Super-Macdonald Polynomials: Orthogonality and Hilbert Space Interpretation
Artikel i vetenskaplig tidskrift, 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.

Författare

Farrokh Atai

Kobe University

Martin Hallnäs

Chalmers, Matematiska vetenskaper, Analys och sannolikhetsteori

Göteborgs universitet

Edwin Langmann

Kungliga Tekniska Högskolan (KTH)

Communications in Mathematical Physics

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

Vol. In Press

Kvasi-invarianter för ändliga Coxeter-grupper och integrabla system

Vetenskapsrådet (VR) (2018-04291), 2019-01-01 -- 2022-12-31.

Ämneskategorier

Algebra och logik

Geometri

Matematisk analys

DOI

10.1007/s00220-021-04166-z

Mer information

Senast uppdaterat

2021-09-10