The Calogero model—anyonic representation, fermionic extension and supersymmetry. (English) Zbl 0941.81606
Summary: We discuss several applications and extensions of our previous operator solution of the \(N\)-body quantum-mechanical Calogero problem, i.e. \(N\) particles in one dimension subject to a two-body interaction of the form \(\frac 12\Sigma_{i,j} [(x_i-x_j)^2+g/(x_i-x_j^2]\). Using a complex representation of the deformed Heisenberg algebra underlying the Calogero model, we explicitly establish the equivalence between this system and anyons in the lowest Landau level. A construction based on supersymmetry is used to extend our operator method to include fermions, and we obtain an explicit solution of the supersymmetric Calogero model constructed by Freedman and Mende. We also show how the dynamical OSp\((2; 2)\) supersymmetry is realized by bilinears of modified creation and annihilation operators, and how to construct a supersymmetric extension of the deformed Heisenberg algebra.
MSC:
| 81V70 | Many-body theory; quantum Hall effect |
| 81Q60 | Supersymmetry and quantum mechanics |
| 81S05 | Commutation relations and statistics as related to quantum mechanics (general) |
| 81T13 | Yang-Mills and other gauge theories in quantum field theory |
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
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