Details
| Original language | English |
|---|---|
| Article number | 93 |
| Number of pages | 31 |
| Journal | Journal of high energy physics |
| Volume | 2024 |
| Issue number | 6 |
| Publication status | Published - 14 Jun 2024 |
Abstract
A new kind of quantum Calogero model is proposed, based on a hyperbolic Kac-Moody algebra. We formulate nonrelativistic quantum mechanics on the Minkowskian root space of the simplest rank-3 hyperbolic Lie algebra AE3 with an inverse-square potential given by its real roots and reduce it to the unit future hyperboloid. By stereographic projection this defines a quantum mechanics on the Poincaré disk with a unique potential. Since the Weyl group of AE3 is a ℤ2 extension of the modular group PSL(2,ℤ), the model is naturally formulated on the complex upper half plane, and its potential is a real modular function. We present and illustrate the relevant features of AE3, give some approximations to the potential and rewrite it as an (almost everywhere convergent) Poincaré series. The standard Dunkl operators are constructed and investigated on Minkowski space and on the hyperboloid. In the former case we find that their commutativity is obstructed by rank-2 subgroups of hyperbolic type (the simplest one given by the Fibonacci sequence), casting doubt on the integrability of the model. An appendix with Don Zagier investigates the computability of the potential. We foresee applications to cosmological billards and to quantum chaos.
Keywords
- Differential and Algebraic Geometry, Discrete Symmetries, Integrable Field Theories, Scale and Conformal Symmetries
ASJC Scopus subject areas
- Physics and Astronomy(all)
- Nuclear and High Energy Physics
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In: Journal of high energy physics, Vol. 2024, No. 6, 93, 14.06.2024.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - A hyperbolic Kac-Moody Calogero model
AU - Lechtenfeld, Olaf
AU - Zagier, Don
N1 - Publisher Copyright: © The Author(s) 2024.
PY - 2024/6/14
Y1 - 2024/6/14
N2 - A new kind of quantum Calogero model is proposed, based on a hyperbolic Kac-Moody algebra. We formulate nonrelativistic quantum mechanics on the Minkowskian root space of the simplest rank-3 hyperbolic Lie algebra AE3 with an inverse-square potential given by its real roots and reduce it to the unit future hyperboloid. By stereographic projection this defines a quantum mechanics on the Poincaré disk with a unique potential. Since the Weyl group of AE3 is a ℤ2 extension of the modular group PSL(2,ℤ), the model is naturally formulated on the complex upper half plane, and its potential is a real modular function. We present and illustrate the relevant features of AE3, give some approximations to the potential and rewrite it as an (almost everywhere convergent) Poincaré series. The standard Dunkl operators are constructed and investigated on Minkowski space and on the hyperboloid. In the former case we find that their commutativity is obstructed by rank-2 subgroups of hyperbolic type (the simplest one given by the Fibonacci sequence), casting doubt on the integrability of the model. An appendix with Don Zagier investigates the computability of the potential. We foresee applications to cosmological billards and to quantum chaos.
AB - A new kind of quantum Calogero model is proposed, based on a hyperbolic Kac-Moody algebra. We formulate nonrelativistic quantum mechanics on the Minkowskian root space of the simplest rank-3 hyperbolic Lie algebra AE3 with an inverse-square potential given by its real roots and reduce it to the unit future hyperboloid. By stereographic projection this defines a quantum mechanics on the Poincaré disk with a unique potential. Since the Weyl group of AE3 is a ℤ2 extension of the modular group PSL(2,ℤ), the model is naturally formulated on the complex upper half plane, and its potential is a real modular function. We present and illustrate the relevant features of AE3, give some approximations to the potential and rewrite it as an (almost everywhere convergent) Poincaré series. The standard Dunkl operators are constructed and investigated on Minkowski space and on the hyperboloid. In the former case we find that their commutativity is obstructed by rank-2 subgroups of hyperbolic type (the simplest one given by the Fibonacci sequence), casting doubt on the integrability of the model. An appendix with Don Zagier investigates the computability of the potential. We foresee applications to cosmological billards and to quantum chaos.
KW - Differential and Algebraic Geometry
KW - Discrete Symmetries
KW - Integrable Field Theories
KW - Scale and Conformal Symmetries
UR - http://www.scopus.com/inward/record.url?scp=85196167114&partnerID=8YFLogxK
U2 - 10.1007/JHEP06(2024)093
DO - 10.1007/JHEP06(2024)093
M3 - Article
AN - SCOPUS:85196167114
VL - 2024
JO - Journal of high energy physics
JF - Journal of high energy physics
SN - 1029-8479
IS - 6
M1 - 93
ER -