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Perturbative Bosonic End Anyon Spectra and Contact Interactions

Giovanni Amelino-camelia A

Published Jan 1, 1994• 11 citations
OpenAlex

Abstract

Bosonic end perturbative calculations for quantum mechanical anyon systems require a regularization. I regularize by adding a specific $\\delta$-function potential to the Hamiltonian. The reliability of this regularization procedure is verified by comparing its results for the 2-anyon in harmonic potential system with the known exact solutions. I then use the $\\delta$-function regularized bosonic end perturbation theory to test some recent conjectures concerning the unknown portion of the many-anyon spectra.

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Bosonic end perturbative calculations for quantum mechanical anyon systems require a regularization. I regularize by adding a specific delta\\deltadelta-function potential to the Hamiltonian. The reliability of this regularization procedure is verified by comparing its results for the 2-anyon in harmonic potential system with the known exact solutions. I then use the delta\\deltadelta-function regularized bosonic end perturbation theory to test some recent conjectures concerning the unknown portion of the many-anyon spectra.
In bosonic and perturbative calculations for quantum mechanical anyon systems a regularization and renormalization procedure, analogous to those used in field theory, is necessary. I examine the reliability and the physical interpretation of the most commonly used bosonic end regularization procedures. I then use the regularization procedure with the most transparent physical intepretation to derive some bosnic end perturbation theory results on anyon spectra, including a three-anyon ground state energy.
We study a many-body system consisting of NNN identical anyons in an external harmonic-oscillator potential. A partial set of exact solutions is obtained by using Jacobi coordinates. For completing the full investigation, we employ a perturbative method to compute those spectra, which have not been analytically obtained. Near the boson limit, we point out that the requirement of the boson being hard core is needed for performing the perturbative calculations, and the first-order NNN-anyon ground-state energies are explicitly presented by using a proper regulation procedure. Up to third perturbative order from the fermion limit, the three-anyon ground-state energy corrections are explicitly presented. Finally we compute the first-order three-anyon ground-state wave function near the fermion limit.
As a continuation of a recent paper, using density and current density as coordinates for interacting bosons at absolute zero, this work deals with the perturbation theory. The zerothorder Hamiltonian is the one shown to yield the Bogoliubov energy spectrum. First- or secondorder perturbation calculations have been made of the three-phonon and the four-phonon vertices for the energy corrections of the low-lying excited states. The results in the long-wavelength limit differ, although only slightly, from the quantum hydrodynamic calculations, and indicate that the perturbed excitation energy ${\ensuremath{\epsilon}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}$ is an odd-power series in kkk if the Fourier coefficients of the two-body potential concerned can be approximated by a constant or an evenpower series in kkk. However, whether this remains so in all higher orders of various perturbations is an open question. Two methods were used in the calculation. The first involves functional representations of the coordinates in the ${\ensuremath{\rho}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}$ representation recently reported. The second method is new, and employs "occupation-number" representations of the current algebra by introducing creation and annihilation operators for "phonons" from density fluctuations. These latter representations enable us to benefit from the advantages of the canonical field theory.