Details
| Original language | English |
|---|---|
| Article number | 085203 |
| Journal | Journal of Physics A: Mathematical and Theoretical |
| Volume | 57 |
| Issue number | 8 |
| Publication status | Published - 12 Feb 2024 |
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 ′ .
Keywords
- Calogero model, Casimir operator, W algebra
ASJC Scopus subject areas
- Physics and Astronomy(all)
- General Physics and Astronomy
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematics(all)
- Statistics and Probability
- Mathematics(all)
- Mathematical Physics
- Mathematics(all)
- Modelling and Simulation
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In: Journal of Physics A: Mathematical and Theoretical, Vol. 57, No. 8, 085203, 12.02.2024.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - A Casimir operator for a Calogero W algebra
AU - Correa, Francisco
AU - Leal, Gonzalo
AU - Lechtenfeld, Olaf
AU - Marquette, Ian
N1 - F C was supported by Fondecyt Grants 1171475 and 1211356. He thanks the Departamento de Física Teórica, Atómica y Óptica at Universidad de Valladolid and Universidad Austral de Chile, where this project was initiated, for its kind hospitality.
PY - 2024/2/12
Y1 - 2024/2/12
N2 - 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 ′ .
AB - 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 ′ .
KW - Calogero model
KW - Casimir operator
KW - W algebra
UR - http://www.scopus.com/inward/record.url?scp=85187256333&partnerID=8YFLogxK
U2 - 10.1088/1751-8121/ad24ca
DO - 10.1088/1751-8121/ad24ca
M3 - Article
VL - 57
JO - Journal of Physics A: Mathematical and Theoretical
JF - Journal of Physics A: Mathematical and Theoretical
SN - 0022-3689
IS - 8
M1 - 085203
ER -