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arXiv:hep-th/9801196 · Progress of Theoretical Physics Supplement

Renormalization-Group Resummation of a Divergent Series of the Perturbative Wave Functions of Quantum Systems

Teiji Kunihiro

Published Jan 1, 1998• 17 citations
arXivHTMLPDF

Abstract

The perturbative renormalization group (RG) equation is applied to resum a divergent series of perturbative wave functions of a quantum anharmonic oscillator. It is found that the resummed series gives the cumulant of the naive perturbation series. It is shown that a reorganization of the resummed series reproduce the correct asymptotic form of the wave function at x → ∞ when the perturbation expansion is stopped at the fourth order. A brief comment is given on the relation between the present method and the delta-expansion method, which is based on a kind of nonperturbative RG equation.

DOI
10.1143/ptps.131.459

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The perturbative renormalization group (RG) equation is applied to resum a divergent series of perturbative wave functions of a quantum anharmonic oscillator. It is found that the resummed series gives the cumulant of the naive perturbation series. It is shown that a reorganization of the resummed series reproduce the correct asymptotic form of the wave function at x → ∞ when the perturbation expansion is stopped at the fourth order. A brief comment is given on the relation between the present method and the delta-expansion method, which is based on a kind of nonperturbative RG equation.
The standard perturbation theory for time-dependent problems in quantum mechanics is reconsidered through renormalization-group methods. This approach justifies a posteriori the theory of quantum resonance given by the multiple-scale analysis of the perturbation series applied to the coupled equations of wave-function amplitudes. The resonance equations for the leading-order amplitude probabilties are then obtained up to second order and a general method for the algorithmic computation of higher-order terms of the perturbation series is given. The three-level model in a monochromatic wave with two resonant states is discussed in the light of the present results.
A thermodynamic perturbation method for the quantum-anharmonic oscillator is studied, based upon the Bloch equation. By this method the density matrix (Green's function) for the anharmonic oscillator can be obtained in a compact form for each order of the perturbation series. It is shown that the density matrix of the anharmonic oscillator can be readily expanded in series of the Hermite polynomials. The various expressions of the expansion series for the density matrix are also derived by a unified method.
We apply the functional renormalization group method to the calculation of dynamical properties of zero-dimensional interacting quantum systems. As case studies we discuss the anharmonic oscillator and the single impurity Anderson model. We truncate the hierarchy of flow equations such that the results are at least correct up to second order perturbation theory in the coupling. For the anharmonic oscillator energies and spectra obtained within two different functional renormalization group schemes are compared to numerically exact results, perturbation theory, and the mean field approximation. Even at large coupling the results obtained using the functional renormalization group agree quite well with the numerical exact solution. The better of the two schemes is used to calculate spectra of the single impurity Anderson model, which then are compared to the results of perturbation theory and the numerical renormalization group. For small to intermediate couplings the functional renormalization group gives results which are close to the ones obtained using the very accurate numerical renormalization group method. In particulare the low-energy scale (Kondo temperature) extracted from the functional renormalization group results shows the expected behavior.
Abstract These introductory notes are about functional renormalization group equations and some of their applications. It is emphasised that the applicability of this method extends well beyond critical systems, it actually provides us a general purpose algorithm to solve strongly coupled quantum field theories. The renormalization group equation of F. Wegner and A. Houghton is shown to resum the loop-expansion. Another version, due to J. Polchinski, is obtained by the method of collective coordinates and can be used for the resummation of the perturbation series. The genuinely non-perturbative evolution equation is obtained by a manner reminiscent of the Schwinger-Dyson equations. Two variants of this scheme are presented where the scale which determines the order of the successive elimination of the modes is extracted from external and internal spaces. The renormalization of composite operators is discussed briefly as an alternative way to arrive at the renormalization group equation. The scaling laws and fixed points are considered from local and global points of view. Instability induced renormalization and new scaling laws are shown to occur in the symmetry broken phase of the scaler theory. The flattening of the effective potential of a compact variable is demonstrated in case of the sine-Gordon model. Finally, a manifestly gauge invariant evolution equation is given for QED.
In classical and quantum systems, perturbation of an evolution equation is often invalidated by secular terms which diverge at late times. The diverging behavior of evolution can be remedied by various techniques of resummation such as the renormalization group or multi-scale analysis. In this paper, we prove that, in a generic quantum-mechanical system, secular terms can be systematically removed to all orders in the Dyson series by the method of improved (renormalized) perturbation. A recurrence relation to provide an explicit method to remove the secular terms is given. As a by-product, we give a simple method to obtain energy eigenvalues and decay rates to all orders of perturbation.
A new formalism of the thermodynamic perturbation theory for the quantum-anharmonic oscillator is proposed. By this method, for each order of the perturbation series, the partition function and the free energy of the system can be obtained in a compact form which is expressed by the temperature Green's function of the harmonic quanta. Based upon a unified method, the various expressions of the perturbation series for the partition function and the free energy are also given. It is shown that the perturbation series for the free energy can be naturally formulated in terms of a cluster expansion.