A Casimir operator for a Calogero W algebra
- authored by
- Francisco Correa, Gonzalo Leal, Olaf Lechtenfeld, Ian Marquette
- Abstract
We investigate the nonlinear algebra W
3 generated by the 9 functionally independent permutation-symmetric operators in the three-particle rational quantum Calogero model. Decoupling the center of mass, we pass to a smaller algebra W 3 ′ generated by 7 operators, which fall into a spin-1 and a spin- 3 2 representation of the conformal sl(2) subalgebra. The commutators of the spin- 3 2 generators with each other are quadratic in the spin-1 generators, with a central term depending on the Calogero coupling. One expects this algebra to feature three Casimir operators, and we construct the lowest one explicitly in terms of Weyl-ordered products of the 7 generators. It is a polynomial of degree 6 in these generators, with coefficients being up to quartic in ℏ and quadratic polynomials in the Calogero coupling ℏ 2 g ( g − 1 ) . Putting back the center of mass, our Casimir operator for W
3 is a degree-9 polynomial in the 9 generators. The computations require the evaluation of nested Weyl orderings. The classical and free-particle limits are also given. Our scheme can be extended to any finite number N of Calogero particles and the corresponding nonlinear algebras W
N and W N ′ .
- Organisation(s)
-
Institute of Theoretical Physics
- External Organisation(s)
-
Universidad de Santiago de Chile
University of Queensland
- Type
- Article
- Journal
- Journal of Physics A: Mathematical and Theoretical
- Volume
- 57
- ISSN
- 0022-3689
- Publication date
- 12.02.2024
- Publication status
- Published
- Peer reviewed
- Yes
- ASJC Scopus subject areas
- Physics and Astronomy(all), Statistical and Nonlinear Physics, Statistics and Probability, Mathematical Physics, Modelling and Simulation
- Electronic version(s)
-
https://doi.org/10.1088/1751-8121/ad24ca (Access:
Open)
