Super-Macdonald Polynomials: Orthogonality and Hilbert Space Interpretation
Journal article, 2021

The super-Macdonald polynomials, introduced by Sergeev andVeselov (Commun Math Phys 288: 653-675, 2009), generalise the Macdonald polynomials to (arbitrary numbers of) two kinds of variables, and they are eigenfunctions of the deformed Macdonald-Ruijsenaars operators introduced by the same authors in Sergeev and Veselov (CommunMath Phys 245: 249-278, 2004). We introduce a Hermitian form on the algebra spanned by the super-Macdonald polynomials, prove their orthogonality, compute their (quadratic) norms explicitly, and establish a corresponding Hilbert space interpretation of the super-Macdonald polynomials and deformed MacdonaldRuijsenaars operators. This allows for a quantum mechanical interpretation of the models defined by the deformedMacdonald-Ruijsenaars operators. Motivated by recent results in the nonrelativistic (q -> 1) case, we propose that these models describe the particles and anti-particles of an underlying relativistic quantum field theory, thus providing a natural generalisation of the trigonometric Ruijsenaars model.

Author

Farrokh Atai

Kobe University

Martin Hallnäs

Chalmers, Mathematical Sciences, Analysis and Probability Theory

University of Gothenburg

Edwin Langmann

Royal Institute of Technology (KTH)

Communications in Mathematical Physics

0010-3616 (ISSN) 1432-0916 (eISSN)

Vol. In Press

Quasi-invariants of finite Coxeter groups and integrable systems

Swedish Research Council (VR) (2018-04291), 2019-01-01 -- 2022-12-31.

Subject Categories

Algebra and Logic

Geometry

Mathematical Analysis

DOI

10.1007/s00220-021-04166-z

More information

Latest update

9/10/2021