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Journal of Mathematical Physics

Supersymmetric embedding of a model of a quantum harmonic oscillator interacting with infinitely many bosons

Asao Arai

Published Feb 1, 1989• 15 citations
Publisher

Abstract

A mathematically rigorous analysis is given on supersymmetric embedding of a model of a one-dimensional quantum harmonic oscillator interacting with infinitely many bosons moving in the s-dimensional space Rs. The model is exactly soluble. By a rigorous explicit construction of supersymmetric quantum field theories, it was proven that the Hamiltonian of the model is supersymmetrically embeddable if it is non-negative. The index of the Dirac–Kähler-type operators associated with the supercharges of the supersymmetric quantum field theories is computed.

DOI
10.1063/1.528417

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Journal of Physics A Mathematical and General•Sep 7, 1994•1 citation
A mathematically rigorous analysis is given on supersymmetric embedding of a model of a one-dimensional quantum harmonic oscillator interacting with infinitely many bosons moving in the s-dimensional space Rs. The model is exactly soluble. By a rigorous explicit construction of supersymmetric quantum field theories, it was proven that the Hamiltonian of the model is supersymmetrically embeddable if it is non-negative. The index of the Dirac–Kähler-type operators associated with the supercharges of the supersymmetric quantum field theories is computed.
A class of exactly soluble models of a one-dimensional quantum harmonic oscillator interacting with bosons moving in the d-dimensional space Rd is considered and the long-time behavior of the two-point function of the oscillator at zero temperature and at finite temperatures is analyzed. It is shown that the two-point functions decay with a power-law respectively as the time tends to infinity and that, in the case where the boson is massless, the two-point function at zero temperature decays faster than those at finite temperatures, while, in the case where the boson is massive, they decay with the same order. Further, the dependence of the decay order on d as well as on the infrared behavior of the one-boson energy and the momentum cutoff function in the interaction Hamiltonian is clarified in each case.
An algebraic view of supersymmetric quantum mechanics is taken, emphasising the couplings between bosonic and fermionic modes in the supercharges. A class of model Hamiltonians is introduced wherein the fermionic (bosonic) operators are canonical and the bosonic (fermionic) ones satisfy a Lie algebra (superalgebra) whose representation theory permits the complete solution of the model in principle. The kinematical symmetry of such models is also described. The examples of one and two bosonic models, with 5U(2) and 5U(3) dynamical algebras respectively, are analysed in detail.
We discuss the superoscillators in thc supersymmetric quantum systems, and give the solution to the system. By making use of the annihilation and creation operators and their q -deformed operators, we also give the q -deformed quantum harmonic oscillator model in the supersymmetric quantum systems. The q -supercoherent states associated with the q -deformed supersymmetric quantum harmonic oscillators are constructed explicitly, and their properties are investigated. The uncertainty relations for q -supercoherent states are discussed as well.
By modifying and generalizing known supersymmetric models, we are able to find four different sets of one-dimensional Hamiltonians for the inverted harmonic oscillator. The first set of Hamiltonians is derived by extending the supersymmetric quantum mechanics with reflections to non-Hermitian supercharges. The second set is obtained by generalizing the supersymmetric quantum mechanics valid for non-Hermitian supercharges with the Dunkl derivative instead of [Formula: see text]. Also, by changing the derivative [Formula: see text] by the Dunkl derivative in the creation and annihilation-type operators of the standard inverted harmonic oscillator [Formula: see text], we generate the third set of Hamiltonians. The fourth set of Hamiltonians emerges by allowing a parameter of the supersymmetric two-body Calogero-type model to take imaginary values. The eigensolutions of definite parity for each set of Hamiltonians are given.
Abstract Liénard-type nonlinear one dimensional oscillator is quantized using van Roos symmetric ordering recipe for the kinetic-like part of the new derived Hamiltonian. The corresponding Schrödinger equation is exactly solved in momentum space via the approach of supersymmetric quantum mechanics (SUSYQM). The bound-states energy spectra and corresponding wave functions are given explicitly in terms of the ambiguity parameters. The limiting case of no deformation agrees exactly with the eigenenergies and eigenfunctions of the ordinary quantum harmonic oscillator.