• DOI:10.1007/s00220-021-04166-z
• Corpus ID: 232222412

Super-Macdonald Polynomials: Orthogonality and Hilbert Space Interpretation

@article{Atai2021SuperMacdonaldPO,
  title={Super-Macdonald Polynomials: Orthogonality and Hilbert Space Interpretation},
  author={Farrokh Atai and Martin A. Halln{\"a}s and Edwin Langmann},
  journal={arXiv: Quantum Algebra},
  year={2021}
}

• F. Atai, Martin A. Hallnäs, E. Langmann
• Published 12 March 2021
• Mathematics
• arXiv: Quantum Algebra

The super-Macdonald polynomials, introduced by Sergeev and Veselov, 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. 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 Macdonald-Ruijsenaars operators. This allows for a quantum mechanical interpretation of the models defined by the deformed Macdonald-Ruijsenaars operators. Motivated by recent results in the nonrelativistic ($q\to 1$) 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.