoided and there are no explicit partial summations of the perturbation series, but similar results are gained by the systematic use of the unitary conditions, of the Lie character, and of a modified t-operator. (auth)
The conventional many-body perturbation theory suffers from three major ills: (i) some terms in the perturbation series may be unbounded; (ii) the series as a whole may not have a limit and (iii) despite convergence,the calculated limit may be false ('bogus' convergence). In renormalizable field theories, as the name indicates, problem (i) can be bypassed. Problems (ii) and (iii) remain. Here we demonstrate that these two problems can also be resolved, by choosing an appropriate Kohn-Sham Hamiltonian as the 'unperturbed' Hamiltonian. For this to be possible, some pertinent ground-state densities must be pure-state non-interacting v-representable. We illustrate our findings by means of two examples.
Abstract Necessary and sufficient conditions for quantum Hamiltonians to be exactly solvable within mean-field (MF) theories have not been formulated so far. To resolve this problem, first, we define what MF theory is, independently of a Hamiltonian realization in a particular set of operators. Second, using a Lie-algebraic framework we formulate a criterion for a Hamiltonian to be MF solvable. The criterion is applicable for both distinguishable and indistinguishable particle cases. For the electronic Hamiltonians, our approach reveals the existence of MF solvable Hamiltonians of higher fermionic operator powers than quadratic. Some of the MF solvable Hamiltonians require different sets of quasi-particle rotations for different eigenstates, which reflects a more complicated structure of such Hamiltonians.
We present a new Lie algebraic method to study the eigenvalues and eigenfunctions of quantum anharmonic oscillators. We consider the Hamiltonians for the simple harmonic and anharmonic oscillator as the two generators of a Lie algebra, whose other generators may be found exactly or up to any order of the parameter involved. Speciflcally, the closed commutator algebra for the quartic anharmonic oscillator is established in a perturbation sense. An element of this Lie group, turning out to be the four-photon operator, transforms the quartic anharmonic oscillator Hamiltonian to the harmonic one in a perturbation sense; thus, facilitating the calculation of the eigenvalues and eigenfunctions of the former.